Suppose a perfectly competitive firm has a cost function described by
TC = 100 + Q2
The industry price is $100.
a. Find the profit maximizing level of output.
b. Is this a short-run or long-run situation? How do you know?
c. Assuming that this firm’s total cost curve is the same as all other producers, find the long-run price for this good
a. To find the profit-maximizing level of output, we need to calculate the marginal cost (MC) and set it equal to the market price. In this case, the cost function is TC = 100 + Q^2, which implies that the average cost (AC) is given by:
AC = TC / Q
AC = (100 + Q^2) / Q
AC = 100/Q + Q
To find the marginal cost (MC), we differentiate the average cost (AC) with respect to quantity (Q):
MC = d(AC)/dQ
MC = -100/Q^2 + 1
Setting the marginal cost (MC) equal to the market price of $100:
MC = 100
-100/Q^2 + 1 = 100
-100/Q^2 = 99
Q^2 = -100/99
Q ≈ 10
Therefore, the profit-maximizing level of output for this perfectly competitive firm is approximately Q = 10.
b. This is a short-run situation because the firm is deciding how much to produce based on its current cost function, which includes fixed costs (100). In the long run, the firm can adjust its capital and other inputs, allowing it to change its cost function.
c. In the long run, a perfectly competitive industry will reach a state of equilibrium where all firms earn zero economic profit. This means that the long-run price (P) will equal the long-run average cost (LRAC). Therefore, to find the long-run price, we need to calculate the LRAC for this firm.
LRAC = TC / Q
LRAC = (100 + Q^2) / Q
Setting LRAC equal to P and rearranging the equation:
P = 100 / Q + Q
Since we know that Q is approximately 10 (from part a), we can substitute this value into the equation to find the long-run price:
P = 100 / 10 + 10
P = $20
Therefore, the long-run price for this good in the industry is $20.
To solve this problem, we need to use the concept of profit maximization for perfectly competitive firms. The profit-maximizing level of output occurs where marginal cost equals marginal revenue, and the firm will produce that quantity if the market price is greater than or equal to the average variable cost.
a. Find the profit maximizing level of output:
The first step is to find the marginal cost (MC) and marginal revenue (MR) functions.
Given the total cost (TC) function: TC = 100 + Q^2
To find the marginal cost (MC), take the derivative of the total cost with respect to the quantity (Q):
MC = d(TC)/dQ = 2Q
We know that in a perfectly competitive market, the price (P) is equal to the marginal revenue (MR). Therefore, MR = P.
Given that the industry price is $100, MR = $100.
Setting MR = MC, we have:
100 = 2Q
Q = 50
Therefore, the profit-maximizing level of output is 50 units.
b. Determine whether this is a short-run or long-run situation:
In this case, we do not have enough information to determine if it is a short-run or long-run situation. The distinction between short-run and long-run is based on the ability of the firm to change its inputs or adjust its fixed costs. Without information about the firm's ability to adjust its inputs or fixed costs, we cannot determine the time horizon.
c. Find the long-run price for this good:
To find the long-run price for this good, we need information about the market and other producers. However, assuming that the total cost curve for this firm is the same as all other producers, we can assume that their average cost curve is also the same.
In the long run, in perfect competition, firms will adjust their inputs to achieve minimum average total cost (ATC). This implies that in the long run, the price (P) will be equal to the minimum ATC.
Since the cost function is TC = 100 + Q^2, the average total cost (ATC) is given by:
ATC = TC / Q = (100 + Q^2) / Q = 100/Q + Q
To find the long-run price, we need to find the minimum point of the ATC curve. To do this, we take the derivative of ATC with respect to Q and set it equal to zero:
d(ATC)/dQ = -100/Q^2 + 1 = 0
-100/Q^2 = -1
Q^2 = 100
Q = 10
Substituting Q = 10 back into the ATC equation, we can find the minimum ATC:
ATC = 100/10 + 10 = 20 + 10 = 30
Therefore, assuming that the firm's total cost curve is the same as all other producers, the long-run price for this good would be $30.