数理经济学研究 2006年第1辑（总第1辑）

数理经济学研究 2006年第1辑（总第1辑） Research of Mathematical Economics No. 1 2011 CONTENTS The Effect of Income and Interest Rates on the Consuming Behavior of Chinese Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural Areas …………………………………………………Zhang Dongyang (5) The Influence of Wealth Gap on China‘s Economic Growth ………………………………………………………Guo Yumei (21) Bundling Sales of Information Goods: Models and Analysis ……………………………………………………Shi Fangning (31) The Introduction and Applications of RDEU Hypothesis …………………………………………Wang Bo, Hong Beiyun (54) Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition Survey ……………………………………………………Zhang Luezhao (72) A Dynamic Model of the Optimal Carbon Tax and Sustainable Development ……………………………………………………Feng Junlong (89) A Study of Chinese Residential Electricity Demand Elasticities …………………………………………………Ren Junqiushi (101) The Factors that Determine a Firm‘s Optimal Allocation of Capital Between Research and Manufacturing ……………………………………Xu Lingjue, Wang Danna (118) A Quantitative Analysis of the Influences of China‘s Inflation to Different Asset-Holding Classes ………………………………………………………Chen Jun (134) Analysis of the Mechanism of ―Dutch Disease‖ ……………………………………………………Wang Qizhi (161) 1 数理经济学研究 2011年第1辑(总第5辑) 目 录 收入和利率对中国居民消费行为的影响——基于中国城乡统计数据的比较 …………………………………………………………………… 张冬洋 (21) 贫富差距对中国经济增长的影响 …………………………………………………………………… 郭豫媚 (47) 信息产品的捆绑销售:模型与分析 …………………………………………………………………… 施芳凝 (31) REDU假说的介绍与应用 ………………………………………………………… 王 勃、洪蓓芸 (54) 收入目标与劳动供给弹性:来自《中国健康和营养调查》的证据 …………………………………………………………………… 张略钊 (72) 一个最优碳税和可持续发展的动态模型 …………………………………………………………………… 冯焌昽 (89) 中国居民电力需求的弹性探析 ……………………………………………………………… 任君秋拾 (101) 决定公司在研究与制造之间资本最优配置的因素 ……………………………………………………… 许龄珏、王丹娜 (118) 中国通货膨胀对不同财富持有阶层影响的定量分析 ………………………………………………………………… 陈 军 (134) 荷兰病作用机制考察 ………………………………………………………………… 王麒植 (161) 2 Research of Mathematical Economics Organizers: Economics and Mathematics Double Bachelor Program Renmin University of China Academic Adviser: Yang Ruilong Liu Yuanchun Zhang Yu Liu Rui Guo Jie Wang Jinbin Zheng Xinye Chen Yanbin Editorial Committee: Li Tianyou Zhang Yanhong Yu Ze Wang Xianghong Cheng Hua Yu Yihua Han Song Instruct Teacher: Cheng Hua Editorial Director: Ren Junqiushi Xu Lingjue Zhang Lingyun Wang Jue 《数理经济学研究》编辑机构 主办单位: 中国人民大学经济学—数学(双学位)实验班 学术顾问: 杨瑞龙 刘元春 张 宇 刘 瑞 郭 杰 王晋斌 郑新业 陈彦斌 编 委: 李天有 章艳红 于 泽 王湘红 程 华 虞义华 韩 松 指导老师: 程 华 编 辑 部: 任君秋拾 许龄珏 张凌云 王 珏 3 4 Research of Mathematical Economics No. 1 2011 The Effect of Income and Interest Rates on the Consuming Behavior of Chinese Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural Areas Zhang Dongyang (School of Economics, Renmin University of China) Abstract: This essay analyzes how to expand domestic demand and how to stimulate consumption. Based on the random walk hypothesis and permanent income hypothesis, consumers are divided into two groups, one with rational expectations and the other regarded as ―short-sighted‖. This paper uses ordinary least squares method and uses data from 2003Q1 - 2008Q2 to estimate the aggregate consumption function of China‘s urban and rural areas. The conclusion is that the number of ―short-sighted‖ consumers in rural areas of China significantly exceeds that in urban areas. Also, this paper suggests that lowering interest rates has a strong stimulating effect on consumption growth and verifies the parameters‘ stability by hypothesis testing. Last, with regard to the statistics obtained through econometric model, final conclusions of this essay and certain corresponding policy recommendations are given out. Keywords: Permanent Income Hypothesis, Uncertainty, Consumption, Interest Rates 1. Introduction In 2008, the financial crisis swept the entire world including China, which imposed an extremely large impact on China‘s economy. On the other hand, China has been under reform and opening-up policy for 30 years until 2008. Along with the ever-deepening reform, Chinese economy system has gradually turned from traditional planned economy to socialist market economy, from extensive to intensive. In the support of the three pillars of its development – investment, domestic demand and foreign trade, the entire national economy follows an orderly and stable development trend. Hence, in face of such a sudden shock, the problem of how to expand domestic demand and stimulate China‘s economy is put on core agenda once again. In the traditional researches, Keynes pointed out that national income influences consuming; 5 The Effect of Income and Interest Rates on the Consuming Behavior of Chinese Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural Areas Friedman and Modigliani put forward permanent income hypothesis and life-cycle hypothesis respectively. These theories have been far-reaching. The basis of these theories, interest rates act as a direct means to influence national income through adjusting saving rates----lowering interest rates helps to increase the income for people‘s consumption----have long been considered as an effective approach to expand domestic demand. However, from the ―Ninth Five-Year‖ to the ―Eleventh Five-Year‖, a lot of scholars, such as Long Zhiming, Zhou Haoming (2000), think that China‘s economic development and consumer behavior showed some kind of ―rebellion‖ phenomenon: on one hand, the economy experiences stable and fast growth and national income increases steadily; on the other hand, along with the ever-deepening reform and opening-up policy, consumer behaviors have changed dramatically, among which what worthies our particular attention is the fact that in spite of the continuously increasing saving percentage, which to some extent reflects decreasing marginal consuming propensity, consumption scale has grown slowly. In fact, it is still too early to draw the conclusion of whether interest rate adjusting has positive effect on national consumption. Thus, this paper will explore the effective factors, such as interest rate and income, and their effectiveness on consumption increase. Then, the problem is raised: what is the factor that coordinates the consuming behavior with the trend of economy growth during the transition period of China? Focusing on this phenomenon, domestic scholars have done a series of researches referring to foreign consumption theories. For instance, under a unified economic logic framework, Yuan Zhigang and Song Zheng (1999) conduct a comprehensive analysis of consuming behavior of urban residents in China in the transition period: based on consumption theories such as Life Cycle Hypothesis, Permanent Income Hypothesis, Precautionary Saving Theory and Liquidity Constraint, they try to find out the reason for the variation in urban consumption behavior and the way to stimulate urban household consumption. Li Yan (1999) makes use of the data from 1978 to 1998 to carry out the empirical analysis which aims to find out the role of interest rates on household savings. The transfer channel for interest rates‘ effect on saving is wealth: the low level of income and the underdeveloped capital markets are the primary obstacles that restrict the interest rates‘ effect on household saving. That is to say, under the environment of low level of income and underdeveloped capital markets, wealth value and wealth gains are not sensitive to 6 Research of Mathematical Economics No. 1 2011 interest rates changes. Wang Hongju and Zhang Huilian (2002) put forward the idea of using modern Western economic theory to analyze the consumption growth in China. They point out the restriction effect of Random Walk Hypothesis, Precautionary Saving Theory and Liquidity Constraint on consumption and then analyze the impacts of these factors on China‘s household consumption. They conclude that Liquidity constraints depressed consumer‘s current spending; moreover, low income, uncertainty and liquidity constraints lead to the shortsightedness of the residents. However, theoretical research in this regard focused mainly on the analysis from 1999 to 2003 and few in recent years. This situation prompts us to refresh our understanding and study the problem of consumer behavior both from the perspective of theory and practice, trace the deep influencing factors behind these consumer behavior and then to give theoretical explanations as well as pragmatically measures to direct the household consuming behavior. The second part of this paper is literature review, pointing out the limitation of traditional consumption theory on explaining consumer behaviors and the feasibility of using Residents Consumption Excess Sensitivity Hypothesis and Random Walk Hypothesis to explain and analyze the consumption problems in current China. The third section reviews the theoretical models based on Random Walk Hypothesis. The fourth part gives out the quantitative results of consumption functions about China‘s urban and rural residents in the transition period with the 2003-2008 quarterly data on China‘s urban and rural residents. And the fifth part takes advantage of the theory and results of quantitative analysis to draw conclusions and policy recommendations are given. 2. Literature Review Consumption function defines the relationship between household consumption expenditure and the variables that determine the consumption. It was first proposed by Keynes (1936) in the book ―Employment, Interest and Money‖. Absolute Income Hypothesis suggests that people‘s current consumption depends on current income and a stable functional relationship exists between them. With the increase of income, consumption will also increase, but the increase of consumption is slower than the increase of income: consumption increment in the proportion of increased income is decreasing ---- that is diminishing marginal consumption propensity, showed 7 The Effect of Income and Interest Rates on the Consuming Behavior of Chinese Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural Areas by cy,，,,,,,,01 Yet, Duesenberry puts forward the Relative Income Hypothesis by negative the conditions of Absolute Income Hypothesis. In his view, consumers are influenced by their surrounding environment from time to time. The ―model‖ of consumption is likely to make the low-income group to get closer to the consumption level of their surrounding people. Hence, he believes that the lower the income is, the high the marginal consumption propensity will be. However, both Absolute and Relative Income Hypothesis only take consumers‘ current income into consideration and thus lack of microeconomic foundations. For that matter, within the framework of neo-classical Friedman and Modigliani extended consumer decision to embrace inter-stage selection and proposed Permanent Income Hypothesis and Life Cycle Hypothesis, respectively. They believed that consumers‘ decision for current expenditure is determined by their life income. Nevertheless, none of these econometric models can escape from Lucas critique. Lucas (1976) pointed out that no matter how the traditional consumption function fits real data, they have no value in policy making decisions, because any change in policy will after all alter the relationship between aggregates in macroeconomic models. Therefore, Hall (1978) proposed a consumption theory based on rational expectations. His idea is that consumption should follow the first-order optimal conditions of a typical forward-looking rational consumer, which is also known as ―Euler Equation‖. Hence, the Random Walk Model gives out the following equation assuming the quadratic utility function: (1) CC,，～,,,E0tttt,1 This means that the changes in consumption are unpredictable, that is the current consumption depends only on the consumption in the last period and independent of any other economic variables. In view of Hall‘s hypothesis, later researchers have done a lot of empirical analysis but end up with no satisfied results. So Cambell and Mankiw (1990) proposed the idea that aggregate consumption function does not necessarily satisfy the general random walk. They put consumers into two categories: one to meet the Permanent Income Hypothesis of rational expectations and the other to spend their current income which , also , 8 Research of Mathematical Economics No. 1 2011 known as Excess Sensitivity. The greater is, the more sensitive consumption will reflect on , , then consumption depends entirely on the current ,,1 , it means the Permanent Income Hypothesis of rational ,,0 expectations is verified. Chinese scholars have made some progress in the research of ―random walk‖ consumption theory. Wan Guoguang (2001) use the model of Hall and the idea of Campbell and Mankiw through the data from 1961 to 1998 in China and come to the conclusion that with the ever-deepening economic reform in China, the consuming behavior of Chinese households has experienced structural change in the early 1980s. They believe that both the rise in the proportion of liquidity constrained consumers and the increased uncertainty are responsible for the low consumption growth rate in China and insufficient domestic demand. Hang Bin and Shen Chunlan (2004) believe that Excess Sensitivity in the sample period is likely to change due to policy changes, so they conducted a varying parameters analysis on the consumer sensitivity of urban residents through data from 1978 to 2002. They come to the results that before 1990 Excess Sensitivity had dropped dramatically, but after 1990, Excess Sensitivity had experienced stead growth, at about 0.64. Also, the substitution effect of changes in interest rates is greater than the income effect; that is to say, consumption expenditure will increase along with the rise in interest rate. Through the analysis of relevant macroeconomic data about rural areas from 1978 to 2003, Zhu Xinkan (2005) concluded that 62.5% of rural household limited by liquidity constraint are short-sighted. By the analysis of residents‘ consumption behaviors on durable and non-durable goods, Liu Hinquan (2003) found that household savings in China have significant ―precautionary savings‖ components and future expected income is extraordinary uncertainty. Zhou Jian through the research into the variable parameter space state model of the consumption function studied the sensitivity of consuming behavior of rural residents in the transition period. Empirical results showed that there exists significant ―precautionary savings‖ motivation among the rural residents. Zhang Xuheng (2007) made use of precautionary saving, Liquidity Constraints Hypothesis and the dynamic modeling method to compare the characteristics of consumer behavior in China‘s rural areas and urban areas. His empirical results implied that the proportion of urban residents who were affected by liquidity constraints is greater than that of rural ones, and rural residents‘ responses to uncertainty and real interest rates are more sensitive than urban people. 9 The Effect of Income and Interest Rates on the Consuming Behavior of Chinese Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural Areas However, these models are based on annual statistics and lack research in recent years. This paper‘s main logic is as follows: the Random Walk Hypothesis and the Excess Sensitive Hypothesis are used as theoretical foundation and the statistics from 2003Q1 to 2008Q2 may be processed according to the model to solve out China‘s consumer function. Then, this paper will go on to verify the effectiveness of interest rates and income on consumption so as to fill the bland area in this research field. 3. Mathematics Model Campbell and Mankiw (1990) pointed out that consumers can be divided into two types: one part meets the Hall's permanent income hypothesis of rational expectations, they maximize the expected utility function: ,s, (2) EUC(1)()，,,tts，0s, where is subjective discount rate, C is the consumption, U satisfies the first derivative to be , Egreater than 0, the second derivative to be less than 0, and is the conditional expectation t rwhich based on the t period information, if the consumption can borrow by interest rates , then the optimal first-order conditions as 1+, (3) =EUCUC''，，，，ttt，1，1r If it is assumed , and marginal utility satisfies the conditions of linear, then the r,, ECC,,,,CE0,0,consumption reduces to the random walk, that‘s to say , or . ttt，1ttt And the other part those who spend their current income, share of , also known as Excess , YY，Sensitivity. If the total income of these two types of consumers, respectively , the 12tt equation can be: (4) ,,,，,,,，,CCCY,,,(1)ttttt12 ,,yHowever, the residual may be correlated with , this formula cannot be directly tt used for estimating model. Alternative methods are mainly divided into two types, one is Hall 10 Research of Mathematical Economics No. 1 2011 (1978), which is to see whether lag period consumption and income can predict the current consumption or not; the other method was proposed by Campbell and Mankiw (1989), by setting a set of instrumental variables to estimate the equation. This paper uses instrumental variables ,method, because any lag and stationary time series may be orthogonal with , so they can be t an effective tool for the candidate instrumental variables. Also a good set of instrumental ,yvariables is also possible to be relevant with , so we need to select some of the lag variables t which can predict future income variables. Because in reality, the number of consumption and income data is closer to log-linear and nonlinear, and therefore we need to adopt all the variables on the number (with lowercase letters to replace the original variable capital letters), replace the ,,,，,cy,,,(1)original regression equation , using the set of instrumental variables to ttt do regression ,,，cX,,ttct (5) ,,，yX,,ttyt Xwhere is the set of instrumental variables, with the dimension , is the number of nk，kt ,,,,,instrumental variables. We can get the solution of vector from the regression function. ,,,,,，，vuRe-use the relationship function , which is constituted by k simple equations ,,u,,to estimate the value of , noted that when k > 3, the equation may have no solution, while Xk < 3, there may be many solutions. adopted the set of instrumental variables which is no t less than three to avoid multiple solutions in this case. And when , we use two-stage least k,3 squares to estimate the solution. 4. The Empirical Results of China's Consumption Function 4.1 Data Selection and Data Processing Based on the permanent income hypothesis of rational expectations models emphasizing the microeconomic foundation, the general recommendation is to use quarterly data. In addition, the annual data model‘s spread is too long to avoid the Lucas critique ---- a change in policy will change the total relationship between macroeconomic models. Monthly data is not a good choice 11 The Effect of Income and Interest Rates on the Consuming Behavior of Chinese Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural Areas because there is a big seasonal problem. Therefore, in accordance with established practice of the total consumption curve model, the paper uses quarterly data for empirical research in China. The time span is from the 1st quarter, 2003 (2003Q1) to the 2nd quarter, 2008 (2008Q2). We need the following three types of data: per capita disposable income of urban residents, real per capita consumption expenditure of urban residents, per capita disposable income, and per capita consumption expenditure of rural households, rural household, and real interest rates. Data is collected from China Monthly Economic Indicators and the People's Bank of China website. (1) Urban per capita actual disposable income. As the National Bureau of Statistics data is presented quarterly cumulative nominal per capita consumption expenditure of urban residents, firstly we need to process accumulate data into quarterly data, that is, the first quarter of each year‘s data keeps unchanged, while the other quarter with this quarter cumulative minus the cumulative data of last quarterly data. Secondly, we also need to deflate the nominal data the actual data‘s deflator. Using the index of disposable income of urban households is a common approach. However, National Bureau of Statistics did not give the index of the quarter of the disposable income of China in order to match the data; we decided to use urban residents‘ price index to calculate the deflator. With multiplication of the quarter‘s urban residents with the annulus data, we can get the fixed base quarter price index of urban residents. The selected base period is December 2002. (2) Real per capita consumption expenditure of urban residents, real per capita cash income of rural households, real per capita consumption expenditure of rural households. In the same way, we handle these data into quarterly data of the urban and rural areas. (3) Real interest rates. ,,，riSince this article takes the quarterly data, and uses the Fisher equation (), I use after-tax deduction of three-month nominal deposit rate minus the current rate of urban consumer price‘s changes, and then real interest rates can be obtained. Thus, descriptive statistics results of the main variables above in the regression analysis are in Table 1 below. 12 Research of Mathematical Economics No. 1 2011 Table 1 Descriptive Statistics Results of the Variables Standard Variables Mean Median Max Min Deviation Urban per capita actual disposable income 2591.643 2526.235 466.1033 3619.182 1960.499 Rural real per capita cash income 887.749 836.417 279.294 1370.027 421.8753 In- urban per capita actual disposable income 7.8452 7.8345 0.1750 8.1940 7.5810 In- rural per capita actual disposable income 6.7393 6.7291 0.3281 7.2226 6.0447 Growth rate of urban per capita actual disposable income 1.3001 2.8210 15.5045 27.0464 -23.9000 Growth rate of rural real per capita cash income 0.6587 0.3493 37.2562 52.6574 -54.6100 Urban real per capita consumption expenditure 1884.996 253.4658 2378.568 1450.421 1922.109 Rural real per capita consumption expenditure 498.7701 477.9494 115.3829 700.0013 296.4340 In- urban real per capita consumption expenditure 7.5528 7.5417 0.1327 7.7743 7.2796 In- rural real per capita consumption expenditure 6.1856 6.1695 0.2392 6.5511 5.6918 Growth rate of urban real per capita consumption expenditure 1.3015 -0.5442 10.9334 17.2387 -17.5775 Growth rate of rural real per capita consumption expenditure 0.7738 12.216 23.67100 25.2864 -39.0509 Urban real interest rates 0.8291 0.4778 1.2581 3.6633 -1.1525 Rural real interest rates 0.6158 0.4674 1.1467 2.8643 -1.3548 If the random process is non-stationary, then usually, it must be very difficult and imprecise to use a simple algebraic model to reflect the past and future time series. Therefore, we usually require that the time series are stationary. Generally we use the ADF test, including the choice of three parameters: intercept, time trend term, the lag periods. In addition, for the traditional econometric theory, people think that the overall economic time series are identified as having a fixed trend, therefore, the general practice is to get rid of overall fixed trend of economics from fixed-trend model, and then the series are stationary which can be analyzed. However, Nelson and Plosser (1982) found that the majority of overall economic time series have a stochastic trend, so I just get rid of the fixed overall trend of economic time series, and did not remove the time series of the stochastic trend. Then the analysis turns to be a big problem. In addition to the amendments to the traditional unit root test, Ng and Perron (2001) have 13 The Effect of Income and Interest Rates on the Consuming Behavior of Chinese Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural Areas modified information criterion for the selection of the most suitable augmented lag phases, called the modified AIC, MAIC and modified SIC, MSIC and so on. In reality, the time series data are often non-stationary processes that cannot be used in the classical regression model, or will cause regression error (Granger, 1987). However, if stable relationship exists between the long-term time series of the same order, regression equation (co-integration) can be made by the classical regression model. From the above theory, in order to ensure the reliability of estimates, we must firstly consider the time series of each variable must satisfy a single whole order. Under the modified cAIC, the method used to test is ADF test. Table II shows the variables of the ADF test, and t yrI(1)I(1) are processes, the consumption of rural areas is process and is a stationary tt series. Use Eviews to do ADF test, with intercept models, with trend models, neither trend nor intercept models, and also both with the trend and intercept model. Test results are as follow: 14 Research of Mathematical Economics No. 1 2011 Table 2 Variable Unit Root Test Results Test form Variable category p-value Conclusion (Intercept, Trend, Lag phases) 0.9562 (0,0,0) (Urban) ctNon- stationary 0.6865 (0,0,0) c (Rural) t0.9848 (0,0,0) y (Urban) tNon- stationary 0.3566 (0,0,0) (Rural) yt 0.0000 (1,1,1) (Urban) ,ct Stationary 0.0001 (0,0,1) ,c(Urban) t 0.0000 (1,0,1) ,c(Urban) t 0.0005 (1,1,1) ,c (Rural) t0.0000 (0,0,1) Stationary ,c (Rural) t 0.0000 (1,0,1) ,c (Rural) t 0.0001 (1,1,1) ,y(Urban) t Stationary 0.0000 (0,0,1) ,y (Urban) t 0.0000 (1,0,1) ,y(Urban) t 0.0013 (1,1,1) ,y (Rural) tStationary 0.0001 (0,0,1) ,y (Rural) t 0.0000 (1,0,1) ,y (Rural) t 0.0020 (1,1,0) r (Urban) t Stationary 0.0003 (1,0,0) r (Urban) t 0.0014 (0,0,0) r (Urban) t 0.0035 (1,1,0) r (Rural) t Stationary 0.0005 (1,0,0) r (Rural) t 0.0006 (0,0,0) r (Rural) t To prove that the time series is stationary, it must meet the test results of three types of models (with intercept model, with a trend term model, both a trend and intercept term model) are stable. So, under the significance level of 0.05, from the p-value of Table 2, we could get ccyy0-order, that all the p-value of (urban), (rural), (urban), (rural) are over 0.05. So tttt every variable is non- stationary. However, after a lag, also from the p-value, every variable can 15 The Effect of Income and Interest Rates on the Consuming Behavior of Chinese Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural Areas pass the test, 1-order steady. Therefore equation (6) can be analyzed by quantitative analysis. 4.2 Econometrics Results The above analysis shows that all of the variables and explanatory variables are stationary time series of regression model, which meets the requirements to establish the mathematics model. Thus, we can use a mathematical model which was established to calculate, using Eviews second-order least squares to get the regression results shown in Table 3: Table 3 Regression Result Adjusted Set of instrumental variables (***) Intercept λ(*) θ(*) 2(**) R Category (OLS) urban rural urban rural urban rural urban rural 1.71 3.42 0.49 0.50 0.50 -4.84 0.59 0.87 None (.43) (.14) (.00) (.00) (.45) (.03) (.00) (.00) ,,,yyy,,4.59 5.23 0.33 0.48 -4.25 -7.01 0.61 0.83 ttt,,,234 (.15) (.13) (.07) (.00) (.13) (.04) (.00) (.00) rrr,,ttt,,,234 ,,,ccc,,4.77 5.28 0.35 0.48 -4.49 -7.10 0.60 0.83 ttt,,,234 (.13) (.12) (.07) (.00) (.12) (.04) (.00) (.00) rrr,,ttt,,,234 ,,,,,yyyyy,,,,ttttt,,,,,234560.79 5.90 0.50 0.48 -0.79 -6.20 0.64 0.82 ,,,,rrrrr(.78) (.09) (.00) (.00) (.72) (.04) (.00) (.00) ttttt,,,,,23456 ,,,,,ccccc,,,,ttttt,,,,,234561.90 5.90 0.47 0.48 -1.93 -6.20 0.63 0.82 ,,,,,rrrrr(.51) (.09) (.01) (.00) (.42) (.05) (.00) (.00) ttttt,,,,,23456 Note: (*) Under probability at 5% confidence level coefficient‘s The probability levels, that‘s t-value. (**) The probability levels under all regression coefficients are 0, F-statistics of the whole equation. (***) Instrumental variables set includes constant item, but save them in table. The first column is a set of selected instrumental variables the second column gives the intercept of two stage least squares (2SLS) regression results. The third column is the consumers‘ 16 Research of Mathematical Economics No. 1 2011 income proportion of total income, of which the consumers satisfy "Rule of Thumb". The fourth column is the interest rate‘s intertemporal elasticity of consumption. The last column is adjusted 2, while each column shows the coefficients of the measurement of regression of urban and R rural areas respectively, values in brackets are coefficient under t test‘s p-value. First, no set of instrumental variables‘ least squares regression results show that intercept and of the town are not significant, which in line with the results of earlier data analysis in , ,yr,,cr,this article; secondly, with a lag of two to six after the instrumental variables and , their least squares regression results of second-order show that urban with intercept and are , ,yr,not significant; finally, containing the lag two to four of the instrumental variables and ,cr,, the second least squares regression results of them are more stable. When considering the income proportion of the consumers satisfying "rule of thumb" is only of total income, although in no set the value of the results of the least squares instrumental , variables and the results with lags two to six after the second-order least squares instrumental ,yr,,cr,variables and of urban and rural areas are nearly equal, due to other poor regression results, they should be abandoned; Containing the lag two to four of the instrumental ,yr,,cr,variables and , after using the second-order least squares, (Urban) is about 0.33, , while (Rural) about 0.48, which means the urban consumers is greater rational expectations , than the consumers in rural areas, more generally, the higher income people get the higher proportion of rational expectations consumers will be (Campbell and Mankiw, 1989), so in order to expand consumption, firstly we should aim at increasing rural people's income, followed by increasing the incomes of the low-income people in urban. Considering about the interest rate‘s intertemporal elasticity, we found that all the results are showing that urban is lower than rural, while with the lag two to four of the instrumental ,yr,,cr,variables and (Urban) is about -4.3, while (Rural) is about -7.0, generally ,, lower interest rates can help to promote growth in consumption; The main reason is reflected in two aspects. Direct utility is mainly reflected in that lower interest rates reduce the opportunity cost of future transactions so that people will choose more current consumption, thus stimulate consumption; From the point of view of indirect utility, as the investment diversifies among urban residents, the risk can be passed on by other means, compared with single investment in 17 The Effect of Income and Interest Rates on the Consuming Behavior of Chinese Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural Areas rural areas, and thus the risk-shift capacity is weak, then a larger intertemporal elasticity appears. 5. Conclusions Through quantitative analysis of this paper, I think we can improve the level of consumption of Chinese urban and rural residents from the following aspects: (1) Lowering interest rates contributes to growth in consumption. Different from previous researches on consumption which come to the conclusion that consumption is not sensitive to interest rates, I believe that lowering interest rates is an effective measure of China‘s consumption. Of course, while lower interest rate brings growth to consumption, we also need to pay attention to the possible cost of the inflation. (2) It is highly recommended to continue sparing no effect on economic growth and improving national income levels, thereby stimulating consumption and thus promoting economy growth. Particularly, in the face of significant "short-sighted" behavior of the rural residents, great efforts should be made to improve the income level of rural population and poor urban population. (3) If the "short-sighted" consumer account for the main component of total consumption, and rapid growth in total income cannot be achieved, then as long as the benefits of consumption growth are greater than the cost of income redistribution, the redistribution of income may become a possible way to stimulate consumption demand. (4) It is suggested to establish a convenient and wholesome credit environment available to the public. In an incomplete market, residents have a great tendency to be "self-insurance", which is expected to increase savings in order to alleviate the impact of the uncertainty of income. Thus, a comprehensive credit market is conducive to smooth their consumption and reduce the uncertainty of the future, thus contributing to the increase in current consumption. Improving the credit market and the social security system are the fundamental way of expanding domestic demand. (5) The government should focus more on accelerating the establishment of market-oriented social security system, strengthening the support for social security funds, and maximizing the possible scope of coverage of the population to eliminate the worries of the residents and to change the residents of their expected future uncertainty. Because of the imperfection of credit 18 Research of Mathematical Economics No. 1 2011 market in our country, the fees in medical, education and other industries will strengthen the residents of their precautionary saving motive, thus inhibit the growth of consumption (6) The government also needs to vigorously promote the development of consumer credit markets to ease residents of their liquidity constraints. To be more specific, in the present circumstances, what we need to do is: firstly, to gradually realize mercerization of the interest rate market, thus to provide both supply and demand sides with a relaxed environment;secondly, to speed up banking reform and simplify loan procedures, and establish wholesome mechanisms of internal constraints; thirdly, to gradually improve individual credit system; and finally, to further improve relevant laws and regulations of consumer credit, thus to maintain the normal order for the credit economy. References 杭斌、申春兰, 经济转型期中国城镇居民消费敏感度的变参数分析, 数量经济技术经济研究第9期, 2004. 李焰, 关于利率与我国居民储蓄关系的探讨, 经济研究第11期, 1999. 龙志和、周浩明, 中国城镇居民预防性储蓄实证研究, 经济研究第11期, 2000. 施建淮、朱海, 中国城市居民预防性储蓄及预防性动机强度: 1999-2003, 经济研究第10期, 2004. 宋铮, 中国居民储蓄行为研究, 金融研究第6 期, 1999. 陶长琪、齐亚伟, 转轨时期中国城乡居民预防性储蓄比较研究——中国城乡居民消费的理论框架及实 证研究, 消费经济第5期, 2007. 万广华、张茵、牛建高, 流动性约束、不确定性与中国居民消费, 经济研究第11期, 2001. 汪红驹、张慧莲, 不确定性和流动性约束对我国居民消费行为的影响, 经济科学第6期, 2002. 袁志刚、宋铮, 城镇居民消费行为变异与我国经济增长, 经济研究第11期, 1999. 朱信凯, 流动性约束、不确定性与中国农户消费行为分析, 统计研究第2期, 2005. 