onsider the Kr an 1 model introduced in Lecture 7. Suppose that there are L identical consumers in the economy, each of whom have the following utility function: where 93 is the consumption of a homogeneous good and g, is the consumption of the 37th variety of a differentiated good. Note that only the differentiated good has different varieties. Here n > 1 is the total number of differentiated varieties available while a > U is a constant. (a) [2 marks] Use the tggency condition between the homogeneous good and the ith variety of the differentiated good to show that the inverse demand for the latter can be written as Pi=Gqsi where p, is the price of the ith variety. You can assume throughout this problem that the price of the homogeneous good, pg, is equal to one. (b) [4 marks] As in the Krugman (1979) model, each variety of the di'erentiated good is produced by a single rm in a monopolistically competitive environment. Each rm must pay a fixed cost, F, to produce a variety while its variable costs are (co - k;)I; where r; is the firm's output. Market clearing implies that I; = Lqi. co > 0 is an initial, constant marginal cost while k, 0 is a constant. This information implies that a firm's problem is to choose the r; and k; that maximizes its profits. Use this profit maximization problem to show that the optimal k; must satisfy the following condition: La- (co - k) = 1+ 8ki 2 (1) (c) [1 mark] The left-hand-side of equation (1) is the marginal benefit (MB) of innovation effort while the right-hand-side is the marginal cost (MC). Plot the marginal benefit and marginal cost functions in a diagram with k; on the horizontal axis. You can assume that (a - Co)L > 2, which ensures that the vertical intercept of the MB line is greater and that 26 > L, which ensures that the MC line is steeper. (d) [5 marks] Use your answers above to solve for the optimal innovation effort, k; . How does k; vary with L? Illustrate the effect of an increase in L on k; using your diagram in part (c) and provide economic intuition for this effect. (e) [3 marks] Use the inverse demand function in part (a) to write the equilibrium price of variety i, p;, as a function of a, Co, and k;. How does an increase in L affect this price