Linear expenditure systems and demand analysis

http://www.jstor.org Linear Expenditure Systems and Demand Analysis: An Application to the Pattern of British Demand Author(s): Richard Stone Source: The Economic Journal, Vol. 64, No. 255, (Sep., 1954), pp. 511-527 Published by: Blackwell Publishing for the Royal Economic Society Stable URL: http://www.jstor.org/stable/2227743 Accessed: 29/06/2008 06:03 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=black. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org. LINEAR EXPENDITURE SYSTEMS AND DEMAND ANALYSIS: AN APPLICATION TO THE PATTERN OF BRITISH DEMAND' I. INTRODUCTION THE object of this paper is fivefold. The first objective is to derive a practical system of demand equations which possess properties usually considered desirable from the standpoint of elementary economic theory. The second is to consider the statistical problems involved in applying the system of equations. The third is to analyse the pattern of demand for consumers' goods in terms of this system on the basis of annual data relating to the United Kingdom over the years 1920-38. The fourth is to compare the results of this system, and systems like it, with the actual state of demand in 1900. Finally, the fifth is to compare the post-war structure of demand with what might be expected from the inter-war relationships under free-market conditions. The information on which this study is based is derived from the investigation of certain groups of consumers' expenditure, quantities bought and prices paid given in [1] and preliminary estimates for the remaining categories of consumers' goods and services which, in final form, will in due course appear in [2]. The essential feature of this investigation is that it is concerned not simply with the demand for individual commodities or groups of commodities considered in isolation but with the demand for all commodities bought by consumers classified into a number of groups. It is convenient at this point to indicate the nature of the notation used in the following sections. Greek letters in the lower case denote scalars. Column vectors are denoted by Roman letters in the lower case and unrestricted matrices are denoted by Roman capitals. Diagonal matrices are represented by the symbol for a column vector surmounted by a circumflex accent. Letters (usually Roman) of the lower case with suffixes are used to denote the elements of vectors or matrices. 1 I should like to acknowledge here the help I have received from a number of friends in the preparation of this paper. In particular my thanks are due to Messrs. S. J. Prais (now at the University of Chicago), J. A. C. Brown and J. Aitchison of the Department of Applied Economics and to Mr. D. A. Rowe who is working with me at the National Institute of Economic and Social Research on the estimates appearing in [1] and [2]. The calculations were undertaken by Miss Potter and Miss Ayling at the National Institute. 512 THE ECONOMIC JOURNAL [SEPT. II. A SYSTEM OF DEMAND RELATIONSHIPS Let p and q denote respectively a price and a quantity vector and let ,u 3 p'q denote total expenditure. Consider the system of relationships 11 A~~~~~~~~~~~1 pq _ pq + b(t- p'q)..***(1 In (1), q may be identified with a vector of quantities to which consumers are in some sense committed and b denotes a vector of constants which sum to unity. Then, on the hypothesis (1), the expenditure on the ith commodity (or group) is equal to a certain basic consumption, qi, valued at current prices plus a certain proportion, bi, of supernumerary income, measured here by total expenditure, t, less total committed expenditure, p'q. From a common-sense point of view (1) may seem a reasonable first approximation to a law of demand. For, on the assumption that tu > p'q, it entails that consumers first use up a certain amount of their income (here total expenditure) in acquiring the consumption vector q at current prices, whatever these may be, and then distribute their remaining income over the set of avail- able commodities in certain fixed proportions given by the elements of b. If q is the null vector, then (1) reduces to Aqz b:b 1t.