Suppose that a competitive firm faces a total cost function c(q)=450+15q+2q^2, and the industry price is 115$ per unit of output.
A) Find the firm's equilibrium output, levels of profit and producer surplus?
B) Is this a short-run or log equilibrium?
C) Above which prices will this firm make a positive economic profit?
D) Suppose due to a lower industry demand, the price falls to 63$ per unit. What supply decision would this firm make?
A) To find the equilibrium output, we need to find where the firm's marginal cost equals the industry price. The derivative of the total cost function is:
MC(q) = 15 + 4q
Setting MC(q) equal to price:
115 = 15 + 4q
100 = 4q
q = 25
So the firm's equilibrium output is 25 units.
To find the firm's profit, we need to subtract its total costs from its total revenue:
TR(q) = 115q
Profit = TR(q) - TC(q)
= 115q - (450 + 15q + 2q^2)
= -2q^2 + 100q - 450
At q=25, profit = -$125
To find producer surplus, we need to find the area below the price and above the supply curve (which is the marginal cost curve in this case) up to the quantity supplied. This is the triangle with base 25 and height (115-15) = 100:
Producer surplus = (1/2) x 25 x 100 = $1250
B) This is a short-run equilibrium because the firm's fixed costs are not included in the marginal cost calculation.
C) To find the price at which the firm makes zero economic profit, we need to set its profit function equal to zero and solve for q:
-2q^2 + 100q - 450 = 0
Using the quadratic formula, we get:
q = (100 ± sqrt(100^2 + 4(2)(450)))/(2(2))
q = 50 ± 5sqrt(34)
So the firm makes zero economic profit at q = 50 - 5sqrt(34) or q = 50 + 5sqrt(34). At any price above this, the firm makes a positive economic profit.
D) At a price of $63, the firm's marginal revenue would be $63 and its marginal cost at q=25 is $115, so it would choose to shut down production in the short run.