Question

Consider a production function of two inputs, labor and capital, given by Q = (√L + √K)2. Let w = 2 and r = 1. The marginal products associated with this production function are as follows:MPL=(√L + √K)L-1/2MPK=(√L + √K)K-1/2

a) Suppose the firm is required to produce Q units of output. Show how the cost-minimizing quantity of labor depends on the quantity Q. Show how the cost-minimizing quantity of capital depends on the quantity Q.

b) Find the equation of the firm’s long-run total cost curve.

c) Find the equation of the firm’s long-run average cost curve.

d) Find the solution to the firm’s short-run cost minimization problem when capital is fixed at a quantity of 9 units (i.e., K = 9).

e) Find the short-run total cost curve, and graph it along with the long-run total cost curve.

f ) Find the associated short-run average cost curve

Answer 1

a) A firm will minimize its cost when the ratio of marginal product of labor and marginal product of capital is equal to ratio of wage and rent that is factor prices. So, the firm will minimize its cost at the point where,

marginal product of labor / marginal product of capital = w / r

[(L + K) / L] / [(L + K) / K] = 2 / 1

K / L = 2 / 1

K = 2L

K = 4L

So, this means that the cost is minimized at the point where K = 4L. Putting this value in the production function above,

Q = (L + K)²

Q = (L + 2L)²

Q = (3L)²

Q = 9L

L = Q / 9 and this shows that cost minimizing quantity of labor depends on Q.

Putting the value of L in the equation of capital or K,

K = 4L

K = 4 X (Q / 9)

K = 4Q / 9 and this shows that the cost minimizing quantity of capital depends on Q.

b) Total cost = (wage x quantity of labor) + (rent x quantity of capital)

= 2L + K is the answer.

c) Average cost is calculated by dividing the total cost function by the quantity produced as it is the per unit cost of producing the output. So, Average cost = Total cost / Q

= (2L + K) / Q is the answer.

d) Since K = 3, putting this value in the production function given in the question,

Q = (L + K)²

Q = (L + 9)²

Q = L + 3

L = Q - 3

L = (Q - 3)²

Total cost = 2L + K

Putting the value of K and L in the cost equation,

Total cost = 2(Q - 3)² + 9

In order to minimize this cost, we differentiate the cost function with respect to Q and set it equal to 0 to get the cost minimizing quantity. So, differentiating this function with respect to Q and setting it equal to 0,

[2 x 2 x (Q - 3)] / 2Q = 0

4(Q - 3) = 0

4Q - 12 = 0

Q = 12 / 4 = 3

Q = 3² = 9 is the answer.