Question

An industry has 1000 firms, each with the production function f(x1; x2 ) x1^.5 x2^.5. The
price of factor 1 is 1 and the price of factor 2 is 1. In the long run, both factors are variable, but in
the short run, each firm is stuck with using 100 units of factor 2.The long run industry supply curve:
Can somebody explain how to solve?

Answer 1

Given that we have an industry where there are 1000 firms, each with the production function Q = x1^.5 x2^.5.

The price of factor 1, x1 = 1 and the price of factor 2, x2 = 1.

In the long run, both factors are variable, but in the short run, each firm is stuck with using 100 units of factor 2.

The long run supply curve of the firm shows the level of output where it produces a certain level of output that is determined by the minimum of AC at which we have LMC = LAC.

From the production function we have MRTS = Ratio of two marginal products

MRTS = 0.5*(x2/x1)^0.5/0.5*(x1/x2)^0.5 = x2/x1. At the optimum mix, firm has MRTS = price ratio

x2/x1 = 1/1

x2 = x1

Use this fact to find the demand functions for x1 and x2

Q = x1^.5 x2^.5.

Q = x1^0.5x1^0.5

x1* = Q

x2* = Q

Hence the demand functions are x1* = Q and x2* = Q

With the budget equation we have C = x1*1 + x2*1, the cost function is C = Q + Q = 2Q. This is the long run total cost function. Marginal cost = Average cost = 2. Since MC = AC = 2 is a constant line, the firm in the long run would supply any quantity given to it.

Hence the industry supply curve is horizontal line at MC = P = $2.