Suppose a monopolistically competitive firm’s demand is given by
P = 100 – 2Q
And its cost function is given by
TC = 5 + 2Q
a. Find the profit maximizing quantity, price, and total profit level.
b. Is this a long-run or a short-run outcome? How do you know?
c. Assuming that the slope of the demand curve remains constant, what will be the long-run equilibrium price and quantity for the firm?
To find the profit maximizing quantity, price, and total profit level, we can use the following steps:
a. Find the profit maximizing quantity and price:
To determine the profit maximizing quantity, we need to find the quantity at which marginal revenue (MR) equals marginal cost (MC). The marginal revenue is the derivative of the total revenue (TR) with respect to quantity (Q), and the marginal cost is the derivative of the total cost (TC) with respect to quantity (Q).
The total revenue (TR) function is given by:
TR = P * Q
Taking the derivative of TR with respect to Q gives us the marginal revenue (MR):
MR = d(TR)/dQ
In this case, P = 100 - 2Q, so TR = (100 - 2Q) * Q = 100Q - 2Q^2.
Differentiating TR with respect to Q, we have:
MR = d(100Q - 2Q^2)/dQ
= 100 - 4Q
The total cost (TC) function is given by:
TC = 5 + 2Q
Taking the derivative of TC with respect to Q gives us the marginal cost (MC):
MC = d(TC)/dQ
In this case, TC = 5 + 2Q, so MC = d(5 + 2Q)/dQ = 2.
Setting MR equal to MC:
100 - 4Q = 2
98 = 4Q
Q = 24.5
Now that we have the quantity, we can find the price by substituting the quantity back into the demand equation:
P = 100 - 2Q
= 100 - 2(24.5)
= 51
Therefore, the profit maximizing quantity is 24.5, the price is 51, and the total profit can be calculated by subtracting total cost from total revenue:
TR = P * Q = 51 * 24.5 = 1251.5
TC = 5 + 2Q = 5 + 2(24.5) = 54
Total profit = TR - TC = 1251.5 - 54 = 1197.5
b. To determine if this is a long-run or a short-run outcome, we need to consider whether there are barriers to entry and exit in the market.
In monopolistically competitive markets, there are usually no significant barriers to entry. If profits exist in the short-run, new firms would enter the market in the long run, increasing competition and driving down prices until profit is eliminated.
It is important to note that this information alone does not provide conclusive evidence regarding the time frame of the outcome. We would need more information about market conditions and industry characteristics to make a definitive determination.
c. Assuming that the slope of the demand curve remains constant, the long-run equilibrium price and quantity can be calculated by finding the quantity at which average total cost (ATC) equals price, given the cost function.
The average total cost (ATC) is the total cost (TC) divided by quantity (Q):
ATC = TC / Q
In this case, TC = 5 + 2Q, so ATC = (5 + 2Q) / Q = 5/Q + 2.
To find the long-run equilibrium price and quantity, we need to find the quantity at which ATC equals the given price, which is 51:
5/Q + 2 = 51
5 + 2Q = 51Q
5 = 49Q
Q = 5/49
Now that we have the quantity, we can find the price by substituting the quantity back into the demand equation:
P = 100 - 2Q
= 100 - 2(5/49)
= 99.7959
Therefore, the long-run equilibrium price is approximately 99.7959, and the long-run equilibrium quantity is approximately 5/49.