朱春燕、臧旭恒, 预防性储蓄理论——储蓄 (消费) 函数的新进展, 经济研究第1期, 2001. Campbell, J. and Mankiw, G., ―Consumption, Income, and Interest Rates: Reinterpreting the Time Series Evidence‖, NBER Working Paper, 1989. Hall, Robert E., "Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence", Journal of Political Economy, 1978, 86(6), pp. 971-987. Hall, R., ―Intertemporal Substitution in Consumption‖, Journal of Political Economy, 1988, 96(2), pp. 339-357. Ng, Serena and Pierre Perron, ―Lag Length Selection and the Construction of Unit Root Tests with Good 19 The Effect of Income and Interest Rates on the Consuming Behavior of Chinese Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural Areas Size and Power‖, Econometrica, 2001, 69(6), pp. 1519-1554. (Responsible Editor: Xu Lingjue) (Proofreader: Wang Jue) 20 Research of Mathematical Economics No. 1 2011 The Influence of Wealth Gap on China’s Economic Growth Guo Yumei (School of Economics, Renmin University of China) Abstract: With the development of China‘s economy, the gap between the rich and the poor is increasingly severe day by day. People hold different opinions on the influence of the gap. This paper wants to use Cobb-Douglas production function as the basis of the model and introduces an income distribution coefficient t to represent Gini coefficient in order to show the positive effect of the gap of wealth. At the same time, we try to prove the positive effect of the gap through ternary linear regression based on China‘s GDP, capital and industrial income from 2003 to 2008. The positive effect of the gap through ternary linear regression based on China‘s GDP, capital and industrial income from 2003 to 2008 will be proved. Keywords: The Gap of Wealth, Economic Growth, Reinvestment, Gini Coefficient 1. Introduction As the reform and opening-up policy goes, China‘s economic makes a big difference. However, the gap of wealth has been widening day by day. China‘s GDP reached 34.05 trillion RMB in 2009 while it is just about 0.36 trillion in 1978, leaping to the third place in the world. The average real economic growth is over 9.8%, which is a rare occurrence in the development history. Though the world economy slowed down in 2008, the economic growth of China was still above 9%. At the same time, as the World Bank said, Gini coefficient of China in 2005 was over 0.42, which broke through the world cordon. The income of the first 20% people who gets the highest income is ten times much more than the income of the last 20%. The gap between rich and poor is a two-edged sword. On one hand, the expansion of the gap of wealth will contribute to economy. As the rule of diminishing marginal consumption, the gap will increase savings which will accelerate capital investment. The more capital is invested, the more rapidly economic will grow. On the other hand, the expansion of economic gap between rich and poor has inhibition and will cause social problems. Again according to the principle of diminishing marginal consumption, the poor consumption ratio is on the increase and rich 21 The Influence of Wealth Gap on China’s Economic Growth consumption ratio is reduced, which will lead to insufficient total consumption demand. Meanwhile, the gap has caused serious social phenomenon such as anxiety. When overnight and night laid-off appear frequently, people begin to come up with uncertainty about the future and are discontent with the gap. Scholars have different opinions on the influence of wealth gap on China‘s economy. Sun Liping says it is the insufficiency of consumption demand that matters instead of increasing reinvestment. Combining static and dynamic analysis method, time series data and section data, Liu Ying thinks about the relationship between income distribution and economic growth of Beijing based on its Gini coefficient and GDP and she finds the positive correlation. But Xiang Shujian finds the unemployment rate is the most prominent among the four explanatory variables of unemployment rate, the first industrial added value of GDP, social relief pension financial expenditure and average wage growth in relation with Gini coefficient through multiple regression models. From this, he concludes that if the income distribution is even or not has nothing to do with economic growth. However, according to current situation of China‘s economic development, a widening gap between rich and poor influences economic growth dominated by promoting effect in general. As we all know, China is in and will remain so for a long time of socialist development stage. Therefore, the effect of capital accumulation and capital reinvestment on economic development is rather obvious. Since 2005, total capital formation of GDP by expenditure has been above 45%. Although many scholars believe that the promoting influence of capital is no longer as significant as it is in 1990s, the negative effect caused by insufficient domestic demand cannot be compared to the positive effect brought by capital. Thus, this paper would make some improvements on the basis of past research. First, we will use Cobb-Douglas production as the basis of the model and introduce an income distribution coefficient t to represent Gini coefficient. With the rule of diminishing marginal consumption, we take the first-order of production function to intuitively see how the widening gap influences economic growth. So far, many researches are based on Solow model which focuses on capital per unit labor and output per unit labor. The author thinks that capital per unit labor is not convenient to describe the gap between rich and poor. Capital per unit labor is the average of capital in labor, which is contradictory to reflect the difference in wealth. Thus, we introduce t coefficient to make the model simple and intuitive. Second, we will take a ternary linear 22 Research of Mathematical Economics No. 1 2011 regression based on China‘s GDP, capital and industrial income from 2003 to 2008 in order to see the correlation between the gap and economic growth. Instead of word describing, we show the comprehensive effect of wealth gap in econometrics method. The structure is as follows: the first part is mainly about the situation and impact of the gap between rich and poor and the author‘s opinions and perspective; in part two, we would build a simple model to find the function of economic growth involving GDP, capital and the gap in order to prove that the expansion of the gap of wealth contributes to the reinvestment of capital and it will promote the economic growth; in part three, we try to prove the positive effect of the gap through ternary linear regression based on China‘s GDP, capital and industrial income from 2003 to 2008 according to the function in the second part; finally, we will comprehensively make some conclusions and properly make suggestions. 2. Theoretical Model From the standpoint of capital accumulation, we try to prove that the expansion of the gap of wealth will contribute to the reinvestment of capital and it will promote the economic growth in the end. We would like to build a simple model to represent the economic environment with the view of that inequality will increase output because Gini coefficient is in direct proportion to the rate of capital reinvestment and the growth rate. Alien from the previous research, we want to make some assumptions as below: Firstly, the society is made up of two groups and each group may have one person or more and we regard each group as a labor unit. But in this paper, we just want to focus on the gap of wealth between the two groups regardless of the gap within each group. Secondly, the two groups produce the same product while we don‘t distinguish wage income or capital income. The way to distribute income between the two groups is on the basis of the ratio of capital each group has. Thirdly, we introduce the t index to represent the gap so that we can indirectly see the relationship between Gini coefficient and capital reinvestment. Fourthly, loan market is unavailable and we don‘t need to take r into account. Lastly, we assume a concave consumption function, which indicates that the richer group has higher marginal propensity to save than the poorer group. In the model, we will begin by considering the Cobb-Douglas production function: 23 The Influence of Wealth Gap on China’s Economic Growth ,,, Y,KL 0,,,1,0,,,1where . Since the society is made up of two labor units, we can change the production function into that: , Y,BK And we assume it is a concave returns to scale production function. Also, we can reach that: Y,Y，Y12 K,K，K12 As we distribute income between the two groups on the basis of the ratio of capital each has is no more than the capital group has. And group has, we assume that the capital group ??the proportion of group ?‘s capital is t so that . Thus, 0,t,0.5 K,tK1 K,(1,t)K2 Y,tY1 Y,(1,t)Y2 KK,tKIf the total capital is at beginning, the group has and group has ??010 K,(1,t)K. So the output or income is distributed as: 20 ,Y,BK00 Y,tY 1,00 Y,(1,t)Y2,00 At this time, each group begins to decide how much to make consumption and how much to reinvest. Due to the concave consumption function, the proportion of income group ? makes reinvestment is lower than that of group ?. If we assume that p is the reinvestment proportion, q is proportion of income to the total income, p is the function of q and Y, , and pqY()pqY()is an increasing function involving . Thus, qY ,K,p(tY)tY100 ,K,p(1,t)Y(1,t)Y,, 200 ,,,K,,K，,K,p(tY)tY，p(1,t)Y(1,t)Y120000 24 Research of Mathematical Economics No. 1 2011 From the production function, we obtain, ,Y,K,L,，,, YKL = 0 and For L = 2, then ,L ,Y,K,, YK KThus, when the capital is , the relationship between reinvestment and output growth is 0 showed below: ,,p(tY)tY，p(1,t)Y(1,t)Y,Y,K0000,,,, (1) YKK000 KThis is a growth function of t when is given. In fact, t represents the gap of wealth. 0 Due to , the gap of wealth is larger when t converges to 0 and smaller when t 0,t,0.5 converges to 0.5. So we can obtain: , , (2) tGGini,()GGini'()0,00.5,,t which means that t is inversely proportional to Gini coefficient and the function is monotonic decreasing. Then, make derivation of function (1): ''2'2,,p(tY)tY，p(tY)Y,p,,,,(1,t)Y(1,t)Y,p(1,t)YY,Y00000000,, ,,,,YK00,, Here, we assume that is linear, pqY() p(qY),a*t*Y，b where a > 0, then, ' p(qY),a Thus, '22,,,,atY，p(tY)Y,a(1,t)Y,p(1,t)YY,Y000000,,,,,,YK,,00 22,,atY,a(1,t)Y,,，p(tY)Y,p,,(1,t)YY000000,,K0 where , then . Thus, 0,t,0.50,t,1,t 25 The Influence of Wealth Gap on China’s Economic Growth 22atY,a(1,t)Y,0 00 Given that is an increasing function involving , pqY()qY ,,p(tY),p(1,t)Y 00 ,,p(tY)Y,p(1,t)YY,0 0000 With all above, we can conclude that '22,,atY，p(tY)Y,a(1,t)Y,p(1,t)YY,,,Y000000,, (3) ,,,0,,YK00,, ,YAnd only when t = 0.5, we can obtain the equation. So, we can learn that is Y0 ,Ymonotonic decreasing about t, which means that the smaller t is, the larger is and with (2) Y0 ,Yand (3), is directly proportional to Gini coefficient. Thus, the expansion of the gap of Y0 wealth will contribute to economic growth. 3. Empirical Research Based on Chinese Economy From the simple model in part 2, we can conclude that, theoretically, the expansion of the gap of wealth would contribute to reinvestment which will result in the growth of economy. YBesides the influence of t index, has an effect on reinvestment next year. Thus, we‘d like to 0 find the empirical evidence of the relation between the gap and economic growth with the data of capital, GDP and Gini coefficient. From the model, we can learn that: ,,p(tY)tY，p(1,t)Y(1,t)Y,Y,K0000,,,, YKK000 tGGini,() Economic growth rate is a function involving GDP, total capital and Gini coefficient. Thus, we try to find the relationship between growth rate and Gini coefficient adopting linear regression. 26 Research of Mathematical Economics No. 1 2011 We would take growth rate as the dependent variable and GDP, total capital and Gini coefficient last year as independent variable. First of all, we will show the method of calculating Gini coefficient. Firstly, population is divided into N groups and was labeled by the average wage of each group on AAA,,,12n lower-to-upper motion. Then we can get the percentage of the first i group‘s total wage named and the percentage of the first i group‘s total population named . and are equal YXYXii00 to zero. After that, we get the equation of calculating Gini coefficient with its definition. Graph 1 Lorenz Curve nn1111，，XXYYXXYXXYY,,,，,,,，，，，，，，，，，，,,,,,,1111,,11iiiiiiiiiii,,2222，，,,11ii Gini,,11 22 Thus, with the wage and population of 19 industries, we can obtain Gini coefficient from 2003 to 2007. By now, specific date is showed below, 27 The Influence of Wealth Gap on China’s Economic Growth No. G (growth rate) K (capital) Y (GDP) Gini 1 0.17710944 0.268341054 135822.8 0.472367 2 0.156738594 0.259603788 159878.3 0.46738 3 0.169662816 0.239086846 184937.4 0.461608 4 0.228814633 0.24841952 216314.4 0.457987 5 0.181464375 0.258545672 265810.3 0.450935 The result of linear regression by SAS is that g = -8.77045 + 0.000018K - 8.23E - 6Y + 19.17453Gini (0.5984) (7.666E-7) (3.965E-7) (1.241567) 2,2N=5, , R,0.99852R,0.99408 2,2From and , we can learn that GDP, total capital and Gini coefficient explain RR almost all the changes of growth rate. The degree of fitting is high enough to guarantee credibility of the regression. By the way, Gini coefficient has a positive correlation with the growth rate that the growth rate will increase about 20% if Gini coefficient increase 1%, for the t-value is 15.44. So we can reject H at significance level of 5%. 0 Summarily, we can make a conclusion based on the regression that the gap will contribute to economic growth. 4. Conclusions From the paper, we can conclude that: (1) Based on the model and regression, the widening gap of wealth will contribute to China‘s economic growth in nowadays; (2) The Gini coefficient in China is about 0.45, which has crossed the international warning line of 0.4. Therefore, the gap of wealth is a two-edged sword. We cannot blindly negative the gap between rich and poor. Of course, the increasingly widening gap may cause serious problems because Chinese regard common prosperity as development goals; meanwhile, complete equality is not desirable as it will slow growth steps down. Thus, state and local government should take 28 Research of Mathematical Economics No. 1 2011 some measures to maintain the widening gap in a modest and relatively stable level. From part three, we can learn that the gap in China is about 0.45. According to the UN organizations, income is absolute average if Gini coefficient below 0.2, while compare average if 0.2-0.3. Gini coefficient of 0.3-0.4 means relatively reasonable and 0.4-0.5 indicates a bigger gap. And income distribution is extremely bad if Gini coefficient is over 0.5. So, the gap is bigger in China. Enjoying the fast economic development, the government should take timely measures, such as tax reform policies, to actively deal with the widening gap and try their best to guarantee the GINI coefficient around 0.3 to 0.4. Together with the fast development, the government also needs to stop the reproduction of economy. However, we wonder if the gap of wealth in China is reasonable. China is a socialist country, with economic development and common prosperity for a common goal. Given that common prosperity is established on the basis of development, to realize common prosperity must develop economy. This will inevitably cause a widening gap between rich and poor. It is easy to understand that in certain period the richer get richer and the poor get poorer. It is said in Bible Matthew: a king gives 1000 coins to three people. After a period of time, the first person changes 1000 coins into 5000 coins and the second person gets 2000 coins while the last person is still holding the 1000 coins. Therefore, the king gets back the 1000 coins and gives it to the first person. It is the famous Matthew Effect, which seems unfair but is reasonable indeed. For one, people who have a lot are almost people who have succeeded. They obtain social recognition and trust, which makes them easier to get more chances. Conversely, people who get little social recognition and trust have less opportunities. At the same time, rich people are more able to make big investments. As the effect of capital accumulation is not simply the sum of unit capital, the expected income of big investment is much higher. This is obviously one of the reasons why the gap is widening. However, from the angle of efficiency and fairness, these reasons are anyhow reasonable. References 郭熙保, 经济发展: 理论与政策, 中国社会科学出版社, 2000. 李实、赵人伟等, 中国经济改革中的收入分配变动, 管理世界, 1998(1): 43—56. 刘长庚、吕志华, 改革开放以来我国居民边际消费倾向的实证研究, 消费经济, 2005(8): 21-4. 29 The Influence of Wealth Gap on China’s Economic Growth 刘红慧, 转型期经济增长与收入分配关系研究, 硕士学位论文. 刘颖, 北京市收入分配与经济增长关系实证研究, 硕士学位论文. 权衡, 收入分配与经济增长的相关性现代分析框架及其争论, 社会科学, 2004年第6期. 王德祥、李建军, 财政分衩、经济增长与外贸依存度----基于1978-2007年改革开放30年数据的实证分析, 开放战略探索三十年回归与展望. 向书坚, 我国城镇居民收入分配差距变化定量分析?, 当代经济科学, 1998(20): 1-5. 杨俊、张宗益, 收入分配与经济增长近代西方理论与实证考察, 重庆大学学报, 2002, 25(10). 叶礼奇, 基尼系数计算方法, 中国统计, 2003(4). 张汉斌、刘庆华, 浅谈贫富差距对经济增长的影响》, 邢台职业技术学院学报, 2000, 17(2). 朱平, 贫富差距的合理性及其限度, 南京师大学报 (社会科学版), 2001(05). Alesina A., Perotti R., ―Income distribution, political instability and investment‖, 1996(06). Alesina A., Rodrik D., ―Distributive politics and economic growth‖, 1994. Barror, ―Government spending in a simple model of endogenous growth‖, 1990(02). Barror, ―Inequality and Growth in a Panel of Countries‖, 2000. Bourguignon F., ―Pareto superiority of unegalitarian equilibria in Stiglitz's model of wealth distribution with convex saving function‖, 1981(49). Deininger K., Squire L., ―Measuring income inequality: A new data-base‖, 1996(03). Deininger K., Squire L., ―Inequality and growth: Results from a new data set‖, 1998. Persson T., Tabellini G., ―Is inequality harmful for growth? Theory and evidence 1994 development‖, 1999. Romer P. M., ―Increasing Returns and Long-Run Growth‖, 1986. Stiglitz J. E., ―The distribution of income and wealth among individuals‖, 1969. (Responsible Editor: Xu Lingjue) (Proofreader: Wang Jue) 30 Research of Mathematical Economics No. 1 2011 Bundling Sales of Information Goods: Models and Analysis Shi Fangning (School of Economics, Renmin University of China) Abstract: This paper studies the impact of bundling sales on merchandising of information goods. Our analysis focuses on seller‘s profits and consumer‘s surplus, as well as the optimal bundle prices. We consider two common bundling types: bundling of single-product in amount and bundling of multiproduct. For the two cases, we develop models to calculate the optimal pricing and its condition, profits and consumer surplus. Our analysis implies that, under certain situation, bundling sales yields more profits, but harms the consumers‘ interest. Keywords: Information Goods, Bundling Sales, Price Discrimination, Optimal Pricing 1. Introduction Information good is a type of commodity aiming at spreading information with its main value derived from the information it contains. In essence, any products which can be digitized are information goods, such as newspapers and magazines, movies and music. Usually, information products also contain information service, that is, the seller provides services like collecting and processing information, or provides a platform where users can share information, such as search service on web, mailbox and MSN. In this paper, we study only information goods which can be digitized and traded, like software, newspapers, database, financial information and research reports and so on, but not information service. Information goods have several characters which make themselves distinctive from other commodities. First, values of information goods depend on individual‘s preference. Different consumers value the same information goods bundle differently. For example, individual‘s knowledge, the ability to use the commodity, the necessity of the information for the person, these factors will all affect individual‘s value of the information goods. Consumers do not consider the costs or the sources of the information but the utility they bring. Second, information goods have the feature of high fixed costs and low marginal costs. Obviously, it determines that, 31 Bundling Sales of Information Goods: Models and Analysis to make a lucrative investment, information goods cannot be sold by its marginal price. This character also means that there exists explicit scale economy, that is, the more the information goods are produced, the lower the average cost is. So it is more proper to bundle a large number of information goods than other commodities. On the other hand, this character also leads to natural monopoly within this industry. Third, information goods have strong network effect. Growing market penetration or access to the products helps improve their function. For instance, more users make software spread faster, and also lead to better function. Bundling sales is a merchandising strategy that is often used by sellers to conduct price discrimination or enhance their monopoly power. It means sellers provide goods in packages, which can be the same goods provided in quantity-dependent packages or two even more different goods provided in a package. Bundling sales is very common in daily life: airline companies provide both single-way and round-trip tickets; banks offer a series of comprehensive service which cannot be divided; restaurants usually provides complete dinners which contains several different dishes; and daily commodities like toothpaste and detergent are sold in small and large packages. As for information goods, the cases of bundling sales are also very popular. In this paper, we divide them into two broad classes: bundling of single-product and bundling of multiproduct. Bundling of single-product means sellers provide the same goods in quantity-dependent packages. When providing commodities like internet access and messages, sellers usually charge in this way. For example, the following tables are price policies of China telecom Beijing Corporation 1for 3G cellphones‘ charge for Internet tariff and messages. 1 Usually when consumers purchase for the next month‘s cell phone services, they will always avoid using the service exceed the amount contain in the bundle. So we see the price policies as bundling sales, but not two-part pricing. 32 Research of Mathematical Economics No. 1 2011 Internet Tariff of the Bundle Basic Tariff (per month) Tariff for Excessive Amount (per month) 0.005yuan/KB,with upper limit 5 yuan 30MB 500 yuan 10 yuan 100MB 20 yuan 200MB 50 yuan 1G 100 yuan 2G sources: Amount of Messages of the Bundle Basic Tariff (per month) Tariff for Excessive Amount (per month) 5 yuan 60 0.10 yuan per message 10 yuan 130 20 yuan 200 50 yuan 300 100 yuan 500 sources: Bundling of multiproduct means the seller provides two or more different goods which are similar to each other in a package. For example, Microsoft sells its office software in bundle; some websites provide financial information and research reports of stock, time-bargain in bundles; sales of music discs and newspapers are also cases of multiproduct bundling. So far, theoretical research for bundling sales concentrated on two aspects. One is to use bundling to conduct price discrimination and extract more consumer surplus. The phenomenon of bundling sales was firstly referred to by Burnstein (1960) and Stigler (1980). After that, utilizing the framework of Stigler, Adams and Yellen (1976) analyzed bundling strategies of the single-product monopolist in a duple-product market. They compared the three ways of bundling: pure components strategy (set the single price on each commodity separately), pure bundling (offer two commodities for sale only in a package), mixed bundling (offer each commodity separately and a package of both). They illustrated that in most situations, various types of bundling can be more or less profitable than unbundling sales. Following this, Schmalensee (1984) extended their model by assuming that the distribution of reservation prices follows a 33 Bundling Sales of Information Goods: Models and Analysis bivariate normal distribution. He confirms that in the mixed bundling case, it is always more profitable than unbundling sales if the two goods are complements. Also extending the Adams and Yellen model, McAfee, McMillan and Whinston (1989) confirmed mixed bundling is often more profitable than pure bundling, and also provided a general sufficient condition for the situation when mixed bundling is optimal in a duple-product case. Salinger (1995) developed a graphical analysis of bundling and illustrates the benefits of bundling depends on the cost of the individual products, its relative magnitude to the reservation prices and the correlation of demands between the products. More recently, McCardle, Rajaram and Tang (2007) studied the impact of bundling products on retail merchandising. They established conditions and insights under which bundling is profitable, and confirmed again that bundling profitability depends on individual product demands, bundling costs, and the nature of the relationship between the demands of the products to be bundled. The other aspect is to use bundling as a market strategy, so that a monopoly firm can enhance its monopoly power as well as extend its power to other newly emerged relevant markets. M.D. Whinston (1990), S. Martin (1999), Carlton and Waldman (2002) have discussed on this problem elaborately and derived some important results. But in this paper, we do not discuss effects of bundling in monopoly power, so we don‘t talk about these theories. However, multiproduct of large bundles has received little attention until recent years. Armstrong (1996) showed that in some certain cases, the optimal pricing in the multiproduct case can be determined using the techniques typically used in the single-product case. He found that the optimal bundle price will exclude some low-demand consumers. Bakos and Brynjolfss (1999) studied the strategy of bundling a large number of information goods. They found that when the marginal cost of information goods is close to zero, as the number of bundling goods increases, the deadweight loss per good and the consumer‘s surplus per good for a bundle converge to zero, and the seller‘s profit per good increases to its maximum value. That is to say, bundling of large amount of information goods is profitable. By assuming that consumer types and consumer valuations are discrete, they also found that, in many cases, consumers can be induced to reveal information about their valuations through their choices by offering them a menu of bundles at different prices. Thus the monopolist may gain profit by pursuing a mixed bundling strategy of offering several bundles, each including a subset of the available information goods. Our paper differs from the previous work in two aspects. Firstly, in this paper, we consider 34 Research of Mathematical Economics No. 1 2011 two different types of bundling sales, bundling of single-product and multiproduct, in terms of price discrimination. For single-product case, we refer to Jean Tirol‘s (1998) model of optimal nonlinear pricing and compare the optimal prices, profits, consumers‘ surplus and net welfare of the total society of bundling and unbundling cases. We get one of our results, that is, for the single-product case, bundling sales can bring more benefits to sellers, but reduce the consumers‘ surplus and total welfare of the society in certain circumstances. For the multiproduct case, by assuming the consumer types and valuations are continuous, we extend Bakos and Brynjolfss‘s (1999) model, which discusses two ways to segment the market and compare them with pure bundling and unbundling sales. We get the other result that under certain circumstances, price discrimination through mixed bundling is profitable and extract the consumers‘ surplus more efficiently than pure bundling and unbundling sales. Secondly, our results imply that, in the single-product case, the effect of bundling is relevant to the difference between the consumers‘ types. The bigger the difference is between the consumers‘ types, the more consumers‘ surplus the seller can extract by bundling sales. In the multiproduct case, the effect of mixed bundling is dependent on consumers‘ average valuation of products in package and the degree of discount the seller set. The two bundling patterns can only efficiently segment the market and make profits when consumers‘ average valuation is not too high and the seller set the discount large enough. This paper is organized as follows. In the second section, we present our model and analysis of sing-product bundling sales. We use this model to determine the optimal bundle prices and prices of unbundling sales, seller‘s profits and consumers‘ surplus. We establish conditions under which bundling is profitable. In section 3, we present our model and analysis of multiproduct bundling sales. We discuss the ―quality discount‖ and ―quantity discount‖ of bundling separately and repeat this analysis. In section 4, we present some intuitive explanations for the results we get. In the conclusion section, we summarize our key results and provide future research directions. 2. Model and Analysis of Single-Product Bundling When provide commodities like internet tariff and cellphone messages, sellers usually provide different bundles containing different amount of products according to various consumer types. Consumers choose the bundles which maximize their utility. In this way, sellers can 35 Bundling Sales of Information Goods: Models and Analysis segment the market and extract more consumers‘ surplus. Now, we develop a model to illustrate that the bundling strategy is more profitable than unbundling sales. To simplify the model, we assume that the sellers only provide two bundles qqwhich contains and amount of goods, aiming at high-demand consumers and 12 low-demand consumers separately. Consumer types are continuous, denoted by taste parameter ,. Usually the information goods tend to have zero or very low marginal costs of ,,,,,,,, production, so in our model, we ignore the effect of marginal costs. 2First, we assume consumers have the following utility function: qU,,V(q),T , if they purchase units; , if they purchase nothing. U,0 ,,,V(0),0V(q),0V(q),0where , and (that is, in this utility function, consumers have qT(q)decreasing marginal utility); is the consumers‘ total expenditure of the units of goods; V(.) is consumer‘s taste parameter, differs from one person to another; is identical for all , the consumers. Then, assume taste parameter follows uniform distribution in , with ,,,,,, ,f(,)f(,)d,,1 being the uniform density. Setting , it is easy to know that ,, 1,f(),. ,,, In this model, we can get the following two propositions in the single-product case: For information goods, a single-product bundling sale is more profitable Proposition 1: than unbundling sale. As long as is large enough, consumers‘ surplus under bundling sale Proposition 2: ,/, is less than that under unbundling sale; at the same time, bundling sales reduce net surplus of the whole society. Now we present the proof. 2.1 Bundling Sales 2 Refered in Jean Tirole, 1994, Industry Organization Theory (7e). 36 Research of Mathematical Economics No. 1 2011 If the seller provides bundle I with q amount of goods and price of T to consumers 11 ,,,,,,,with taste parameter , and provides bundle II with q amount of goods and price 02 T to consumers with taste parameter , in which , then the profit of of ,,,,,,,,,,/220 the seller is: ,,0 (1) ,,Tf(,)d,，Tf(,)d,112,,,,0 The seller will maximize this profit under two kinds of constraints. The constraint of the first kind requires consumers to be willing to purchase, that is, the individual rationality constraints. ,,,VqT()0,,,,,,,, if ; 011 ,VqT()0,,, if . ,,,,,,,220 That is, ,(),,0VqT,11 (2) ,(),,0,VqT022, The constraint of the second kind requires that the consumers do not exercise personal arbitrage. These are known as ―incentive-compatibility constraints‖. In particular, the high-demand consumers should not want to consume the low-demand consumers‘ bundle and vice versa. So, ,,,,,,,,V(q),T,,V(q),T, if ; 01122 ,V(q),T,,V(q),T, if . ,,,,,,,22110 From these constrains, we get, T,T21,, (3) 0V(q),V(q)21 ,Substituting for in (1) using the formula (3), we yield the function of profit in term of 0 TTqq, and ,: 1212 37 Bundling Sales of Information Goods: Models and Analysis ,,,,,,,TfTf,(),(,)，(),(,)11020 ,,，，，，T,TT,T12121,,TT,,,，,,,,,,,,12,,VqVqVqVq(),()(),(), (4) ，，，，2121,, 2，，，，TT,121,T,T,,,,,21V(q),V(q),,,21，， ,TTDifferentiating in respect to and separately, we have 112 ，，,,2(T,T)11121，，,,,,，,,2,,, 0,,,TV(q),V(q),,,,,,121，， ，，,,2T,T，，11121，，,,,,,,,,2, 0,,,TV(q),V(q),,,,,,221，， ,,1,0It is obvious that always satisfies. So combining with the inequality (2), the ,T1 seller will choose T,,V(q). (5) 11 ,,1,0Setting , we get ,T2 1,,,/2,,T,V(q),V(q),,，,V(q), ; (6) 022112 3TTThe seller maximizes his profit at this point. Substituting for , in function (4) 12using (5) and (6), we can derive the maximum profit: 22,,,(,2) ,,,V(q),,V(q)max121,,,,4(,)4(,) Consumers‘ surplus is: 2,,,213 The second-order condition is satisfied, for ,,02,T，，,,,,,,V(q),V(q)22138 Research of Mathematical Economics No. 1 2011 ,,0,,,,,,,,,,CS,V(q),T,f()d，V(q),T,f()d11122,,,,0 ,,10，，,,,,,,,,,,,,V(q),V(q),f()d，V(q),(V(q),V(q)),V(q),f()d112211,,,,,,02，， 21111111,,222,,,,,,,,,,,V(q),(，,，,)，V(q),(,),,100200,,222222,,, 2221,,,,,4,，，,,V(q)，,V(q),,,,12，，8,,, 2.2 Unbundling Sales pT(q(,)),pq(,)Assume the seller sets price at for per unit of goods, then , and U(q),,V(q),pqq(,)each consumer chooses to maximize his own utility: . ,,,U(q),,V(q),p,0p,,V(q)Setting, we get, the profit is ,. ,,pq(,),f(,)d,2,, pThe seller chooses to maximize this profit. ,,，，,,q2p，，,q,f(,)，pf(,)d,,0Setting , we can get the optimal value of . ,,,,,p,p，， Consumers‘ surplus is, ,, CS,,(V(q,(),T(q,())f(,)d,,(,V(q(,)),pq(,))f(,)d,2,,,, 2.3 Comparison 21(1),,q,V(q),1,qTo simplify the model, let , and , ()Vq,2 (1) If the products are sold in bundles, the profit is, 39 Bundling Sales of Information Goods: Models and Analysis 22,,,(2),,()(),,Vq,,Vq121,,,,4()4(),, 2222,,,2q,q2q,q(2),2211,,,,224,(,)4(,,),, 4qqq,1q,0The seller choose and to maximize the profit, that is, , . We get 2112 the maximum profit: 22,,1, ,,,,max124(,,,)8(,,,) surplus is, Consumers‘ 2221,,,,CS,4,,,V(q)，,V(q),,,,，，112,,8,，， 22,,1,,,2，，8,,,16(,,,) p,,(1,q)(2) If the products are sold separately, the adverse demand function is , the profit 2,,,,,pp(lnln),,,,,,,,,,pqf()dp(1)f()dpis . 2,,,,,,,, ,,,,,,2(ln,ln)p,2,1,,0p,Setting , we get , ,p,,2ln(,/,), ,,,,,,,,,,,,,. max22ln(,/,)4ln(,/,)4ln(,/,) Consumers‘ surplus is, 2,,12,qq,,,,,,,((())())()(),,,,,CSVqpqfdpqd2,,,,,,2, ,,13(),,,(),，,48ln(,/,) 4 We do not take the cost of production into consideration, so the values ofanddo not affect the result we qq12derive. 40 Research of Mathematical Economics No. 1 2011 (3) Now compare seller‘s profit with consumers‘ surplus under the two cases. 22,,2(,),,,,,,ln(/),Setting , we get . ,,,,,,0212,,,,8(,)4ln(/),Rearrange the inequalities, we have 2. 2(1,,/,)，ln(,/,),0 2Let‘s define , , so F(x),2(1,x)，lnxx,,/, 11,F(x),,4(1,x)，,4x，,4. xx 1,F(x),0,/,x,,Obviously, always satisfies for , so we have 2 1F(x),F(),0. 2 ,,,That means , so bundling sales is always more profitable than unbundling sales. 12 Now compare the consumers‘ surplus. 2，，,,,13,，，,,CS,CS,,(，),,0,,Setting . 1248,,,,16(,)ln(/),,，， Rearranging the inequalities, we get 22. (7) ,,，，6(,/,,1),3,/,,4,ln(,/,),0 22G(x)Let‘s define , . Differentiating by G(x),6(x,1),(3x,4)lnxx,,/, x, we have 142,G(x),12(x,1),6x,lnx,(3x,4),,9x,6x,lnx，,12. xx ,，，GxIt‘s easy to know that is always less than zero when . So as long as x,2,/, CS,CSis large enough, inequality (7) can be satisfied, which means . 