Question

A firm produces output according to the following function: q = f (L, K) = L^1/2 K^1/4 . The

cost of labor is $8 per hour and the rental cost of capital is $2 per hour.

a. Determine the returns to scale for this function.

b. Suppose the firm wishes to produce at cost $96. How much capital and how

much labor does the firm employ?

c. Derive the short-run cost function with optimal amount of K from part b.

d. Suppose that there are 80 firms in a market, each with the cost function from part

c. Derive the short-run market supply curve.

e. Suppose the market demand is QD = 1600 - 30p Find the equilibrium market

quantity and price.

f. How much output will each firm produce? How much profit is each firm

making?

Answer 1

(a) Increasing the input by a times, we have or or . Hence, increasing the input by a times increased the output by times. Hence, there is decreasing returns to scale.

(b) The problem here is to maximize the production for given cost. The optimal input combination would be where or or or or . Putting it in the cost function or or , and since , we have . These are the required labor and capital.

(c) In the short run, K would be fixed at 16. Hence, the cost of production would be or . Also, the production function would be or or or . Putting it in the cost of production, we have or is the required short run cost function.

(d) The individual firm supply would be where price is equal to marginal cost, ie , ie or or . If there are 80 firms, then the market supply would be as or or .

(e) The equilibrium price would be where the quantity demanded is equal to quantity supplied, ie or or , and at this price, equilibrium quantity demanded would be or units.

(f) Total quantity supplied is 640 units. There are 80 firms. Each firm would supply 640/80 or 8 units. This can be confirmed with the individual supply curve or units.