Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1968 The mpire ical estimation of substitution terms from demand analysis Joseph Salim Fuleihan Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Economic Theory Commons Recommended Citation Fuleihan, Joseph Salim, "The mpe irical estimation of substitution terms from demand analysis " (1968). Retrospective Theses and Dissertations. 3548. https://lib.dr.iastate.edu/rtd/3548 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. This dissertation has been 69-9859 microfihned exactly as received FULEIHAN, Joseph Salim, 1935- THE EMPIRICAL ESTIMATION OF SUBSTITUTION TERMS FROM DEMAND ANALYSIS. Iowa State University, Ph.D., 1968 Economics, theory University Microfilms, Inc., Ann Arbor, Michigan THE EMPIRICAL ESTIMATION OF SUBSTITUTION TERMS FROM DEMAND ANALYSIS Joseph Sallm Fulelhan A Dissertation Submitted to the Graduate Faculty In Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subjects Economics Approved : Signature was redacted for privacy. In Chaise of mjor Work Signature was redacted for privacy. Head of Major Department Signature was redacted for privacy. raduate College Iowa State University Ames, Iowa 1968 11 TABLE OP CONTENTS Page INTRODUCTION 1 THEORETICAL DISCUSSION 2 Classical Demand Theory 2 Substitution terms 5 Properties of substitution terms 11 Importance and implications 13 Modern Demand Theory 17 Geometric interpretation of substitution effects 22 EMIRICAL ESTIMATION OF SUBSTITUTION TERMS FROM UNCONSTRAINED DEMAND EQUATIONS 26 Procedure 26 Estimating structural equations 26 Estimating substitution terms 33 Results ^9 Beef and pork 50 Beef, pork, and broilers 62 Beef, pork, broilers, and mutton 6? Red meat end poultry meat 76 Eggs 80 Eggs and cheese 83 Eggs, cheese, and breakfast cereals 90 Margarine and butter 95 ill i^age Selected vegetables 100 Coffee 104 Pood, pleasure goods, durables and others 105 Summary of results 108 ESTIMATING DEMAND FUNCTIONS BY LINEAR PROGRAMMING 118 Statement of the Problem 118 The use of linear programming in regression analysis 119 Results 136 Beef and pork 137 Beef, pork, and broilers 139 Beef, pork, broilers, and mutton 1^5 Red meat and poultry meat 152 Eggs and cheese 155 Eggs, cheese, and cereals I60 ESTIMATING DEMAND FUNCTIONS BY QUADRATIC PROGRAMMING I70 Statement of the Problem 170 Results 174 Beef and pork 17^ Red meat and poultry meat 175 SUMMARY AND CONCLUSIONS 177 BIBLIOGRAPHY I8I ACKNOWLEDGEMENTS 186 APPENDIX 187 1 INTRODUCTION Economists have always been Interested in the degree of substltutability In consumption that exists between various commodities. They have even devised measures to estimate sub­ stltutability, such as the cross elasticity of demand. The concept of substitution effects (or substitution terms), Is fairly recent. It was developed by the Russian economist Eugene Slutsky (35). Other economists who followed him developed the concept further and expounded its mathemati­ cal properties. Today, substitution terms play a key role in modern demand theory. Strange as it may seem, no economist, with the recent exception of Barten (2), has attempted to verify or refute the theoretical characteristics attributed to substitution terms. The empirical testing of the Slutsky conditions is one of the objectives of this dissertation. After the more important properties have been tested and verified, methods of estimating demand functions of substitute commodities satisfying the Slutsky conditions will be discussed. Specifically, demand functions will be estimated by applying both linear and quad­ ratic programming techniques, subject to inequality constraints which preserve these Important properties of substitution terms which have been verified by empirical evidence. 2 THEORETICAL DISCUSSION Classical Demand Theory^ The classical theory of consumer behavior assumes that the consumer derives utility from the commodities he consumes. Utility, u, is a function of the amounts of various commodities Xi,,,.,Xn consumed. The utility function is assumed to be cardinal and to have first- and second-order partial deriva­ tives. If u represents the consumer's utility function, then (1.1) u = u (X^,,,,,X^), The consumer is faced with a fixed budget or money income M, and given commodity prices P^, or n (1.2) M = 2 P.X. 1 = 1 n. •1=1 ^ ^ The consumer is assumed to maximize u, subject to the Income or budget constraint. In other words, he maximizes the expression n (1.3) F = u (X^ X^) - X (M P^X^) where X is a Lagrange multiplier. First-order conditions for utility maximization require that the partial derivatives of the Lagrangean expression F ^A detailed discussion of demand theory Is found in Allen (1), Hicks (21), Kuenne (2?) and Samuelson (33). 