Consumer Demand—Theory

Abstract

The neoclassical theory of consumer behavior is the conceptual basis for the demand analysis framework formulated in this book. In this chapter, general comparative static results using the primal approach are derived in order to account for all factors affecting consumer behavior. The main result is the Fundamental Demand Matrix of Consumer Demand (Barten, 1964), giving comparative static results for Marshallian, Hicksian, and Frischian demand functions for changes in prices, income, and preference shift variables. Comparative static results are derived for both general quasi-concave preferences and strictly concave preferences. The main results are the Slutsky equation and general restrictions of consumer behavior. For Frischian demand functions, specific and general substitution effects are derived. Preference shift variables are shown to be proportional to price effects. Comparative static results are also derived using duality theory, and the theory of inverse demand functions is presented.

Problems

• 2.1 Using the Kuhn-Tucker Theorem, derive the Marshallian demand functions for the linear utility function, $u = v\left( {q_{1} , q_{2} } \right) = \alpha_{1} q_{1} + \alpha_{2} q_{2}$.

• 2.2 Verify the elasticity form of the general restrictions, Eqs. (2.17)–(2.18).

• 2.3 Show in the two good case that diminishing MRS does not imply diminishing MU, and vice versa. (Hint: note that the indifference curve in this case is $u_{0} \equiv v\left[ {q_{1} ,\phi \left( {q_{1} } \right)} \right], q_{2} = \phi \left( {q_{1} } \right).$ The MRS is $- \left. {\frac{{dq_{2} }}{{dq_{1} }}} \right|_{{u = u_{0} }} = \frac{{v_{1} }}{{v_{2} }}$. Diminishing MRS means $\frac{dMRS}{{dq_{1} }} < 0$.)

• 2.4 Prove that the matrix $\lambda U^{ij}$ is the specific substitution matrix obtained by income-compensating price changes holding the marginal utility of income constant.

• 2.5 Using the Envelope Theorem, derive: (a) Roy’s Identity, Eq. (2.34); (b) Shephard’s lemma, Eq. (2.37); and (c) Hotelling’s Theorem, Eq. (2.40).

• 2.6 Consider the Cobb-Douglas utility function, $u = v\left( \varvec{q} \right) = \mathop \prod \limits_{j = 1}^{n} q_{j}^{{\beta_{j} }}$.

  • (a) Derive Marshallian, Hicksian, and Frischian demand functions using the duality relationships in Eqs. (2.31)–(2.40). Specifically, verify that condition (2.41) holds.

  • (b) Show that the same results would be obtained if the utility function had either the form (i) $u\text{ = }v\left( \varvec{q} \right)\text{ = }\sum\nolimits_{{j\text{ = }1}}^{n} {\beta_{j} \log q_{j} }$ or (ii) $u\text{ = }v\left( \varvec{q} \right)\text{ = }\prod\nolimits_{{j\text{ = }1}}^{n} {q_{j}^{{\beta_{j} \text{/}\sum\nolimits_{{k\text{ = }1}}^{n} {\beta_{j} } }} }$. Why is it the case that the resulting demand functions are the same with these two utility functions?

• 2.7 Show that the elasticity form of the Antonelli equation is $f_{ij} = f_{ij}^{*} + f_{i} w_{j}$, where $f_{ij}$ is the elasticity of the ith normalized price with respect to the jth quantity (flexibility of ith price with respect to jth quantity), $f_{ij}^{*}$ is the compensated price flexibility, $f_{i}$ is the scale flexibility (i.e.,, relative change in price to an equal proportional change in all quantities), and $w_{j}$ is the budget share as defined before. Compare and contrast this form with the Slutsky equation, $e_{ij} = e_{ij}^{*} - e_{i} w_{j} .$