12 ,，CS,，CSAs for the total social surplus and , 1122 41 Bundling Sales of Information Goods: Models and Analysis 22,,,,,,13(),,,,,,()，CS,,CS,，,,，，1122,,,,,,,,488()16()4ln(/)ln(/),, 2,,,311(),,,,,(，)，4816,(,)ln(,/,), ,，CS,,,CS,0Setting , we get 1122 22 ，，，，，，2(,/,,1)，4ln,/,,,/,ln,/,,0 22Define , . T(x),2(x,1)，4lnx,xlnxx,,/, So, 24,x4 ,T(x),4(x,1)，,lnx,2x,3x,4，,lnx,2xxx 4,,T(x),1,,2lnx,0 2x ,xT(x),0T(x),0When is large enough, we have ,, that is, as long as is ,/, ,，CS,,，CSlarge enough, there is . 1122 So, we can know the quantity-dependant bundling of single-product is profitable. Moreover, as long as the value of is large enough, bundling sales will harm the ,/, consumers‘ surplus and even reduce the total surplus of the society. 3. Model and Analysis of Multiproduct Bundling In this section, we consider bundling strategies for multiproduct. Bundling of multiproduct is also a widely-used market strategy. For example, music disc is a kind of bundle which contains certain amount of songs; newspaper, like music disc, is also composed of various kinds of information. Bakos and Brynjolfss (1999) illustrated that under the condition that consumer valuations are independent, if the sellers bundle large amount of information goods together, the law of large number implies that the total willingness to pay the bundle of consumers can be similar. By setting the bundling prices, the sellers provide their information goods at the average willing to pay of consumers. Thus as the number of bundling goods increases, the deadweight 42 Research of Mathematical Economics No. 1 2011 loss per good and the consumers‘ surplus per good for a bundle converge to zero, and the seller‘s profit per good increases to its maximum value. Based on this result, they consider two ways for the seller conducting price discrimination. One of them is ―quality discount‖, that is, to remove or degrade a feature that disproportionately affects high-demand consumers. By offering appropriately-priced bundles with and without this feature, the seller may be able to infer consumers‘ expected valuation for the bundle based on their purchasing decisions. The other way is ―quantity discount‖, that is, to leave certain items out of some bundles. For example, the seller can offer an ―economy‖ bundle at a reduced price that is a subset of ―premium‖ bundle. By assuming the consumer types and consumer valuations are discrete, Bakos and Brynjolfss found that, when the type of consumers satisfies certain conditions, the two bundling patterns can efficiently segment the market and thus make more profit. In this section, we extend Bakos and Brynjolfss‘s model to a continuous case and prove that under certain circumstances, those bundling strategies can still be more profitable, and extract the consumer‘s surplus more efficiently than pure bundling and unbundling sales. ,,Suppose consumers‘ types change continuously, which is denoted by . follows the f(,)f(,),1,,0,1uniform distribution on an interval . Let be the density and . The higher ,the value of , the higher the demand of consumers is. Consumers have either a high or a low vvvaluation for each information good, respectively denoted by and . Consumers of type HL ,1,,(,),(,)vv value each good at with probability , and at with probability . HL ,(,),(,),,,,0,1Assuming that is strictly increasing on the interval . When , . ,,00 ,,,，，，，，，，，,,,01,,,,,,,Additionally, we suppose and . Then we can know, when ,bundling large amount of goods, consumers of type value each good in the bundle at an ，，，，，，,,v，1,,,vaverage . HL If the seller conduct unbundling sales, he may set a high price and selling only to the high vdemand consumers separately; that‘s to say, he chooses the price so that the profit is H 1,S,0, and the consumers‘ surplus is . If not, he may set a low ，，，，,,,,f,d,,vs1sH1,0 ,v,,vprice instead and sell to everyone. In this case, the price is , profit is , and the sLL2 43 Bundling Sales of Information Goods: Models and Analysis 1consumers‘ surplus is . ，，，，，，S,v,v,,f,d,sHL2,0 If the seller conducts pure bundling sales, the seller‘s profit is 15, ，，，，，，，，，，,,,,maxf,d,,,,v，1,,,v,,v，1,,vBHL0H0L,, Consumers‘ surplus is: 1,,,,,,,,,，,,，,Svvfdvv,,，，，，，，，，，，,,11BHL0H0L,0 1，，,，，，，,,vv,,d,,HL0,,,0，， Now we will discuss the following two ways of price discrimination. 3.1 Quality Discount Let the seller offer a ―degraded‖ bundle where the value of goods for the high-valuation ,,(,),(,)vconsumer is reduced to intended for consumers of type . Assuming that is H ,(,),,,(,),1,,0,1strictly increasing on the interval , and when ; when ,,00 ,,,(,),0,(,)v,,(,)v. Also satisfies , . ,,1LH In this model, we can get the following proposition: 1When the two conditions, and Proposition 3: ，，，，，，,,,,d,,max,,,12,0 ,,,,,v，，000L,,,,,，， are satisfied, conducting price discrimination by quality discount ,,vH,,00 will bring the seller more profit and reduce the consumers‘ surplus at the same time. 6,,,,0,1,,, selecting arbitrary , we know that for every , there are two Proof:010 kinds of constraints. To satisfy the individual rationality constraints, we have, ,,d,B5,，，，，，，，， By differentiate , we get ,,, so set , that ,,1,,,,,,,v,v,v,0,,0BHLLd, means the seller choose to set a low price and sell to everyone. 6 Case of can be analyzed in the same way. ,,,10 44 Research of Mathematical Economics No. 1 2011 ，，，，,,p(,),,(,),,v，1,,,v 000H0L ，，，，,,p(,),,(,),,v，1,,,v. 111H1L To satisfy the incentive-compatibility constraints, we have, ，，，，，，，，，，，，,,,,,(,),,v，1,,,v,p,,,(,),,v，1,,,v,p,00H0L001H0L1 ，，，，，，，，，，，，,,,,,(,),,v，1,,,v,p,,,(,),,v，1,,,v,p, 11H1L110H1L0 ，，p,，，The seller will choose and p, as large as possible which satisfies the four 01 constraints. So we get ，，，，,,p(,),,(,),,v，1,,,v 111H1L ,,,,,,,,,,,,,p(),()，,()v，1,v，，，，，，，，，，,,,,000101H1L ，，，，，，，，,,,v，()v，,,,()v,v,,,,,,,,,,,,L00H101H1L ,Because is chosen arbitrarily, so 1 ，，,,，，，，，，p,(),,,v，,(,),,v，min,,,,,,(,),,v,,,v0L00H0HL ,,,,,0,0 ，，，，，，,,T,,,,,(,),,v,,,vSetting , we have 0HL ,T,,,，，，，，，，，，，，，,,,,,,,v，,,,,,,,v,,,v H0HL,, 2T, ,,,,,,，，，，,,，，，，，，，，,,，，,,,,,v,v，2,,,,v，,,,,,,,vHLH0H2,, ,,2,,,v，，,T000L,,,,,，，Obviously, , and if , we can always have ,002,,v,,H,,00 ,T,0. ,, ,,,,,v，，000L,,,,,，，So, when the condition is satisfied, the corresponding price ,,vH,,00 for the bundling is: ,,,,,,,,,,,p(),v，()v，,()v,v,,,,，，LH00H0L ，，，，,,，，,1,,v，,,，,(,),,,,,v0L000H 45 Bundling Sales of Information Goods: Models and Analysis The seller‘s profit is: 11,,,,,,,,,,,,,,,,pfd,1,v，，(),,vfd，，，，，，，，,,，，,,，，dLH0000,,00 1，，，，，，,1,,v，,,v，v,,(,),,,,,d,LHH0000,0 Price discrimination must be more profitable for the seller than selling the goods separately or in bundles. It means the following two inequalities should be satisfied: ,,,,,,,max，，,,,,,,, and ds1s2dB This is the case when the following two conditions are met: 11vLdd，，，，，，，，,,,,,,,,(1，,),,1,,,,,,, 00001,,00vH 11 ，，，，，，，，,,,,d,,,,,d,，1,,,,,0002,,00 1So, when the quality discount satisfies and ，，，，，，,,,,d,,max,,,12,0 ,,,,,v，，000L,,,,,，，, the price discrimination can make bigger profits to the seller. ,,vH,,00 Then consider the change of consumers‘ surplus. When the seller conduct quality discount, the consumers‘ surplus is: 1,,,,,,,,,,，，，，，，，，，，,,,,1S,v，,v,pfddHL,0 11，，,,,,,,,,,,,,,,,,，，，，，，,,，，，，，，，，,,11,v，,vd,,v，v，v,d0000HLLHH,,00,,，， 1,,，，，，,vv,,d,,,,,,,00HL,0,, So, the change is: 11,,，，,,,,,,,,,S,S,v,vd,,v,vd,，，，，，，，，,,dBHLHL000,,,,00,,，， 1,,，，，，,,,1v,,d,,,,0,,H00,0,, 46 Research of Mathematical Economics No. 1 2011 11,,,,,,,,,,S,S,v,vd,,v,vd，，，，，，，，,,dsHLHL200,,00,, 11,,，，，，，，,,,1v,,d,,v,,d,,,,0,,HL00,,00,, That means the consumers‘ surplus is reduced by bundling sales of quality discount. Compare total surplus of the society in the three cases, we can know, 11* ，，，，，，S，,,v，v,,,,,,d,,v,,,d,ddLHL,,00 1,, ，，，，S，,,S，,,v，v,v,,d,ssBBLHL22,0 ,,*S，,,S，,S，,Obviously, we get >. s2s2BBdd Now we can know, in the case of price discrimination, the consumers‘ surplus is decreasing, that means, the seller extracts more consumer surplus by quality discount. Additionally, in this way, bundling sales of quality discount also leads to decreasing of total surplus of the society. 3.2 Quantity Discount The quantity discount is similar to the quality discount. Suppose the seller can segment the market according to the consumers‘ demand type, and provide an incomplete bundle that ,，，,,contains a fraction of the goods in the full bundle, intended for consumers of type . The values of goods for the high-valuation consumer and low-valuation consumer are reduced to ,,，，，，，，，，,,v,,v,,,,,0 and separately. Assuming that is strictly increasing, and . HL ,(,),,,(,),1, when ; , when . ,,0,,10 In this model, we can get the following proposition. 1When the two conditions and Proposition 4: ，，，，，，,,,,d,,max,,,12,0 ,,,，，00,,，，, are satisfied, conducting price discrimination by quantity discount will bring ,,0 the seller more profit and reduce the consumers‘ surplus at the same time. ,,,,0,1,,, Set arbitrary , for every , to satisfy the individual rationality Proof:010 constraints, we have, 47 Bundling Sales of Information Goods: Models and Analysis ，，，，，，,,p(,),,(,),,v，1,,,,,v 000H00L ，，，，，，,,p(,),,(,),,v，1,,,,,v 111H11L To satisfy the incentive-compatibility constraints, we have, ，，，，，，，，，，，，，，，，,,,,,(,),,v，1,,,,,v,p,,,(,),,v，1,,,,,v,p,00H00L001H01L1 ，，，，，，，，，，，，，，，，,,,,,(,),,v，1,,,,,v,p,,,(,),,v，1,,,,,v,p, 11H11L110H10L0 ，，p,，，The seller will choose and p, as large as possible which satisfy the four 01 constraints. So we get, ，，，，，，,,p(,),,(,),,v，1,,,,,v 111H11L ,,,,,,,,,,,,,p(),()，,()v，1,v，，，，，，，，，，,,,,000101H1L ，，，，，，，，，，,,,v，()v,v,,,,()v,v,,,,,,,,,,L00HL01HL1 ,Because is chosen arbitrarily, we have 1 ，，，，，，，，，，p,(),,,v，,(,),,v,v,max,,,,,,(,)v,v,,0L00HL0HL ,,,,,0,0 ，，，，，，,,T,,,,,,,,Set , then, 0 ,T,,，，，，，，，，，，,,,,,,,,，,,,,,,, 0,, 2T, ,,,,,,，，，，，，，，，，，，,,，，,,,,,,,2,,,,，,,,,,,,02,, ,2,,，，,T,T00,,，，,,0Obviously, , and if , we can always have . ,002,,,,,,0 ,,,，，00,,，，,So, when the condition is satisfied, the corresponding price for the ,,0 bundling is: ,,,,,,,,,,p(),v，()(v,v),(),(v,v)，，,,,,LHLHL00 ，，，，,(v,v),(,),,,,(,),，,,，vHLL000 The seller‘s profit is: 48 Research of Mathematical Economics No. 1 2011 1,,,,,,pfd，，，，d,0 1，，，，，，，，，，,v,v,,，v，v,v,,,,,,d,HL00LHL0,0 11,,,,,,,,,,,,,,,,pfd,1,v，，(),,vfd，，，，，，，，,,，，,,，，dLH0000,,00 1，，，，，，,1,,v，,,v，v,,(,),,,,,d,LHH0000,0 Price discrimination must be more profitable for the seller than selling the goods separately or in bundles. It means the following two inequalities should be satisfied: ,,,,,,,max，，,,,,,,, and . ds1s2dB This is the case when the following two conditions are met: 11vvHLdd，，，，，，,,,,,,(,，),,,,,,,,, 0001,,00vvvv,,HLHL 11 ，，，，，，,,,,d,,,,,d,，,(1,,),,0002,,00 1So, when the quantity discount satisfies and ，，，，，，,,,,d,,max,,,12,0 ,,,，，00,,,，，, the price discrimination can make bigger profits to the seller. ,,0 Then consider the change of consumers‘ surplus. When the seller conduct quantity discount, the consumers‘ surplus is: 1,,,,,,,,,,,,S,v，,v,pfd，，，，，，1，，，，，，,,,,dHL,0 1,,,,,,,,,,,,,,,,,,，，，，，，,v，,v,v,v,，,vd,,,,，，，，，，，，，，,,1 000HLHLL,0 1,,，，，，,vv,,d,,,,,,,00HL,0,, So, the change is: 11,,，，,,,,,,,,,S,S,v,vd,,v,vd,，，，，，，，，,,dBHLHL000,,,,00,,，， 1,,，，，，，，,,,1v,v,,d,,,,0,,HL00,0,, 49 Bundling Sales of Information Goods: Models and Analysis 11,,,,,,,,,,S,S,v,vd,,v,vd，，，，，，，，,,ds20HL0HL,,00,, 1，，，，，，,,,1v,,d,,,,v,v,00H00HL,0 bundling sales of quantity discount. That means the consumer‘s surplus is reduced by Compare total surplus of the society under the three cases, we can know, 1* ，，，，，，S，,,v，v,v,,,,d,ddLHL,0 1,, ，，，，S，,,S，,,v，v,v,,d,ssBBLHL22,0 *,,S，,,S，,S，,Obviously, we get >. s2s2BBdd In the case of quantity discount, just like the quality discount case, the consumers‘ surplus is decreasing, as well as total surplus of the society. 4. Explanations In the single-product bundling case, we can explain our results intuitively. Proposition 1 and Proposition 2 tell us the bigger the difference is between the consumers‘ types, the better the price discrimination is, because the seller can extract more consumers‘ surplus by mixed bundling in our model. This result does not contradict with our intuitive sentiment, since the seller will have no incentive to conduct price discrimination if the consumers‘ valuation is identical and they have the same preference. But at the same time, bundling sales cause the total welfare of the society to decrease, which can be seen from the model. qIn unbundling sales, consumers choose to purchase goods so that ,,T(q),p,,V(q). This means a consumer‘s marginal purchase equals his marginal utility. But in the case of bundling sales, we have, ,,,,,,,,,,T(q),,V(q),,V(q), 0111 1,,,,T(q),,V(q),,V(q),,V(q), ,,,,,,,2202202 This means the consumer‘s marginal utility exceed his marginal purchase, that is, the seller induces the consumers to purchase certain amount of goods which is lower than the optimal 50 Research of Mathematical Economics No. 1 2011 amount for the society. We can see, bundling sales for single-product distort the product distribution of the society. It violates the conditions for distributive efficiency and lowers the net welfare of the whole society. In the multiproduct bundling case, the seller use bundles of quality discount and quantity discount to screen consumers by market segment. These bundling strategies induce consumers to reveal their characteristics and preferences. They can also be seen as mixed bundling, the essence of which is also to sort consumers of various types, attract low-demand consumers and extract consumers‘ surplus as much as possible. So, the result in our paper is corresponding with which have been derived earlier by Adams and Yellen (1976)Schmalensee (1982)McAfee et al (1989), ，， that is, in most cases, mixed bundling is more profitable than pure bundling. Proposition 3 and ,(,),(,)，，,,Proposition 4 indicate that if is small enough, and , are large enough, price discrimination is profitable. Intuitively, when the average valuation of consumers is not too high, the two bundling patterns can efficiently segment the market and make profits if the seller set the quality discount and quantity discount large enough. This result does not contradict with our intuitive sentiment, too. 5. Conclusions This paper studies the impact of bundling sales on merchandising of information goods. We put our emphasis on the difference between consumers and pricing strategy of firms since they are important factors that affect seller‘s profits and consumer‘s surplus. This paper discusses only about information goods, but the results we get are just similar with that of common goods. We consider two familiar bundling types: bundling of single-product in amount and bundling of multiproduct. In the single-product model, we assume the seller provides two kinds of bundles, which contain different amount of goods, and the consumers choose the one which maximizes their utility. Using the model, we compare the seller‘s profit, consumers‘ surplus and net welfare of the society of bundling sale and unbundling sale. Our result is that a single-product bundling sale is more profitable than unbundling sale, but consumers‘ surplus and net welfare of the society decrease. At the same time, the bigger the difference is between the consumers‘ types, the better the price discrimination is. In the multiproduct case, we extend Bakos and Brynjolfss‘s 51 Bundling Sales of Information Goods: Models and Analysis (1999) model and discuss two bundling patterns: quality discount and quantity discount. We prove that when the consumers‘ type is continuous, under certain circumstances, those mixed bundling strategies can be more profitable, and extract consumer surplus more efficiently than pure bundling and unbundling sales. Also, we present the conditions when price discrimination is profitable: When the average valuation of consumers is not too high, the two bundling patterns can efficiently segment the market and make profits if the seller set the quality discount and quantity discount large enough. This paper provides several new avenues for future research. First, for many information goods, there may exist ―usage barrier‖, that is, different users may have different costs when using the same goods. This barrier will surely affect sales of information goods to some extent. Second, this paper is based on the setting that the seller is a monopolist in the market. If we consider the existence of competitors, different results may be discovered. Third, most information goods are ―experience goods‖, meaning consumers can evaluate the goods only after they have purchased them. Some consumers who have purchased certain products in the first phase may be unwilling to buy them again in the second phase. So, if we develop a dynamic model of bundling sales of information goods, valuable results can also be derived. In conclusion, we believe that the methods developed in this paper provide a simple and useful framework to understand and analyze in the context of information goods merchandising. Bundling of information goods is a problem which calls for further study. References 曹洪, 2004, 捆绑销售的社会福利分析, 学术研究, 2004年第2期, 40-43. 泰勒尔, 1997, 产业组织理论, 中国人民大学出版社. Adams, W.J., Yellen, J., 1976, ―Commodity bundling and the burden of monopoly‖, Quarterly Journal of Economics, 40, 475-498. Armstrong, M., 1996, ―Multiproduct nonlinear pricing‖, Econometrica, 64(1), 57-75. Bakos, J.Y., Brynjolfsson, E., 1999, ―Bundling information goods: Pricing, profits and efficiency‖, Management Science, 45(12), 1613-1630. Bakos, J.Y., Brynjolfsson, E., 2000, ―Bundling and competition on the Internet‖, Marketing Science, 19(1), 63-82. Carlton, D.W., Waldman, M., 2002, ―The Strategic Use of Tying to Preserve and Create Market Power in 52 Research of Mathematical Economics No. 1 2011 Evolving Industries‖, The RAND Journal of Economics, 33(2), 194-220. Choi, J.P., 2003, ―Bundling new products with old to signal quality, with application to the sequencing of new products‖, International Journal of Industrial Organization, 21, 1179-1200. Jensen, S., 2008, ―Two-part tariffs with quality degradation‖, International Journal of Industrial Organization, 26, 473-489. Martin, M., 1999, ―Strategic and welfare implications of bundling‖, Economics Letters, 62, 371-376. McAfee, R.P., McMillan, J., Whinston, M.D., 1989, ―Multiproduct monopoly, commodity bundling, and correlation of values‖, Quarterly Journal of Economics, 103, 371-383. McCardle, K.F., Rajaram, K., Tang, C.S., ―Bundling retail products: Models and analysis‖, European Journal of Operational Research, 177, 1197-1217. Mirman, L.J., Sibley, D.S., 1980, ―Optimal Nonlinear Prices for Multiproduct Monopolies‖, The Bell Journal of Economics, 11(2), 659-670. Salinger, M.A., 1995, ―A Graphical Analysis of Bundling‖, The Journal of Business, 68(1), 85-98. Schmalensee, R., 1982, ―Commodity Bundling by Single-Product Monopolies‖, Journal of Law and Economics, 25(1), 67-71. Schmalensee, R., 1984, ―Gaussian demand and commodity bundling‖, Journal of Business, 57, S211-S231. Sibley, D.S., Srinagesh, P., 1997, ―Multiproduct Nonlinear Pricing with Multiple Taste Characteristics‖, The RAND Journal of Economics, 28(4), 684-707. Spence, A.M., 1980, ―Multi-Product Quantity-Dependent Prices and Profitability Constraints‖, The Review of Economics Studies, 47(5), 821-841. Whinston, M.D., 1990, ―Tying, Foreclosure, and Exclusion‖, The American Economic Review, 80(4), 837-859. (Responsible Editor: Zhang Lingyun) (Proofreader: Wang Jue) 53 The Introduction and Applications of RDEU Hypothesis The Introduction and Applications of RDEU Hypothesis Wang Bo, Hong Beiyun (School of Economics, Renmin University of China) The passage gives us a brief introduction of the RDEU theory and its two simple applications. At the Abstract: end of the passage, the author discusses the disadvantages of the RDEU theory. Expected Utility, Rank-dependent Expected Utility, Nash Equilibrium Keywords: 1. Preface The Expected Utility (EU) hypothesis, established by Von Neumann and Morgenstern (1944), evaluates uncertain prospects according to their expected level of ―satisfaction‖ or ―utility‖. The EU hypothesis is the predominant descriptive and normative model of choice under uncertainty in economics, which provides the analytical underpinnings for many game theories on the aspects of insurance and financial decisions. However, through numerical simulations and realistic experiments, many experts found the evidence like Allais paradoxEllsberg paradox and ， Common Ratio effect, showing that individuals may systematically depart from its predictions and then Quiggin (1982) came up with the rank-dependent expected utility (RDEU) hypothesis. In the first part of the passage we give a brief introduction of EU hypothesis. The second part is the introduction of RDEU hypothesis, following with 2 simple applications of it in the third part. Then we make some simple improvement of RDEU hypothesis mentioned in the last part of the passage. Most of the mathematics analysis is list in the Appendix. 2. The Introduction of EU The expected utility theory requires the following axioms: (1) Completeness: for any two distributions and , either , or . xxxx,xx,xx~ 12121212 54 Research of Mathematical Economics No. 1 2011 (2) Transitivity: if , and , then . xx,xx,xx,322131 (3) Mixture Continuity: if , then there exists the only p (0 < p <1) such that x ~ xxx,,2321 . xpxpx~(1)，,231 (4) Independence: for any two distributions and , if and only if xxxx,1212 for all p (0 < p ? 1) and for all pxpxpxpx，,,，,(1)(1)x.13233 Under these four axioms, we can find a utility function that meet: if , xx,U(.)12 ; else if , . UxUx()(),xx,UxUx()(),121212 The formal representation of the objects of choice, and hence of the expected utility preference function, depends upon the set of possible outcomes. When the outcome set is finite, we can represent any probability distribution over this set by its Xxx,{,,}1n vector of probabilities (where ) and the expected utility Ppp,(,,)pprobx,()1nii preference function takes the form . When the outcome set VPVppUp()(,,),,,1nii consists of the real line or some interval subset of it, probability distributions can be represented by their cumulative distribution functions (where ), and the Fxprobx()(),,,F(.) expected utility preference function takes the form VFUxdFx()()(),. , In fact, the expected utility hypothesis of behavior towards risk is the hypothesis that the individual possesses (or acts as if possessing) a ―Von Neumann-Morgenstern Utility Function‖ or ―Von Neumann-Morgenstern Utility Index‖ defined over some set X of {}UU(.)i alternative possible outcomes, and when faced with alternative risky prospects or ―lotteries‖ over these outcomes, will choose the prospect that maximizes the expected value of or . {}UU(.)i Since the outcomes could be alternative wealth levels, multidimensional commodity bundles, time streams of consumption, or even non-numerical consequences (such as a trip to Paris), this approach can be applied to a tremendous variety of situations, and most theoretical researches in the economics of uncertainty, as well as virtually all applied work in the field (e.g., insurance or investment decisions) is undertaken in the expected utility framework. 55 The Introduction and Applications of RDEU Hypothesis 3. The Introduction of RDEU There is much evidence that individuals may systematically depart from its predictions. Many people disagree to the fourth axioms as it‘s different from the choice of a real person. As a result, many alternative models developed, among which the Rank-Dependent Expected Utility (RDEU) model is the most widely used, and arguably the most empirically successful, generalization of Expected Utility (EU). Firstly, if the outcome set is finite, we make that is better than , Xxx,{,,}xx1nij if i > j. Now we can represent any probability distribution over this set by its vector of probabilities (where ). Ppp,(,,)pprobx,()1nii Secondly, we define a new function-----emotion function : ,()p , , , ,(.):[0,1][0,1],,'()0p,,(0)0,,(1)1, If , it means the person has an optimistic emotion; else if , it means ,"()0p,,"()0p, the person has a pessimistic emotion. Usually, we set (Quiggin, 1982). As a result, if r > 1, it means optimistic ,()ppr, emotion; else if 0 < r < 1, it means pessimistic emotion; else if r = 1, the decision is not affected by the emotion, which is the same as the EU theory. Thirdly, we can define the rank-dependent expected utility. jj,1()()()iiiWpwpwp()()() ,, ,,ijiikkkk,,11 When the outcome set consists of the real line or some interval subsets of it, probability distributions can be represented by their cumulative distribution functions (where F(.) ), and the rank-dependent expected utility preference function takes the Fxprobx()(),,, VFUxdFx()()(()),,form . , 4. The Applications of RDEU 56 Research of Mathematical Economics No. 1 2011 4.1 The First Application: Battle of the Sexes with Mixed Strategies In the Battle of the Sexes Game, a husband (A) and wife (B) are planning a vacation. A prefers mountain locations; B prefers the seaside. Both players prefer a vacation spent together to one spent apart. The payoffs in the table reflect these preferences. Graph 1 Battle of the Sexes with Mixed Strategies Suppose that the spouses in the problem tire of constant bickering about vacations and decide to let ―chance‖ decide. Specifically, suppose A decides to choose his mountain strategy with probability and seaside with probability . Similarly, suppose B chooses her 1,pp mountain strategy with probability and seaside probability . Given these probabilities, q1,q the outcomes of the game occur with the following probabilities: mountain-mountain, ; pq mountain-seaside, ; seaside-mountain, ; and seaside-seaside, pq(1),(1),pq . (1)(1),,pq A‘s expected utility is then given by EUpqpqpqpq()2(1)0(1)0(1)(1)1,，，,，，,，，,,，A 1(31),,，,qpq 1Obviously, A‘s optimal choice of p depends on B‘s probability, q. If , utility is q,3 57 The Introduction and Applications of RDEU Hypothesis 11maximized by choosing . If , A should opt for p = 1. And when , A‘s p,0q,q,33 2expected utility is no matter what value of p is chosen. 3 For spouse B, expected utility is given by EUpqpqpqpq()1(1)0(1)0(1)(1)2,，，,，，,，，,,，B 22(32),,，,pqp 22When , B‘s expected utility is maximized by choosing . When , q,0p,p,33 2utility is maximized by choosing . And when , B‘s expected utility is independent q,1p,3 of what q she chooses. According to the analysis above, we get three Nash equilibriums. Two of these we can see obviously: , and , represent the joint vacation strategies. And p,0p,1q,1q,0 21, is a Nash equilibrium with mixed strategies. q,p,33 Now, we deal with this problem through the RDEU theory instead of the EU theory. Many books have given the proof that there exists the Nash equilibrium with mixed strategies depending on the EU theory if there are finite people and the set of the plots is finite for each people. The following theorem will show that it is the same depending on the RDEU theory. If the set of players is finite and the set of sections is finite for each player, there Theorem: exists a Nash equilibrium depending on the RDEU theory. The proof will be shown in the Appendix. rWe assume that the emotion function of A is and the emotion function of B is wpp(),A s. wqq(),B The utility function of A is 58 Research of Mathematical Economics No. 1 2011 s20qrrVpp,,(1)()()As011,q rsrs 2(1)(1),，,,pqpq rsrs,,,，pqpq 31 The utility function of B is s10qrrVpp,,(1)()()Bs021,q rsrs 2(1)(1),，,,pqpq rsrs 3221,,,，pqpq As A will maximize his utility, the partial differentiation will be zero: 1,V1rsrrs,,,111A*s3(31)0,,,,,rpqrprpq q,(),p3 As B will maximize his utility, the partial differentiation will be zero: 1,V2rsssr,,,111B*r32(32)0,,,,,spqsqsqp p,(),q3 We can get several conclusions: 1121**sr(1) , is a Nash equilibrium with mixed strategies. Obviously, it depends p,q,()()33 on r and s. (2) If the emotion factor will not affect the decision of the husband and wife, which means , the RDEU theory can reach the same result as the EU theory. rs,,1 (3) As r increases, which means the husband has a strong desire to the seaside, p* will increase. In this case, if s > 1, which means the wife will consider the decision of the mountain, especially when s > 1.9, the husband and wife will probably make the same decision to go to the mountain. (When s > 1.9, q* will be more than 0.56, which means the wife has a probability which is more than 56% to go to the mountain) (4) If the wife is disgusted about the mountain, which means s < 1, even when the husband has a strong desire to the seaside, which means r >> 1, they will probably make a different decision. (5) The same analysis can be taken on the seaside strategy. Obviously, the RDEU makes the conclusion more clear and precise. 59 The Introduction and Applications of RDEU Hypothesis 4.2 The Second Application: Insurance Premiums with Moral Hazard Firstly, we give two theorems. Theorem 1: If is a distribution function of the random variable, and Fx(), , then P(0)0,,, ( is an arbitrary utility function) Gx()GxdFxGFGxdFx()()(0)(0)()(),，,,xx,,00 Theorem 2: If is a distribution function of the random variable , and is a Fx(),()x, strictly increasing continuous function: [0,1]?[0,1], π(0)=0, π(1)=1, then is a ,(())Fx distribution function. The proof would be shown in the appendix. We assume that the premium rate is , the quantity of the insurance you buy is π, the , wealth you have at the beginning is , the loss is x, the money you can get if the accident w0 happen is which is a function of x, the utility function of the consumer is , the sx()ux() emotion function of the consumer is , the utility function of the insurance company is wx()1 and the emotion function of the insurance company is . We also assume that the wx()vx()2 behavior of the consumer is measured by a, and the cost of the consumer to behave as a is , ca() which is a function of a. Obviously , we can see that . ca'()0, In the paper, we only consider two special conditions. The first condition : After signing the contract with the insurance company, the consumer‘s behavior will only have an effect on the quantity of the loss but the probability of the accident. As consumer‘s behavior will not affect the possibility of the accident, we assume that the possibility of the loss is , which is a constant, and the distribution function of the loss is q . Gxa(,) So the consumers will maximize their utility by (())((,))()uwsxxdwGxaca，,,,,,01max, sx,(),,x0 60 Research of Mathematical Economics No. 1 2011 According to the theorem 1 and 2, it‘s the same as w(1)()(())((,))(),,，，,,,quwuwsxxdwGxaca,,,,1001max, sx,(),,x0 (Notice that , and ) s(0)0,Gaq(0,)1,, To simplify the model, we define the conditional distribution function , which means the possibility of the loss of x in case that you FxaGxaGa(,)(,)/(1(0,)),, have known that the incident must happen. In general, r, so . ()()()wqqwqwq, ()wqq,1212 As a result, wGxawFxaGa((,))[(,)(1(0,))],,11 ((,))(1(0,))()((,)),,,wFxawGawqwFxa1111 In this case, the model can be simplified as (1)()()(())((,))()wquwwquwsxxdwFxaca,,，，,,,,,,,10101max, sx,(),,x0 So the insurance model is (1)()()(())((,))()wquwwquwsxxdwFxaca,,，，,,,,,,,10101max,(),sx,x0, .. (1)()(())((,))(0)stwqwqvsxdwFxav,，,,,,,,222,x,0 argmax(1)()awquw,,,，10,, ()(())((,))()wquwsxxdwFxaca，,,,,,101,0x, In the second equation, means the minimum utility the company will suffer. If the v(0) company gets a utility less than , he will quit the market. By assuming that the insurance v(0) ,market is completely competitive, ―‖ can be replaced by ―=‖. The third equation means that the consumer will maximize its utility by choose a, so we can make a partial differentiation on a. According to Mirrlees (1974) and Holmstrom (1979), the third equation can be replaced by 61 The Introduction and Applications of RDEU Hypothesis ,wFxafxa'((,))(,)1,,wquwsxxdxca()(())'()，,,, 10,,ax,0 (To simplify the model, we assume that has a density function here) Fxa(,)fxa(,)So the insurance model should be (1)()()(())((,))()wquwwquwsxxdwFxaca,,，，,,,,,,,10101max,sx(),,x,0 .. (1)()(())((,))(0)stwqwqvsxdwFxav,，,,,,,,222, x,0 ,wFxafxa'((,))(,)1dxca,'() ()(())wquwsxx，,,10,,,a,x,0 As a result, the Lagrangian function is LsxwquwwquwsxxdwFxaca((),)(1)()()(())((,))(),,,,,,,,，，,,,10101,x,0 [(1)()(())((,))]，,，,wqwqvsxdwFxa,,,,,222, x,0 ,wFxafxa'((,))(,)1)] [()(())'(，，,,,wquwsxxdxca10,,,,,ax,0 The first order differentiation condition of is , ,,,,，,,，,,,,,,wquwwquwsxxdwFxa(1)'()()'(())((,))10101,x,0 wqwqvsxdwFxa(1)()'(())((,)) ,，,,,,,,,,222, (1) x,0 ,wFxafxa'((,))(,)1wquwsxxdx()'(())，,,10,,,,,,ax,0 The first order differentiation condition of s(x) is wquwsxxwFxafxa()'(())'((,))(,)，,,,,101 ,,wqvsxwFxafxa()'(())'((,))(,) ,,,22 (2) ,wFxafxa'((,))(,)1，，,,,wquwsxx()'(())0,,,10,a (Notice that it depends on calculus of variationsthe proof will be shown Appendix) ， Combining with the equation (2), we can simplify the equation (1) as 62 Research of Mathematical Economics No. 1 2011 wq(1),2,,,uw'(),, 0wq(1),1 We can see that the insurance premium is irrelevant with the loss x, which means ,,,, is a constant, so that and are in an inverse proportion, which we can get as a same result ,, through the EU theory. By simplifying it, we can get that uwsxx'(())，,,,,wqwFxa()'((,)),022,fxa(,)wFxa''((,)) vsxwqwFxa'(())()'((,)),,,a1111(,)，，Fxa,,afxawFxa(,)'((,))1 In the case of EU theory, if we give some assumption, we can get a relationship of s(x) and x. However, it‘s much harder in the RDEU case, as we don‘t know whether the consumer and the company are optimistic or pessimistic. If we assume that the company and the consumer have a same emotion function, uwsxx'(())，,,,,,0,fxa(,)wFxa''((,)) vsx'(()),,,a11(,)，，Fxa,,afxawFxa(,)'((,))1 wFxa''((,))1Fxa(,)As and exist here, it‘s so hard to analyze. As a result, the awFxa'((,))1 conclusion we get through EU cannot be applied now. We cannot get an accurate relationship of s(x) and x here. The second condition: After signing the contract with the insurance company, the consumer‘s behavior will only have an effect on the probability of the accident but the quantity of the loss. As consumer‘s behavior will not affect the quantity of the loss, we assume that the , which is a function of a. Obviously, . The possibility of the accident is qa'()0,qa() distribution function is equals to . Gxa(,)qaFx()() Similar to the condition 1, the insurance model should be 63 The Introduction and Applications of RDEU Hypothesis (1())()(())(())(())()wqauwwqauwsxxdwFxca,,，，,,,,,,,10101max,(),sx,x,0 .. (1())(())(())(())(0)stwqawqavsxdwFxv,，,,,,,,222,0x, '(1())'()()'(())'()(,,,，，wqaqauwwqaqauws())(())'()xxdwFxca,,,,,10101,,,x,0 The Lagrangian function is (1())()(())(())(())()wqauwwqauwsxxdwFxca,,，，,,,,,,,10101max,(),sx,x,0 [(1())(())(())(())(0)]，,，,,wqawqavsxdwFxv,,,,,222,0x, +['(1())'()()'(())'()(()),,,，，,,wqaqauwwqaqauwsxxdwFxca(())'()],10101,,,,,,0x, The first order differentiation condition of is , ,,,,，,,,,,,,,wqauwwqauwsxxdwFx(1())'()(())'(())(())10101,0x, ，,，,[(1())(())'(())(())] (3) wqawqavsxdwFx,,,,,222,0x, ，,,,，,,,,,,,,,,['(1())'()'()'(())'()'(())(())]0wqaqauwwqaqauwsxxdwFx10101,0x, The first order differentiation condition of is sx() wqauwsxxwFxfx(())'(())'(())()，,,,,101 ,,wqavsxwFxfx(())'(())'(())() (4) ,,,22 +'(())'()'(())'(())()0wqaqauwsxxwFxfx，,,,,,,101 Combining with the equation (4), we can simplify the equation (3) as ,wqa(1()),2uw'(),,,, 0wqawqaqa(1())'(1())'(),,,,11 We can see that the insurance premium is irrelevant with the loss x, which means ,,,, is a constant, so that and are in an inverse proportion, which we can get as a same result ,, through the EU theory. By simplifying it, we can get that 64 Research of Mathematical Economics No. 1 2011 uwsxx'(())，,,,,wFxwqa'(())(()),022,wqa'(()) vsxwFxwqa'(())'(())(()),,,1111'()，qa,wqa(())1 Similarly to the condition 1, we can get nothing through this equation, however, if we assume that the consumer and the company have a same emotion function, which means , then we can see that ww,12 uwsxx'(())，,,,,wqa(()),02,wqa'(()) vsxwqa'(())(()),,,111'()，qa,wqa(())1 The right side is irrelevant with x. by assuming that the company is risk neutral, we can get that is constant, as a result, should be constant, so that v'(.)uwsxx'(())，,,,,0 is constant. We can get a same result through the EU theory. sxx(), 5. An Improvement of RDEU 5.1 The Violations of Monotonicity Violations of monotonicity come from an experiment. People can pick up the ball in a box while the revenue and the probability had been shown in the table. People were asked to choose C or D. Here is the table: 90% White 6% Red 1% Green 1% Blue 2% Yellow A 0 45000 30000 -15000 -15000 B 0 45000 45000 -15000 -10000 White Red Green Yellow 90% 6% 1% 3% C 0 45000 30000 -15000 90% 7% 1% 2% D 0 45000 -15000 -10000 We can see that the two tables are really the same and table 2 only combines the choice of table 1. In the table 1, B is obviously a better choice. Similarly, D is better than C. But in fact, 65 The Introduction and Applications of RDEU Hypothesis 58% of the people choose C, which can‘t be interpreted by the EU theory. Suppose we have n results . If i > j, then is not worse than . The xx,,xx 1nij probability of is . We break up into and , and the probability is and xpxxxpiiii1i2i1 . According to the RDEU theory, the changing utility in the violation is pi2 wUxwUxwUx()()()，,iiiiii1122 ,，,wUxwUxwUx()()() iiiiii12 ,，,()()wwwUxiiii12 ww，ii12 ,，，，，，,，，，，，,,(...)