The characterization of demand and excess demand functions, revisited
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The characterization of demand and excess demand functions, revisited Marwan Aloqeili1 Received: 12 March 2019 / Accepted: 11 August 2020 © Associazione per la Matematica Applicata alle Scienze Economiche e Sociali (AMASES) 2020
Abstract In this note, the characterization problem of individual demand and excess demand functions is revisited. It is assumed that the individual’s income is price dependent. When the income function is homogeneous of degree one, we show that similar conditions characterize both demand and excess demand functions. Keywords Individual demand · Excess demand · Homogeneous functions · Slutsky matrix JEL Classification C02 · D11
1 Introduction The integration problem in consumer theory has attracted the attention of a large number of scholars since the earlier work of Slutsky (1915). The Marshallian demand function of a certain consumer is the maximizer of his utility function under the budget constraint. In this standard setting, the Marshallian demand function is characterized by: (i) symmetry and negative semi-definiteness of Slutsky matrix, (ii) budget constraint, (iii) zero-homogeneity. In a recent work, the author, Aloqeili (2014), extended the classical individual model by considering a price-dependent income function. Conditions that characterize individual’s demand functions when individual’s income is price dependent were given. Namely, symmetry and negative semidefiniteness of an extended Slutsky matrix, together with Walras law, characterize individual demands. In the extended model, the consumer maximizes utility u(x) under the budget constraint p x = w( p) where p is the price vector, w( p) is the income function, and prime denotes transposition. The problem needs special treatment when the individual’s income function
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Marwan Aloqeili [email protected] Department of Mathematics, Birzeit University, P.O. Box 14, Birzeit, Palestine
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M. Aloqeili
is homogeneous of degree one (henceforth 1−homogeneous). In Aloqeili (2014), the author considered the nonhomogeneous case in which the demand function is not 0−homogeneous in prices p. Only the mathematical integration problem was solved for the homogeneous case in which the demand function x( p) is 0−homogeneous and the Lagrange multiplier λ( p) is −1−homogeneous. A substitution matrix with undetermined vector v( p) that satisfies the condition p v( p) = 1 has been given. In the standard setting, the vector v( p) is the negative of the derivative of demand function with respect to income. More details in terms of substitution and income effect are given below. The necessary and sufficient condition for mathematical integration is the symmetry of this matrix. In this paper, we concentrate on the homogeneous case. The vector v( p) can be fully determined if the consumer’s utility function is homogeneous. We prove also the negative semi-definiteness of the substitution matrix. To achieve this purpose, we perform (fundamental) matrix analysis of demand systems as in Phlip (1974). On the other hand, we give conditions t
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