Suppose that there are 1,000 consumers in the economy, each one with $10 of income. There are only two goods in the economy, X and Y. The price of good X is P and the price of good Y is $1. One group of 500 consumers (group A) like X more than Y and the other 500 consumers (group B) like Y more than X. The utilities of each type of consumer are
UA(X, Y) = 2X + Y
UB (X, Y) = X + 2Y
Firms can each produce good X with production function F (L, K) = 10(√L + √K)
The price of capital is R = $4 and the price of labor is W = $1. There is also an avoidable fixed cost of $20.
a. Derive the demand functions for good X for consumers of type A and B. Call these XA (P) and XB (P) respectively.
b. Derive the market demand function for good X, X(P), and sketch a graph of it.
c. Derive the cost function, average cost function, and efficient scale of a firm.
d. Supposing that the firms operate in a perfectly competitive market with free entry, determine the long-run equilibrium price and amount bought and sold, and the number of firms operating in the market.
e. Suppose that the price of capital increases to R = $80. If in the short run, capital is a fixed input and the $20 fixed cost is sunk, what is the new short-run competitive equilibrium price?
f. What is the new long-run competitive equilibrium price?