Question

Suppose Company A’s production function for good x is q = 2?^(1/2)?^(1/2), where K and L refer to the inputs for capital and labor respectively. The capital rental rate (r) is $2, and the hourly wage rate (w) is $8. In short run, the amount of capital is fixed at K = 25.

(a) Calculate Company A’s short-run variable cost in term of q.
(b) Derive Company A’s short-run total cost C(q).
(c) Derive Company A’s short-run average fixed cost AFC(q), short-run average variable cost AVC(q), and short-run average total cost ATC(q).
(d) Derive Company A’s short-run marginal cost MC(q)

Answer 1

q = 2K^1/2L^1/2

q = 2KL

It is given in the question that K = 25 and wage rate = w = 8 and rental rate = r = 2. So, putting the value of K = 25 in the production function above,

q = 225L

q = 2x 5 x L

q = 10L

L = q²/100

a) Since, capital is fixed in the short run and it is only the labor which is changing,

the variable cost of the company = wage x labor = 8 x (q²/100) = 2q²/25 is the answer.

b) Total Cost = wL + rK that is [(wage x units of labor used) + (rent x units of capital used)]

= 2q²/25 + 2(25)

= 2q²/25 + 50 is the answer.

c) Fixed cost is the cost which does not change with the change in the level of output that is it is not dependent on the output. Average fixed cost is calculated by dividing the total fixed cost by q,

Total fixed cost = 50

So, average fixed cost = 50/q is the answer.

Total variable cost depends on the level of output and changes with the change in the level of output. Average variable cost is calculated by dividing the total variable cost by q,

Average variable cost = 2q²/25q = 2q/25 is the answer.

Average total cost = average fixed cost + average variable cost

So, average total cost = 50/q + 2q/25 is the answer.

d) Short run marginal cost is calculated by differentiating the short run total cost function with respect to q.

So, short run marginal cost = 4q/25 is the answer.