Suppose the demand curve for a monopolist is Qd = 500 – P, and the marginal revenue function is MR = 500 -2Q. The monopolist has a constant marginal and average total cost of $50 per unit.
A. Find the monopolist’s profit- maximizing output and price.
B. Calculate the monopolist’s profit.
C. What is the Lerner Index for this industry? The Lerner Index for this industry is
To find the monopolist's profit-maximizing output and price, we need to determine the level of output at which marginal revenue equals marginal cost (MC).
Since the monopolist's marginal revenue function is MR = 500 - 2Q and the marginal cost is constant at $50, we can equate MR and MC:
500 - 2Q = 50
Solving for Q gives us:
2Q = 450
Q = 225
So, the monopolist's profit-maximizing output is 225 units.
To find the price, we can substitute the value of Q into the demand curve:
Qd = 500 - P
225 = 500 - P
Solving for P:
P = 500 - 225
P = 275
So, the monopolist's profit-maximizing price is $275 per unit.
To calculate the monopolist's profit, we need to find the total revenue (TR) and total cost (TC) at the profit-maximizing output.
Total revenue is calculated as:
TR = Price * Quantity
TR = $275 * 225
TR = $61,875
Total cost is calculated as:
TC = Cost per unit * Quantity
TC = $50 * 225
TC = $11,250
Profit is calculated by subtracting total cost from total revenue:
Profit = TR - TC
Profit = $61,875 - $11,250
Profit = $50,625
So, the monopolist's profit is $50,625.
The Lerner Index for this industry measures the degree of market power that the monopolist has. It is calculated as:
Lerner Index = (Price - Marginal Cost) / Price
In this case, the price is $275 and the marginal cost is $50. So, the Lerner Index is:
Lerner Index = ($275 - $50) / $275
Lerner Index = $225 / $275
Lerner Index ≈ 0.818
Therefore, the Lerner Index for this industry is approximately 0.818.