In Section B of the text, we developed a model of tastes for diversified goods — and then applied a particular functional form for such tastes to derive results, some of which we suggested hold for more general cases.
B: We first introduced a general utility function representing such tastes in equation (26.33) before working with a version that embeds the sub-utility for y goods into a Cobb-Douglas functional form in equation (26.34). Consider now the more general version from equation (26.33).
(a) Begin by substituting the budget constraint into the utility function for the x term (as we did in the Cobb-Douglas case in the text).
(b) Derive the first order condition that differentiates utility with respect to pi.
(c) Assume that the number of firms is sufficiently large such that terms in which yi plays only a small role can be approximated as constant. Then use your first order condition from (b) to derive an approximate demand function that is just a function of pi and a constant. What is the price elasticity of demand of this (approximate) demand function?
(d) Set up firm is profit maximization problem given the demand function you have derived.
Then solve for the price pi that the firm will charge.
(e) True or False: The equilibrium price p∗ = −c/ρ we derived in the text for the Cobb-Douglas case does not depend on the Cobb-Douglas specification.
(f) Recalling our Chapter 15 discussion of treating groups of consumers as if they behaved like a “representative consumer”, what form for the utility function might you assume if you were concerned that the Cobb-Douglas version we used in the text might technically not satisfy the conditions for a representative consumer? Would the implied equilibrium price differ from the Cobb-Douglas case?