(...)ppppppppp121121211iiiii,, (...)(...)ppppppp，，，，,，，，,,1211121iii,, ,，，，，，,，，，(...)(...)pppppppp,,12112121iiii,, ,，，，，,，，，(...)(...)ppppppp121121iii,,,, ,wi So the choice will similarly not change according to the RDEU theory although we have spitted the event. 5.2 An Improvement of RDEU Now, we give an improvement of RDEU so that it can explain the violations of monotonicity. In fact, we make our decision by the conditional possibility instead of the objective possibility. So we can correct the RDEU theory in the following ways: wp,,()11 j,1pj ww,,(1)(),,jij,1i,11,p,ii,1 It means that you select a choice in case that you can‘t choose a better one. In this case, 66 Research of Mathematical Economics No. 1 2011 ww，ii12 j1,i1,ppji1,,，,,(1)()(1)()www,,,,jii1ij,,11ji11,,1,p1,，pp,i,ii1 i,1i1, i1,pi(1)(),,,ww,ji,i1,j,11,pj,j1, And the violation also changes , when j > i. As a result, the choice may change, which wj means that now violations of monotonicity can be interpreted by RDEU depending on the conditional possibility. In fact, if we consider the application 1 through RDEU depending on the conditional possibility, the answer will not change. Proof: In the application 1, n = 2, which means there are only two outcomes and , xx12 and the probability is and . pp 12 Assume is better than . Notice that: xx12 p2wpwww,,,,,,,() ()(1)1 112111,p1 As , we get a same result as before. pp，,112 So the answer of the first application needn‘t to be changed. 6. Further Research We give an improvement of RDEU theory which depends on the conditional possibility at the end of the passage. Further research is to put this model into a continuous way so that it can be applied in more situations. 67 The Introduction and Applications of RDEU Hypothesis Appendix The Proof of Theorem Firstly, we will give out the meanings of the symbols. We define a finite simultaneous game as a tuple , where N is a set of {,{},{}}NSUiiplayers, and for each player is a set of plots and is the utility function. SUii ()()iiFor each person i (), he can make several choices , the probability {,,}xxiN,1m mi()k()k()()()()iiiithat he chooses is . = , which means the xSp{,...,|1,0}pppp,,,ijj1mjji,1j plots set of person i. mmn1(1)()nUpwuxxWpWp(,)(,,)()(),, which means the utility of the ,,11iinjnj1n,,11jj1n person i depending on the RDEU theory. jj,1()()()iiiWpwpwp()()() ,, ,,ijiikkkk,,11 where means the emotion function of person i. w i Now, I will give you the proof of the theorem. We denote that Proof: **FppUppUpp(){|(,)max(,)},, iiiiii,,,iiipi n tm,FpFp()(),, =[0,1], S,,iii,i,1 t(1) is a compact convex set. S tttThen is a reflection : . As S is a compact convex set, S is a compact F(.)SS, convex set too. Fp(),,(2) . S is a continuous function and is a closed set. So can be maximized at Upp(,)Uiii,i68 Research of Mathematical Economics No. 1 2011 , which means that . As a result, . SFp(),,Fp(),,ii,ii, (3) is a convex set. Fp() ()iUppcWp(,)(),,,iiijjj, where is irrelevant with , which can be seen as a cpjiconstant here. Obviously, is nonnegative here. We denote . cAicc,,{|max}jijj We can see that if , then equals zero if . Or, qqqFp,,{,,}()qjA,1mii,ji ***, where (,see (2)), , A,,UqpUqp(,)(,),qq,,(0,0,...,1,0,...,0)iA,*iiii,,i . which is in contrast with the definition of Fp()ii, pFp,()If , then ii, ()()()iiiUppcWpcWpcWp(,)()()(),,，,,,,iijjjjjjjjj ,,jAjA ()iNotice that if , ; if , . So jA,jA,cc,p,0*jji ()()iiUppcWpcWp(,)()(),，,,,iijjjjjj,,jAjA ()()()iii,，,,cWpcWpcWpc()()()****,,,jjjjjjiiii,,jAjA ,,Fp() ii, Notice that if , , so ; if jA,, . jA,,,,,0,,,,,,，,,(1)0cc,*jjjjjjiAs a result, (,)()()UpcWcW,,,,，,,iijijjij,jAjA,, ()()(),，,,cWcWcWc,,,****,,,ijijijiiiijAjA,, So, , from which we can know that is a convex set. ,,Fp()Fp()ii, (4) F is upper semi-continuous. 69 The Introduction and Applications of RDEU Hypothesis Assume nnnn ,(),,,,,,,,,,,,nF nnn'' (,)(,)asUU,,i,,,,,iiiiii,, setn,, '' (,)(,)wegetUU,,iiiiiii,,,,,,, i ()soF,,iii,,, As a result, . ,,,F According to Kakutani Theorem, , , so is a Nash equilibrium. ,，,stF..,,, The Proof of Theorem 1 Proof: {:0}{:0}{:0} {:0}{:0}xxxxxxxxxx,,,,,,,,,, ()()()()()(),,，GxdFxGxdFxGxdFx,,, xxx,,,000 ,,, ()()(0)(0)(0)(0) ((0)0)GxdFxGPGPP,,,,,,,x,0 The Proof of Theorem 2 Proof: () Fxisadistributionfunction (1)lim()0,,Fxx,,, (2)lim()1Fx,x,，, (3),lim()(),,xFxFx00，xx,0 (4)() is increasingFx (x) is a continuous function, lim(())(lim())(0)0 (1)？,,,,,,FxFxxx,,,,,, lim(())(lim())(1)1 FxFx,,, (2),,,xx,，,,，, 70 Research of Mathematical Economics No. 1 2011 0,0, (), (y)-(F(x)),,，,,,,ifyFxthen,,,,,,00 ,lim()(),,xFxFx00，xx,0 0, , ()()？，,,,,,ifxxthenFxFx,,,1010 ,,,,(()) lim(())(()) (3)FxsoFxFx,, (())andthenFx,00,，xx,0 x>y,()(),so (())(()) (4),,,FxFyFxFy,, (1)(2)(3)(4), (()) accordingtoFxisadistributionfunction , The Proof of Equation 2 Proof: x2 ()(,,')AfLxffdx,,x1 () max ifAfisimizedatf0 ,,LdL then, ,,Fdxf' The EulerLagrange equationthisis, theresult if (2), we weputitintheequationcanget References , . 何凯洁基于信息修正的非期望效用模型和保险市场均衡问研究 , . 钟桦道德风险下的最优保险模型研究 Nicolson, Intermediate microeconomics 9th edition. Mark, J.M., ―Expected utility hypothesis‖. Matthew, J.R., ―Risk aversion in RDEU‖. Xiong Guoqiang, n-person, ―Non-expected Utility Games Model and Its Application Based on RDEU Theory‖. (Responsible Editor: Xu Lingjue) (Proofreader: Ren Junqiushi) 71 Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition Survey Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition Survey Zhang Luezhao (School of Economics, Renmin University of China) This article extends the work on labor supply theory of behavioral economics. It proposes some new Abstract: views of the model of behavioral labor supply theory, for example the effects on labor supply elasticity that are made by income target. It estimates the elasticities of two kinds of labor working in urban informal sector and compares them. The samples are derived from China Health and Nutrition Survey. The empirical results confirm the predictions of the model. However, the article doesn‘t avoid the ―division bias‖, which should be improved in the future. 1. Introduction In recent decades, labor economics has developed a lot. In labor economics, labor supply theory can be divided into two branches: the neo-classical labor supply theory and behavioral labor supply theory. 1.1 Neo-classical Theory of Labor Supply When neo-classical labor supply theory is used to analyze labor supply behavior, consumer theory is applied to labor economic. Here, an individual has two kinds of special commodities, which can give him utility: consumption and leisure. Individuals‘ time in a day is limited, which can be used for work to get income for consuming, or for leisure. Individual consumption is the opportunity cost of leisure he gives up. The opportunity cost of leisure is the consumption he gives up. Labor market price (wage) w is the opportunity cost of leisure time monetization, said: for each additional unit of leisure time, one should give up w units of consumption. In this way, the problem faced by individuals is, when a budget constraint is given, how to allocate time between consumption and leisure resources to achieve their utilities maximization. Labor supply can be regarded as the opposite of leisure demand. The core of this analysis is: 72 Research of Mathematical Economics No. 1 2011 labor supply responses to different wages. It is generally believed that the increase of labor market prices (wages) would reduce the demand for leisure, increasing labor supply, which is called substitution effect, that is, if wage increases, workers will substitute leisure time with labor, and thus labor supply increases. The decrease of labor market prices (wages) will increase the demand for leisure, reducing labor supply, that is, if wage decreases, more workers will replace the labor with leisure. This is the substitution effect. In addition, when leisure is a normal good and income increases, people will have more leisure time and consumption increases, which accordingly reduces the labor supply. This is the income effect. The individual's response to wage changes depends on the relative size of the two kinds of effects. If the income effect is greater than the substitution effect, in the labor market, wage growth will reduce labor supply. If the income effect is smaller than the substitution effect, wage growth will increase labor supply. For a family, the individual‘s goal is to achieve the family's overall joint utility maximization. But different members face different wage levels. At this point, each family member‘s labor supply not only relies on their own wage, but also on other family members‘ wages. 1.2 Theory of Behavioral Labor Supply In recent years, the behavioral economics penetrates into the various fields of economics where the original theory is modified or refined. After the introduction of psychological and behavioral perspective, labor economics has resulted in the behavioral labor economics. The standard neo-classical labor supply theory predicts that: the increase in wages will enable workers with more labor, when the wages are low, the wage increase will allow workers to substitute leisure with labor, where income effect is greater than substitution effect. So, we can get an upward-sloping labor supply curve. A lot of empirical research evidences show that it is true for the participation of the marginal changes in labor supply. However, the daily wages and daily labor supply don‘t show a clear positive correlation. On the contrary, some studies have shown that intertemporal wage elasticity is negative, indicating that the higher daily wage leads to less labor supply. For example, Camerer et al and Farber respectively studied the relationship between daily income and daily working hours in New York City taxi drivers. They took regression of daily 73 Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition Survey working time and the daily wage and found that taxi drivers cross-elasticity of substitution is negative, but, according to neo-classical labor supply theory, this cross-elasticity of substitution should be positive. Neo-classical theory of labor supply usually attributesthis negative cross-elasticity of substitution in labor supply to the measurement errors. But it is not a good explanation, since Camerer et al and Farber have conducted a calibration of the measurement errors. And different from the assumption in the neo-classical economic theory that individual is fully rational, behavioral economics theory suggests that a person‘s rationality is limited. And, compared with the final result, the individual may pay more attention to the decision-making process and the relative results. For example, during wage negotiations, workers tend to focus on the average wage levels of the group he stays in, rather than the equilibrium wage level in the labor market. Specifically in the labor supply, the difference between neo-classical labor supply theory and behavioral labor supply theory is the introduction of three concepts: Loss Aversion: The hate an individual has for one unit of loss is stronger than the happiness an individual has for one unit of revenue. Income Target: workers may not make labor supply decisions based on the utility maximization in entire life cycle, but in a relatively short period of time, such as daily or weekly. In this short period, workers consider achieving this income target. Reference Dependent Preferences: This is the greatest difference between neo-classical theory and the behavioral economics. Neo-classical theory suggests that people's preferences are stable. The reason why people's behavior will change is that the constraints faced by the individual have changed. And behavioral economics think people‘s preferences depend on their reference point. Individuals evaluate the income change not according to how much income they earned, but according to the gap between the goals and their income. When the income is less than this goal, the individual will regard it as the loss and failure, and when income is higher than this goal, the individual will regard it as income and success. 1.3 Main Points Lewis proposed dual economic structure in the study about economic development of developing countries. He divided a country's economy into the traditional economic sector in 74 Research of Mathematical Economics No. 1 2011 rural areas and modern sector in urban areas. Along with economic development, the surplus labor of the traditional sector will continue to shift to the modern sector of the economy, this process will last until surplus labor of the traditional sector was completely absorbed by the modern sector of the economy. Further, Todaro proposed that the city's modern economic sector can also be divided into two parts. One is the formal sector, where wages are higher and the working hours are relatively fixed. Another part is informal sector, where wages are low, work is unstable, working hours are relatively flexible. Workers can choose the time of labor freely there. Labor force shifting from rural to urban areas can generally only be employed in the informal sector. Few workers have access to the city's formal sector employment. This article describes China's labor supply issues under the framework of the behavioral theory of labor supply. I discussed when the labor supply elasticity is positive and when it is negative. This article also discusses the impacts on labor supply elasticity made by daily income target, which didn‘t exist in previous research. Since those who works in a company taking 8-hour per day is clearly not free to decide their own working hours, so this article will mainly focus on working hours of those in the informal sector. This paper examined changes of working time in the family business and domestic service as wage increases from 1993 to 2006 in China's 10 provinces. And, the samples will be separated into two categories: urban and rural according to registration residence type. Section 2 describes the model of this article that will be used, including the form of theoretical analysis and specific econometrical forms. Section 3 is a description of the statistics and empirical analysis. Section 4 gives the summary and conclusions of this article. 2. Labor Supply Model In our model, labor is an existence which can lead to two results. On one hand, the increase of the labor will increase workers‘ labor income, allowing workers to consume more goods and buy more services, which enable workers to obtain a higher utility. On the other hand, the increase in labor would make workers physically and psychologically feel more tired and weary, increasing the disutility of workers. Workers‘ disutility‘s increase with labor time‘s increase and the marginal utility of leisure‘s increase are the both sides of a coin in the neo-classical labor 75 Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition Survey supply theory. v(wl，y,r)We denote to represent the utility workers get from labor income where wr is an individual‘s wage, is his daily working time, y is his non-labor income and is his l income target. According to the assumption of behavioral economics, individuals‘ sensitivity to income changes is decreasing. Relative to the reference point, i.e. our income target, regardless of income increase or decrease, its marginal effect is diminishing. This is a very different point between the neo-classical theory and behavioral economics. In the neo-classical theory, for a risk-averse individual, the increase in the marginal utility of income must be diminishing, the revenue for the loss of marginal disutility must be increasing at any point. Thus, the form of utility function our model used is different from that in the neo-classical theory. Neo-classical theory uses Von Neumann-Morgan Stein utility function, which is smooth, increasing and concave everywhere (Figure 1), while in behavioral economics, the individual's utility is described with Kahneman-Tversky utility function-a function which is smooth and increasing everywhere except the reference point. That is to say, on the left of reference point it is a convex function, and on the right side of the reference point it is a concave function (Figure 2). This also indicates that the individual‘s marginal utility of wealth increases before reaching the reference point, and only after a reference point his marginal utility of wealth decreases. Figure 1 Von Neumann - Morgan Stein Utility Function 76 Research of Mathematical Economics No. 1 2011 Figure 2 Kahneman-Tversky Utility Function To simplify our analysis, we can assume that in the long run, individuals‘ daily income must be higher than his income target. (This is well understood. If a person simply cannot achieve his daily revenue target, he might wonder whether the daily revenue target is set too high to achieve. So he will decrease his daily income goals.) Thus, in our analysis, after the individual‘s work in a single day, he must be on the right side of the reference point in the utility function. We consider the utility function on the right side of the reference point. The utility function is increasing, so vwlyr'()，,the derivative of the utility function is greater than 0. And marginal utility is vwlyr'()，,decreasing with the increase in l, derivative of is less than 0. For a further vwlyr"()，,derivative of , what will happen? In fact, it can‘t always equal to 0. If it equals to vwlyr"()，,0, will be a negative constant, then the marginal utility eventually becomes vwlyr"'()，,negative, which is incompatible with common sense. So will not equal to 0 (whether it is greater than 0 or less than 0 has respective meaning). I will discuss it in the vwlyr"'()，,appendix. And, we believe that in normal circumstances, is greater than 0. cl()Let us consider the disutility of labor. We use to indicate the disutility labor produced, cl()l is an individual‘s working hours in a day. Obviously, are incremental as l increases, that cl'()cl()is, the marginal disutility is positive, so the first derivative of , i.e. is positive. Marginal disutility is also increasing, because, with the increase in labor time, the uncomfort individual felt would increase faster and faster. For example, after one hour‘s work, the disutility 77 Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition Survey that one more minute makes is so small that we cannot feel. However, when we finish 20 hours of work, one more minute work will make us tired and no longer willing to continue to work, cl''()cl'()which shows marginal disutility is increasing. So the derivative of , i.e. is greater than 0. To simplify our analysis, here we assume that the increasing rate of marginal disutility is cl''()cllcl'()"(),cl''()constant, and thus is a positive constant, and . So the derivative of cl()will be 0. In short, is an increasing and convex function. To sum up, we can list the properties of utility function and the disutility function as follows. vwlyr'()0，,, vwlyr"()0，,, vwlyr"'()0，,, cl'()0, cl"()0, cl"'()0, cllcl'()"(), u,v(wl，y,r),c(l)Next, we define the value function of an individual as . Then the individual faces the following question when he makes labor decision: Maxuvwlyrcl,，,,()() subjecttol024,, Because when l is large, the increasing speed of marginal disutility will be faster than the increasing speed of marginal utility, thus the optimal solution to the planning problem L must be taken in time to . Then, first-order condition is: 024,,l ,u,0 ,L wv'(wL，y,r),c'(L)i.e. (1) This is an implicit function where L is the dependent variable and w, r is the independent variables. We can get worker's labor supply function as follows: LLwr,(,) (2) 78 Research of Mathematical Economics No. 1 2011 2.1 Worker's Labor Supply Elasticity The partial derivative of L with respect to w may be: ,，,，，,LvwLyrwLvwLyr'()"(), 2,,，,wcLwvwLyr"()"() So labor supply elasticity is: 2wLwvwLyrwLvwLyr,，,，，,'()"() (3) e,,2LwLcLwLvwLyr,,，,"()"() We can deform equation (3) like: wvwLyrvwLyr'()"()，,，,ewL,，[1] 2LcLwLvwLyrvwLyr"()"()'(),，,，, So the elasticity is wvwLyr'()，,,ewL,,(1) (4) 2LcLwLvwLyr"()"(),，, vwLyr"()，,,,,where is the coefficient of absolute risk aversion. It evaluates the vwLyr'()，, individual‘s risk preferences and level of risk preferences. For a risk aversive individual, his ,,,0. The more the individual averts risk, the larger will be. v(wl，y,r)cl()By the properties of and which are described earlier, we can tell wvwLyr'()，,,0. Then the sign of elasticity will depend on whether 2LcLwLvwLyr"()"(),，, 1,wL, is positive or negative. This can explain why at lower income levels labor supply ,elasticity is likely to be negative. Because if is very large, that is, the individual is extremely averse to risk, even if w is very small, elasticity will still be negative. That is, even wages are low, relatively higher wages will lead to reduction in working hours. 2.2Changes of Labor Supply Elasticity Resulting From the Changes of Income Targets In equation (3), the partial derivative of e with respect to r is: 79 Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition Survey 2,，,ewLvwLyrcLcL"'()'()"(),222,r (5) ，，cLwLvwLyrwvwLyrcL'()"()["()"()],，,，,,，， Find specific derivation in Appendix 1. 222Since,,，，wvwLyrcL"()"()0，,,,cLwLvwLyr'()"()0,，,,，， ,e2vwLyr"'()，,, the sign of will depend on the sign of . If wLcLcL'()"()0,,r ,e,evwLyr"'()0，,,vwLyr"'()0，,,,0, we have . If , we have . Here, we ,0,r,r vwLyr"'()0，,,assume that (reasons are presented in Appendix 2). Then, e decreases when r increases. Therefore, we can get the following conclusions: When the labor supply elasticity is positive, the higher the income target is, the smaller the labor supply elasticity will be. When the labor supply elasticity is negative, the higher the income target is, the greater the labor supply elasticity will be. For these conclusions, we can explain them like this. There is no doubt that if wages remain unchanged, the higher the income target, the longer the working time. In fact, on both sides of ,，,LwvwLyr"(),equation (2), the partial derivative of L with respect to r is . 2,，,,rwvwLyrcL"()"() ,L,0Obviously, . ,r For the two kinds of people, the ones with high income targets work longer, the ones with low income target work shorter. With an additional unit of wage, income of people who work longer will increase much more, but income of people who work shorter will increase only a little. If their labor supply elasticity is positive, the workers working longer can achieve utility maximization by increasing working time only a little, while the workers working shorter hours have to increase much more work before they can obtain additional utility maximization. If their labor supply elasticity is negative, the workers working longer are able to reduce 80 Research of Mathematical Economics No. 1 2011 much more labor to obtain utility maximization, while the workers working shorter time can‘t reduce too much labor in order to obtain utility maximization. 2.3Econometric Model Used in this Article To estimate the elasticity of labor supply, we use the following form of measurement equation: y,,，,Wage，,Type*Wage，, it01it2ititit Since it is difficult to get the data of income target, I choose the variable household yWageregistration type as its proxy variable. is the daily working hours of individual i, itit Type,is the hourly wage, and is the individual‘s household registration type. is a itit ,stochastic error term. This is a panel equation so t denotes the time. represents the elasticity 1 and represents the effect on elasticity made by household registration type. ,2 3. Empirical Analysis 3.1Data Source Chinese CDC and the North Carolina University conducted China health and nutrition survey (CHNS) in year 1989, 1991, 1993, 1997, 2000, 2004, 2006. A total of 19,000 samples were taken from China's nine provinces (Guangxi, Guizhou, Heilongjiang, Henan, Hubei, Hunan, Jiangsu, Liaoning and Shandong). This detailed survey, covers family relations, property, work, income, health, diet and so on. This article uses this statistics. For the research‘s need, I will restrict the samples to those in the urban informal sector employment whose jobs are family and individual business, commercial, household workers, workers in informal enterprises, carpenters, nurses, tailors, family clinics, repairers of electrical appliances and so on. All of them are the so-called ―self-employers‖. The common feature of these workers is that they are able to freely decide their own working hours. Considering the limitation of years, we choose the samples from year 1993, 1997, 2000, 2004 and 2006. Because of this restriction, the amount of samples satisfying our requirements is reduced to 633. 81 Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition Survey 3.2Statistics Description Table 1 Variables and their means Standard Vaviables Unit Means Min value Max value deviation Labor time Hours per day 8.95577 2.62497 1 18 Hourly wage Yuan per hour 4.649300 11.43677 0.41 176.22 0 represents urban and Household registeration type 0.396524 0.48956 0 1 1 represents rural Annual income yuan 7755.387 8980 835.668 98400 In Table 1 we can find that the average working time of self-employer is 8.95577 hours per day, nearly 1 hours more than the usual employee. Average hourly wage is 4.6493000 yuan per day, minimum value is 0.41, maximum value is 176.22 and standard deviation is 11.43667, which show that polarization among the self-employee is rather severe, income of minimum level can only cover the basic need and the maximum level exceeds the social average level. In our samples, 39.6524% of the workers is rural and the other is urban. The average income is 7755.787 yuan, which is lower than the average wage of all the workers. 3.3Empirical Results Table 2 Empirical Results Independent variables Coefficent Standard deviation p-value Ln (hourly wage) -0.1829 0.02225 <0.0001 Interactive item 0.069357 0.0294 0.0189 Constant 2.258651 0.1616 <0.0001 Joint significance test F-value: 2.04 <0.0001 2R 0.6357 From Table 2 we can find that two independent variables explain the labor supply hour significantly. We can find working time will decrease by 0.1829% if the wage increases by 1%. This shows that labor supply elasticity of urban workers is -0.1829. The coefficient of interactive item shows that labor supply elasticity of rural labor is 0.069357 less than urban labor. According to the estimation result, labor supply elasticity is likely to be negative though the 82 Research of Mathematical Economics No. 1 2011 hourly wage level of self-employer is low. It supports that some people‘s labor supply elasticity is likely to be negative although their wage level is low. Furthermore, labor supply elasticity of rural labor is significantly smaller than that of urban labor. Combined with the assumption we mentioned before that the income target of rural labor is comparatively low, the estimation results support the second conclusion of the model: if the labor supply elasticity is negative, the higher the income target, the larger the labor supply elasticity. 4. Summary and Conclusions This article has done a certain amount of expansions for the behavioral labor supply theory. It discusses another reason why the labor supply elasticity is positive or negative, i.e. individual's coefficient of absolute risk aversion. This article discusses how the income target affects the size of labor supply elasticity based on Kahneman-Tversky utility function and theory of reference dependent preferences. In addition, when doing empirical research, this paper examines performances of the two different labor forces in the urban informal sector of China according to the division of household registration type, which have never been done before. For empirical results, this paper examines the labor supply elasticity in urban informal sector and found that the elasticity is negative, which is contrary to neo-classical labor supply theory, but consistent with the predictions of behavioral labor supply theory. In addition, the article also compares the sizes of the elasticities of urban labor supply and rural labor supply in informal sector, verifying the forecast of the model about the effects on elasticity made by the income target. This article still has some shortcomings, which are reflected in the following aspects. (1) Assumptions of the model are too strict, especially in which increasing rate of marginal disutility is constant. This assumption needs further empirical examination. (2) For the regression results in Table 2, some coefficients need to be explained more. (3) There is a measurement error. The hourly wage in this article is computed by dividing annual income by annual working hours where there may be a "division bias‖. For these shortcomings, I will make further improvements in the future, for example, to discuss how the labor supply model will change when increasing rate of marginal disutility is non-constant. Also, I will use more detailed "field data", or use instrumental variables to estimate the individual's hourly wage, and thus avoid the "division 83 Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition Survey bias". Despite above shortcomings, the expansion of the behavioral labor supply theory is worth considering. And it shows it is necessary and meaningful to divide the labor in the informal sector into two types and conduct a comparative study. 84 Research of Mathematical Economics No. 1 2011 Appendix 1 ,eThe Derivation of ,r ,,,LLL222["()"()"'(1)]['()"()]cLwvwLyrwLvwcLwLvwLyr，，,，,,，,,e,,,rrr,2 2,r，，cLwLvwLyr'()"(),，,，， ,,,LLL222["()"()"'(1)]['()"()]cLwvwLyrwLvwcLwLvwLyr,，,,,,，,,,,rrr,22，，cLwLvwLyr'()"(),，,，， ,,,LLL222wvwLyrcLwLvwLyrcLwLvwLyrcLw"()'()"()"()"'()'().(1)，,,，,，，,,,,,rrr,22，，cLwLvwLyr'()"(),，,，， LcLcL"()'(),Since , the first two items of numerator offset. Differentiate equation (2), ,，,LwvwLyr"(),can be obtained. Substitute it into equation (6), we can get: 2,，,,rwvwLyrcL"()"() 2,，,ewLvwLyrcLcL"'()'()"(),222,r ，，cLwLvwLyrwvwLyrcL'()"()["()"()],，,，,,，， Appendix 2 vwlyr"'()，,About vwlyr'()，,We know that, after the reference point, is a decreasing function and 85 Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition Survey vwlyr"'()，,always positive, and is its second derivative. fx()fx"()Therefore, we consider a general function and its second derivative . fx()If is a decreasing function and it must always be greater than 0, when x is large fx"()0,. Therefore, enough, the image is shown as Figure 3. This is a convex function, so vwlyr"'()0，,, is a good and reasonable assumption. Another evidence comes from the precautionary savings effect (see Kimball, 1989) in macroeconomics. In macroeconomics, the premise of the existence of precautionary saving effect is that the third derivative of utility function is positive. 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Todaro, ―A Model of Labor Migration and Urban Unemployment in Less Development Countries‖, American Economic Review, 1969, Vol.59: 138-148. 87 Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition Survey (Responsible Editor: Xu Lingjue) (Proofreader: Ren Junqiushi) 88 Research of Mathematical Economics No. 1 2011 A Dynamic Model of the Optimal Carbon Tax and Sustainable Development Feng Junlong (School of Economics, Renmin University of China) With the development of the industrialized society, global warming caused by the increasing emission Abstract: of greenhouse gases has become a threat to the whole human beings. Under this pressure, governments discuss every possible solution, among which levying carbon tax is one of the most considered way. In a view of sustainable development, this paper discusses the optimal tax rate within a dynamic frame. Besides, the optimal reforestation subsidy and the relationship between this subsidy and forest stock are also studied as forest is a social common capital absorbing CO, which can be either felled for present profit or reforested for future welfare. 2 Theoretically, this paper is a development of the Ramsey dynamic model, while for the reality significance this paper provides some positive ideas for the policy making. Carbon Tax, Reforestation Subsidy, Imputed Price Keywords: 1. Introduction Nowadays, global warming has become a threat to the whole mankind. One of the fundamental reasons is the emission of greenhouse gases, among which CO is the first to blame. 2 However, as a sign of modern industry, the emission of CO is a result of the economic 2 development, and even the utility of consumers increases with CO emission since almost all 2 consumption is connected with it. Therefore we have a trade-off in the decision of CO emission. 2 Meanwhile, forests provide woods as a vital capital for an industrialized economy. However, on the other hand, forests make great contribution to CO absorbing. According to a report from 2 IPPC (1991a, 1992, 1996a, 2001a, 2002), if we keep the rate of felling trees and releasing CO, 2 the average global temperature by the end of 21 century will be 3-6 degrees higher than that before the Industry Revolution. So, we also meet a tradeoff between felling trees and reforestation. 