Suppose that U(C, y) = C + v(y), where v(y) = y1-α/(1- α) with α ∈ (0, 1). Suppose also that new varieties can be introduced at the fixed cost μ.
(a) Consider the allocation determined by a social planner who also controls prices. Characterize the number of varieties that a social planner would choose to maximize the utility of the representative household in this case.
(b) Suppose that prices are given by (12.15). Characterize the number of varieties that the social planner would choose to maximize the utility of the representative household in this case.
(c) Characterize the equilibrium number of varieties (at which all monopolistically competitive variety producers make zero profits) and compare this number with the answers to the previous two parts. Explain the sources of differences between the equilibrium and the social planner's solution in each case.