Question 1:

A consumer consumes goods x1 and x2 and has a utility function u(x1; x2) =(x^4).(x^6) . Prices of goods x1 and x2 are p1 and p2, respectively.

(a) Derive demand functions for goods x1 and x2 for this consumer.

(b) Calculate own price elasticities , income elasticities, and cross-price elasticities of demand for the two goods.

(c) There are 1,000 individuals on the market, and they all have this same utility function and income of $1; 000. Derive the market demand function for goods x1 and x2.

Question 2:

Production function for good x1 is f1(k; l) = (k^0.3)(l^0.7), where k is capital and l is labor, whose prices are r = 0.5 and w = 1, respectively.

(a) Assuming that capital is the fixed input, derive the short-run cost functions: total, marginal, average, and average variable costs. What are the fixed costs?

(b) In the short-run, there are small and large firms in the market. The former have capital ks = 100 and latter kl = 1000. Derive and fully specify firm supply curves for both types of firms.

(c) Assuming that there are 100 firms, 10 of which are large with k = 1000, and 90 are small with k = 100, derive the market supply.

(d) Derive the long-run cost functions: total, marginal and average.

(e) What is the firm supply and what is the market supply in the long-run?

Question 3:

In this part, use the results for market demand for short-run and long run market supply of good x1 obtained in parts one and two. When a change (e.g. income or taxes) is introduced in one part, assume they hold in the subsequent parts of the question.

a. Find the short-run equilibrium price and quantity of x1 and plot the demand and supply curves. What are the profits of small and large firms?

b. Suppose that wages and income of each individual increase by 10%. Plot the new demand and supply curves and find the new equilibrium.

c. Suppose that the government imposes a 10% tax on consumption of good x1. Plot the demand and supply curves and find the new equilibrium.

d. Find the price and quantity of x1 in the long-run equilibrium.

e. Summarize your findings. How is the long-run different from the short-run? How does the increase in prices and wages affect the market? How does the tax affect the market?