# How much does each consumer consume?

In a competitive market there are m consumers and n rms. Show more Problem 2 (25 pts) (Explain your answers!) In a competitive market there are m consumers and n rms. Consumers and rms take prices as given. Each consumer has income $M and a utility function u(q; z) = z + 200q ?? q2 20 where q (with price p) is the output of the industry and z (with price 1) is dollars spent on all other goods. All n rms are identical with cost function c(q) = q2 10 + 20q + F. Suppose xed costs are F = 4; 000. (a) Find the demand function of each consumer for the industrys output q. What is the market demand function Qd(p)? (b) Find the supply function of each rm. What is the market supply function Qs(p)? (c) Suppose m = 1000 and n = 20: Draw a graph of the market supply and demand and nd the equilibrium price and quantity on this market. How much does each consumer consume? How much does each rm produce? Do rms make a prot or loss in equilibrium? (d) Given your answers in (c) what do you expect to happen in the future in this market if rms can freely enter and exit? With free entry/exit what would be the long-run industry equilibrium price? Quantity produced per rm? Number of rms? What would happen if m = 100 instead? Problem 3 (20 pts) (Explain your answers!) A rm has two variable inputs and a production function f(x1; x2) = p2x1 + 42. (a) Draw the isoquants corresponding to output of 3 and output of 4. (b) Suppose the output price is p = 4 the price of input 1 is w1 = 2 and the price of input 2 is w2 = 3. Find the prot-maximizing input and output quantities that this rm would use. Hint: be careful and look at the isoquants in (a) before you try to take derivatives. (c) At the prices given in (b) nd the rms cost function c(y) for producing y units of output. Then show that if you solve the problem max y py ?? c(y) you obtain the same solution for the optimal output y as what you obtained in part (b). Why is this the case? Show less