inter-temporally

inter-temporally A Two-Sector Adaptive Economizing Model , Of Economic Growth Xiaochuan Song San Diego Mesa College, San Diego, CA 92111, USA Expectation errors in this model lead to over or under estimates of future outputs, wages, and interest rates; therefore causing over or under investment in capital goods, continuing fluctuations in consumer goods and capital accumulation. Growth cycles are generated by the agent?s adaptive economizing behaviors, and incomplete knowledge. Key words: adaptive economizing, endogenous growth cycle, complex dynamics. 1. Introduction Growth cycles in this paper are driven by inter-sector flows of capital and labor that arise endogenously from adaptive economizing behavior. Specifically, adaptive expectation overestimates the future rate of return when “too little” capital is allocated in the capital goods sector while underestimating the rate of return when “too much” capital is allocated in the consumer goods sector. As a result, a significant amount of capital will flow from the consumer goods sector to the capital goods sector. This results in overshooting the equilibrium resource allocation, eventually causing a reversed flow of resources the next period. Such ill found anticipations could lead to continuing growth fluctuations, which bear some resemblance to the real world. We first derive a value function and an optimal policy function for Uzawa?s two- sector economy. Then, a modified model based on a more realistic “adaptive economizing” assumption is presented. We show that the optimal growth trajectory can be derived if perfect foresight is assumed and that growth cycles can emerge if perfect foresight is replaced by adaptive expectations. The Golden Rule of capital accumulation can be generated only if economic agents do not discount the future. Complex dynamics of capital accumulation depends on the preference parameter as well as capital intensities of the two sectors. 2 2. Optimal Growth In Uzawa?s model a representative agent maximizes the present value utility of the entire consumption stream. The two sectors, consumer goods and capital goods sectors, labeled as 1 and 2 respectively, use a single grade of labor and a single type of capital good. The two inputs are smoothly substitutable for each other in each sector and are freely transferable from one sector to the other. Labor supply grows at a constant rate n, and capital goods depreciate at a fixed rate ,. Both exougenously determined labor supply and irrevocably existing capital stock are inelastically offered for employment. Both sectors use neoclassical technology with the standard Inada conditions. Let cCLkKL,,/,/tttttt kKLlLL,,/,/itititiitt 112lli，,,,,,12 where is consumers good, is capital good, is labor supply, and KKK,LL,CLttt1212 are capital stocks and labor allocated in sector 1 and 2 respectively. Let u be utility function, production function and , then, the ,, discount factor with 01,,fi discounted optimization problem in per capita terms is ,tMaxuc,(),,tt,0 ,stcfkl..()tt111 1,，,1 k[()()]fklk ,，1222ttt1，n ，,lklkk1122ttt ,,00kgivenc,,0t where uuuand'(),','',000,，,,, ffff'(),'(),',''.0000,，,,,,,iiii 3 Let the state variable be defined as and control variables where kklck, , and , 'ii is the capital stock that the agent decides to carry into the next period. Then Bellman?s k' equation is vkMaxufklvk(){[()](')}.,，,111cklk,,,'ii After some substitutions, LOLOlkknk'()()，,,11,,12 vkMaxuf(){[,,f](')}.lvk，,MP121MPcklk,,,'iilll112NQNQ The first order condition is 1, ucfkfknvk'()'()'()()'(').1，,,1122 The envelope condition is ,1 vkucfkfk'()'()'()['()()].,，,11,112 Because are strictly concave, it follows that is strictly uandf (.) (.) vk () i ,1concave ( is a decreasing function of k). Therefore, as a solution to the value function, f2 the optimal policy function, , is a nondecreasing function of k. kk'(),, There is a maximum capital stock such that, if,forall,then, ,thus ctllkkk,,,,,0010,,,t1212 1 kMk,,fkk，,,()[()()].1tttt，12，n1 ' Because '(),'(),''theequationffff0000,，,,,,,,,2222 ,,,1 k,[()()]fkk，,1,21，n , hasauniquepositivesolution ,and kkconvergestokast,,.t , We can construct a system . Since is nondecreasing in is a kk,k,(0,k)fttt2 monotone, bounded sequence, converging to a limit point . This limit point kast,,,does not depend on initial conditions. At the limit point, kkk,,, both first order tt，,1condition and envelope condition hold, we have 4 knk()()knk()()11，,,,11，,,,11,,ucfkf'()'()'()()['()'()['(nucfkf)()]].11，,，1,,,112112ll22 Collecting terms, LO,,11 ，,,,1MP1,，nfn，'(.)