The demand curve for a monopolist is Qd = 500 - P and the marginal revenue function is MR = 500 - 2P. The monopoloist 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?
First, (And I hope this clears up confusion rather than add to it), I like to re-arrange the equation such that P is a function of Q. (When you graph a demand function, you always put P on the y-axis, Q on the x-axis). Re-arranging terms in your demand function. P=500-Qd. Total Revenue is P*Q = 500Q - Q^2. So, marginal revenue is MR=500-2Qd.
a) always always always. MC=MR.
So, 500-2Qd = 50. Solve for Qd. I get Qd=225. Plug this into the demand equation. I get P=275.
b) Total profit is Qotal revenue - Total costs. So profit=(275*225) - (50*225) = $50,625
c) the Lerner index is (P-MC)/P or (275-50)/275 = .8182
a. To find the monopolist's profit-maximizing output and price, we need to set the marginal revenue equal to marginal cost and solve for the output level.
Since the marginal cost is constant and given as $50 per unit, we can set it equal to the marginal revenue:
MC = MR
$50 = 500 - 2P
Solving for P:
2P = 500 - $50
2P = $450
P = $225
Now, we can substitute the value of P into the demand equation to find the quantity:
Qd = 500 - P
Qd = 500 - $225
Qd = 275
Therefore, the monopolist's profit-maximizing output is 275 units, and the price is $225.
b. To calculate the monopolist's profit, we need to subtract the total cost from the total revenue.
Total revenue (TR) is given by the equation:
TR = P * Qd
TR = $225 * 275
TR = $61,875
Total cost (TC) is calculated by multiplying the marginal cost by the quantity:
TC = MC * Qd
TC = $50 * 275
TC = $13,750
Profit (π) is 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.
c. The Lerner Index measures market power and is calculated as (P - MC) / P. In this case, we can substitute the values into the formula:
Lerner Index = (P - MC) / P
Lerner Index = ($225 - $50) / $225
Lerner Index = $175 / $225
Lerner Index = 0.778
Therefore, the Lerner Index for this industry is approximately 0.778.
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.