Question

The production function of a competitive firm is given by f(x1, x2) = x1^1/3 (x2 − 1)^1/3 The prices of inputs are w1 = w2 = 1. (a) Compute the firm’s cost function c(y). (b) Compute the marginal and the average cost functions of the firm. (c) Compute the firm’s supply S(p). What is the smallest price at which the firm will produce? 2 (d) Suppose that in the short run, factor 2 is fixed at ¯x2 = 28. Compute the short run total, average and marginal cost functions and the firm’s supply. What is the smallest price at which the firm will produce? (e) Suppose that the market demand is given by D(p) = 240/p. Compute, for the long run, the equilibrium: price, output of each firm, and number of firms. (The output of a firm does not have to be an integer number, but the number of firms must be an integer.)

Answer 1

The production function is given as . The cost of production is or .

(a) The MRTS would be as or or . The optimal input combination would be where or or .

Putting this in the production function, we have or or or , and since , we have or . These are the conditional input demand. Putting these in the cost of production, we have the cost function as or or .

(b) The marginal cost would be as or or . The average cost would be as ro or .

(c) The VC would be as , and the AVC would be . The supply curve for the firm would be where MC=p, sucht that MC>AVC. As can be verified, since , we have MC>AVC for all y>0. The supply function would be as or . The minimum price at which the firm can produce would be the minimum of AVC, which is where MC=AVC. We have and , and MC=AVC would only be where y=0. At y=0, we have the minimum of AVC where . Hence, the firm would produce for p>0.

(d) For x2 fixed at 28 in the short run, we have the production function as or or , and we have or as the short run conditional input demand. The short run total cost function would be as or .

The marginal cost would be as , and the average cost would be as . The short run supply would be where MC=p, provided MC>AVC. We have AVC as , and MC>AVC for all y>0. Hence, the short run supply would be as or or . The smallest price would be where MC=AVC, and since and , we have MC=AVC where y=0. Hence, the minimum AVC would be , meaning that for p>0, the firm produces.

(e) In the long run, the firms produces at minimum of AC, which would be where or or or or or , and the minimum of AC would hence be . Hence, the long run equilibrium price would be p=3. The output of each firm would be y=1.

The long run quantity demanded would be units. For each firm produces y units, we have the total quantity supplied would be , and since quantity supplied would be equal to quantity demanded, we have or , ie there would be 80 firms.