To find the monopolist's profit maximizing output and price, we need to equate marginal revenue (MR) with marginal cost (MC). Since we are given the marginal revenue function (MR = 500 - 2P) and the constant marginal cost (MC = $50 per unit), we can set them equal to each other:
500 - 2P = 50
Solving this equation for P will give us the profit maximizing price. Let's proceed with the calculations:
500 - 2P = 50
-2P = 50 - 500
-2P = -450
P = -450 / -2
P = $225
So the monopolist's profit maximizing price is $225 per unit.
To find the monopolist's profit maximizing output, we can substitute the price (P = $225) into the demand curve equation (Qd = 500 - P):
Qd = 500 - 225
Qd = 275
Therefore, the monopolist's profit maximizing output is 275 units.
Now, to calculate the monopolist's profit, we need to calculate total revenue (TR) and total cost (TC). Total revenue is given by the price multiplied by the quantity (TR = P * Q), while total cost is the constant marginal cost multiplied by the quantity (TC = MC * Q). Let's calculate:
TR = P * Q
TR = $225 * 275
TR = $61,875
TC = MC * Q
TC = $50 * 275
TC = $13,750
The monopolist's profit, represented by π, is given by the difference between total revenue and total cost:
π = TR - TC
π = $61,875 - $13,750
π = $48,125
Therefore, the monopolist's profit is $48,125.
Lastly, we can calculate the Lerner Index for this industry. The Lerner Index measures the market power of a firm and is calculated as (P - MC) / P. We already know the price (P = $225) and the constant marginal cost (MC = $50). Let's calculate the Lerner Index:
Lerner Index = (P - MC) / P
Lerner Index = ($225 - $50) / $225
Lerner Index = $175 / $225
Lerner Index ≈ 0.778
So the Lerner Index for this industry is approximately 0.778.