Question

Suppose a competitive firm has a short-run cost function: C(q) = 100 + 10q − q^2 + q^3 , where q is the quantity of output.

1. Is this a short-run or a long-run cost function? Explain.

2. Find the firm’s marginal cost function: MC(q).

3. Find the firm’s average variable cost function: AVC(q).

4. Find the output quantity that the firm AVC at the minimum. Does the MC increasing or decreasing before the quantity. And does the MC increasing or decreasing after the quantity?

5. Suppose the firm maximizes profit, and the market price is $40. How much would the firm produces? How about if the market price is $20.

6. Plot MC, AVC, and the firm supply curve. Label firm’s profit, revenue and cost when the market price is at $40.

Answer 1

C(q) = 100 + 10q − q^2 + q^3, This cost function is of the short run as there is a constant value in the function of 100. The fixed or constant value indicates that there are some fixed costs associated with fixed variables. The variables are fixed only in the short run and in the long run all factors are varied.

Marginal cost is the first derivative of the cost function. That is MC = dC/dQ = 10 - 2q + 3q^2

Variable Cost(VC) is that part of the cost function which is dependent on output. So

VC =  10q − q^2 + q^3, AVC = VC/Q

AVC = 10 - q + q^2

for max or min AVC we put dAVC/dQ= 0

dAVC/dQ = -1 + 2q = 0

2q = 1 or q = 1/2

we have MC = dC/dQ = 10 - 2q + 3q^2

MC at q = 1/2, MC = 10-2*1/2 + 3*1/4 = 39/4 = 9.75

For MC decreasing or increasing we take derivative of MC

dMC/dQ = -2 + 6q = 0, at q=1/2, dMC/dQ = -2+6*1/2 = 1 which means that MC function is decreasing at q = 1/2

6q = 2, q = 1/3,(MC is minimum at q=1/3)

For the firm profit max condition is P=MC

for P=40

40 = 10 - 2q + 3q^2

3q^2 - 2q - 30 = 0

q = 3.51

for P = 20

20 = 10 - 2q + 3q^2

3q^2 - 2q - 10 = 0

q = 2.19