Consider the production function

Consider the production function Q = LK. Suppose that the price of labor equals w and the price of capital equals r. Derive expressions for the input demand curves.

 

From the tangency condition, we get

 

 

Substituting into the production function yields

 

 

This represents the input demand curve for .  Since

 

 

we have

 

 

 

This represents the input demand curve for .

 

 

5. A cost-minimizing firm’s production function is given by Q = LK. The price of labor services is w and the price of capital services is r. Suppose you know that when w = $4 and r = $2, the firm’s total cost is $160. You are also told that when input prices change such that the wage rate is 8 times the rental rate, the firm adjusts its input combination but leaves total output unchanged. What would the cost-minimizing input combination be after the price changes?

 

Using the tangency condition, with the original input prices: . So, K = 2L. Also, using the information on total costs,  Combining these two equations, we get (L, K) = (20, 40). Therefore the firm produces 20*40 = 800 units of output.

After the prices change, even though we don’t know the numerical values of the input prices, we can still answer the question using the fact that we’re told w = 8r. The tangency condition implies that  so K = 8L. Also, we have . This implies that the optimal input combination is (L, K) = (10, 80).