The Income-Demand Curve: Implicit Function and Data Analysis Methods
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The Income‑Demand Curve: Implicit Function and Data Analysis Methods Shuji Takahashi1 Accepted: 9 September 2020 © The Indian Econometric Society 2020
Abstract By using implicit function, utility and demand functions with inferior goods are developed. It has an advantage in tractability and extendability. But, the systematic data fitting method is not developed yet. This article presents a discussion of this fitting method and other matters: (1) a utility and demand function suitable for data fitting; (2) a systematic method for the data fitting; and (3) a 48-good utility function that generated income demand curves that are well fitted to 48-good data of the Japanese Household Survey 2018. Keywords Inferior goods · Income-demand curve · Utility function JEL Classification C4 · D01 · D11 · D2
Introduction Many computable utility functions have been developed for dealing with inferior goods. A simple one is that reported by Epstein and Spiegel (2000). Moffatt (2002) reports an approach using an indifference curve representing the Giffen case. An x–y good indifference curve function y = f (x, u) is explained by Sasaki (1992). Takahashi (2019a, b) describes employment of the implicit function. Using the implicit function and its optimization, it realizes both tractability and extendability. In this article, based on Takahashi (2019a, b), we discuss the systematic data fitting method. The trial and error method, with testing of various parameters repeatedly, often produces demand curves well fitted to the data. Nevertheless, such a method is unrealistic for a multi-good model because minuscule changes in a parameter affect many other demand variables in complicated ways. Therefore, a systematic method is desired to untangle these complexities. Although some compromises * Shuji Takahashi [email protected] 1
Institute of Economics and Management, Kanto-Gakuin University, Kanazawa ku, Mutsuura Higashi 1‑50‑1, Yokohama, Japan
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Vol.:(0123456789)
Journal of Quantitative Economics
by which some parameters are not derived from the data and so on, the estimated utility function generates an income–demand curve that is well fitted to the data. The remainder of this paper comprises the following sections. Section 2 presents formulation of the utility function and derivation of the compensation demand function. The fitting technique and a two-good example are presented in Sect. 3. A 48-good utility function fitted to Japanese Household Survey Data, 2018 is presented in Sect. 4. The conclusion is given in Sect. 5.
The utility function and its demand function Letting u and xk , ( k = 1, … , n ) respectively denote the utility level and the quantity of the k-th good, then the utility function used here is a generalized version of Takahashi (2019b) as
u = Σnk=1 gk (u)xk ak ,
(1)
where 0 < ak < 1 . The gk (u) is a positive and non-increasing function in u , that is gk (u) > 0 and g�k (u) ≤ 0.1,2 The budget constraint is m = Σnk=1 pk xk , where m and pk respectively denote the income and )the price of the k-th g
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