3 with respect to the quantity of each of the n commoclltiers, unci with respect to k, be equal to zero. This yields (n + l) equations in (n + 1) variables (n X*s and X), The last equation guarantees that the budget constraint is met, (1-4) = izib - & *1 = 0 = - A ?! # 1 J" 9 = -§x; - t pR = 0 = "n - ^ Pn ^ p n where Ui = , the marginal utility of . Transposing the second term of the first n equations to the right-hand side, and dividing the 1^^ by the equation gives u/ P, u. P. where u^ and Uj are the familiar marginal utilities. Accord­ ing to classical economists, marginal utility, as well as total utility, may be measured cardinally. The result (1.5) states that in equilibrium, the ratio of the marginal utilities of any two commodities should equal their price ratio. 4 (1.5) can also be expressed as U.6, 2^ = = = ^1 ^2 ^n-1 ^n which are Marshall's results that in equilibrium the ratio of marginal utility to price should be equal for all commodities. These (n-l) equations and the budget constraint equation pro­ vide (n) equations for determining the quantities To guarantee that utility is a maximum, second-order con­ ditions must hold. In other words, not only should du = 0, but also d 2u < 0, or n (1.7) du = Z u, dX. = 0 i=l ^ ^ P n n (1.8) d^u = Z Z u, ,d X, d X, < 0 J=1 1=1 ^ J where u^^^ is the second partial derivative of u. It can be seen that u^^ is symmetrical with respect to both i and j, i.e., Uj^j = Uj^. (1,8) is a quadratic form subject to a linear side-relation. For (1.8) to be negative definite, the deter­ minants 5 (1.9) 0 u. u. u. u. u. Ur f » » » f ###U.n U U Ui 12 Ui U12 u^3 Ui 12''" '^in u ^2 ^21 22 ^22 ^2] ^2 "^21 "22'' "^2n ^3 "31 U32 ^n %1 ^n2""%n must be alternatively positive and negative. The last determinant shall be denoted by U, which is the determinant of the first and second partial derivatives of u, and shall be used to denote the cof actors of u^ and in U respectively. Substitution terms Demand functions can be derived from the analysis of utility maximization. More specifically, they are obtained by solving the first-order equilibrium conditions for the un­ knowns , The consumer's demand functions for commodities in terms of prices and income are given by the first-order equi­ librium conditions (1.4), or Ui = X Pj n Z P,X, = M. 1=1 ^ ^ 6 To get the change In demand when income varies, prices constant, we differentiate these equations with respect to income; to get the effect of a price change on demand, we differentiate with respect to price. First consider the effect of an increase in Income M, prices remaining unchanged. For the sake of clarity, write the first-order equilibrium conditions as (1.10) + PgXg +...+ P^X^ = M - + u^ = 0 - XPg + Ug = 0 Upon differentiating these equations with respect to M, we get (1.11) ()X^ ^n mr = ^ - ^1 Tm ^ ^1 ^ ""12 TiT + o>X ^^1 o" %2 aXn . - Fn ITw + ^nl "SIT + %2 TIT + %n TT" = ° By rearrangement, the first elements of the last n equa­ tions can be expressed as P^(- -j-^), ^2^' f • °O 9' p ^n^ " . Such a rearrangement enables us to express the matrix of coefficients in the equations in symmetric form, since ^ij' *ji' 7 ZPhlis system contains (n+l) equations In (n+l) variables* A ^4 X— "5^)f and. the n variables ^ ^ , The equations can be <) X. j \ :solved, for the variables "5"]^ and for the variable ( - -y-^) as a VbyMprod-uct , SoLving the equations by Cramer's rule gives (Cl .12,) 0 ,P. 1 ^2 !•„ 1-1 i+1 n pi 0 "ll "l2 »ln ^11 ^1,1-1 *1,1+1 *ln •^2 "l2 "22 "2n ^2 *12 *2,1-1 ° *2,1+1 *2n "in "2n l^n *ln *n,i-l ° *n,i+l *nn u Because 3j = ^ in (1,10), we can write (1,12) as (1.13) u.. d.X. u, u u n 0 h. hdL 1 u dM X \ '"'X \ • • • k i+1 ...X h .X *11 *l2'''*in \ *11''"*1,1-1 ° *l,l+l''*ln u. 0 A *21 *22'''*2n *21•••*2.1-1 *2,1+1^•*2n w# vO • • w# •• # 0 • # «• U* # u u.n n 1" *nl *n2 unn *nl'''*n,l-l *n,i+l***nn M Hence àx u "5M • 7 ° i "i or dX U, ,2 U (1-14) tm" = r • r = ^ û" Since the equilibrium conditions say nothing about the ax, sign of "57î~ could be positive or negative.