89 A Dynamic Model of the Optimal Carbon Tax and Sustainable Development After the Copenhagen Conference in 2009, to levy carbon tax has become a hot topic worldwide. Then it‘s important to decide the optimal tax rate which will exactly cover the negative externality for firms and make no dead-weight loss. And as reforestation calls for money, part of the tax can be transformed to be subsidies for it. Therefore, this paper is dedicated to the study of the optimal carbon tax rate and the optimal subsidies, as well as the relationship between subsidies and the forest stock. In this field, Keeling (1983), Takahashi et al. (1980) and Uzawa (2005) have developed some fundamental models. But these models took only CO into consideration and the factors like forests were treated as 2 exogenous. And for simplicity, some of them only discussed discrete cases of two periods. For forests as a social common capital, Solow (1974a, b), Sen (1982), Page (1997) made great contribution to solving the very problem within a dynamic framework. This paper will develop a dynamic model in long run equilibrium. Forests as a social common capital will be treated endogenous. The Ramsey-Cass-Koopmans dynamic model is applied and developed here to work out the optimal tax rate as well as the subsidies. This paper is within a framework of Uzawa‘s theory of social common capital, while some evolutions are made so that the model is much more realistic. The rest of the paper is organized as follows: divided into 3 parts, section 2 presents basic model, where the propositions to be proposed are formally derived and proved; section 3 presents the main results of this paper and my concluding remarks. 2. The Dynamic Model 12.1 Consumers First, we study the consumers‘ behavior. To simplify the problem, we discuss a case where utility can be aggregated as a whole. We consume that there is only one goods in the market and the price is given. Then, we derive the utility function, uucaVV,(,,,) (1) cwc caVwhere is consumption is the CO emission in consumers‘ life, is forestry stock and ，2cw 1 In this paper, to focus mainly on the target problems, we omit discuss of constrains and the first order conditions for consumers and producers. Nonetheless, the analysis is already set within a frame of equilibrium, so the result derived will satisfy the omitted conditions automatically. 90 Research of Mathematical Economics No. 1 2011 V is CO stock. 2c (,)caVVObviously, in reality, the utility function should be independent for , , . The cwc reason is that in a short period, the stock of either CO or forests don‘t affect consumers‘ daily 2 choice. Therefore, the utility function can be ucaVVVVuca(,,,)()()(,),,, (2) cwccwc ()V,,()Vwhere and are the extent to which the consumers are affected by a marginal cw increase of CO and forests, which can be referred to as the impact index. Therefore they satisfy 2 the following conditions: ''',,,()0,()0,()0VVV,,, (3) ccc ''',,,()0,()0,()0VVV,,, (4) www uca(,)Meanwhile, we assume that is neo-classical. c To simplify the discussion, we introduce the impact coefficient. We define '',,()()VVcw,,,(),()VV,, cw()()VV,,cw as the impact coefficient which means the relative rate of the marginal change in the impact index due to the marginal change in CO or the forest stock (Uzawa, 2005). From the definition, we 2 have ,,()0,'()0VV,, cc ,,()0,'()0VV,, ww 2.2 Producers Here, we divide the producers into two sectors. One is all the industries connected with woods, including felling trees and reforestation. As forestry is social common capital, we assume there is only one designer in this sector to decide the amount to fell and the amount for reforestation. The other is the firms producing the only goods as mentioned in 2.1. And we assume this market is competitive. 2.2.1 Forestry 91 A Dynamic Model of the Optimal Carbon Tax and Sustainable Development Before we come to the problem of decision making, an important fact should be introduced. Since trees can regenerate by themselves, it‘s necessary to introduce a regenerating function '',()VVV,,,()()0VV,,,()V, then we have , and there exist , such that . ww,0wwww Now if we add felling and reforestation to the model, a dynamic function for the forest stock can be: , (5) VVYX,，,,()wwVw Ywhere is the amount of reforestation and is the amount felled. XVw We assume the cost for felling trees is connected with the stock of the trees and the amount CXV(,)to be felled, so the cost function for felling trees satisfies the following conditions: w CXVCXVCXV(,)0,(,)0,(,)0,,, wXwVww 2CCCCCC,,,,0,,0,0 (6) XVXXVVXXVVXVwwwwww And the cost for reforestation is also connected with the stock as well as the amount to be BYV(,)reforested, so the cost function for reforestation satisfies the following conditions: Vww BYVBYVBYV(,)0,(,)0,(,)0,,, VwYVwVVwwVwwww 2BBBBBB,,,,0,,0,0 (7) YVYYVVYYVVYVVwVVwwVVwwVwwwwwww ,Therefore the profit of this sector, to be denoted as , is derived as: w ,,,,pXBYVCXV(,)(,) (8) wwVwww 2.2.2 Other Firms Now we‘ll turn to the firms producing the only goods for consumers. As we have assumed, 2pthe price is given. And we assume CO emission per product of each firm equals, then we 2g have: ,,,,,,,pxa (9) gp ,a,where is denoted to be the th firm‘s profit, is the CO emission by firms, is ,2,p 2 In reality, this assumption is satisfied. We only convert the product into carbon unit. 92 Research of Mathematical Economics No. 1 2011 carbon tax per unit of emission. As for the whole sector, ,,,,,g (10) , Therefore, the profit of this sector can be derived as ,,,,，,,，,,,,,()(,)(,)pxapXBYVCXV,gwgpwVww (11) w, 2.3 The Imputed Price and Social Optimum Now, as we have discussed both sides of the market, we can study the optimal choices of consumers and firms. But before that, it‘s necessary to introduce a new concept, i.e. the imputed price. The imputed price means the marginal welfare change due to marginal change of CO or 2 ,,,forest. Denote the imputed price of CO and forest to be . Therefore, the optimal carbon tax 2 and reforestation subsidies should be equal to these factors‘ imputed price, so that they could reflect the very welfare change and cause no dead weight loss. And for simplicity, we assume that in an equilibrium condition, the society‘s utility maximization and the profit maximization of producers could be reached together, and they are connected by the following equation: ,,,u (12) ,where is the inverse of marginal utility of income (as assumed, equals profit). So if we convert the imputed price to discuss utility, we have: ,,,, (13) ,,,, (14) Now, as there are two aspects to discuss, the optimal carbon tax together with the optimal subsidies for reforestation, and the relationship between the subsidies and forest stock, we‘ll study a symmetric problem from the utility maximization and profit maximization. As mentioned above, such two maximizations are equal and only two sides of a unified equilibrium. To discuss them separately is only a convenient way to solving the problems. 2.3.1 Social Utility Maximization In order to achieve sustainable development, we maximize the following Ramsey utility 93 A Dynamic Model of the Optimal Carbon Tax and Sustainable Development function: U ,t,, (15) Uuedt,t,0 uWe should guarantee each is maximized and the integral converges. t The increase of CO in the atmosphere satisfies the following equation: 2 , (16) VaV,,,cw where ,aaa,，,cp (17) , Therefore we derive the imputed utility function at time t: HVVucaVYXaV,，，,,,,,,,,,()()(,)(())() (18) tcwtctwVttttwtw The Hamiltonian Equation should be ,,Ht,,,,,, (19) tt,Vwt ,,Ht,,,,, (20) tt,Vct with the transversality conditions: ,,tlim,Vedt (21) twtt,, ,,tlim,Vedt (22) tctt,, ,,3The steady state should be . ,,,,0tt From (19) (20) (13) (14) (11) (2), we derive ,()V()V,1*cw,，, (23) (),',,,,,, ,()V*c,, (24) ,,, 3 Here, the realistic meaning of the steady state can be interpreted that policies are relatively steady and not flexible with time. 94 Research of Mathematical Economics No. 1 2011 Then it‘s easy to derive the expressions of the optimal carbon tax and the optimal reforestation subsidies: ,()Vvv*c, (25) ,,, ,()V1vv*c,，(())V,,, (26) w',,,, Here we have the following conclusions: The optimal carbon tax should be proportional to firms‘ profit. It‘s also in Proposition 1: direct ratio with people‘s CO coefficient and is the inverse of utility discount rate. 2 Firms should take the responsibility of reforestation, which is proportional to Proposition 2: profit and affected by the coefficient of CO and forestry. 2 2.3.2 Producer‘s Profit Maximization Now we‘ll turn to the other side of the market to study the dynamic relationship between subsidies and forests stock. First, we construct a profit maximizing problem. Similarly we have ,t,, (27) Pedt,,t,0 where is denoted to be the total profit with sustainable development. P Then maximize the profit of each time t with the converging conditions. From (11) we get ,,HpxapXBYVCXVVYXaV,,，,,，，,,,()(,)(,)(())(),,,,,,tgttptwttVtwttwttwtVttttwtww, (28) YXTake partial differential of , and set them to be 0we get ，Vttw pC,，, (29) wXtt B,, (30) Yttw In the mean time, with the Hamiltonian Equation, we have , (31) ,,,,,,,,，，,['()][(,)(,)]VCXVBYVtwttVtwtVVwttwww with the transversality conditions being 95 A Dynamic Model of the Optimal Carbon Tax and Sustainable Development ,,tlim0,Ve, (32) twtt,, ,To simplify the model, we‘ll fix at first and make it changeable after we analyze the t steady state characters. So here it‘s a given parameter at time t. Then we‘ll focus on a dynamic (,,)XYVoptimum time-path to be denoted as and we need to find a positive continuous tVtwtw ,time-path of the imputed price . With the differential equations (31) and (5), under the t 4V,constraints of (29) (30), we‘ll derive the following phase diagram of and : w , B A'C E *, C'B' A 0O V Vww V,Figure 1 The Phase Diagram of and w From figure 1, we can get the following conclusions: If the price of woods increases, curve AA’ shifts upward with a slight upward Proposition 3: shift of curve BB’. Therefore, the long run equilibrium forest stock will decrease which calls for an increase of reforestation subsidies. If time discount rate increases, curve BB’ shifts downward leading to a Proposition 4: 4 The derivation of the diagram is proved in detail in the index. 96 Research of Mathematical Economics No. 1 2011 decrease in both long run forest stock and reforestation subsidies. If carbon tax increases, curve BB’ shifts upward so that the long run forest Proposition 5: stock and reforestation subsidies increase. 3. Conclusions This paper develops some basic models for sustainable development. In a dynamic view, the optimal carbon tax together with the optimal subsidies for reforestation, and the relationship between the subsidies and forest stock are discussed. Through analysis, the paper gives five conclusions rearranged as follows: The optimal carbon tax should be proportional to firms‘ profit. It‘s also in Proposition 1: direct ratio with people‘s CO coefficient and is the inverse of utility discount rate. 2 Firms should take the responsibility of reforestation, which is proportional Proposition 2: to profit and affected by the coefficient of CO and forestry. 2 If the price of woods increases, curve AA’ shifts upward with a slight Proposition 3: upward shift of curve BB’. Therefore, the long run equilibrium forest stock will decrease which calls for an increase of reforestation subsidies. If time discount rate increases, curve BB’ shifts downward leading to a Proposition 4: decrease in both long run forest stock and reforestation subsidies. If carbon tax increases, curve BB’ shifts upward so that the long run forest Proposition 5: stock and reforestation subsidies increase. Nevertheless, further work should be done in order to perfect the model. The optimal tax rate and subsidies derived in this paper are connected with some physical and psychological ,()V,()Vparameters like , and . To get the average level of these parameters, some ,cw statistic work is required. 97 A Dynamic Model of the Optimal Carbon Tax and Sustainable Development Appendix The Derivation of Figure 1 First, take differential of (29) and (30): C11Xvw,,，,,[]dXdVdpd (33) wwCCCXXXXXX BYV1Vww,,，,[]dYdVd (34) VwwBBYYYYVVVVwwww With the consumption of (6) and (7), we derive ,Y,Y,X,X,XVVww00,,,00,, , , , 0,,V,p,,,V,,www The conditions of steady state are 0**,()0VYX，,, (35) wVw 00*0*0['()][(,)(,)]0,,,,,,，，,,VCXVBYV (36) wttVtwtVVwttwww ,Take differential of (35) and (36) ( is given), and with (33) (34), we get t 10,, ,,0C,,,,dVdp,,,,XX1112ww,, 37 ,(),,,,,,*,,,C,d,,,dXV,2122,,,,w,,,,,C,,XX,, where BCYVXvVwww,,,,，,'0 11CBXXYYVVww 11,,，,0 12CBXXYYVVww 98 Research of Mathematical Economics No. 1 2011 22BBB,CCC,VVYYYVXXVVXVwwVVVwwwwwww,,''0,，,, 21CBXXYYVVww BCYVXVVwww,,,,,,，,,(')0 22CBXXYYVVww It‘s obvious that ,,,,,,,,0 11221221 V,Now, we‘ll discuss the relationship between and . w d,,11()0,,,(1) ,V,0dVw,w12 ,Then, is an upward sloping curve denoted to be AA’. On the right side of AA’, V,0w ,, while on the left side of AA’, . V,0V,0ww d,,22()0,,,(2) ,,,0dV,w21 ,, ,,0,,0Then, is a downward sloping curve denoted to be BB’. Above BB’, while , ,,0below BB’, . Therefore E is a saddle point and only one pair of solution paths, CE and C’E exists. References Aghion, P. and Howitt, P., ―A Model of Growth through Creative Destruction‖, Econometrica, 60, 1992, pp. 323-352. Keeling, C.D., ―The Global Carbon Cycle: What We Know from Atmospheric, Bio-Spheric, and Oceanic Observations‖, Proceedings of Carbon Dioxide Research, Science and Consensus, United States Department of Energy, Washington, DC, , 1983, pp. 3-62. ? Page, T., ―Sustainability and the Problem of Valuation‖, in R. Costanza (ed.), Ecological Economics: The Science and Management of Sustainability, New York: Columbia University Press, 1997, pp. 58-74. Ramsey, F.P., ―A Mathematical Theory of Saving‖, Economic Journal, 38, 1928 pp. 543-559. 99 A Dynamic Model of the Optimal Carbon Tax and Sustainable Development Sen, A.K., ―Approaches to the Choice of Discounted Rate for Social Benefit-Cost Analysis‖, in R.C. Lind, K,J. Arrow, and G.R. Cores. Discounting for Time and Risk in Energy Policy, Baltimore: Johns Hopkins University Press, 1982, pp. 325-353. olow, R. M., ―Intergenerational Equity and Exhaustible Resources‖, Review of Economic Studies, 41, 1974a, S pp. 29-45. Solow, R. M., ―The Economics of Resources, or the Resources of Economics.‖ American Economic Review, 64, 1974b, pp. 1-14. Uzawa, H., Economic Analysis of Social Common Capital, Cambridge University Press, 2005, pp. 49-57, 257-292. Uzawa, H., ―Imputed Price of Greenhouse Gases and Land Forests‖, Renewable Energy 3, 1992a, pp. 499-511. (Responsible Editor: Zhang Lingyun) (Proofreader: Xu Lingjue) 100 Research of Mathematical Economics No. 1 2011 A Study of Chinese Residential Electricity Demand Elasticities Ren Junqiushi (School of Economics, Renmin University of China) As energy plays a more and more significant role in the economy, electric power as an extremely Abstract: important kind of energy has received more and more attention. This paper measures short-term residential electricity demand elasticities of Beijing as the representative of China in two ways: econometric analysis (using the ELES model) and terminal-energy-need method (adopting a double-log demand function). Both methods suggest that the income elasticity of Chinese residential power demand is about 0.5, while the self-price elasticity is around -0.3. Further, this paper put forward approaches to estimate long-term elasticities by improving these two methods. The findings of this research can help better inform public policy makers and regulators about the responsiveness of residential electricity consumers to price and income changes, as well as scholars who would like to research more in-depth. Residential Electricity Demand, Income Elasticity, Price Elasticity, Econometric Analysis, Keywords: Terminal-Energy-Need Method 1. Introduction 1.1 Research Meaning and Significance Energy resources have served as a strong constraint for long-term economic development of China, and even of the whole world. As the United Nations Millennium Development Goals Report of 2009 noted, for the realization of the world‘s economic sustainable development, every country must try to change its economic development approach so that the globe could break this constraint. Among all the energy resources, electricity occupies a ponderable status. In fact, the electric power, which is an important kind of fuel as well as a clean secondary source, has had a far-ranging appliance in different sectors of society. Moreover, in recent years, Chinese residents‘ electricity consumption has been continuously increasing and occupying a significant weight in the total consumption of electric power. Between 2005 and 2007, the annual consumption of 101 A Study of Chinese Residential Electricity Demand Elasticities residents hits 282.48 billion degrees, 325.16 billion degrees and 358.40 billion degrees respectively, with an average annual growth rate of 12.7%; in the next few years, the average annual growth rate has maintained a high bit at 10%, and residents‘ electricity consumption 1weighed 12.86%, 10.04% and 12.53% of the total electricity consumption respectively. Thus, taking into account the short-term requirement of supply-demand balance and the long-run energy substitutability, we need to analyze more in-depth the demand, especially the demand of residents, for power. Currently, the abroad has already had many studies exploring residential electric power demand, but few researches have considered residential electricity demand in Asian countries, especially in the developing ones. As a matter of fact, most researches about Asian electricity market have focused on the industrial demand merely. This paper will study the electricity demand of Chinese residents based on the representative city Beijing, thus fill up this blank. In the study of residential electric power demand, a very important and fundamental subject is to measure demand elasticity. Theoretically, elasticity is not only a basic economic concept but also an indispensible factor in demand description; practically, no matter what we aim at—understanding Chinese electricity market, judging whether it would be effective to control residents‘ power using by adjusting the price of power, or estimating how much more burden is put on residents when electricity price increases a certain amount—we cannot reach our goal without studying elasticity first. Therefore, this paper is meaningful when it puts forth effort to measure the electric power demand elasticity of Chinese residents. 1.2 Overview In retrospect, many models and theories have been proposed to research the demand of electricity. The most commonly used methods in studying residents‘ electric power demand are econometric analysis, terminal-energy-need method and artificial intelligence. Econometric analysis aims at describing the interdependent or causal relationships among different economic variances according to a set of simultaneous equations derived from certain economic theory and hypothesis. If we could set up simultaneous multi-equation econometric model which would describe the relationships between electric power demand and relevant economic variables accurately, we would be able to estimate the elasticity of power demand 1 Statistics are obtained from 102 Research of Mathematical Economics No. 1 2011 accordingly. (See T. R. Lakshmanan and William Anderson, 1980; Xavier Labandeira, José M. Labeaga and Xiral López-Otero, 2009) The basic thought of terminal-energy-need estimation is to figure out the consumption of fuel through terminal equipment. As for residents, we can regard equipment such as lighting, cooking, home appliances (television, refrigerator and washer), heating and air conditioners as terminators, thus estimate resident power consumption based on their universality rates, occupation coefficient and fuel outgo rates, etc. (Tsuyoshi Ueno, Ryo Inada, Osamu Saeki and Kiichiro Tsuji, 2005). This thought averts from sophisticated mathematical deduce; moreover, it is able to resolve problems such as the inevitable (sometimes rather large) deviation from reality caused by econometric models, which endows it a higher precision when being used to estimate elasticity. A famous representative of artificial intelligence—artificial neural network has been successfully applied to analyzing short-term electric power market, and valued by scholars at the early part of 1990s because of its outstanding study function, ability to handle the non-linear relation between input and output variables and shorter time needed in the calculation process (see Abdallah Al-Shenri, 1999). However, to make use of an artificial intelligence tool to steer demand analysis, we need a great deal of training data, and the accuracy would be seriously compromised when historical data was limited; furthermore, this tool lacks the ability to study uncertainty or to handle misty information. Therefore, this paper will abandon artificial intelligence tool, but make use of the two previous methods: econometric analysis (using the ELES model) and terminal-energy-need approach (adopting a double-log demand function), according to data between 1999 and 2008 got from Beijing Statistical Yearbook (compiled by Beijing Bureau of Statistics and National Bureau of Statistics Survey Office in Beijing, and published by China Statistics Press), to estimate short-term income elasticity and self-price elasticity of residential electricity demand in Beijing, arrive at a conclusion and then analyze this conclusion. This paper is structured as follows: section 1 is an introduction; section 2 is the application of econometric analysis method, in which the specific ELES model is introduced to estimate short-term elasticity of demand; section 3 is based on the thought of terminal energy needs, in which a double-log demand function is built to re-estimate elasticity, and the results are compared with those of the ELES model; section 4 includes conclusions and analysis; section 5 103 A Study of Chinese Residential Electricity Demand Elasticities discusses some probable directions of future development, in which the two ways described in this paper are improved so that they can be used to estimate long -term elasticities of demand. 2. Extended Linear Expenditure System Model 2.1 Development of Model Beginning from the 1980s, scholars have developed and applied the Almost Ideal Demand System (AIDS): nM,,,,,，，loglogp(1,2,,)in, (1) ,iiijjip,1j ,,,,where is the consumption expenditure of the ith commodity; and are 、ijiii pparameters needed estimation; is the price of the jth good; represents the total Mj expenditure; p is the general price index, and is defined as follows : 1pppp,，，,,,loglogloglog (2) ,,,0kkklkl2kkl Since (1) (2) are non-linear equations, the parameter estimations are too troublesome. Thus ,logpDeaton and Mulbauer (1980) proposed to make use of the Stone indexes: to ,kk replace (2), which give rise to (3): nn ,,,,,,，，,logloglogpMp (3) ,,iiijjijj,,11jj ,,,,where . jij However, empirical research found that the parameters in the AIDS often failed to satisfy the coefficient constraints required by demand functions (see Thomas J. Lareau and Joel Darmstadter, 1983). To deal with this problem, a lot of scholars began to introduce the linear expenditure utility function proposed by Klein and H. Rubin: nn UUXbXr,,,()ln() (4) ,,iiiii,,11ii This function assumed that the utility of commodity i could be represented by the logarithm 104 Research of Mathematical Economics No. 1 2011 Xrof the difference between actual demand and basic demand , thus total utility function ii bcould be calculated by adding up utility functions of all commodities; represented the i marginal budget share of the commodity i. Among all the theories derived from this utility function, a rather famous one is the Linear Expenditure System (LES) proposed by English economist R. Stone: n PXPrbCPr,，,()(1,2,,)in, (5) ,iiiiijj,1j PXwhere is the consumption expenditure of the ith good or service; C is the total ii n PCPX,consumption expenditure of all goods and services, and ; is the market price ,iiii,1 brof the ith good or service; is the basic demand of the ith good or service; is the marginal ii budget share of the ith good or service, namely, the ratio of additional expenditure leaving basic n b,1expenditure outside used to consume the ith good or service. Obviously, . ,i,1i Again, because the parameters of LES model are nonlinear, which makes it difficult to estimate them, later economist Lluch revised the LES model with income Y replacing total *bbudget expenditure C, and marginal propensity to consume taking the place of marginal i n*bb,1budget share . As total expenditure is less than total income, we have . Thus the ,ii,i1 Extended Linear Expenditure System (ELES) follows: n*PXPrbYPr,，,()(1,2,,)in, (6) ,iiiiijj,1j The Extended Linear Expenditure System not only explains the impact income and prices bring on residential consumption pattern, but also considers varies residential consumption expenditure as interrelated and mutually constraining behaviors. In addition, it divides people‘s demand into basic needs and beyond-basic needs, in which basic needs have nothing to do with the level of total expenditure, and residents would not arrange beyond-basic needs according to corresponding marginal propensity to consume unless basic needs were satisfied. Theoretically, 105 A Study of Chinese Residential Electricity Demand Elasticities this model makes it possible to analyze residential basic needs, marginal propensity to consume and different kinds of demand elasticity. 2.2 Application of the Model Due to the fact that the Extended Linear Expenditure System itself is a function of demand, elasticity can be obtained by closing into corresponding elasticity formula. XY,,. For income elasticity, based on the formula: , we have ,YX, n，，*bYPr(),,ijj,,**XbYbY,YY,,1jiii,,,..r (7) ,,，,,iiYXYPXPXV,,,,iiiiii,,，， VPX,where . iii XP,,. For own price elasticity of demand, according to the formula: , there is ,PX, n，，*bYPr,,ijj,,,XPP,j1,iii,,,..,,，riii,,PXPPX,,iiiii (8) ,,，， *nbPr*iii()(1)1,,，,,,,PrYPrb,iijjiVVj,1ii Adapting the ELES model to calculate residential electricity usage, we are able to establish the regression equation of electric power consumption: nn***PXPrbYPrPrbPrbY,，,,,，()() (9) ,,eeeeejjeeejje,,11jj where the parameters need estimation are: Pr(1) The basic demand for electricity ; ee n Pr(2) The total basic demand ; ,jjj,1 106 Research of Mathematical Economics No. 1 2011 *(3) The propensity to consume b. e *Among the parameters above, b can be obtained from regression equation e n*PrPrVabY,，, but the estimation of and requires us to establish econometric ,jjeeeeej,1 equations of all residential consumptions. The classification of statistical yearbooks gives us a wonderful guidance: foods, clothing, household equipment and service, medical care, transport and communications, education and recreation, housing, and miscellaneous items. We mark each of them successively with a number ranging from 1 to 8. And because electricity is included in housing, it is possible to remove it from the residence. In the final analysis, we are able to establish 9 econometric equations: *(1,2,,8)i, (10) VabY,，iii * VabY,，eee Obtaining a and b through regression, and then according to n a,in,1i, (11) Pr,iin*,1i1,b,i,1i and n，，a,j,,,1j*,, (12) Prab,，eeeen,,*1b,,j,,,1j，， we find the way to estimate all the necessary parameters. Based on the data between 1999 and 2008 gained from Beijing Statistical Yearbook, after calculation, this paper gets the following OLS regression equations: 107 A Study of Chinese Residential Electricity Demand Elasticities Table 1 OLS Regression Equations in ELES Application 2Item Regression Equation R Foods C = 0.1502Y + 1295.1 0.9857 1Clothing C = 0.0562Y + 93.386 0.9763 2Household equipment and C = 0.0206Y + 490.8 0.8259 3service Medical care C = 0.0571Y + 29.543 0.8456 4Transport and communications C = 0.1125Y - 476.71 0.8286 5Education and recreation C = 0.079Y + 536.02 0.7461 6Housing(without electricity fee) C = 0.0368Y + 76.823 0.8218 7Miscellaneous items C = 0.017Y + 221.4 0.6998 8Electricity fee C = 0.0069Y + 107.45 0.8180 e 2 The smallest in this table is 0.6998, moreover, items except for the miscellaneous or R 2education and recreation have larger than 0.81, which means that the data fit relatively well. R Bringing the regression coefficients into (11) (12) (7) (8), we have the following results: (1) The residential marginal propensity to consume power is around 0.0069; (2) The basic power need is about 137.16 yuan per person per year; (3) The income elasticity of demand is about 0.533 ; (4) The self-price elasticity is -0.38 approximately. 3. Terminal-Energy-Need Method 3.1 Brief Introduction of the Method and Model Terminal-Energy-Need method aims at calculating energy consumption through terminal equipment. Its principle is to determine the energy consumption from the number of devices, the utilization rate of appliances and the efficiency of energy usage, rather than the quantity of energy consumption itself. Generally speaking, residential electricity terminal is considered to include lighting, appliances, heating, air conditioning and so on. The elasticity of residential power demand can be estimated on the basis of knowledge about the power usage of typical devices. This method is directly based on the user's demand, thus avoiding the more complex theoretical analysis and enigmatic problems such as the failure of solving mathematical models, so that the accuracy of estimation is improved. Building on Terminal-Energy-Need method, and making use of double-log model, this 108 Research of Mathematical Economics No. 1 2011 paper will go on measuring the elasticity of residential electricity usage. The reason why a double-log model is used here is that it is not only easier to explain, but also more logical. The specific model is set as follows: lnlnlnlnlnQaaYaWaPaPu,，，，，， (13) 12345ef where Q is the annual electric power consumption per capita (in degrees); Y is the annual income per head (in yuan); W is the total annual electricity consumption of three representative Presidential power terminals (lighting, TV, air conditioning) per year per capita (in degrees); e Pis the consumer electricity price index; is the fuel retailing price index; and u represents the f random error. aThe income elasticity of power demand can be measured by , and the self-price elasticity 2 acan be got by estimating . 4 3.2 Numerical Calculation lnQlnPIn order to determine the relation between and , , as well as lnWlnYe lnP, I begin with adopting SPSS18.0 to make multiple regression and at the same time f diagnose multi-collinearity. The result is displayed in Table 2. aTable 2 Multiple Regression and Multicollinearity Diagnosis Non-standardized Standardized Multicollinearity coefficient coefficient statistics t Sig. Standard B Tolerance VIF deviation (Constant) -2.985 0.491 -6.077 0.002 Income 0.933 0.086 0.912 10.794 0.000 0.064 15.613 Appliance -0.044 0.044 -0.041 -1.006 0.361 0.276 3.625 Electricity -0.181 0.139 -0.068 -1.307 0.248 0.170 5.870 price Fuel price 0.270 0.181 0.142 1.492 0.196 0.050 19.811 a. Dependent variable: electricity Observation finds that the regression results are inconsistent with common sense: the coefficient of three typical power consumption terminals was negative. Further observation reveals that the variance inflation factors are relatively great, especially in terms of income and 109 A Study of Chinese Residential Electricity Demand Elasticities fuel prices, each of which has a VIF of 15 or more. These results show that there exists significant multicollinearity in the regression equation. In view of this, I apply the commonly used method in eliminating multicollinearity--Ridge regression to fix the results of multiple regression. Table 3 displays the standardized regression coefficients of independent variables under different values of k. From Table 3, we can see that the regression coefficients change quickly when k varies from 0 to 0.25, which is the abnormal change caused by multicollinearity. However, when k reaches 0.25, the ridge regression coefficients become stable, indicating that multicollinearity has been 22almost eliminated. Reference to the value of , one can find that =0.988 when k = 0.25, RR which is still a great value. Thus I select the ridge parameter k = 0.25. 110 Research of Mathematical Economics No. 1 2011 Table 3 Standardized regression coefficients under different ridge parameters 2k R Income Appliance Electricity price Fuel price 0.00 0.999 0.928 0.067 -0.050 0.051 0.05 0.997 0.666 0.261 -0.077 0.095 0.10 0.995 0.521 0.325 -0.078 0.137 0.15 0.992 0.445 0.341 -0.075 0.144 0.20 0.991 0.402 0.340 -0.074 0.196 0.25 0.988 0.344 0.338 -0.084 0.186 0.30 0.985 0.337 0.334 -0.083 0.181 0.35 0.982 0.331 0.330 -0.083 0.177 0.40 0.978 0.324 0.326 -0.082 0.173 0.45 0.974 0.318 0.323 -0.082 0.170 0.50 0.970 0.313 0.319 -0.081 0.166 0.55 0.965 0.307 0.316 -0.081 0.162 0.60 0.961 0.302 0.312 -0.081 0.157 0.65 0.956 0.297 0.309 -0.080 0.154 0.70 0.952 0.292 0.306 -0.080 0.149 0.75 0.947 0.287 0.302 -0.079 0.145 0.80 0.942 0.283 0.300 -0.079 0.141 0.85 0.937 0.279 0.297 -0.078 0.137 0.90 0.932 0.275 0.294 -0.078 0.133 0.95 0.927 0.271 0.291 -0.077 0.129 1.00 0.922 0.267 0.288 -0.077 0.126 When k = 0.25, redo the ridge regression, and non-standardized coefficients are obtained, as below: Table 4 Non-standardized regression coefficients of independent variables aaaa2345 0.501 0.417 -0.297 0.207 111 A Study of Chinese Residential Electricity Demand Elasticities Results displayed in Table 4 means that the income elasticity of power demand is nearly 0.501, while the self-price elasticity is -0.297 approximately. 4. Conclusions and Analysis A comparison with the elasticities calculated from ELES model (0.533 and -0.38) shows that the results derived from terminal energy needs (0.501 and -0.297) are smaller; This may be due to the reason that terminal-energy-need method focuses on three representative appliances, and some power consumption projects (such as dryer) whose usage is more flexible are not taken into account for the purpose of building a more simplified model. But because the differences between these two methods are relatively small, and since no perfect method is proposed, this paper considers the differences between results as understandable. In other words, we can conclude that the income elasticity of Chinese residential electricity demand is about 0.5, and the self-price elasticity is in the -0.3 or so. Compared with overseas studies, China's short-term price elasticity is at an intermediate level, while the income elasticity is slightly larger than that of abroad (James A. Espey and Molly Esper, 2004). The possible explanation is that since most families in China are rural households whose economic conditions are not very good, combined with the traditional concept of regarding thrift as a virtue, most families consume electricity frugally in everyday life, and the usage rate of household appliances is not very high; that is, appliances are not fully utilized. When income increased in a short term, residents are able to make use of appliances more frequently, thus the potential increase in electricity consumption is relatively large. In addition, a comparison of the income elasticity and price elasticity reveals that the previous is clearly greater than the latter, indicating that the major reason for a significant increase in residential electricity consumption is the rise of income. Although electricity price and income have been increasing simultaneously in recent years, the impact revenue brings on power consumption is larger than the influence brought about by price. A relatively small price elasticity also shows that the policies cannot achieve significant control effect in the short term which try to reduce the power consumption by raising the price of electricity so that energy can be saved and environment can be protected, although it is true that to judge the effectiveness of such policies we need to further measure the long-term elasticity. 