A) To find the firm's equilibrium output, we need to set marginal cost (MC) equal to the industry price (P):
MC = c'(q) = 15 + 4q
115 = 15 + 4q
4q = 100
q = 25
So the equilibrium output is 25 units.
To find the firm's profit, we need to subtract total cost from total revenue:
total revenue = Pq = 115(25) = 2875
total cost = c(q) = 450 + 15(25) + 2(25)^2 = 1875
profit = total revenue - total cost = 2875 - 1875 = 1000
To find producer surplus, we need to find the area below the industry price curve and above the marginal cost curve up to the quantity produced:
producer surplus = (115 - 15)(25)/2 - (2)(25)^2/2 = 1375
B) This is a short-run equilibrium because we are given a specific cost function and not all inputs can be adjusted.
C) To find the minimum price at which the firm makes a positive economic profit, we need to set profit equal to zero and solve for P:
Pq = c(q)
P(25) = 450 + 15(25) + 2(25)^2
P = 125
So above a price of $125 per unit, the firm will make a positive economic profit.
D) If the price falls to $63 per unit, the firm's profit-maximizing output will be where MC = P:
MC = 15 + 4q = 63
4q = 48
q = 12
So the firm would reduce its output to 12 units.
Suppose that a competitive firm faces a total cost function c(q)=450+15q+2q^2, and the industry price is 115$ 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?
all question must be done without calculus
3 answers
how did you get this MC = c'(q) = 15 + 4q
To find the marginal cost (MC), we take the derivative of the total cost function (c(q)) with respect to q:
c'(q) = 15 + 4q
The first term in this expression comes from the constant term in the total cost function, which disappears when we take the derivative. The second term comes from the linear term in the total cost function, and the third term comes from the quadratic term in the total cost function. Taking the derivative with respect to q eliminates the quadratic term, as the derivative of q^2 with respect to q is 2q.
c'(q) = 15 + 4q
The first term in this expression comes from the constant term in the total cost function, which disappears when we take the derivative. The second term comes from the linear term in the total cost function, and the third term comes from the quadratic term in the total cost function. Taking the derivative with respect to q eliminates the quadratic term, as the derivative of q^2 with respect to q is 2q.
1 answer