Problem 2 (25 pts) (Explain your answers!) In a competitive market there are m c

ID: 1099190 • Letter: P

Question

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

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 industry's 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)?

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 + 4x2.

(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.

output. Then show that if you solve the problem max

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?

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