112 Research of Mathematical Economics No. 1 2011 Furthermore, such a small price elasticity of demand also shows that the burden adds to residents which is caused by a rise in electricity price cannot be ignored. 5. Possible Directions for Future Development This paper studies only short-term elasticity of residential electricity demand, but in fact long-term elasticity is also very important. In particular, to policy makers who make effort to determine a electricity price which can encourage residents to save power and reduce energy consumption, whether the policies they make will receive the intended effect is largely depends on long-term elasticity. Moreover, since long-term elasticity is more significant than short-term one in general case, the effect it brings out will be more apparent. Therefore, I will provide some possible directions in this section for further research on long-term elasticity. For the ELES model, theoretically, it is possible to dynamize it so that it can be used in estimate long-term elasticity. But the model derived in this way will be extremely complicated in parameter estimation, and there may be no satisfactory solution under all parameter constraints. Due to the additivity of utility functions, we can separate utilities; that is, isolating energy consumption from total consumption system and build a dynamic model considering only the energy consumption system. We may establish the power demand function following the dynamic form of Stone-Geary model proposed by Pollak in 1970: n,i，，uqq,,,,， (14) ，，,(1),eitiiit，，,1i uqwhere is the total utility function; is the consumption of the ith energy in period t; eit qt,1, is the consumption of the ith energy in period ; is the basic need coefficient of it(1),i ,,the ith energy; is the dynamic adjustment factor of the ith energy; and is the marginal ii budget share of the ith energy. Supposing that the energy expenditure constraint on utility maximization is: n (15) mpq,,etititi,1 113 A Study of Chinese Residential Electricity Demand Elasticities pwhere is the total energy expenditure in period t; is the price of the ith energy in mitet qperiod t; is the consumption of the ith energy in period t. it Maximizing the utility function (14) under the constraint (15), we get the following demand function: n,，，iqqmpq,，，,，,,,, (16) ，，,itiiitetitiiit(1)(1),,,,pi,1，，it whose corresponding expenditure function is: n，，mpqmpq,,,,,,，，,， (17) ，，，，,ititiiitietitiiit(1)(1),,,,i,1，， mwhere is the expenditure on consuming the i th energy during period t. it From formula (17), one can see that the expenditure on the ith energy is still divided into two parts: one is the basic need part, namely, the first part on the right-hand side of formula (17), pq,,，; the other is the additional need expenditure, as the second part on the right ，，itiiit(1), n，，,,,mpq,，side, . ，，,ietitiiit(1),,,i,1，， For the terminal-need-method, because residents are able to change the stock of household appliances in the long run, there is no need to suppose a fixed stock in interpretation of the change in power consumption. Thus model (13) is modified to exclude W. The long-term model can be set as follows: lnlnlnlnQaaYaPaPu,，，，， (18) 1234ef Similarly, due to the existence of multicollinearity, we can apply ridge regression to fit the model in actual operation. Only when we have a comprehensive understanding of residential electricity demand elasticity of not only the short term but also the long run, are we able to get a full range of knowledge about the characteristics of China‘s residential power consumption, which is the foundation of developing effective policies and researching more in-depth. 114 Research of Mathematical Economics No. 1 2011 Appendix The Derivation of Formula (16) Setting up the Lagrangian expression nn,i，，，，L=qqmpq,,, (19) ,，，,，，,,itiietitit,1，，it,,，，,1,1，，ii q(with respect to and ) equal to zero, Then setting the partial derivatives of L,it which gives rise to: ,,,1ji，，，， (20) qqqqp,,,,,,0,，,，,,，，，，,jtjjjtiitiiitit(1)(1),,，，，，ji, (1,2,,)in, and n mpq, (21) ,etititi,1 From (20), we have ,Bi,，，qq,, (22) itiiit(1),p,it ,in，，where . Bqq,,,,，，，,itii,1it，，，，,1i Therefore nn，，,Bipqpq,，，,,,,itititiiit,,(1),p,ii,,11it，， nn B,,,，，pq，，,,,,,iitiiit(1),,,ii,,11,,, m,et 115 A Study of Chinese Residential Electricity Demand Elasticities Thus nB,,，,,mpq (23) ，，,etitiiit(1),,i,1 Finally, n,，，iqmpqq,,，，， (16) ,,,,，，,itetitiiitiiit(1)(1),,,,pi,1，，it References Beijing Statistical Yearbook, China Statistics Press, 2000-2009. United Nations Millennium Development Goals Report of 2009. Abdallah Al-Shenri, ―Artificial Neural Network for Forecasting Residential Electrical Energy‖, International Journal of Energy Research, June 1999, 23(8), pp. 649-661. Angus Deaton and John Muellbauer, ―An Almost Ideal Demand System‖, The American Economic Review, June 1980, 70(3), pp. 312-326. Constantino Lluch, ―The Extended Linear Expenditure System‖, European Economic Review, April 1973, 4(1), pp. 21-32. Ferda Halicioglu, ―Residential electricity demand dynamics in Turkey‖, Energy Economics, March 2007, 29(2), pp. 199-210. George Hondroyiannis, ―Estimating residential demand for electricity in Greece‖, Energy Economics, May 2004, 26(3), pp. 319-334. James A. Espey and Molly Esper, ―Turning On the Lights: A Meta-Analysis of Residential Electricity Demand Elasticities‖, Journal of Agricultural and Applied Economics, April 2004, 36(1), pp. 65-81. L. R. Klein and H. Rubin. ―A Constant-Utility Index of the Cost of Living‖, The Review of Economic Studies, 1947-1948, 15(2), pp. 84-87. Richard Stone, ―Linear Expenditure Systems and Demand Analysis: An Application to the Pattern of British Demand‖, The Economic Journal, September 1954, 64(255), pp. 511-527. Robert A. Pollak, ―Habit Formation and Dynamic Demand Functions‖, The Journal of Political Economy, Part I July-August 1970, 78(4), pp. 745-763. Thomas J. Lareau and Joel Darmstadter, Energy and Household Expenditure Patterns, Washington, D.C.: Resources for the Future, Inc., 1983. T. R. Lakshmanan and William Anderson, ―Residential energy demand in the United States: A regional 116 Research of Mathematical Economics No. 1 2011 econometric analysis‖, Regional Science and Urban Economics, August 1980, 10(3), pp. 371-386. Tsuyoshi Ueno, Ryo Inada, Osamu Saeki and Kiichiro Tsuji, ―Effectiveness of Displaying Consumption Data in Residential Houses. Analysis on How the Residents Respond‖, ECEEE 2005 Summer Study Proceedings, pp. 1289-1299. Xavier Labandeira, José M. Labeaga and Xiral López-Otero. ―Estimation of Elasticity Price of Electricitywith Incomplete Information‖, FEDEA Working Paper No. 09-18, 2009. (Responsible Editor: Zhang Lingyun) (Proofreader: Xu Lingjue) 117 The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and Manufacturing The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and Manufacturing Xu Lingjue, Wang Danna (School of Economics, Renmin University of China) This paper is based upon the endogenous growth theory model written by Philippe Aghion and Peter Abstract: Howitt (1992). Assuming drastic innovation, the model explains how a firm with limited endowment chooses the optimal amount of resources used in research. We apply the model in analyzing the exogenous factors affecting the choice made by producer, who is also innovation motivator. Firstly, given capital endowment, rental rate, market competitiveness, time preference, innovation arrival rate, productivity increase level and monopoly power, we construct a model from which the optimal amount of capital used in research derives. Then we conduct comparative-statics analysis of these exogenous factors and give both mathematical and economical explanations of the result. Furthermore, we apply the conclusions to explain the innovation inadequacy in China. Finally, we give appropriate policy recommendations. Innovation, Creative Destruction, Comparative-Statics Analysis Keywords: 1. Background After China joined the WTO, the government‘s preferential policies would be gradually cancelled. Firms are confronted with a competitive market featured in variability and uncertainty. With intense competitions, international environment and ever-decreasing product lifecycle, innovation plays an increasingly important role in a firm‘s existence and development (Fujita, 1997; Nakaha ra, 1997). In the process of modern production which is distinct in resource scarcity, new products just spring out on end. However, the crucial creative contents often come from transnational corporations of developed countries rather than domestic companies. Even if in some departments, manufacturing of these innovative products is going on domestically, we have to buy patents or technologies which are very costly. From the late 19th century, China‘s high value-added products in manufacturing have been climbing continuously, but still it is only a 118 Research of Mathematical Economics No. 1 2011 quarter of America‘s. Actually, America manufactures the most highly value-added products owing to its ever-increasing investment in innovation in the field of IT and biochemical industries. Considering the circumstance we have been in, it is high time that the firms of China emphasized more on the innovation in order to promote their core competencies. There is no doubt that innovation is crucial to both firms and the whole nation, what we care now turns out to be how firms‘ innovation decision can be affected. However, the variables that are faced by firms have been very complex. If we could find out the comparatively decisive parameters among these variables, we can be able to explain the firms‘ strategy in innovation from these dimensions. Furthermore, if social planner can change some of these parameters, there will be a possibility that the governor indirectly influence firms‘ decision through policy-making. 2. Literature Review The factors that affect the input of resources in research have been explained in multi-dimension. In the field of management, the effort in research chosen is determined by the entrepreneur‘s innovative spirit, ability in creating new inventions, corporate culture and intra-incentive mechanism. Joseph Alois Schumpeter (1950) defined entrepreneur as an organization‘s leader with innovation ability and positively evaluated the effect of entrepreneur. The innovation ability refers to human capital involving the strategy, innovative spirit, confidence, determination and specific skills. In another word, the human capital is the core of innovation in a corporate. Yu Jin (2008) considers the absence of innovative spirit of entrepreneur as the reason for the failure in research. Cai Bing (2006) views the lack of self-conducted innovation capability in a corporate culture crucial, whereas Wang Bingjie and Wang Jingyu (2010) relate the quality of staff, the ability in creating new invention and the goal of the firm to the process of innovation. From the point of Industrial Organization, the succession of new product research will lead to a certain level of production differentiation. In the new product market, the research promoter will be the only supplier and thus attains complete monopoly rent. The chase of profit finally determines the input in research. So, to some extent, a higher degree of monopoly power will create more monopoly rent which gives the firm more incentive to innovate (Wang Junfeng, Zhou Shaodong, Zhu Quanzhen, 2010). 119 The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and Manufacturing Considering the policy making, the research and development (R&D), especially fundamental research, owns the property of positive externality, which the firm does not take into account. Meanwhile, the non-competitiveness feature also lures the firm to be a free-rider instead of striving in innovation. These two effects tend to reduce the firm‘s incentive to invest in R&D of new products. Consequently, public sectors ought to take the responsibility of supporting research, through making reasonable and sound policies like guaranteeing patent and franchise right. Zhang Jingyi (2004) explores the cause of inadequate innovation in small and medium-sized enterprise and found that the indistinctive definition of property plays the key role. Cao Yan (2005) focuses on the property rights involving human capital and owes the lack of innovation to the absence of it. Anthony Bartzokas (2003) emphasizes the role of venture capital as a core element of the institutional framework that supports innovation dynamics in developing countries. There is also a group of research explaining the reason of innovation insufficiency with Institutional Economics. Research from Jia Liqun and Wang Yingluo (1994) perceive the pursue of innovation is determined by the ―potential‖ of its initial state. Potential will be distributed to multi degrees of freedom, and variant path will lead to the final goal of a firm, in which the path without innovation will be preferred by the firm because of less cost and risk. 3. Inadequacies of the Existing Literature and Our Point of View Most existing research starts from the practical side in interpreting the factors that influence firms‘ innovation decisions. Rarely are there economical models explaining the phenomenon theoretically. In this article, we utilize part of the work by Aghion and Howitt, and get a standard formula with a series of exogenous parameter. Then we discuss how the resource will be allocated between researching and manufacturing when these parameters change. Finally, empirical analysis is made on the basis of data of China and policy recommendations are given. 4. The Model 4.1 Assumptions We assume a random sequence of quality improving innovations that result from research 120 Research of Mathematical Economics No. 1 2011 activities. For simplicity, in the model, the new inventions make old technologies or products obsolete. In another word, we take the innovations drastic. This assumption is partly true with reality. For example, we can imagine the flow of reform in the field of software. Windows 7 has been the latest invention among Windows series, and hardly can we find Windows 2000 nowadays. ,The economy is populated by a continuous mass of capital K, and represents the time preference. The economy only consists of two sectors, the intermediate good sector and the final consumption good sector whose production depends on the intermediate good. We denote the intermediate good x and the final consumption good y. The market of y is perfectly competitive, whereas the producer of x is a monopolist. The relation between x and y is defined as: y,AF(x) (1) '''where , the parameter A represents the productivity of x. F(x),0,F(x),0 Innovations consist of the invention of a new variety of intermediate good that replaces the old one. It is the obsolescence process that raises the productivity parameter A, by the constant ,,1Afactor, . That is, denoting as the productivity level after the tth innovation takes t place, t (2) A,,A,,A，10tt Awhere is the initial value given by history. 0 The capital supply is divided into two parts. It can produce the intermediate good x, one unit xof K for one unit of x, so the equals the capital used in manufacturing during the tth t innovation instant. And it can also be used in research. So , where x is the amount of K,x，n capital in manufacturing and n is the amount of capital in research. Research produces a random sequence of innovations. The poisson arrival rate of ,,(n)innovations in the economy at any instant is , where >0 is a parameter indicating the , ,(n)productivity of the research technology. is a constant-returns, concave production ',(0),0,(n)function. It satisfies and . Both and are given by the ,(n),0, ,,(n)technology of research. The arrival rate means that given the number of capital used in 121 The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and Manufacturing ,,(n)research at some instant, the amount of innovations arrive naturally. Here the assumption of Poisson process well ensures the no memory feature between the two adjacent research instants. Given the assumption of the model above, we can construct the model next. 4.2 Production Decision Made by the Producer of the Intermediate Sector t,1,2,？There is a sequence of discrete period, , representing the tth innovation, where the variable t does not refer to real time, but rather to the sequence of innovations. During the tth innovation interval, we denote the production quantity of the intermediate xvgood , the rental rate of the capital used in manufacturing , the profit of the monopolist tt ,xp(x)An, the price of , the productivity parameter , the capital used in research tttttt Vand the value of the tth innovation . t In this part, our discussion only concerns the tth innovation, i.e. t is given. xRecalling that the market of y is perfectly competitive, the price of is determined by t ,ytp(x), tt,xt xyThis is the equation that decides the amount of used in producing . The left hand tt xside (LHS) is the marginal cost of and the right hand side (RHS) is the marginal revenue of t x (we may wish to set the price of y equals 1). According to the production function in t xequation (1), the price of is determined by t ,y'tp(x),,AF(x) (3) tttt,xt xAccordingly, the monopolist producing chooses the output targeting t 'max,,{p(x),v}x,{AF(x),v}x (4) tttttttttxt Take first order condition of the equation (4), we get 122 Research of Mathematical Economics No. 1 2011 ,,"'t,AF(x)x，AF(x),v,0 tttttt,xt vtr,We denote , which has the economical meaning of effective rental rate. tAt Rearranging the above equation, we get "' (5) r,r(x),F(x)x，F(x)tttttt or x,x(r) (6) ttt As a result, the monopoly rent generated from the drastic innovation is 2"~ (7) ,,,AF(x)x,A,(r(x))ttttttt 2''"~where . Noting that we have assumed that , so here the ,(r(x)),,F(x)xF(x),0tttt profit must be positive. 4.3 Research Decision Made by the Producer of the Intermediate Sector As was stated in the last part, if tth innovation takes place, there will be a positive monopoly rent, therefore impelling the producer to allocate part of its capital in research, which means n,0xn. This part will explain how the producer of chooses the optimal amount of . ttt VWe should be careful that the value of the (t+1)th innovation is not the complete t monopoly profit. Allowing for the possibility of another innovation occurs after the (t+1)th V,,Vinnovation succeeds, there should be . Consequently, must satisfy the following t，1t，1t equation, ,V,,,,,(n)V (8) t，1t，1t，1t，1 The LHS of equation (8) is the net expected present value of the (t+1)th innovation, which ,equals the RHS, that is, the profit attainable by the (t+1) intermediate good monopolist t，1 minus the expected loss that will occur when the (t+1)th innovation is replaced by a new ,,(n)innovator, which it does at the expected rate . t，1 Rearranging the equation (8), we get 123 The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and Manufacturing ,t，1V, (9) t，1,，,,(n)t，1 With more usage of capital in research, given fixed amount of capital endowment, less capital will be available for manufacturing. This trade-off will balance the distribution of limited capital by the arbitrage condition ' (10) v,,,(n)Vttt，1 The LHS of equation (10) is the value of using one more unit of capital in manufacturing whereas the RHS is the expected value of investing one more unit of capital in research — the 'V,,(n)flow probability of an innovation times the value . That is, the equation (10) t，1t represents the condition that the marginal value of manufacturing must equal to the marginal value of research. VSubstituting the in (10) with (9) and combining equation (2), we can get t，1 ~,,(r(x))'tt,,r,(n) (11) tt,，,,(n)t，1 4.4 Steady-State equilibrium The steady-state equilibrium is simply defined as a stationary solution with K,x(r)，nx(r),K,nr,r,n,n,,t,1,2,3？. Note that and , we can get tt ~ˆˆ,,r(K,n)(r(K,n)), (12) 'ˆˆ,,(n),，,,(n) This is the equation that determines the optimal amount of research that the firm chooses. ,Specifically, consider the Cobb-Douglas example: with a linear F(x),x,0,,,1 ,(n),nresearch technology . Correspondingly, the equation (3)(6)(7)(12) will be transferred to v,,1t (3)‘ ,()px,Ax,tttt, 124 Research of Mathematical Economics No. 1 2011 1rt,,1 (6)‘ (),()xrtt2, ,1, (7)‘ ,vx,ttt, ,1,,,ˆ()(K,n) (13)‘ ,1,ˆ,，,n ,1,,,()K The condition for to be positive is . ˆ,n1,, 5.Comparative-Statics Analysis We will go further to discover the economic meaning of equation (12). ~ˆˆ,,r(K,n)(r(K,n)), (12) 'ˆˆ,，,,(n),,(n) The RHS of equation (12) stands for the marginal benefit of research — the expected present value of the intermediate goods‘ profit. The LHS of equation (12) stands for the marginal cost of research — the expected amount of rental rate that has to be paid to increase one unit of innovation arrival rate. We denote marginal benefit and marginal cost of research as follows, ~,,ˆ(r(K,n))ˆb(n), (13) ˆ,，,,(n) ˆr(Kn),ˆc(n), (14) 'ˆ,,(n) ~drdxd,'''ˆˆ,0,0,0Noting that , , , and , we get ,(n),0,(n),0ˆdrdxdn ~,ddrdx'~,,,,,,,,ˆˆ[，(n)],(n)d ˆdrdxdnˆb(n),,02ˆˆ,,,dn[，(n)] 125 The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and Manufacturing drdx''',,,,ˆˆ(n),(n)rd ˆdxdnˆc(n),,0'2ˆˆ,,dn[(n)] r(x)=,r(x)=0c(0),b(0)limlimAssume , and (we do not consider x,,x,0 c(0),b(0), since a firm will not innovate under this situation). Given the assumption, we can obtain the marginal benefit curve and marginal cost curve as illustrated in Graph 1. Graph1 Marginal Benefit Curve and Marginal Cost Curve ˆ,，,,(n)From equation (13), we find a lower discount rate or a higher profit ~,ˆˆ,(r(K,n))b(n) or a higher level of productivity advance will shift upward and thus ˆincrease (Graph 2). n 126 Research of Mathematical Economics No. 1 2011 ˆˆc(n)b(n)Graph2 Holding constant, any change that can shift upward will increase ˆ n ˆr(K,n)From equation (14), we find a higher effective rental rate or a lower innovation 'ˆc(n)ˆˆarrival rate will shift upward and thus decrease (Graph 3). ,,(n)n 127 The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and Manufacturing ˆˆb(n)c(n)Graph 3 Holding constant, any change that can shift upward will decrease ˆn 5. Empirical Analysis on the Basis of Reality in China In the rest of this article, we are going to give some possible explanations of the innovation inadequacy in China by applying what we have concluded before. ˆ,，,,(n)5.1 Firms in China has a High Discount Rate . A reasonable explanation is related to State owned enterprises (SOE). As is well known, in China, these enterprises make up a large part of all firms. Consequently, their decisions of innovation will greatly influence the total amount of innovations. Most managers in state-owned enterprises are not concerned about the consequence of their policies after they leave. They just focus on their firms‘ performances when they are still in charge. As a result, they have little incentive to invest in innovation whose value is accomplished in the future. In other words, they are not patient enough to wait for the return of innovation. This impatience can be characterized 128 Research of Mathematical Economics No. 1 2011 by a high discount rate. Another reasonable explanation is the variability of government‘s policy. There isn‘t a sound system to guarantee the returns of innovation. If a relevant policy can be altered at will, how dare firms innovate? How can firms attach high value to something uncertain in the future? As a consequence, variability of government‘s policy results in a high discount rate. ,5.2 Firms in China has a Lower Level of Productivity Advance . Compared to other countries, we find that invention patents only account for a small percentage of all patent applications (the other two kinds of patentisdesign patent and utility ,model patent). It means the level of productivity advance in China is comparatively low (Table1). Table 1 Three Kinds of Patent Applications Accepted and the Percentage of Inventions Item Country Percentage of Inventions Inventions Utility Models Designs Total (%) China 194,579 223,945 298,620 717,144 27.13 USA 231,588 ** 15,463 247,051 93.74 Japan 330,110 29,621 367,448 89.84 7,717 France 14,743 121(2007) 4,093 18,836 78.27 Australia 2,821 1,024 2,727 6,572 42.92 Italy 8,588 1,184 11,753 73.07 1,981 Switzerland 1,594 ** 1,123 2,717 58.67 Austria 2,298 682 805 3,785 60.71 Czech Republic 712 1,107 288 2,107 33.79 Denmark 1,634 218 183 2,035 80.29 Korea 61,115 16,971 52,786 130,872 46.70 Russian Federation 27,712 10,483 2,356 40,551 68.34 Source: World Intellectual Property Indicators (2010) 'ˆ5.3 Firms in China has a Lower Innovation Arrival Rate . ,,(n) A fundamental breakthrough will lead to a wave of innovations and thus increase innovation 129 The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and Manufacturing arrival rate. Fundamental breakthrough often derives from basic research. As a result, to some degree, the more the basic research the higher the innovation arrival rate. Data in Table 1 evidently shows that compared to other countries, China‘s investment in basic research is extremely low. The inadequacy of basic research can be the cause of a low innovation arrival rate. Table 2 International Comparison of R&D Activities Item By types of research (%) Country year Experimental Basic Research Applied Research Development China (2007) 4.7 13.28 82.01 USA (2006) 18.56 23.12 58.31 Japan (2005) 12.65 22.18 65.17 France (2005) 23.70 38.99 37.31 Australia (2004) 23.17 38.14 38.69 Italy (2005) 27.71 44.40 27.89 Switzerland (2004) 28.70 33.32 37.98 Austria (2002) 17.80 37.6 44.60 Czech (2006) 29.32 24.06 46.62 Denmark (2005) 18.71 28.08 53.22 Korea (2006) 15.15 19.86 64.99 Russian Federation (2003) 15.10 15.60 69.40 Source: China Statistical Yearbook on Science and Technology 5.4 Patent Protection System in China is Imperfect To some extent, having a patent means enjoying a certain degree of monopoly power. Since a patent prevents others from utilizing the innovation without payment, other competitors will lose competitiveness or at least be in a disadvantageous position in the market. In this way, innovator can be regarded as having some monopoly effect. The monopoly power can be measured by the value of price minus marginal cost divided by price. We can examine the monopoly effect in the linear Cobb-Douglas example. Recalling equation (3)‘ and (13)‘ 130 Research of Mathematical Economics No. 1 2011 v,,1t (3)‘ ,()px,Ax,tttt, ,1,,,ˆ()(K,n) (13)‘ ,1,ˆ,，,n ,pvpp,,tttt,,1,,from (3)‘, we can easily acquire the monopoly power, that is . The t ,degree of monopoly power is measured inversely by the parameter . From equation (13)‘, we can easily find that, the higher the monopolistic degree is, the higher the marginal benefit is. As we have mentioned before, here a firm‘s monopoly power should be interpreted as the ability to gain revenue from its own innovation. A perfect patent protection system can exactly grant the utmost to gain revenue, that is, the strongest ―monopoly power‖, which motivate the firms to innovate. However, piracy is rampant in China, or in other words, patent protection is so weak in China. Since innovation may be plagiarized or copied by other firms, the innovator cannot enjoy the complete outcome of its innovation. As a result, firms would have less incentive to innovate and the innovation tends to be insufficient. 6. Policy Recommendations (1) Construct a reasonable and sound system to evaluate the performances of SOEs‘ managers. This evaluation system should include assessment of innovation performances. (2) Try to keep relevant policy stable. Policy makers should let firms operate in a less uncertain environment and ensure their investments‘ safety. (3) Pay more attention to the development of basic science and the education of basic science personnel. Attach more importance to basic research and invest more resources in this field. (4) Encourage more invention-oriented innovations by preferential policy, rewards and so forth. (5) Establish a more effective policy to protect patent. Make sure that firms can enjoy their achievements and be free from plagiarism. (6) Give preferential interest rate to R&D loans and at the same time set up a supervision system 131 The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and Manufacturing to guarantee that the loans are used appropriately. 7. Conclusions This paper studies the factors that determine a firm‘s optimal allocation of resources between research and manufacturing. We discover that the amount of research input moves in the opposite direction of discount rate and effective rental rate whereas it moves in the same direction with profit of innovation product, productivity advance level and innovation arrival rate. Through empirical analysis, we find that there are mainly four reasons for the innovation inadequacy in China. Firstly, absence of innovation performance evaluation of SOE‘s managers and the variability of government‘s policy result in a high discount rate. Secondly, level of productivity advance is small since firms focus more on design and utility improvement rather than invention. Thirdly, innovation arrival rate is low due to the inadequacy of basic research. Fourthly, patent protection system is imperfect. At last, we have to point out that this model could be extended in several ways. For example, introducing labor markets and a variety of intermediate goods into this model will make it closer to the reality. The assumption of drastic innovation can also be relaxed. After learning Econometrics systematically in next year, we can also use this tool to do more empirical analysis. References Jin Xiangrong, Yu Dongyun, ―Industrial Characteristics, Innovative Efficiency and Regional Economic Growth‖, Journal of Zhejiang University, 2010 (5), pp. 73-81. Li yuan, Wang Yingluo, ―Exploration on the Components of Enterprises‘ Technology Innovation Motivation‖, Scientific Management Research, 1994 12(4), pp. 44-45. M. Ishaq Nadiri, ―Innovation and Technological Spillovers‖, NBER Working Paper No. 4423, 1993. N. L. Stokey, ―Learning by Doing and the Introduction of New Goods‖, Journal of Political Economy, 1988 96(4), pp. 701-717. Philippe Aghion, Peter Howitt, ―A Model of Growth Through Creative Destruction‖, Econometrica, 1992 60(2), pp. 323-351. Wang Junfen, Zhou Shaodong, Zhu Quanzhen, ―Research on the Relationship between Market 132 Research of Mathematical Economics No. 1 2011 Concentration and Innovation Efficiency of Non-incumbent Enterprises‖, Science &Technology Progress and Policy, 2010 27(14), pp. 81-85. Wu Hanhong, ―The Theory of Industrial Organization‖, Beijing: Renmin University of China Publishing House, 2007. Yu Ji, ―Research on Enterprises‘ Technology Innovation Motivation‖, Study & Exploration, 2007(6), pp. 113-175. (Responsible Editor: Xu Lingjue) (Proofreader: Zhang Lingyun) 133 A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes Chen Jun (School of Economics, Renmin University of China) Abstract: Based on the knowledge of the China‘s economic development and the characters of China‘s inflation welfare cost, the paper constructed a Bewley type model with market incompleteness and individual heterogeneity, by which the paper has an analysis of the China‘s inflation welfare cost and the differences of the inflation welfare cost to different groups. The calibration results of the model show us that there are great costs of the inflation to the society as a whole; however, different groups are related to different inflation welfare costs. Generally speaking, if the inflation is very low, the welfare loss is smaller than the wealth loss, but from the wealth distribution viewpoints, the inflation worsen the China‘s wealth distribution. The wealth share of the poor declines with the increase of the inflation while the one of the wealthy increases with the increase of the inflation. Therefore, the wealth Gini index increases with the increase of the inflation. Keywords: Inflation, Incomplete Market, Bewley Type Model, Welfare Cost, Wealth Distribution, Individual Heterogeneity 1. Introduction Inflation is one of the core problems concerned by the government all over the world. It is the common dream for each government to achieve a high growth rate with a low inflation rate. In order to achieve the goal, the central banks all over the world emphasize the role of stabilization of price; therefore, the average inflation rate in the developed government falls from 9% in 1980s to 2% in 2005 and that of China has fallen from more than 10% in 1995 to 1.8% in 2005 (Chen and Ma, 2007). However, with the economic development, there is a trend to increase for the inflation rate in China. Although the subprime crisis has suppressed the increase of price, the stimulus policy in China has brought about the hidden danger of the inflation in the future. Under these circumferences, what‘s the rate of inflation should be in China? How the 134 Research of Mathematical Economics No. 1 2011 Chinese government should do to keep a relatively sound inflation rate? The government are faced with there problems. However, in order to solve the problems, the government should know the mechanism of inflation formation in China; what‘s more, the quantitative analysis of inflation to the macro economy is needed. From the traditional research view, the quantitative analysis on inflation in China is usually based on the Taylor‘s Rule and the Phillips Curve. However, the accurate estimation of Taylor‘s Rule should be based on the accurate estimation of the output gap, but it is impossible to have an 1accurate estimation on the output gap in China. At the same time, when the macro economy is faced with uncertainty, the Phillips curve cannot give us estimation of optimal inflation rate and relevant monetary policy (Zhao, 2006). Comparing with the two traditional methods, the quantitative analysis of inflation welfare cost can give us the estimation of optimal inflation rate, hence more rational policy suggestion can be acquired through the method. The analysis of inflation welfare cost is to know about the inflation through estimating the loss of family and the whole society caused by the inflation. Since the increase of price would change the behavior pattern of individual and family, inflation would cause deadweight loss. Through estimating the inflation welfare cost, the government can acquire the optimal inflation rate of the economy and make more reasonable measures; however, the welfare cost is usually covered and very difficult to estimate (Chen and Ma, 2007). With the development of macroeconomics, economists have offered several methods to estimate the inflation welfare cost. The consumer surplus method developed by Bailey (1956) is the first step for economists to estimate the inflation welfare cost. According to the method, inflation welfare cost should be represented by the area under the inverse money demand curve. With the definition, Bailey estimates that the inflation welfare cost under the rate of 6% in the USA is about 0.3%-1% of national income. Although the method keeps in accordance with the definition of consumer surplus in the microeconomics and hence provides a convenient method to estimate the inflation welfare cost, there is a fatal disadvantage: when the nominal interest rate is changed, the money 1 Specifically, there are several reasons that make the accurate estimation of China‘s output gap impossible: (1) there is no united definition for the output gap (Guo and Jia, 2004), therefore, different standards would be adopted for different researches; (2) there is no united method to estimate the output gap and each method has its own disadvantages. On one hand, time series methods such as detrended method consider relatively few factors, hence ignoring supply side (Xie and Luo, 2002); on the other hand, production function method require higher quality of data, though it can avoid the disadvantage of time series methods, but we cannot get there data in current China (Guo and Jia, 2004). 