()112NQ 1. whichcanbeusedtosolveauniqueoptimalsteadystatek, Consider the standard functional forms ,, uccfAkfBk()ln,.,,,1122 2Assuming capital goods last one period only, Bellman?s equation becomes /1,,,,1 vkMaxAlklknBlvk(){ln[('()/)](')}.,,，，1,122cklk,,,'ii Applying undetermined coefficient method, or iterating on Bellman?s equation, starting from , we can derive an optimal policy function and value function. Substituting vk(),00 3the efficient labor allocation l into the value function, we have kn()，,,,1,11 Note that, The optimal steady state capital accumulation ff'(.)'[,],.22ln()()11，,,,,2 depends on the capital goods sector production function, labor allocation, population growth, depreciation 1,Ifandthenfkrnr'(),().,,,,,0,1rate, and discount factor. Day obtained the same result 22, for a one-sector adaptive economizing model. If which is the result derived by ,,,,11,,then r Sargent (1987). 2 Capital intensity of each sector depends on the technology of each sector and the equilibrium wage- f()kiirental ratio ,, which can be uniquely determined from the equilibrium condition ,,,ki'f()kii ,i(Uzawa 1961, Solow 1961). For Cobb-Douglas technology, k,. In this example, ,,,, ,12i1,,i ,,13/,/,23if,,thenkkkk,,,,,/,,224.kk,. Take another example, 12121221 ，k(),,k13 Efficient labor allocation can be derived by substituting k,k into l,, thus, (),12,,()，k，k(),,12 ,ki2, .l, For Cobb-Douglas production functions, . This is just the capital k,l,,,112i1,,,()，k,i2 5 ,1,2,B()1,,,,,,,1,,1vkA()()[lnln,,，1,,，,，,,,ln()ln]ln.1,,，k1,1，n11,,,,,,,, The optimal capital accumulation is 1,,B,()1,, k,k().,,tt，1，n1 Since ,,,,1,,kastkkandcconvergesforanypositivethesteadystatearet0 ,B()1,~111//,,,,,k,[](),,,，n1 ,B()1,1111,,,//,,,,~cA,,[]().1,,,,,，n1 All other steady state control variables can be calculated readily. The two-sector optimal growth model describes how an economy would work if it worked perfectly. However, our economic agents are bounded, adaptive economizers. They economize the trade-off between current consumption and future standard of living based on current income, wealth, rates of return, and a rough estimates of future values instead of perfect knowledge about the technologies, the feedback structures of the entire future. They adjust their estimates, decisions, and behaviors step by step, period after period, as time goes on and subsequent events unfold instead of behaving in exactly the same way forever in order to satisfy a principle of consistent intertemporal optimality. By explicitly incorporating these salient features of real world into the model, we get a two-sector adaptive economizing model. 43. Adaptive Economizing elasticity in the capital goods sector. The economic intuition follows clearly: given the equilibrium input price ratio, the optimal resource allocation rule is to allocate more labor to the sector in which labor is relatively more productive. 4 This section is based on the one-sector adaptive growth model developed by Day and Lin. 6 Suppose our agents do not know the production functions and technologies of the next period. They must therefore estimate the rate of return of current investment and the wage rate of the next period to make optimal choices about capital and labor allocations between the two sectors. Since plans need to be revised when the next period of time arrives, and when a new rate of return and income are realized, which most likely are different than before and previous anticipations, our boundedly rational agents reduce the infinite horizon problem to a two period problem in practice: the trade-off between present consumption and the future standard of living. This kind of adaptive economizing model requires less information and computational capability and reflects the economizing activity more or less as we experience it. Let u(c) be utility function of current consumption and ,,t01Wcucucwhere(')(')('),,,,,,,,,,, ,1,,,1t be the utility function of the anticipated level of well being c' that could be sustained in perpetuity. Assuming that tastes are constant over time, W (c') is the present discounted flow of future utilities of a constant future standard of living, as evaluated by the current generation. c' The sustained future standard level of living must account for the maintenance of capital stock and the need to provide an endowment for net additions to the work force, where are anticipated income, wage rate, and cynkywrk''()','''',,,，,，,ywandr','' rate of capital return in the next period. The maximization problem then becomes Maxucuc[()(')]，, stcfkl..