(2) The vector fiq has as elements the expenditures on individual commodities or groups. Equation (2) states that each of these expenditures is proportional to t. The system (1) underlies the Klein-Rubin constant utility index of the cost of living [3], and its properties have been ex- amined by Samuelson [4]. It may be written in the form q i-1{bk + (bi' - I)p} . . . . (3) where i is the unit vector, I is the unit matrix and c - This expression when premultiplied by fi reduces to (2) if c is the null vector. If (3) is premultiplied by p it yields a system in which ex- penditures on individual commodities are expressed as linear functions of total expenditure and prices. Such a system is here termed a linear expenditure system. The system (3) is a res- tricted form, since in it the coefficients of the price ratios depend only on 2m - 1 independent parameters instead of the m2 - 1, where m is the number of commodities, which would be involved in the general case. However, it is the most general linear expenditure system which is compatible with three conditions 1954] LINEAR EXPENDITURE SYSTEMS 513 frequently imposed on demand systems in theoretical work. These conditions may be stated as follows. (i) Additivity. This implies that the sum of the expenditures as given by the system is identically equal to total expenditure, or that p'q- .(4) (ii) Homogeneity. This implies that for each commodity the sum of the total expenditure (or income) elasticity and all the price elasticities is identically zero, or that 4-1(a[k + Ap) _= O . . * **(5) where a is the vector of quantity derivatives with respect to income, A is the matrix of quantity derivatives with respect to price and 0 is the null vector. (iii) Symmetry of the substitution matrix. This is the Slutsky condition, namely that S, S.(6) where S - 4j(ai' + A-)[ . .* (7) In order to see with what added restrictions (3) may be re- garded as the most general system of its kind which satisfies (i), (ii) and (iii), let it be rewritten in the more general form pq -by + Bp ... (8) The additional restrictions needed if (i) is to be satisfied by this system are obtained by premultiplying (8) by i'. This will yield on the left-hand side p'q the sum of the expenditures, and on the right-hand side 1t, as required, if and only if i'b= 1 .(9) and i'B '. . (10) In other words the bi must sum to unity, and the elements in the columns of B must in each case sum to zero. Condition (ii) also gives no difficulty, since for (8) Cz,-p-lb . . . , . *(11) and A -:[B p-'(bk + Bp)] . . . (12) where Bp denotes a diagonal matrix constructed from the vector Bp. If these values are substituted into the left-hand side of (5) there results q-'{ft'lbt + p-'[B - p-f(b - + Bp)]p} _ . . (13) 514 THE ECONOMIC JOURNAL [SEPT. Thus condition (ii) places no further restrictions on b and B. It will now be shown that condition (iii) restricts the form of B. From (7), (11) and (12) it follows that S 4- i-:h1{bi' + [B - p-(btu + Bp)]4-1}[ q- fi-Il{bi' + [B - . . . . . (14) From this expression the nature of the restriction on B if (iii) is to be satisfied can be seen. For in (14) the last term on the right-hand side is the product of diagonal matrices, and so symmetric, whereas the first two terms are not. If these terms are pre- and post-multiplied first by q and then by I and if the scalar ,u is neglected, there results bi'pq + Bp which may be written as bq'pi + Bp5 A substitution for q from (8) into this expression yields b(b'p + p'B')p-Ilp+ Bp which, since the first term is symmetric, can be reduced, for present purposes, to bp'B' + Bp or bi'fiB' + Bfi If this expression is to equal its transpose then Bpi(ib' -I)-=(bi' -I)piB' * (15) whence B =(bi' -I)c. (16) as in (3), where c is a vector of constants. Equation (16) gives the most general form of B which is independent of p. It would be possible to replace c in (16) by pD, where D is a symmetric matrix but this would make B depend on p. It will be seen that (8) with B restricted as in (16) cannot, for practical purposes, describe a system which contains inferior or complementary groups of commodities. For the ijth element of 8, sij say, may from (14) be written s=j =$ {bi + (bi -aSij)cj 83} - i (bi - 3ij)(l + cjlqj) piqi piqip qj 1954] LINEAR EXPENDITURE SYSTEMS 515 where 8ij is Kronecker's delta. For practical purposes it may be assumed that supernumerary income, (, + p'c), is positive. If sii is to be negative, as in theory it must be, then 0 < bi < 1. This rules out inferior goods. If such goods are ruled out, then bibj is necessarily positive, whence all off-diagonal elements of S must be positive. This rules out complementary goods. As already indicated, if c is the null vector, the system reduces to (2). In this case the various elasticities are of a peculiarly simple form. Let a* and A* denote respectively the vector of quantity elasticities with respect to total expenditure and the matrix of quantity elasticities with respect to prices. Then if B O a qjlapi Aq-lp-lbp - . .......