135 A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes demand curve should be required to keep unchanged when calculating the welfare cost; this would further require that the marginal utility of real balance should be independent from the demand of other commodities, therefore, the method cannot give us an accurate estimation of 2inflation welfare cost (Laidler, 1990). In order to surmount the disadvantage, lots of general equilibrium methods are developed. Specifically, Cooley and Hansen (1989) first constructed a one-sector stochastic growth model with CIA (Cash-In-Advance). With the model, they estimated that the rate of 10% would cause a 0.38% of gross product. Later on, with the development of MIU (Money-In-Utility) model, lots of MIU models are adopted to estimate the inflation welfare cost. For example, Lucas (2000) raised a general equilibrium model with MIU and showed us that the inflation rate of 10% would cause a loss of about 1.3% of gross consumption to the macro economy. In order to investigate the inflation in a further step, lots of economists try to portray the micro mechanism of keeping money. For example, McCallum and Goodfriend (1987) assume that both time and money can not only provide transaction service and also substitute for each other, hence constructing the money demand function from the microcosmic point; while Logos and Wright (2005) raised a general equilibrium model from the view of money search and estimated that the rate of 10% would cause a loss of about 3%-5% of gross consumption. Although there have been lots of researches based on these methods, but most of them are based on the representative-agent model. In the representative-agent model, all agents have the same preferences and inflation expectation, therefore, the influences of inflation to all the agents would be the same; however, the influences in reality would be obviously different. Doepke and Schneider (2006a) and Meh and Terajima (2008), with the help of micro data, investigate the redistribution effect of inflation in U.S. and Canada, respectively. The results show us that even the moderate inflation would cause great wealth transfer from the olds to the youth. Based on these facts, more and more scholars try to investigate the influences of inflation under the heterogeneous-agent frame. For example, Erosa and Ventura (2002) introduces two kinds of families such as the poor families and wealthy families and two kinds of transaction methods such as cash transaction and credit transaction; the estimation results show us that the wealthy families would be faced with less loss due to greater frequency to adopt the credit transaction. Cysne (2006) introduces the families with different production efficiency and transaction efficiency into the shopping time model and the results show us that the welfare cost would be 2 Numerical explanation about the disadvantage of the method can be seen from ?mrohoro?lu (1992). 136 Research of Mathematical Economics No. 1 2011 different from the traditional shopping time model. Doepke and Schneider (2006b) and Meh et al. (2010) take the different portfolios of the individuals with different ages and different asset-holding classes into consideration and investigate the influences of unexpected inflation shock to different asset-holding classes. Algan et al. (2009) investigates the influences of inflation to macro economy and different asset-holding classes from the labor supply view; the results show us that inflation would cause that the wealth transfers from the wealthy to the poor, hence increasing the labor supply of the wealthy and the increasing of the gross labor supply. In current China, wealth distribution has been worsened and has become the bottleneck of the economic development (Li et al., 2000; Li et al., 2005; Chen et al., 2009). With the change of wealth distribution, the portfolios of different asset-holding classes would be different, too. (Li et al., 2005; Chen, 2008) Since different assets would have different yields in the inflation, the inflation would influent the wealth distribution and the behavior pattern. Therefore, there would be great distinctions of inflation welfare cost for different asset-holding classes. Therefore, only with the help of heterogeneous-agent model can we know about the influences of inflation in China. At the same time, for the developing countries such as China, there would be fewer financial products due to the limit of laws (Li and Han, 2010); therefore, the market incompleteness would be more serious in developing countries (Li and Li, 2004; Zhu and Liu, 2009; Huang et al., 2009). The incomplete market would influence the inflation welfare cost (?mrohoro?lu, 1992;Chiu and Molico, 2008;Algan et al., 2009); therefore, we should take the market incompleteness into consideration when we investigate the influences of inflation in China. In order to investigate the influences of China‘s inflation with a model with incomplete market and heterogeneous agents, the paper raised a Bewley-type model. This kind of model is developed first by Bewley (1983), and then developed in a further step by Huggett (1993), Aiyagari (1994) and Krusell and Smith (1998). In the model, the agents are faced with idiosyncratic income shock and the incomplete market make the agents cannot fully insure against the income risk. In order to smooth the consumption, the agent would self-insure through keeping the money, capital or adjusting the labor supply. Since the shock history of each agent would be different, the asset holding and the portfolio would be different. In a further step, we can conclude that the influences of inflation to different people would be different, too. Therefore, the model would endogeneize the inflation heterogeneity. What‘s more, since asset holding and 137 A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes labor supply would change with the inflation rate, the gross capital and gross labor supply would be different, too. This means that gross production would be influenced by the inflation; therefore, the paper links the inflation and gross product. With the dynamic general equilibrium model, the paper has an investigation into influence heterogeneity of inflation to different asset-holding classes. The remainder of this paper is organized as follows: in section 2, the model would be introduced; the calibration procedure would be given in section 3; in section 4, the results of the model are given; with the analysis of the model, the paper gives some suggestions about the monetary policy, and this can be seen in section 5. In the appendix, some technical details are introduced. 2. The Model In order to have an investigation into the China‘s inflation welfare cost, the paper raises an Bewley-type model with idiosyncratic risk and borrowing constraints according to the work of ?mrohoro?lu and Prescott (1991), ?mrohoro?lu (1992), Algan and Ragot (2010). However, there are great distinctions with the model in the paper and the models in ?mrohoro?lu and Prescott (1991) and ?mrohoro?lu (1992): (1) the model in ?mrohoro?lu and Prescott (1991) and ?mrohoro?lu (1992) is in fact an endowment economy where there is no production sector, therefore, the link between the inflation and gross product are ignored. The paper, however, introduces the production sector into the model; therefore the inflation would influence the gross 3product and the consumption decision in the model. (2) The model in ?mrohoro?lu and Prescott (1991) and ?mrohoro?lu (1992) only takes the saving function of money into consideration; however, the paper raises a MIU model according to Sidrauski (1967), hence introducing the trade medium function and the consumption smoothing function of money into the model. (3) The paper introduces the leisure into the utility function; hence, the choice of leisure would be a method to smooth the consumption (Heathcote et al., 2009). (4) There is only one kind of asset in ?mrohoro?lu and Prescott (1991) and ?mrohoro?lu (1992) while in the model economy in the paper, there is productive capital except the money. From the whole model setting, the model in 3 During the investigation of inflation welfare cost, the link between the gross product and the inflation should be taken into consideration. This is, in fact, very important. More details can be seen in Gomme (1993). Temple (2000) gave a good literature survey about the relationships of inflation and the gross product. 138 Research of Mathematical Economics No. 1 2011 the economy is very similar to the Algan and Raogt (2010), except for the differences of 4government; however, the aim of the paper is different from that of Algan and Ragot (2010). 2.1 Problems for the Family The model is composed by continuous families standardized to be 1 and one-period utility function form of each family is the same; however, the production efficiency, the real balance and asset holding of each family are different. Assume further that the one-period utility of each family mainly comes from the consumption, leisure and real balance. Therefore, the objective function that each family is faced with is as follows: ,t (1) max,,EUcmpl,，，,0tttt0t, where is the subjective discount factor, means the expectation based on the E01,,,0 information in Period 0. , and are the consumption, labor supply and real clmptttt balance in period. The function is the one-period utility function and the function is U(),t differentiable and continuous for each variable. In order to make the utility function coincide with the balanced growth and satisfy the nature of utility function listed above as well as to reduce the calculation cost, the paper, referring to Castañeda et al. (2003) and Lucas (2000), assumes that the one-period utility function is as follows: 1,,11,,,,,,，，cmpl11111,,,,,,，,,～ ,，，，，，，，，tt,,,，， (2) Ucmpl,,,，，,tttt1,,,ln1ln11,1cmpl，,，,, ,，，，，，，，，,,,,,tt, where is the constant relative risk aversion coefficient, and are weight parameter of ,,, consumption and leisure, respectively. is the curvature of leisure in the utility ,,,1，， function. From the one-period utility function in (2), we can see that the function satisfies the conditions that the first-order derivative is larger than 0 and the second-order derivative is smaller 4 The aim of Algan and Ragot (2009) is to investigate the influences of inflation to the gross saving in the economy while the aim of the paper is to investigate the inflation heterogeneity, hence taking the influences of inflation to the gross capital and gross product into consideration. 139 A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes than 0, this means that the utility function satisfies the law of diminishing marginal utility. 2.2 Problems for the Firm Assume that the market is competitive and the production technology is represented by Cobb-Douglas production function; therefore, the whole market can be seen as a factory. Denote the gross capital in period as and gross labor as , then the gross production in period KLttt can be expressed as: t ,,1, (3) YAKL,，，，，ttt When the market is competitive, the price of capital and labor are given by the marginal production; therefore, the wage rate and interest rate are given, respectively, as follows: ,,, wAKL,,1, (4) ，，，，，，ttt and ,,,,11 (5) rAKL,,，，，，ttt 2.3 Exogenous Driving Processes There is no aggregate uncertainty in the model, but the agents are faced with idiosyncratic efficiency shock. Assume that the logarithm of the efficiency shock follows the AR(1) process. If the production efficiency in period is , then the motion function can be expressed as: ,tt 2lnln,~0,,,,,,,,， N (6) ，，tttt,,1 2where , a normal distribution with a mean of 0 and a variance of, represents the stochastic ,,t, shock of production efficiency in period . t Once we know the motion function, we can adopt the Tauchen (1986) to discretize the ,stochastic process and acquire the transitional probability matrix and stable distribution. P 2.4 Problems for the Government According to the literatures, the fiscal policy and monetary policy would play a key role in the inflation welfare cost, however, we do not pay much attention on the influences of fiscal policy and monetary policy to the inflation welfare cost; therefore, the paper simplifies the role of 140 Research of Mathematical Economics No. 1 2011 5the government in the model economy. Referring to the method in Freeman et al. (2010) and Henriksen and Kydland (2010), the paper assumes that the government only controls the currency issue. The motion function of money stock in the model economy can be expressed as: MM,，1, (7) ，，tt,1 ,tMwhere is the gross nominal money stock in period , and is the increase rate of t nominal money stock. Generally speaking, the government would have some income when issuing the currency. In order to avoid the government policy to the model results, the paper assumes that government would spread out the income evenly to each family. Since the gross number is standardized to be 1, the transfer acquired by each family is the gross income of the government. Assume that the ,ttransfer of each family acquired in period is , and then the balanced budget constraint t equation can be expressed as: ,,,M (8) tt,1 2.5 Insurance Market The insurance market in the model is incomplete. Because of the incompleteness of the insurance market, the family cannot fully insure against the risks through the insurance market; however, the family can hold money or productive capital or adjust the labor supply to smooth the consumption. What‘s more, the paper assumes that agents can only lend or borrow productive capital and the nominal money holding must be non-negative. 2.6 Bellman Equation From the assumptions above, we can see that the family income in the model economy can be divided into three parts: wage income, capital income and the transfer from the government; the expenditure can be divided into two parts: consumption and saving, where the saving can be both money and productive capital. 5 For more information about the government monetary policy and the inflation welfare cost, see Meh et al. (2008); for more information about the government fiscal policy and the inflation welfare cost, see Gomme (2008). 141 A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes at Assume that the productive capital holding of one family in period is , the nominal t mtmoney stock is , the depreciation rate is and the price of commodities in period . Then ,t,1 the budget constraint equation can be expressed as: pcpammpwlpra，，,，，，，,,,,1 (9) ，，ttttttttttttt，,11 , Replace the government transfer by Equation (8). At the same time, denote the real t mMtbalance holding and the gross real balance in period as and , respectively. What‘s tt ,tmore, assume the inflation rate in period is , then the budget constraint can be expressed t as: cammMwlra，，,，，，，，,,,,,11 (10) ，，，，，，ttttttttttt，,,111 From Equation (10), we can see that the saving of the family can be both money and productive capital; however, once entering into the next period, the forms of the asset holding have no influences on the consumption and saving decision in the next period. Therefore, in order tto reduce the number of state variable, the wealth in the beginning of period can be expressed as: qmra,，，，,11,, (11) ，，，，ttttt,1 Since there is no aggregate uncertainty, the joint distribution of the individual states should be stable under the equilibrium; hence, the joint distribution would not change with time. Therefore, we can denote that the joint distribution of asset-holding and the productive efficiency ,,q,as . At the same time, when the joint distribution is acquired, the gross labor supply, ，， gross capital and the gross real balance would be acquired. Considering the factors, the state variable of the family should be asset holding and productive efficiency, the families make the optimal consumption, optimal asset holding, optimal real balance and the optimal labor supply. Denote the value function to be , then the Bellmen function can be expressed as: V VqUcmlEVqq,max,',','|,,,,,,，，，，，，，，，，， (12) ,,，，caml,',', stcamqMwl..''1 ，，,，，，,,, (13) ，， qmra''11',，，，,,, (14) ，，，， 142 Research of Mathematical Economics No. 1 2011 (15) m'0, aaa',, (16) ,, awhere and denote the lower limit and the upper limit of asset holding, and aa'm' represent the real productive capital and real balance in the next period. The family solves the dynamic optimum problem and acquires a set of optimal policy function based on the state cqaqmqlq,,',,',,,,,,,variables . ，，，，，，，，,, 2.7 Recursive Competitive Equilibrium Vq,, The recursive competitive equilibrium is composed by the optimal value function , ，， cqaqmqlq,,',,',,,,,,,the optimal policy function , the joint distribution of ，，，，，，，，,, rw,,,q,state variables , the price of capital and labor and the macroeconomic ，，,, KLM,,,,variables . All the parts should satisfy the conditions as follows: ,, (1) The gross labor, gross capital and gross real balance should be acquired through aggregating the decisions of all the families, that is to say: Kad,, (17) , and Lld,,, (18) , and Mmd,, (19) , It should be noted that the equation (18) is also the clearing condition for the labor market and the equation (19) is the clearing condition for the money market. ,,q,(2) Given the stable joint distribution , aggregate capital , aggregate labor , ，，KL wraggregate real balance, wage rateand interest rate, M cqaqmqlq,,',,',,,,,,, should be the solutions of the equations (12)-(16). ，，，，，，，，,, (3) The product market clears, that is to say: 143 A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes 1,,, (20) cqdqKALK,,1,,,,,，,，，，，，，，，, (4) The balanced budget constraint is satisfied, that is to say, Equation (7) is satisfied. ,,q,(5) The joint distribution of state variables should be stable and the motion function is: ，， ,,00,,,,,,,,,qaaqmmqddqd,1'',1'','|''P,,,,, (21) ，，，，，，,,，，，，，，，，,,,,00Q,,q,,,, 1,where the function is an indicator function, when the variable is true, the value of the ，， ,P,,'|function is 1, otherwise, it is 0. represents the probability of transfer from ，， ,Qproductive efficiency in the current period to the efficiencyin the next period. is the ,' asset-holding space and is the efficiency space. At the same time, in order to guarantee that , the wealth space and the efficiency space are static, the equation (21) should be always satisfied 00qQ,,,，,for any point in the space . ，， Since the recursive equilibrium is stable, the joint distribution would not change with time. While from equation (6), we can see that the gross nominal money is growing at a constant rate; therefore, under the recursive equilibrium, the growth rate of the model economy should coincide ,,,with the inflation rate, that is to say, the equation should be satisfied. 2.8 First-order Condition With dynamic program method, the first-order condition can be expressed as: 111111,,,,,,,,,,,,，，，，，，，，，，,,11,,，，，，cmErcm'1''',,,，, (22.a) ，，，，，，，，，， 11111,,,,,,,,,，，，，，，，，,,,,111,,,，，，，,,cmcm'1',,，，，，，， (22.b) 1，，，，，，1111,,,,，，,,,, ,Ecm'''，，，，,,,,1，，，, ,,,,,,11，，，，,,11,,，， (22.c) ,,,1',,lcmw，，，， camqMwl，，,，，，''1,,, (22.d) ，， From the first-order condition, we can see that the equation (22.d) is in fact the budget 144 Research of Mathematical Economics No. 1 2011 constraint of the family. (22.a) is the Euler function of productive capital, where the right hand represents the marginal cost of holding productive capital while the left hand represents the marginal return. Equation (22.b) is the Euler function of money holding where the right hand is the marginal cost of holding money while the left hand of the equation is the marginal returns of l,0,1holding money. Equation (22.c) is the first-order condition for labor supply. Only if ,, can the equation be satisfied. Otherwise, agent will choose the corner solution and the optimal policy will be given by the equations (22.a), (22.b) and (22.d). 3. Calibration In order to solve the model, there are several parameters to be set at first. Specifically, these ,,parameters include: subjective discount rate , relative risk aversion coefficient , weight ,,,parameter and , the curvature of leisure , depreciation rate , output elasticity of , ,capital , technical progress and the upper limit as well as the lower limit of the A ,asset-holding space. What‘s more, the autocorrelation coefficient and the standard deviation , should be given to estimate the motion function of productive efficiency. During the , calibration, the paper assume the technical progress is 1 at first; for other parameters, the paper refers to some research literatures and adopts the data sets in reality to make the results of the model fit the reality including wealth distribution, income distribution, the ratio of capital to output and the ratio of money to capital as well as the average labor supply. Considering there is few data about the China‘s wealth distribution, it become inappropriate to adopt the monthly or quarterly data to calibrate the parameter. Therefore, the model period is set to be 1 year. At the same time, the inflation rate is assumed to be 3%, which is the average China‘s inflation rate during the year 1995 and the year 2007. All the calibration results are shown in Table 1. 145 A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes Table 1 Calibration Results Parameters Calibration Preference Parameters: ,Relative Risk Aversion 1.5 ,Weight of Consumption 0.98 ,Weight of Leisure 1.35 Curvature of Leisure 1.5 , Depreciation Rate 0.86 , Motion Function of Efficiency: ,Autocorrelation Coefficient 0.98 Standard Deviation 0.254 ,, Technical Parameters: Technical Progress 1 A ,Output Elasticity of Capital 0.45 Depreciation Rate 0.052 , Space of Asset Holding aLower Limit of Productive Capital 0 Upper Limit of Productive Capital 25 a Lower Limit of Money 0.001 Upper Limit of Money 5 3.1 The Preference Parameters , The relative risk aversion coefficient is taken from other literatures directly while the other preference parameters are calibrated to make the results of the model fit certain model targets. Specifically, referring to Chen et al. (2009), the paper set the relative risk aversion coefficient to be 1.5, which is in accordance with the value in the relative literatures. , For the subjective discount rate , the paper calibrates it with the help of China‘s ratio of capital to output. Because there is no capital stock data in the dataset, we have no choice but to estimate the gross capital stock to acquire the ratio of capital to output. With the method developed by Wang and Yao (2003) and Liu et al. (2009), the paper estimates the gross capital stock in China from the year 1978 to the year 2007; at the same time, the gross product is represented by the real GDP. With these data, the ratio of capital to output is about 2.45. In order to generate a relatively higher value of capital to output ratio, the subjective discount rate should be very small. Finally, the subjective discount rate is determined to be 0.86. , We choose the weight parameter of consumption so as to match China‘s ratio of money 146 Research of Mathematical Economics No. 1 2011 stock to gross output. According to the data from the website of Chinese Economic Information, the paper estimates that the average share of M0 to the gross output from the year 1990 to the ,year 2009 is about 13.7%. In order to match the value, the weight parameter of consumption is determined to be 0.98 at last. There are still two preference parameters to calibrate. Since there are few literatures on the labor supply and leisure choice in China, it is quite difficult to calibrate the other two parameters with China‘s datasets. Considering the form of one-period utility function and the calibration in Castañeda et al. (2003), the paper choose the weight parameter of leisure and the curvature of leisure so as to make the share of average working time to the gross labor endowment is about ,1/3. After trying several times, the paper finally sets the weight parameter to be 1.35 and the ,curvature of the leisure to be 1.5. We can see that the calibration results are different from Castañeda et al. (2003), but all the parameters are greater than the value 1. 3.2 The Motion Function of Productive Efficiency There are few empirical literatures on the China‘s productive efficiency, let alone the motion function of the individual productive efficiency. In order to overcome the problem, the paper adopts the calibration method to estimate the motion function of individual productive efficiency. ,,The autocorrelation coefficient and the standard deviation are set so as to match the , wealth distribution and the wage distribution in reality. According to the estimation from survey data conducted by Ao Er Duo Investment Center in the year 2005 and the year 2007, the wealth Gini index lies in the interval between 0.56 and 0.58, therefore, the paper sets the Gini index to be 0.57 at last. At the same time, according to the survey data, the wage Gini index lies in the interval between the value 0.42 and the value 0.45; hence, the paper sets the wage Gini index to be 0.435 and takes it as one target to calibrate. With the help of two calibration targets, the paper ,,finally sets the autocorrelation coefficient to be 0.98 and the relevant standard deviation , is 0.254. The results coincide with the calibration results of Chen et al. (2010). After estimating the motion function of productive efficiency, the continuous process can be discretized with the method developed by Tauchen (1986). Specifically, the paper sets the interval width to be 1 and the number of points to be 5. 147 A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes 3.3 Technical Parameters Since the technical progress is standardized to be 1, there are still depreciation rate and , ,output elasticity to calibrate. According to Chen et al. (2009), the estimations of output elasticity of capital in different literatures are quite different due the limit of the availability of data in China. Referring to Chen et al. (2009), the paper sets . Considering that the ,,0.45 estimation is quite different, the paper would investigate the influences of the output elasticity of capital to the inflation welfare cost in the robust analysis. For the depreciation rate , the paper adopts the method in Chen et al. (2009) and average , the depreciation rate during all the years with the help of the data in Chow and Li (2002). Finally, the depreciation rate is set to be 0.052. 3.4 Space of Asset Holding The space of productive capital should satisfy two conditions: first of all, the lower limit of the productive capital should be larger than the natural borrowing constraint so as to guarantee that the model economy is with incomplete insurance market; secondly, the upper limit of the space of productive capital should be set so as to make the productive capital choice of all the agents lie in the interval. In order to improve the accuracy of the inflation welfare cost, the paper should keep the space of the productive capital unchanged; therefore, the upper limit of the space should be larger. After trying several times, the paper finally set the upper limit of productive capital to be 25. At the same time, in order to simplify the computation procedure, the paper assumes that each agent cannot borrow, this means that the lower limit of the space is 0. For the space of money holding, on one hand, the form of one-period utility function means that the money holding should be larger than 0; on the other hand, the low value of ratio of money stock to the output means that the money stock should be very few. Therefore, the paper sets the lower limit of the money stock space to be 0.001 and the upper limit is set to be 5. What‘s more, since the gross asset holding is taken as the state variable, the space of gross asset holding is very important to the computation. Because the gross asset holding is determined by the money holding and the productive capital holding, the choice of the gross asset holding should reply on the space of money capital and the productive capital; however, if the space of gross asset holding is determined directly by the equation (14), the upper and lower limit of the gross asset holding would change with the change of interest rate, this would increase the 148 Research of Mathematical Economics No. 1 2011 computation cost. Considering all the factors above, the paper sets the lower limit of the space to be the sum of the lower limit of money holding and productive capital holding, while the upper limit of the gross asset holding is set to be the sum of upper limit of the money holding and upper limit of productive capital multiplied by 0.9. 4. Results With the help of the solution methods given in Appendix 1, the paper solves the dynamic stochastic general equilibrium model with Matlab Programming. We can acquire the optimal policy function, the macroeconomic indexes and the inflation welfare cost. The section is divided into two parts: in the part 4.1, the paper introduces the main results of the model when the inflation rate is 3% so as to compare the calibration results and the data in reality; in part 4.2, the influences of inflation to different asset-holding classes are introduced. 4.1 The Numerical Nature of the Benchmark Model As the starting point of the analysis, the paper first consider the reasonability of the calibration in the benchmark model and see whether the model results in the benchmark model fit the macro economy in reality. Table 2 shows us the macroeconomic index and relevant targets. Table 2 The Comparison of Model Results and the Macroeconomic Index in Reality Macroeconomic Index Model Results Model Targets Ratio of Capital to Output 2.5343 2.45 Ratio of Money Stock to Output 13.51% 13.7% Ratio of Working Time to Labor Endowment 32.51% 33% Wage Gini Index 0.4394 0.435 Note: The capital stock is estimated with the help of the original capital stock in the year 1978 and the method developed by Wang and Yao (2003); the ratio of working time to labor endowment is in accordance with the eight-hour working day; the estimation of wage Gini index relies on the survey data from Ao Er Duo Investment Center. From the results in Table 2, we can see that the macroeconomic index in the model results fit the model target very well. Specifically, the ratio of capital to output in the model economy is 2.53, while the ratio in reality is 2.45. This means that the ratio in the model economy is higher than the model target; however, the gap is very small, it is only 0.08. The ratio of money stock to 149 A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes output and the share of working time to the labor endowment are also quite close to the model target. The share of working time in the model economy is 0.49% lower than the share in reality while the ratio of money stock to output is 0.19% lower than the model target. We can also see that the model economy fits the model target quite well in the wage Gini index. At the same time, for a heterogeneous-agent model, it should be guaranteed that the model results should fit the wealth distribution in reality with some degree (Krusell and Smith, 1997); therefore, before we can investigate the heterogeneity of inflation, we should see that whether the wealth distribution generated by the model fits the reality very well. The wealth distributions in the model economy and in reality are shown in Table 3. However, only the wealth distribution in reality is chosen as the model target while the shares of asset holding for different asset holding classes are not chosen as the model targets. Table 3 Comparison of Wealth Distribution in the Model Economy and in Reality Gini 1% 5% 10% 20%-30% 30%-40% 40% Index wealthiest wealthiest wealthiest wealthiest wealthiest poorest 2005 0.56 8.62% 25.88% 39.21% 30.84% 18.25% 7.62% Data in Reality 2007 0.58 8.74% 23.32% 36.79% 34.39% 16.94% 4.67% Model Results 0.5759 3.87% 17.52% 32.42% 40.86% 19.22% 3.16% Note: Data in reality is from Ao Er Duo Investment Center, the results can be seen in Chen et al. (2009). From the results in Table 3, we can see that the wealth distribution fits the data in reality quite well. Specifically, according to the calibration procedure listed in Section III, the model target of the wealth Gini index is set to be 0.57, while the Gini index in the model economy is 0.5759, which is very close to the model target and lies in Gini indexes in the year 2005 and the year 2007. From the share of asset holding for different asset-holding classes view, we can see that although the share of the 1% wealthiest and the 5% wealthiest in the model economy is lower than the data in reality, the model economy fits the reality for the share of 10% wealthiest, 20%-30% wealthiest and 40% wealthiest. Considering that there is no particular mechanism to generate a higher share of the wealthy and we do not take the share of different asset-holding classes as the model targets, we can say that the model economy fits the wealth distribution in reality quite well. This means that we can use the benchmark model to study the inflation heterogeneity. 