1. Consider a pure monopolist with short-run total cost function given by
STC = 1000 +200 Q + 12.5 Q2. Suppose also that this firm faces an inverse market demand function given by P = 800 – 20 Q.
a. How much should this firm produce and what price should it charge in order to maximize profits?
b. How much economic profit (or loss) would this firm earn?
c. What is the value of this firm’s demand elasticity at its profit-maximizing price?
d. What is the Lerner Index for this firm? LI = -1/ Ed = 0.30
2. A monopolist has two plants, A and B, with respective marginal cost functions given by MCA = 10 +QA and MCB = 10 + 2QB. It faces a demand curve given by P = 70 – 1/6 Q.
a. What is the expression for this firm’s marginal cost function?
b. How much will this firm produce in order to maximize profit?
→ MC = MR
c. What price will it charge at that quantity?
d. How much output should be produced in the two respective plants
3. A pure monopsonist faces a market supply curve given by the expression P = 10 + Q. The marginal value curve for the input is given by MV = 60 – ½ Q.
a. Find how much this optimizing monopsonist will purchase of the input.
b. What price will it pay for the input?
To answer the questions, we need to follow the steps below:
a. To maximize profits, the monopolist should produce where marginal cost (MC) is equal to marginal revenue (MR).
1. Calculate marginal cost (MC):
MC = d(STC) / dQ.
MC = 200 + 25Q.
2. Determine marginal revenue (MR) by differentiating the inverse market demand function (P) with respect to quantity (Q):
MR = d(P) / dQ.
MR = 800 - 40Q.
Set MC equal to MR and solve for Q:
200 + 25Q = 800 - 40Q.
65Q = 600.
Q = 9.23 (approximately).
Now substitute Q back into the market demand function to find the price (P):
P = 800 - 20Q.
P = 800 - 20(9.23).
P = 616.15 (approximately).
Therefore, the firm should produce approximately 9.23 units and charge a price of approximately $616.15 to maximize profits.
b. To find the economic profit (or loss), we need to calculate total revenue (TR) and subtract total cost (TC):
1. Calculate total revenue (TR):
TR = P * Q.
TR = 616.15 * 9.23.
TR = $5,686.08 (approximately).
2. Calculate the short-run total cost (STC) using the given cost function:
STC = 1000 + 200Q + 12.5Q^2.
STC = 1000 + 200(9.23) + 12.5(9.23)^2.
STC = $2,915.69 (approximately).
Subtract the total cost from the total revenue to find the economic profit (or loss):
Economic profit = TR - STC.
Economic profit = $5,686.08 - $2,915.69.
Economic profit = $2,770.39 (approximately).
Therefore, the firm would earn an economic profit of approximately $2,770.39.
c. The value of the demand elasticity at the profit-maximizing price can be calculated using the formula:
Elasticity (Ed) = -dQ / dP * (P / Q).
From the inverse demand function, we already have:
dQ/dP = -1/20 and P/Q = 616.15/9.23.
Substituting these values into the elasticity formula:
Ed = -(-1/20) * (616.15/9.23).
Ed = 0.3333 (approximately).
Therefore, the demand elasticity at the profit-maximizing price is approximately 0.3333.
d. The Lerner Index (LI) can be calculated using the following formula:
LI = -1/Ed.
From the given information, we know that LI = 0.30.
Therefore, the Lerner Index for this firm is 0.30.