Question

9.     a. Suppose that a firm’s production function is q=9x^1/2 in the short run, where there are fixed costs of $1000, and x is the variable input whose cost is $4000 per unit. What is the total cost of producing a level of output q? In other words, identify the total cost function C(q).

b.   Write down the equation for the supply curve.

c.   If price is $1000, how many units will the firm produce? What is the level of profit? Illustrate your answer on a cost curve graph.

NEED GRAPH FOR C

Answer 1

9.

a.

The firm’s production function is q=9x^1/2 in the short run

That means, x^(1/2) = q/9

x = (q/9)^2

Fixed cost = $1000

Variable Cost = 4000x

Thus, the total cost function is:

Total Cost = Fixed Cost + Variable Cost

Total Cost = TC= 1000 + 4000*(q/9)^2

Thus, the total cost of producing the q level of output = C(q) = 1000 + (4000/81)q^2

b.

The supply curve is the marginal cost curve portion above the minimum average variable cost

Marginal Cost = MC = dTC/dQ = 8000q/81

Average Variable Cost = AVC = (4000*(q/9)^2)/q = (4000/81)q

Thus, the average variable cost is at its minimum at q=0

Thus, the supply curve equation is: P = 8000q/81

c.

P =1000

1000 = 8000q/81

q = 1000*81/8000 = 10.25

Thus, the firm would produce 10.25 units when the price is $1000

The level of profit = Total Revenue - Total Cost

= 1000*10.25 - (1000 + 4000*(q/9)^2)

=1000*10.25 - (1000 + 4000*(10.25/9)^2)

=4061.728395