(18) and A * A_ IA --I ... ... (19) since from (2) b - (20) Finally, in this case it can also be seen that S-ii' b .(21) III. THE ESTIMATION OF THE PARAMETERS IN (8) In the applications of the following section attention is con- centrated on (8) with B as given in (16). A naive model, corre- sponding to (2), is given if c is the null vector and a more sophis- ticated model, corresponding to (1) is given if c is any other vector of constants. These models are termed simple systems. The naive and more sophisticated descriptions can be applied to individual commodities, and their combination also yields a system, though not one which satisfies either (i) or (iii), which is here termed a mixed system. If c is assumed equal to the null vector so that B in (8) is the null matrix, the vector b can readily be estimated by taking the least squares regression of piqi on y (without correction for means) in each of the m equations. But the estimation of the parameters in (8) if c is not the null vector cannot be performed by taking the equations one at a time since each element of c enters into each 516 THE ECONOMIC JOURNAL [SEPT. equation. It seems in fact to be most convenient to adopt the following iterative procedure. A provisional estimate of b, say b*, may be obtained as above and, with this available, (8) with B as given in (16) may be written for period t (t = 1, . . ., n) in the form yt Xtc + u .(22) where yt (qt -. .t) .(23) Xt - (bi' I)pt .(24) and ut denotes a vector of disturbances. Since the elements of yt are dimensionally similar, being in fact all adjusted expenditures, it does not seem unreasonable to estimate the elements of c by minimising the unweighted sum of squares of the disturbances in all equations over all time periods. Thus let Y {Y, -, y}. - (25) X _ {xi * * * ) Xn} * * * * . (26) and u _ {,ul - - un} . . . . . (27) so that the whole set of mn equations for m commodities over n time periods can be written in the form y = Xc. + u. (28) Then the least squares estimate c* of c is given by c* (X'X)-X'y .(29) With this estimate of c it is now possible to form the equations WtZtb + vt .(30) where wt pt(q, + c*) .(31) Zt- (1t + c*'pt)I . (32) and vt denotes a vector of disturbances, and then to form new estimates of b, b** say, from the equation w =Zb +v .(33) where W- {W1....wn}. (34) Z - {A) Z Zn} * * * * * (35) and V - {Vl, ...., Vni . (36) 1954] LINEAR EXPENDITURE SYSTEMS 517 These new estimates take the form b** - (Z'Z)-'Z'w .(37) where Z'Z= . . . . . . .(38) Thus in estimating b, as opposed to c, the commodity groups are treated one at a time. Given these new estimates of b, it is now possible to re-estimate c and to continue the process until stable estimates of b and c are eventually reached. The estimation of the parameters in a mixed system provides no additional difficulty. Provided a criterion can be established for deciding which commodities are to be described by the naive model and which form a subset with price interactions as in the more sophisticated model, then the procedure is as follows. First, the bi for those commodities which are to be described by the naive model are calculated. Second, the actual expenditure on this subset is subtracted from ,t to yield say ,p* and (8) with B as given in (16) is fitted to the remaining commodities with ,* in place of , and without reference to the prices of the first subset of commodities described by the naive model. It can be seen that the above method is not affected if year- to-year changes in the variables are used in place of their levels. It can readily be adapted to the case in which all the variables are divided through by ,. The effect of such transformations of variables is not examined in this paper. IV. AN APPLICATION OF LINEAR EXPENDITURE SYSTEMS, SIMPLE AND MIXED In this section the results are given of analyses of a system of six commodity groups among which the total of consumers' expenditure per equivalent adult has been divided. The methods used were described in the last section and the data to which they were applied relate to the United Kingdom over the years 1920 to 1938. The groups chosen were as follows: (1) meat, fish, dairy products and fats; (2) fruit and vegetables; (3) drink and tobacco; (4) household running expenses such as rent, fuel and light, non-durable household goods and domestic service; (5) durable goods of all kinds such as clothing, household durables and vehicles together with transport and com- munication services; (6) all other consumers' goods and services. 