150 Research of Mathematical Economics No. 1 2011 4.2 The Influences of Inflation With the help of method in Appendix 2, we can acquire the overall inflation welfare cost and the welfare cost heterogeneity for different asset-holding classes. First of all, we can see the overall inflation welfare cost to the society. The results are shown in Table 4. Table 4 Overall Inflation Welfare Cost%() Inflation Rate 0 1% 2% 3% 5% 7% 10% Inflation Welfare Cost 0 -0.46 0.83 4.19 5.2 5.44 6.15 From Table 4, we can see that the overall inflation welfare cost is relatively high. Specifically, when the inflation rate is 2%, the overall inflation welfare cost is about 0.83% of the gross consumption, while when the inflation rate is 4%, the inflation welfare cost rises in a further step to 2.37%. When the inflation rate is 10%, the inflation welfare cost is as much as 6.15% of the gross consumption. This means that the loss to the society due to the inflation in China is quite high. At the same time, Table 4 further shows us that we should allow a relatively low inflation rate when we have inflation control measures. The results show us that when the inflation rate is 1%, there is no welfare cost but welfare gain. The welfare gain is about 0.46% of the gross consumption. Therefore, the optimal inflation rate in China should lie in the interval between the values of 1% and 2%, which coincides with the conclusions of Bewley (1983). That is to say, the optimal inflation rate according to the Friedman Rule is not optimal any more. Taking the market incompleteness in reality into consideration, and considering that the market incompleteness would cause the difference of marginal substitution for different individuals and the complete market would assume that the marginal substitution is same for each agent in the model economy (Li and Han, 2010), it is reasonable for us to believe that the inflation welfare cost calculated through the Bewley type model with incomplete market and the heterogeneous agents would be more accurate for the Chinese economy. In order to have a deeper understanding on the inflation welfare cost, the paper adopts the calculated inflation welfare cost in Table 4 and the consumption and production data in reality to acquire the inflation welfare cost in terms of money. The results can be seen in Table 5. 151 A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes Table 5 China‘s Inflation Welfare Cost in the Years 2000-2009 Real Consumption Inflation Rate Welfare Cost Welfare Cost Welfare Cost Year (100 Million RMB) (%) (Consumption %) (GDP, %) (100 Million, RMB) 2000 39105.70 0.4 -0.2780 -0.1096 -108.7091 2001 42756.11 0.7 -0.4233 -0.1662 -180.9749 2002 48524.09 -0.8 0.4099 0.1640 198.8875 2003 51893.58 1.2 -0.4000 -0.1547 -207.5774 2004 57267.56 3.9 5.4656 2.0341 3129.9906 2005 65988.80 1.8 0.3450 0.1253 227.6554 2006 75280.79 1.5 -0.1470 -0.0519 -110.6951 2007 85124.05 4.8 5.2902 1.7755 4502.2287 2008 102443.53 5.9 5.0053 1.7291 5127.5767 2009 133613.70 -0.7 0.3926 0.1553 524.5081 Note: Data is from the website of Chinese Economic Information. The gross consumption is represented by the total volume of retail sales of consumption. Both GDP and consumption have been adjusted by the CPI and the base year is the year 2000. The inflation welfare cost in each year is acquired through the cubic spline interpolation. Column 4 represents the welfare cost in terms of the gross consumption, Column 5 represents the welfare cost in terms of GDP and the Column 5 represents the inflation welfare cost in terms of money. From Table 5, we can see that the inflation has been very low before the year 2006 since entering the new century. However, the inflation began to rise since 2006 and the trend did not stop until the influences of the subprime crisis. A glance at the 10 years, we can see that there is no hyperinflation. Even in the year 2008 when the inflation rises to its peak, the inflation rate is only 5.9%. However, the inflation welfare cost is very large. In 2007, the inflation welfare cost is as much as 5.3% of the gross consumption and the absolute amount is as much as 450 billion RMB. The inflation welfare cost in 2008 is also as much as 5% of the gross consumption and 1.73% of GDP; what‘s more, the absolute amount is as much as 513 billion RMB, higher than the absolute amount in 2007. Therefore, it is quite necessary to take effective measures to have the inflation under control. What‘s more, we can see from Table 5 that the results are different from the results generated by the model with complete markets: the calibration results show us that when the inflation is quite low, the inflation is good for the economic development and the Friedman Rule is not always optimal. Therefore, the market incompleteness would influence the relationship between the inflation and the macro economy; hence we should try our best to incorporate the market incompleteness into the model. However, it is not enough to investigate the overall influences of the inflation to make rational measures. What‘s more important is to have an investigation into the inflation 152 Research of Mathematical Economics No. 1 2011 heterogeneity to the different asset-holding classes. Table 6 shows us the inflation welfare cost to different asset-holding classes. Table 6 Inflation Welfare Cost Heterogeneity Inflation Rate 1% 2% 3% 5% 7% 10% The Poor: 1% 0.12 -0.72 -0.45 -0.62 -0.25 1.08 5% 0.09 -0.56 -0.30 -0.45 -0.10 1.29 10% -1.19 0.43 1.85 1.72 2.11 3.47 20% -1.50 0.73 0.68 2.87 4.50 2.17 30% -2.40 0.40 1.28 4.32 5.10 4.56 The Wealthy: 1% 0.77 -0.66 0.07 -0.70 1.48 2.90 5% 3.14 1.23 3.32 1.81 2.67 1.71 10% 2.96 0.43 2.45 0.88 2.84 1.96 20% 1.16 0.04 2.65 0.74 1.85 2.37 30% 1.61 0.52 4.61 2.21 3.01 3.60 From Table 6, we can see that there are great distinctions of inflation welfare cost to different asset-holding classes. For example, when the inflation welfare is quite low, the inflation would usually benefit the 1% poorest and the 5% poorest while the most agents in the economy would suffer great welfare loss. Therefore, even the policy would cause a moderate inflation to the economy; the government should also evaluate the different influences to the poor and the wealthy. When evaluating the inflation heterogeneity to the agents, there is one important index to consider: the influences of inflation to wealth distribution. Table 7 and Figure 1 give us the wealth distribution under different inflation rates. 153 A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes Table 7 Inflation and Wealth Distribution Inflation Rate 0 1% 2% 3% 5% 7% 10% Wealth 0.563 0.558 0.567 0.576 0.582 0.581 0.585 Distribution Wage 0.448 0.437 0.447 0.439 0.444 0.442 0.440 Distribution The Poor: 1% 0.007 0.007 0.005 0.006 0.005 0.005 0.005 5% 0.036 0.036 0.027 0.028 0.027 0.026 0.025 The Wealthy: 1% 3.602 3.656 3.678 3.866 3.866 3.785 3.757 5% 16.966 16.792 16.941 17.520 17.469 17.393 17.614 In terms of Tenths 1 0.077 0.082 0.061 0.062 0.057 0.053 0.050 2 0.240 0.253 0.229 0.191 0.157 0.133 0.138 3 0.897 1.018 0.797 0.781 0.623 0.623 0.592 4 2.276 2.443 2.183 2.126 1.896 1.810 1.731 5 4.536 4.671 4.492 4.328 4.241 4.260 4.129 6 8.081 7.924 7.888 7.322 7.213 7.301 7.340 7 12.322 12.283 12.357 11.904 11.970 12.292 11.954 8 17.342 17.344 17.414 17.089 17.269 17.547 17.365 9 23.040 23.189 23.318 23.774 23.996 23.778 23.850 10 31.188 30.792 31.262 32.424 32.576 32.202 32.850 Note: Line 2 and Line 3 represent the wealth Gini index and wage Gini index, respectively. Others represent the share of wealth for each asset-holding class, the unit is 1%. 154 Research of Mathematical Economics No. 1 2011 1 05%10%0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 00.10.20.30.40.50.60.70.80.91 Figure 1 Inflation and Wealth Distribution From Table 7, we can see that inflation would worsen the wealth distribution. Specifically, with the increase of inflation, the wealth Gini index would increase; the wealth share of the poor 6would decrease while the share of the wealthy would increase. On the Lorenz curve, we can see that with the increase of the inflation rate, there is a trend to convex to the lower right. This can be seen from Figure 1. From Figure 1, we can see that the Lorenz curve under the inflation rate of 5% would convex more to the lower right than the Lorenz curve without inflation while convex less to the lower right than the Lorenz curve under the inflation rate of 10%. For China, the worsening of the wealth distribution has limited the development of the economy and harmonious of the society and it is very important to control the wealth distribution. Since inflation would influence the wealth distribution, we should take the effects into consideration when we have some measures to control the inflation. 5. Conclusions Although there have been lots of economic models to investigate the China‘s inflation welfare cost, most of them are based on representative-agent models and the models with 6 For more details, see Appendix 3. 155 A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes complete market (Chen, 2003; Chen and Ma, 2007; Zhao, 2008; Wu and Cui, 2006). The complete market usually assumes that the marginal substitutions for different individuals are same, which contradicts with the reality. In China, the insurance market is not quite fully developed; therefore, agents cannot fully insure against all kinds of risks through the insurance market. This means that it is quite difficult to smooth the consumption, which would lead to the differences of marginal substitution for different agents and different periods; hence, the individual decision would change and thereafter the inflation welfare cost would change. Therefore, we should try to incorporate the market incompleteness into the model when studying the China‘s inflation welfare cost. What‘s more, under the circumferences that the wealth distribution has been worsened, it is not enough to investigate the overall influences of inflation; in fact, it would be helpful for making rational measures to taking the heterogeneity into consideration. Based on the consideration, the paper constructs a Bewley model with production sector and with the help of the model, the paper has an investigation into the inflation welfare cost and the influences of inflation to different asset-holding classes. After computation, we can find that inflation would cause great influences to the macro economy from the overall viewpoint and the overall inflation welfare cost is quite large. At the same time, the paper investigates the influences of inflation to the wealth shares and welfare costs of different asset-holding classes. Numerical results show us that there are great distinctions of inflation to different asset-holding classes. When the inflation is quite moderate, inflation would benefit the poor while causing great welfare cost to the wealthy; however, this does not mean that the inflation would improve the wealth distribution condition. Contrarily, the inflation, in fact, has played a role of transferring the wealth from the poor to the wealthy: with the increase of the inflation rate, the wealth share of the poor would decrease while the wealth share of the wealthy would increase. This means that inflation does not improve China‘s wealth distribution but worsen the distribution. Therefore, with the Bewley model, the paper combines the heterogeneity and the market incompleteness into one model to study the inflation and hence provides new insights of the China‘s welfare inflation. However, there are still lots of work to do to improve the work. First of all, it is very important for policy to do transition dynamic analysis on the inflation welfare cost (Gomme, 2008; Burdick, 1997). It would be surely quite helpful to investigate the heterogeneous influences of transitional dynamics of inflation. Secondly, it is quite important to investigate the 156 Research of Mathematical Economics No. 1 2011 influence heterogeneity of unexpected inflation shock (Doepke and Schneider, 2006a; Meh et al., 2010). I am sure it would be helpful to study the problem under the framework of Bewley type model with incomplete market and heterogeneous agents. Thirdly, Comparing with other developed countries, there is dualistic structure in urban and rural areas. Not only the income and wealth structures in rural and urban areas are different, but also the consumption and behavior habits of rural and urban residents are different. What‘s more, there is dualistic structure for the inflation in China (Chen and Chen, 2009). How to combine all there heterogeneity into one model? This would be one project full of difficulty. However, I think it would be quite helpful to know more about the dualistic structure in rural and urban areas and the influences of China‘s inflation to the macro economy. 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(Responsible Editor: Ren Junqiushi) (Proofreader: Wang Jue) 160 Research of Mathematical Economics No. 1 2011 Analysis of the Mechanism of “Dutch Disease” Wang Qizhi (School of Economics, Renmin University of China) Abstract: This article describes the mechanism of ―Dutch Disease‖ by setting natural resources into the production function and creating an independent sector, the resource exploiting sector. With the help of the Ramsey Model and the general equilibrium analysis framework, I will try to explain the influence of the ―Dutch Disease‖ to the manufacturing sector of the economy and consumption-investment decision making. At last, I‘ll give out some advice on how to avoid it. Keywords: Dutch Disease, Ramsey Model 1. Introduction In recent years, the relation between natural resources and economic growth draws increasing attention among the public. Natural resources are the material basis and prerequisites of economic growth, while its abundance will affect the country‘s gross economic growth to some extent. In short, there is a positive correlation between the two，but there also exists such phenomenon of so-called ―resources curse‖ in the relationship between the two. For a long time, some economists believe that natural resources are the foundation of economic growth and have played a positive role. While other economists argue that natural resources may hinder economic because of the facts that those countries with abundant resources generally can‘t get rid of growth ―resources curse‖. Resources occupy an extraordinary position in sustainable development. When it comes to the ―Resources curse‖, the Dutch disease is a typical one. During the 1960s, the Netherlands, which was already one of the main exporting countries of manufactured goods, found a great amount of gas in the North Sea. Dutch government strongly support that area, thus its exporting leapt, and so did its international balance of payment, the whole economy showed all signs of prosperity. Robbert van Eerd (2010) describes the situation at that time like this: 161 Analysis of the Mechanism of “Dutch Disease” The year 1959 saw the first exploratory gas drillings in the Dutch province of Groningen, which would turn out to be one of the largest gas fields in the world. Initially, the recoverable deposits were estimated to be 50 billion cubic meters. The estimates were repeatedly revised upwards, and by 1963, when the first concessions were granted, it was believed that the total reserve could be as large as 1,100 billion cubic meters. By 1967, estimates stood at 2,200 billion cubic meters (to illustrate how large this volume is, last year‘s natural gas consumption in the 1whole of the Netherlands was about 50 billion cubic meters). But the booming gas sector hurt the other part of the Dutch economy badly, such as the agricultural sector and some other industrial sectors which also reduce the competitive ability of them. In the 1970s, Netherlands suffered a severe inflation thus reducing exports of manufactured goods and citizen‘s income. Here is a description of the situation at that time: The wage pressure in the primary industry leads to wage pressure in other sectors of the economy. As workers seek out the highest wages, wage demands in the secondary sector need to mirror those in the primary sector (extraction and related industries). The apparel industry, for 2example, declined from 60,000 to 20,000 jobs between 1968 and 1978. When resources are abundant, the resources are often "curse" instead of "bless" the economy. The "Dutch Disease", which means that a country digging out considerable amount of resources would firstly boom the economy and then drag the economy into recession, this phenomenon was first brought to light by Corden (1984). Corden divided a country's economy into three sectors----tradable manufacturing sector, tradable resources sector and non-tradable goods sector, the resources sector exert influence on other sectors (in the model, the manufacturing sector and service sector), in which mechanism works through wages and exchange rate, thus affects the country's long-term sustainable development. But the analysis is more focused on the manufacturing sector in its long-term changes, while ignoring the whole economy as an organic whole. In the analysis of economic growth factors, neoclassical growth theory ignored the elements of natural capital, but focusing only on capital and labor (including human capital), neglecting the restriction of resource constraints. Hence, Solow-Swan (1956) established neoclassical growth model which laid the cornerstone for economic growth theory, and the model maintains 1 Robbert van Eerd, 2010, Of Dutch Disease and Other Ailments, The Bologna Center Journal of International Affairs. 2 The same as above. 162 Research of Mathematical Economics No. 1 2011 as the orthodoxy for a very long time. Solow Model uses the "new classical" production function in which capital and labor can be smoothly replaced, which means given a certain amount of labor, with the increase in capital stock, the law of diminishing marginal returns of capital ensures an equilibrium economic growth rate . Variance between the particular equilibrium value and the actual growth rate disappear due to lack of investment return expectations. All the above guarantee the conclusion of steady economic growth. However, as we know, the savings rate in the Solow model is exogenous, which doesn‘t quite fit the fact, so Cass-Koopmans (1965) pulled the analysis of the consumer optimization into the new classical economic growth model, which makes the savings rate endogenous, thus establishing a more general economic growth model. Cass-Koopmans referred to the mathematical form used in Ramsey‘s paper (1928), in which Ramsey constructed a dynamic optimization model, as well as using the classical calculus of variation to get the famous ?Keynes-Ramsey rule‘ on deciding the savings rate. Anyway, Ramsey‘s paper laid the foundation of research on accumulation and growth issues. Although the savings rate achieved became endogenous, Ramsey-Cass-Koopmans model consists with Solow-Swan growth model in the long-term occasion, namely, long-term economic growth rate depends only on the exogenous factors of population growth and technological progress. To fix this situation, in the twentieth century, mid-eighties, new economic growth theory of the neoclassical model came into being. Theoretically, Romer (1990) firstly put forward a model which makes the economic growth depending on endogenous technological progress rate. Around the same time, Lucas (1988) proposed a model on human capital in which the endogenous growth of human capital explains the economic growth. Although the growth theory is kept in the process of being modified and improved, basically speaking, the input factors has always been only two, capital and labor, without any factor of natural resources into consideration. Let‘s think, when resources exhausted, no state, even those with the best equipment and personnel, can put into production, thus, resources is certainly a significant restriction to a country's economic growth. Stiglitz (1974) investigated in the economic growth pattern under exhaustible resource constraints, the model shows that the scarcity of natural resources can be offset by technological progress, in other words, as long as keeping a positive rate of technological progress, aggregate output will not fall. In this paper, I put the Ramsey growth model into Corden‘s analytical framework, equate 163 Analysis of the Mechanism of “Dutch Disease” resources with labor and capital as input factors, and separate resource-exploiting department from the economy, thus lead to a complete study concerning what would happen to small countries‘ economy once abundant resources are exploited, meanwhile my study also shows that the price variance of resources in international market can produce the same effect as to the small countries‘ economy. The second part of my paper describes the basic model, the third part is my study on the mechanism of ―Dutch disease‖ with the use of my model, and the last part ( the fourth one) shows my policy suggestions as well as the deficiency of this paper. 2. Basic Model The economy is divided into 3 parts: household, manufacturing sector and resource exploiting sector. Household hopes to maximize its inter-temporal utility and makes consumption decisions; the manufacturing and resource exploiting sectors both seek maximum profits. We assume that labor and capital markets are perfectly competitive. Assume that the product of the manufacturing sector has no access to export while the resource market is open and any amount of resources can be sold at the world price . P 2.1 Manufacturing Sector (MS) Assume that the MS has the following production function: where represents quantity of production, represents technique, K stands for the total AQ1 Lcapital stock, is the capital stocked in the MS, stands for the labor in the whole (1),vK society, is the labor working in MS, R is the resources used in MS. (1),uLd The manufacturing sector seeks maximum profit: wrwhere is the wage in the MS, is the interest rateand is the depreciation, the ，,QQ 164 Research of Mathematical Economics No. 1 2011 first-order condition is: 2.2 Resource Exploiting Sector (RES) Assume that the RES is labor intensive and its production function is: 01,01,,,,vu where R represents the total amount of resources that are exploited, can be used to A2 measure the quality of one country‘s resources: when is large, little inputs can exploit large A2 amounts of resources; when is small, exploiting will cost more. represents the labor in AuL2 the RES. RES is facing an open resource market with a world price P. And MS also seeks maximum profit: The first-order condition is: 2.3 Household Household has the utility function: 165 Analysis of the Mechanism of “Dutch Disease” and could make the investment decision: (1) where represents the household‘s total assets, represents wage represents interest ， rate is consumption. ， When the control variable is consumption, household‘s utility-maximizing decision making can be expressed by the current-value Hamiltonian: The first-order condition is: and ‘s movement function is: So we get the Ramsey‘s rule : (2) and the transversality condition: lim[()()]0att,,t,, 3. Analysis and Applications 3.1 Equilibrium When this economy reaches the equilibrium, families and firms are faced with the same interest rate and wage level expressed here, (3) 166 Research of Mathematical Economics No. 1 2011 (4) (5) Because of our assumption of closed capital market, families‘ assets is equal to the society‘s capital stock . Besides, the market is perfectly competitive, which means all firms get zero profit, so the yield of the economy equals the sum of wage and interest. Combining (1) (4) and (5), we get: (6) If we transform both sides of (3) (4) (5) to their Logarithm and then differentiate them with 3respect to time, we find that (7) (8) (9) 3.2 Steady State In the Ramsey model, it was proved that this economy has a steady state. When the population is not growing, the growth of consumption and capital stock is zero. On the condition of TVC and making the growth of equal to 0, from (2) we can calculate: (10) which corresponds to the vertical line in Figure 1. Then after we make the growth of equal to zero, from (6) we can understand: (11) This equation illustrates a situation that is similar to the Solow-Swan Model. We can differentiate the right side of this equation with respect to to get the ―golden rule‖ level of Fcapital stock. To clarify this, we can assume that the ―production function‖ of the economy is , .3 represents , and the same way goes for other variables. gccc 167 Analysis of the Mechanism of “Dutch Disease” which satisfies , then FKLQKLRKL(,)(,)(,),， (11) which corresponds with the solid curve in Figure 1. , = 0 cc , = 0 k *MPK K k gold Figure 1 Figure 1 illustrates that, under the condition of different capital stock, households should select different consumption strategies in order to move to the steady point through stable arm. Now I‘ll show how this works: When the economy starts with small capital stock, if households select the consumption lower than the amount steady arm points out, which actually is a phenomena of higher saving rate, . it‘s trajectory eventually crosses locus. After crossing, keeps on rising while c,0 starts to decline. At last, the path will hit the horizontal axis. Similarly, if households select the . higher consumption rate, the trajectory will cross the locus. After intersecting, k,0 continues to rise while starts to decline. So, in finite time it will hit the vertical axis. 3.3 Exogenous Shocks 168 Research of Mathematical Economics No. 1 2011 Because the finding of new resources is unpredictable, this article assumes that it‘s a exogenous shock to the economy. When new resources with high quality are found, it means the average quality of domestic resources rises which indicates a rise of in the production function of RS. So we can get more resources with the same amount of input in RES. Consider the influence of this shock to the steady state. The finding of new resources will lead to a change in domestic industrial Proposition 1: pattern, which means RES will be relatively booming while MS will be relatively shrinking. 4We‘ll try to prove this by observing the changes in the labor market From (8) we can reason that: Then we use and get: When has an exogenous growth , the wage in RES must rise. And because the labor R market is competitive, it has a tendency to flow from MS to RES. So must be positive, and that indicates (12) That means RS grows faster than MS. So undoubtedly, that effect will induce the decline of the proportion of MS and increase the proportion of RES. This is evidence for what was explained earlier that ―RES will be relatively booming while MS will be relatively shrinking‖. Hence, we‘ve proved proposition 1, and we could call this effect a pull-out effect. 4We can get (11) as well by using the same way to analyze the capital market through the change of interest rate. 169 Analysis of the Mechanism of “Dutch Disease” c ,, cc= 0 = 0 01 B * C1 ,* C0k= 0 1A , k= 0 0 **K K k 01 Figure 2 The finding of new resources will pull the economy to a new steady state Proposition 2: where c and K are higher, thus the utility is higher than the formal one. As we assume the production function is linear homogeneous, and the markets are perfectly competitive, households‘ revenue Y equals the sum of wages and interests, hence (13) Combining (13) with (11), . From the equation above we can see that the locus of also moves upward in the k,0 phase diagram. . From (10) we can know, when r rises, will move right, shown in Figure 2. Variables c,0 with subscript 0 show the initial situation of steady state, while 1 shows the situation after shock. We can find the equilibrium point moves from A to B, and the equilibrium consumption and ****capital stock also move from to and to . So compared with the initial CKCK1010 situation, we can now enjoy more consumption and capital stock. Because utility is an increasing function of c, social utility also increases. We can call this effect the growth effect. 170 Research of Mathematical Economics No. 1 2011 Proposition 3: When world price increases, pull-out effect and growth effect will also P occur. Essentially, no matter if the increase is in or , the economy will react the same way. PA2 Both shocks lead to the same result, that being more outputs with same inputs. So the procedure has no actual difference from each other and can be omitted here. 3.4 Empirical Analysis Proposition 1 tells us that when the quality of domestic resources increases, RES will occupy a bigger part of this economy by pull out some part of MS. Meanwhile, proposition 2 shows us consumption and capital stock also increase. So, if we assume that a country suddenly finds a lot of resources with high quality, then the pull-out effect and growth effect will both happen and the economy will enjoy a booming time. But because the exploited resources are always from the high-quality or low-cost resources, after exploiting for a while the average quality of domestic resources will decline instead of continuing to increase. When no other high-quality resources are found, the reversal effects of both effects will happen. At that time, the economy will suffer a declining time, and the speed of decline is related to that of resource quality. One thing needs to be noticed, the MS may still shrink compared with the initial condition (before new resources‘ finding). In real life, due to the imperfect substitution in the labor force between two sectors, when labor flows into the RES, the efficiency of the new workers must be lower. Besides, the cost of management may augment. Those will all lead to a decrease in the production efficiency (which is the opposite effect of the shock‘s), which will reinforce the reversal effects. Besides, the works may not find new jobs at once, so the unemployment undoubtedly will occur. All in all, the imperfection of market leads to the decay of the economy. So we can see that, the fact of the Netherlands between 1960s and 1970s mainly matches the result we conclude from this model. But, there are still some points that I should mention: (1) Because the manufacturing sector of the Netherlands is not in a closed market, selling gas must have an impact on its exchange rate to force it to appreciate which will hurt the MS more. This model can explain this by a little modification that opens the market of manufactured products. But it will not fundamentally change the conclusion of this paper, so I didn‘t do this work. (2) Because the market in the real world cannot adjust as swiftly as it‘s imagined in the model, such 171 Analysis of the Mechanism of “Dutch Disease” as the change in wages, labor‘s transfer, and the economy‘s moving to the steady point will all take some time, so the real economy shows some lagging and mixed characters. That is, some phenomena are not showing immediately or showing some combined results. 3.5 Policy Advices The ―Dutch Disease‖ makes the manufacturing sector decay in the country with rich resources. The problem is, when the manufacturing sector goes down, the country‘s economy is essentially done in the long run and the manufacturing sector has more externalities like technological innovation and nurturing great managers. Moreover, the resource exploiting sector mostly needs lower human ability than the manufacturing sector, so if the manufacturing sector fades, the outflow of human talents is inevitable. More importantly, the manufacturing sector can develop and sustain in the long run while the resource exploiting sector has an inherent limit, which is its reserve. Once all resources are exploited, the resources sector will disappear. So it could be really dangerous for one country‘s sustainable development when the manufacturing sector goes down. Due to the huge uncertainty of the resource sector, the economy will be quite unstable when it takes a great part in it. When the production of the resource sector is undulating, it will influence the economy so badly that may cause some social problems. So, an effective method to control the consequence of the ―Dutch Disease‖ means a lot to both economic growth and society stability. Firstly, the government can tax on the exporting resources. If we do so, the domestic price of resources will stay low which will reduce the cost of the manufacturing sector. At the same time, 5tax makes it possible for the economy to stick to the optimal exploiting path, not too much or too quickly. Secondly, the government can invest in new resource developing activities. When the current resources‘ exploiting cost is high enough, the economy can turn to new resources. That action can also reduce the instability of the economy induced by resource production. At last, the advice I mentioned above is coping with the pull-out effect. So these kinds of actions reach their goal by sacrificing current consumption and investment. So if the government is going to do so, take that cost into consideration. 5 More details in Stiglitz (1974), etc. 172 Research of Mathematical Economics No. 1 2011 4. Concluding Remarks This paper uses the general equilibrium to illustrate that the finding of new high-quality resources may cause two effects to its economy; pull-out effect and growth effect, which will influence the growth of this economy in steady state. And then provides some advice of coping with the negative effects and keeping the economy sustainable. Generally speaking, this paper shows three things that are thought to benefit the economy ordinarily: (1) Rich in resources may ―bless‖ and ―curse‖ one economy at the same time. (2) Opening is not always a good way to benefit one economy. Sometimes, other sectors may sacrifice to the open sector. Meanwhile, it will make the economy less stable. (3) The growth of one economy doesn‘t always show the prosperity. Sometimes it shows the loss of potential development. This paper also has some shortcomings. First of all, in reality the capital market is not open. Because the including of this factor will bring much more variables and more changes, thus more difficulties. So I‘m not going to work it in this paper. Secondly, the analysis of this model is not thorough enough. Because of the complexity of the mathematical procedure, this paper mainly focuses on the difference between the two steady states (before and after the shock), instead of the procedure of moving towards the new steady state(including speeds of convergence , path of convergence, etc). Besides, this model omitted the human capital and knowledge capital. Because the inclusion of these two factors would not change the conclusion of this paper, I didn‘t take these into consideration. But when it comes to the research of the real world, speeds of convergence or path of convergence, they are essential. Finally, the development of new resources can be added in this model. When the world price is higher than a certain level, high price may stimulate the development of substituent resources. That may influence the investment decision and the steady state. 173 Analysis of the Mechanism of “Dutch Disease” References R.M., Solow, ―A contribution to the theory of economic growth‖, The Quarterly Journal of Economics, Vol.70, No.1 (Feb., 1956), pp, 65-94. F.P., Ramsey, ―A Mathematical Theory of Saving‖, The Economic Journal, Vol. 38, No. 152 (Dec., 1928), pp. 543-559. P.M., Romer, ―Endogenous Technological Change‖, The Journal of Political Economy, Vol. 98, No. 5, Part 2: The Problem of Development: A Conference of the Institute for the Study of Free Enterprise Systems (Oct., 1990), pp. S71-S102. R.E., Lucas, Jr. ―On the mechanics of economic development‖, Journal of Monetary Economics, Volume 22, Issue 1 (July, 1988), pp. 3-42. W.M., Corden, ―Booming Sector and Dutch Disease Economics: Survey and Consolidation‖, Oxford Economic Papers, New Series, Vol. 36, No. 3 (Nov., 1984), pp. 359-380. J., Stiglitz, ―Growth with Exhaustible Nature Resources: Efficient and Optimal Grows Paths‖, The Review of Economic Studies, Vol. 41, 1974. (Responsible Editor: Ren Junqiushi) (Proofreader: Zhang Lingyun) 174