Consumer Demand—Empirical Analysis I

Abstract

This chapter discusses empirical implementation of the systems of demand functions approach presented in the last two chapters. I address questions of functional form specification and statistical specification. The functional form question has been a long-standing issue in the area of consumer demand analysis. There are scores of functional forms and a comprehensive review on the possibilities, strengths, and weaknesses is not possible in this book. A brief review is given of the most significant functional forms in the literature. The approaches to functional form specification are based on: (i) direct derivation from the utility function, (ii) directly specified demand functions, (iii) derivation using duality theory applied to locally flexible functional forms, and (iv) derivation from globally flexible functional forms. The focus is on the two well-known workhorses in econometric analysis: the Rotterdam model (RM), and the almost ideal demand system (AIDS). Additional discussion centers on extensions of these two models to more general models which are currently finding popularity. These models include the CBS and NBER differential demand systems, and the EASI demand system.

Problems

• 4.1 Show that in order for the double-log demand system to satisfy the general restrictions of consumer behavior all own-price elasticities must be −1, all cross-price elasticities must be zero, and all income elasticities must be 1. (Hint: Consider the expression for $\log w_{i} = \log q_{i} + \log p_{i} - \log y.$ The Engel aggregation requires all income elasticities be constant with this functional form, implying all expenditure shares must be constant. Substitute for the double-log demand function in the above expression and differentiate with respect to prices and income.)

• 4.2 Show that the LES demand functions, Eq. (4.2), can be derived from the Stone-Geary Utility function, $u = \mathop \sum \limits_{j = 1}^{n} \beta_{j} { \log }(q_{j} - \gamma_{j} )$. Explain why the demand system does not admit inferior goods. Show that all uncompensated cross-price effects are complements but all compensated price effects are net substitutes.

• 4.2 Show that the LES demand functions, Eq. (4.2), can be derived from the Stone-Geary Utility function, $u = \mathop \sum \limits_{j = 1}^{n} \beta_{j} { \log }(q_{j} - \gamma_{j} )$. Explain why the demand system does not admit inferior goods. Show that all uncompensated cross-price effects are complements but all compensated price effects are net substitutes.

• 4.3 In empirical work, it is often convenient to work with share equations, particularly when one is interested in estimating substitution between closely related goods. Show that such a model, $w_{i} = \alpha_{i} + \mathop \sum \limits_{j = 1}^{n} \gamma_{ij} log p_{j}$, can be rationalized as a consumer demand function when preferences associated with the indirect translog function are homothetic. Also show that this model can be derived as a special case of the AIDS model.

• 4.4 Show that for the parametric specification of the absolute price version of the RM that Eq. (3.30) used to test for weak separability is exactly $\frac{{c_{il} }}{{c_{jm} }} = \frac{{b_{i} b_{l} }}{{b_{j} b_{m} }} \forall i,j \in G;l,m \in H, G \ne H$. Show how specification of the RM would be changed when these restrictions are imposed. Indicate how the RM would be estimated with these restrictions imposed and how the test for separability would be conducted.

• 4.5 Derive the Marshallian price elasticities and income elasticities from the Indirect Translog Model, Eq. (4.3). Compare with the elasticities for the RM and AIDS.

• 4.6 Derive the expenditure and price elasticities (both compensated and uncompensated) for the CBS and NBER models and compare with the elasticities derived from the RM and AIDS.