518 THE ECONOMIC JOURNAL [SEPT. The first step was to apply simple systems to these commodity groups with results shown in the following table. TABLE I Alternative Siinple Systems for Six Commodity Groups Naive model More sophisticated model a (1, 2, 3, 4, 5, 6). b. c. r. b. j . r. 1 0 18 0-88 0-12 -14 0*97 2 005 0-88 004 - 3 0-97 3 0 11 0 95 0-06 -10 057 4 0Q19 0-23 0-23 -11 0O81 5 0*23 0 95 0 30 -12 0 97 6 0-24 0-98 0-25 -15 0 97 a The vector of mean prices for the period is p = {0.97, 104, 1-02, 1-02, 1-05, 1-02}. Accordingly, the values of committed expenditure, ? per equivalent adult, at mean prices are -pc = {14, 3, 10, 11, 13, 15}. The mean values of total expenditure are pq = {18, 5, 12, 20, 24, 25}. In this table the figures in the columnns headed r are the sample correlations between the actual and the calculated expenditures. It can be seen that on the whole the more sophisticated model which involves twelve parameters provides somewhat higher correlations in almost all cases than the naive model, which in- volves only six parameters. For one commodity group in each set, the correlation using one model is markedly superior to that obtained from the other. Thus for group 3, drink and tobacco, the naive model yields a correlation of 095, while the more sophis- ticated model yields a correlation of only 057. There are two main reasons for this effect. In the first place the value of b3 in the naive model is much larger than the corresponding value in the more sophisticated model. In the second place, in the latter the value of -p'c is rather large, with the result that supernumerary income, (,u + p'c), is very different from total expenditure, P. In fact, supernumerary income moves very like total expenditure at constant prices and not like total expenditure at current prices. If the two sets of correlations are compared it can be seen that the more sophisticated model provides clearly the better descrip- tion of actual expenditures for commodity groups 1, 2 and 4 and clearly the worse one for group 3. For groups 5 and 6 the correla- tions are high on either model and there is not much to choose between them. It is these facts that suggest the possible advantage of a mixed system employing the naive model for certain groups and the more 1954] LINEAR EXPENDITURE SYSTEMS 519 sophisticated model for the remainder. Such a combination of the systems is not compatible with a symmetric substitution matrix, but represents a possible type of behaviour which can readily be expressed in words. Thus suppose groups 3, 5 and 6 are described by means of the naive model while groups 1, 2 and 4 are described by means of the more sophisticated model. Under this scheme, consumers lay out certain fixed proportions of their total expenditure on groups 3, 5 and 6, while for the remaining groups they purchase a certain fixed quantity at current prices and then spread the balance of their expenditure over these remaining groups in certain fixed proportions. In the following tables the results obtained from three such mixed systems are given. The first, referred to as (1, 2, 4), in- volves the description of groups 1, 2 and 4 by the more sophis- ticated model and 3, 5 and 6 by the naive model. The second, referred to as (1, 2, 4, 5), involves the description of groups 1, 2, 4 and 5 by the more sophisticated model and 3 and 6 by the naive model. FPinally, (1, 2, 4, 5, 6) has only 3 described by the naive model. TABLE II Alternative Mixed Schemes for Six Commodity Groups (1,2,4). (1,2,4,5). (1,2,4,5,6). b. c. r. b. c. r. b. c. r. 1 0 14 -14 0.99 0 13 -17 0 98 0 13 -16 0 97 2 0 05 - 3 0 96 0 05 -4 0 95 0 05 - 4 0*94 3 0 11 0.95 0 11 - 095 0 11 - 0.95 4 0 22 -12 0 94 0 17 -17 0 93 0.19 -15 0 97 5 0 23 - 095 0 30 -19 0.99 0 30 -16 0*98 6 0 24 - 098 0 24 - 098 0 22 -19 0*97 The figures in the b-columns call for some explanation. For purposes of comparison with one another and with the corre- sponding columns of Table I they have been made to sum to unity. If there were no errors in the cases described by the naive model this would be quite straightforward, since ,* would be exactly proportional to ,. The existence of such errors is ignored and the original bi based on the more sophisticated model, which themselves sum to unity, are multiplied in the above table by unity minus the sum of the bi based on the naive model. In Table II there are no really bad descriptions as judged by the correlations. On the whole the three-set model involving only nine parameters is as good as the other two. 520 THE ECONOMIC JOURNAL [SEPT. V. A TEST OF THE SYSTEM : CONSUMPTION IN 1900 From (3) it follows that if p is held constant and ,u is allowed to vary, then q h +g, . (39) where h = A