(),111 11k'[()()],fklk，, ,2221，n lklkk，,1122 1cwrknkwfklk''''()'''[()()],,，,，,，，,222,,, klk,,rn'(),，11. whereSinceimpliesand,',.,(),k,fklkklk,,,0022222111，nl2 The Lagrangian equation is 7 , ,,,,，，,ucucklk()(')()11 where is the Lagrangian multiplier. , Whenthe first order condition withrespectisklktok,,,,,0112 ucfkucfk'()'()''(')'().,,,1122 This Euler equation represents the standard efficiency condition: the marginal rate of substitution between two periods is equal to the anticipated marginal rate of transformation. That is, on an adaptive economizing path, the marginal utility of using capital in the consumer goods sector and allocating its return to current consumption equals the discounted expected marginal utility of using capital in the capital goods sector then allocating its return to future consumption. Rewrite the Euler equation, ucfk'()'()11 ,',.ucfk'(')'(),22 Given the utility functions and discount factor, large thus implies small or large , ,'kk12 or both. Therefore, an overestimated net rate of return could cause the overallocation of capital in the capital goods sector and vise versa. When , ,,,,00,,klkork112 ucfk'()'()11 ,,,,,,,,ucfkucfkor'()'()''(')'(),'0.1122ucfk'(')'(),22 '()'()ucfk11,Since Thus, a negative net rate of return lim,,,000,'.ask2k,02,,''(')'()ucfk22 implies zero capital allocation in the capital goods sector. While producing capital goods for the next period does not appear to be worth the sacrifice in current consumption, our agents allocate all capital to the consumer goods sector and produce consumer goods only. Therefore, no savings occur and capital decumulates. As long as ,' is finite, there is always a positive solution satisfying the first order condition, which is a function of fkwand, , ,,','. 2 hfkwif(,,',',),',,,,0R2 k,S2if00,',,T 8 It is worthwhile to compare the first order conditions of our model ,1 with that of the optimal growth model ucfkfkuc'()'()'()''('),,,1122 vk'('),1, ucfkfk'()'()'(),.11221，n ,iiiiiiiiDefine then ,,,,vkucwherecynkwrnk(')max(),()[()],,,,，,，,，,ik,1i 11222333 ,,,,,,,vkucrnucrnucrn'(')'()[()]'()[()]'()[()]....,,，，,，，,， Suppose our agents have perfect foresight about the entire future and want to keep iithe same sustainable consumption level and rate of return forever, namely, , ccrr,,',' for all i, then ,,vk'('),vkucrnor'(')'(')['()],,,，,,,,''(').uc 11,，n, Our adaptive economizing model has exactly the same first order condition as the optimal rowth model does in the steady state with perfect foresight assumption! g Unfortunately, our economic agents can not foresee the entire future and they do not have a complete knowledge of the feedback structure of the infinite horizon problem. Therefore, they do not behave in exactly the same way, period after period. Most likely, they will make suboptimal resource allocations based on adaptive economizing expectations about future technology, rate of return, and wage rate. Suppose our agents use naive adaptive expectations, namely ,rn,，() rrfkwwfkfkk''(),'()'(),',,,,,,,,,.22222221，n The capital accumulation equation becomes 1R,,[(())()],fhklkifrn，,,，122tt|，n1. kk,,,()Stt，1,,1|kifrn,,，,t，n1T 4. Some Dynamics of Capital Accumulation It is obvious that the trajectory is bounded by 9 1 kMk,,fkk，,,()[()()].1tttt，12，n1 Assuming the same neoclassical technology,, a ffff'(),'(),',''0000,，,,,,,2222 fixed point exists 1mmm k,[()()].fkk，,1,21，n m. AlltrajectorieswithinitialconditionremainkKkinK,,:(,)00 To prove the existence of steady state solutions, zero is a trivial one. If a nontrivial (1,),solution exists, it must occur in the positive investment phase, because, . ,1(1，n) Thus, the steady state condition is ~~aa ()nkfhkl，,,(()).22 In a positive steady state, enough capital must be allocated in the capital goods sector to produce enough capital in the next period to restore worn-out capital stock, plus augment it sufficiently to endow net additions to the population. To determine asymptotic stability of balanced growth, we must evaluate the 5derivative of in a steady state. The first order condition becomes ,()k a~11ucfk'()'(),，,，rn()na11~ ,,,，，,orr().n,1，nucfk'(')'(),,,22 Note that ~aoaooo'~ limrrkkwhererfkn,,,,，,lim,().,2,,,,,, The adaptive economizing becomes optimal in the sense of the Golden Rule of consumption as inter-temporally optimal model does if agents do not discount the future. Let?s take the total derivative of the first order condition with respect to k 5 Capital and labor are freely transferable across sectors, which equalizes the marginal products between the two sectors in a steady state. 10 ucfklh''()'()(')1,112 , RUfkh''()'22,,，，,，，,uckfkhfklfklk''(')''()'[('()())('()())11] ,,,,SV2222222221，nTW fkh''()'22，uc'(').，1n 1~a'At the steady state ccnkfklnfklh',,(),,，,，,,,,(),()()'. Assuming 222222, , we have ff'(.)'(.),again12 ufh'''(.)'~a2flh'(.)(')1,,,,，,，，fkhhn''(.)()'()1. 122un''()1， Solveforh' n，, h',.uf'''(.)~a2，,，fkhfl''(.)()'(.),2121un''()， The stability condition is 1 ,,,1,,'()['(.)'