A general expenditure system for estimation of consumer demand functions
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A general expenditure system for estimation of consumer demand functions Simona Bigerna1 · Carlo Andrea Bollino1 · Maria Chiara D’Errico1 Received: 8 January 2018 / Accepted: 17 April 2019 © Springer Nature Switzerland AG 2019
Abstract The class of flexible functional forms for the utility and cost function has been characterized by the pioneering work of Gorman (Some Engel curves. In: Deaton A (ed) Essays in the theory and measurement of consumer behaviour. Cambridge Univ. Press, Cambridge, 1981), known as the Gorman polar form. Despite several decades have elapsed, the economic literature has not found the most general functional form that satisfies Gorman’s theorem. This note provides a new general theoretical and parametric formulation of demand functions, labeled general expenditure system (GES), satisfying the Gorman requirement that the Engel curve cannot exceed a polynomial of third degree in expenditure. Estimates show that the GES is a significant generalization of previous popular flexible functions. Keywords Integrable demand functions · General expenditure system · Gorman polar form JEL Classification D01 · D11 · C30
1 Introduction The theory of demand states the theoretical restrictions for a system of equations, which have to be derived from rational consumer behavior. The literature has addressed several issues relative to the utility function specification, such as functional flexibility, functional separability and Engel function curvature. In reality, these have been largely equivalent ways to define and discuss different mathematical * Carlo Andrea Bollino [email protected] Simona Bigerna [email protected] Maria Chiara D’Errico [email protected] 1
Department of Economics, University of Perugia, Via Pascoli 20, 06123 Perugia, Italy
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Economia Politica
formulations to represent the fundamental problem in economic theory, namely the rational choice of the optimal vector of consumption quantities, given the vector of prices, the expenditure and the preferences (Barnett and Serletis 2008). Operationally, there are four equivalent ways to represent consumer choice according to duality theory, namely, utility maximization, cost minimization, minimization of the distance function and derivation of demand functions from the indirect utility function (Deaton and Muellbauer 1980; Blackorby et al. 1978; Chavas and Baggio 2010). In this paper, we adopt the last method, assuming that consumers’ preferences can be represented with the indirect utility function and use Roy’s identity to derive the Marshallian demand functions. In this framework, the necessary conditions to parametrize a mathematical function, consistent with the restrictions of the economic theory, are set in the Gorman’s theorem (1961, 1981). This theorem states the so-called Gorman polar form in the literature. However, despite several decades have elapsed, the economic literature has not found a unified general functional form that satisfies Gorman’s theorem. The aim of this pape
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