Question

Consider the city of Fruitville. There are m inhabitants of Fruitville and they wish to consume only two kinds of goods, bananas (b) and tangerines (t). The preference ordering for each consumer can be represented by the utility function U(t, b) = t1/2b1/2. Each consumer has income M. The citizens of Fruitville are bureaucrats— hence their limited diet! The price of tangerines is pt, that of bananas, pb. The marginal utility of bananas is MUb = (1/2)b-1/2t1/2 and for tangerines MUt = (1/2)b1/2t-1/2. The production function for tangerines is t = AK1/2L1/2, where t is the quantity of tangerines produced, K is the amount of land used, L is the amount of labour used and A is a productivity parameter. The marginal product of land is MPK = (1/2)AL1/2K-1/2 and the marginal product of labour is MPL = (1/2)AL-1/2K1/2. Let w represent the wage rate and r the rental rate of land. There are n firms currently producing tangerines. Each currently uses K0 units of land. In the short-run it is prohibitively expensive to change the amount of land utilized. Assume perfect competition.

a) Find the demand curves for tangerines and bananas for a representative consumer.

b) Find the market demand curve for tangerines.

c) Find the short-run cost function for tangerines.

d) Find the short-run marginal cost function for tangerines.

e) Find the supply function of a representative firm producing tangerines.

f) Find the market supply function for tangerines.

g) Find expressions for the equilibrium price and quantity of tangerines.

Suppose that the parameters of the model have the following values: m=100, A=1, M=100, w=4, n=20, K0 =5, and r=1.

h) What is the equilibrium price and equilibrium quantity of tangerines? How much does each firm supply? What are the profits of each firm? How many tangerines does each consumer purchase?

i) With care and precision, graph the equilibrium found in (h) using two diagrams. In one diagram illustrate the market equilibrium. In the second diagram show the equilibrium position of a representative firm. On this second diagram make sure you indicate the profit maximizing output for the firm and the profit earned.

j) Is the industry in short-run or long-run equilibrium or both in part (h)? WHY? If the industry is not in long-run equilibrium, explain the adjustment process that will occur.

Answer 1

j)

As each firm has identical cost curves and there is free entry and exit into the market, the positive economic profit attracts new firms to the market. The entry of new firms increases market supply and given demand causes the price to fall. The price falls until it equals to minimum ATC. By the same argument, the negative economic profit prompted many firms to exit the market causing price to rise. The equilibrium occurs where price equals to ATC and MC. This happens at the minimum point of ATC curve.

In this case each firm earn above normal profit. Then there will be entry in the market. This will increase aggregate supply and the price would fall to minimum of ATC. Each firm will produce less output and the market quantity would rise due to increase in the number of firms.