()],kflh，,,11221，n or ,,,,n2n， '.,,h'(.)'(.)flfl2222 Substituting h'into the preceding inequality, the right hand side becomes uf'''(.)~a2，,，,, fkhflfl''(.)()'(.)'(.),212221，un''() hence u'~a,,,, (hkcef),sin''.02，un''()1 Since the left-hand side of the preceding inequality is negative, the condition is ~ahk,readily satisfied if . That is, if the capital goods sector is more capital intensive, the ~asystem becomes stable and converges to a steady state. However, if has to be hk,,, 11 big enough to stabilize the system; otherwise, the system is not stable. This is the same as the result derived by Uzawa (1963). However, the mechanism of Uzawa?s model is: “The rate of adjustment in real wage is inversely related to the quantity of „involuntary? unemployment.” Thus, the inflexibility of factor prices leads to possible instability. The mechanism of our model is: the rate of adjustment in real rate of capital return is inversely related to the quantity of capital stock. Therefore, anticipation errors and impatience of adaptive economizers generate growth cycles. Rewrite the left-hand side of the stability condition 2n，,，,n, .,,'''(.)uf~'(.)fla222''(.)()'(.)，,，fkhfl,2121''()un， ~~aaNote that when hk,, the condition is readily satisfied. When hk,, if is big , enough to make the denominator positive, the condition is still satisfied. However, if is , not big enough to dominate the denominator, it has to be small to satisfy the right hand side of the inequality. We can summarize the dynamics of our adaptive economizing model as following: If capital goods sector is more capital intensive, the system is stable and converges to steady state. If consumer goods sector is more capital intensive, has to be either big , enough or small enough to obtain stability. Since h is not a constant function and it changes continuously due to inter-sector flow of capital and labor caused by our agent?s adaptive behavior, the system is intrinsically unstable. Periodic or chaotic trajectories can emerge, depending on the coupling of time preference and periodic changing capital intensities. Consider the same Cobb-Douglas production functions and utility functions as before but with naive expectations, ,,1. rrfkBkwwfkfkkBk''(),'()'()(),,,,,,,,,,1222222222 ,Assume that Wccn(')ln',,,,,,,,10 then . ,,,',',,,,，rcwBkl122 The first order condition becomes 12 w,,,1, ，,,().BklkklBk22222, 6Substituting w and r into the first order condition, assuming ,=1, collecting terms, we have Substitute this back into the capital accumulation equation klk,,.22 . kBk',, The steady state condition becomes as shown before. Comparing with the optimal B,,1 ,LOBl,,2growth model solution, , then . Again in this If,,,1,,nokBk',,k',kMP1，nl2NQ special case, our adaptive economizing model generates the optimal trajectory. Day (1994) parameterized the comparative dynamics of the adaptive economizing model by showing different combinations of preference parameter and capital elasticity. uHe argued that, for each , there exists such that k is unstable whenever ,0,,,(,)01 uu and asymptotically stable whenever . Our example is the extreme case - ,,,,,, the steady state exists and the trajectory converges when , = 1. The reason is that when capital elasticity is 1, the marginal product of capital no longer depends on capital stock; therefore, the mechanism of our system breaks down. No matter how much is invested each period, the expected rates of return remain the same; it is equal to B, the technological factor. Even though agents do not know production functions, they know that the rate of return is constant; therefore, they invest the same amount of capital forever. This is equivalent to perfect foresight where no anticipation errors exist and nothing can drive the system away from the steady state but exogenous shocks. References 宋小川，2003:《非均衡的经济动态模型》，《经济研究》第7期。 6 Technically speaking, this is equivalent to normalizing labor as one in this example, or to assuming the capital goods sector uses capital only. 13 Day, R. H. 1982. “Irregular Growth Cycles.” American Economic Review, 70, 406-414. Day, R. H., and T. Y. Lin. 1992. “An Adaptive, Neoclassical Model of Growth Fluctuations.” In A.Vercelli, and N. Dimitri (eds.), Macroeconomics: A Survey of Research Strategies. Oxford: Oxford University Press. Day, R. H. 1994. Complex Economic Dynamics. Cambridge: MIT Press. Lin, T.Y. 1988. Studies of Economic Instability and Irregular Fluctuations in a One-Sector Real Growth Model. Ph.D. dissertation, University of Southern California at Los Angeles. Sargeant, T. J. 1987. Dynamic Macroeconomic Theory. Cambridge: Harvard University Press. Song, X. 1998. 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