The problem states that a monopolist faces a market demand curve given by $P = 1000 - 5Q$. The monopolist produces at a constant marginal cost of $MC = ATC = $100$. We need to find the profit-maximizing output level, the price at that output level, and the monopolist's maximum profit.

1. Problem Description

The problem states that a monopolist faces a market demand curve given by $P = 1000 - 5Q$. The monopolist produces at a constant marginal cost of $MC = ATC =$100$. We need to find the profit-maximizing output level, the price at that output level, and the monopolist's maximum profit.

2. Solution Steps

To find the profit-maximizing output level, we need to find the quantity where marginal revenue (MR) equals marginal cost (MC).

First, we need to derive the total revenue (TR) function. Total revenue is price times quantity:

$TR = P \cdot Q = (1000 - 5Q) \cdot Q = 1000Q - 5Q^2$

Next, we find the marginal revenue (MR) by taking the derivative of TR with respect to Q:

$MR = \frac{dTR}{dQ} = 1000 - 10Q$

Now, we set MR equal to MC to find the profit-maximizing quantity:

$1000 - 10Q = 100$

$10Q = 900$

$Q = 90$

Now we can find the price at this output level by plugging the value of Q back into the demand equation:

$P = 1000 - 5Q = 1000 - 5(90) = 1000 - 450 = 550$

Finally, we can calculate the monopolist's maximum profit. Profit is total revenue (TR) minus total cost (TC). In this case, average total cost (ATC) is equal to marginal cost (MC).

$Profit = TR - TC = (P \cdot Q) - (ATC \cdot Q)$

$Profit = (550 \cdot 90) - (100 \cdot 90) = 49500 - 9000 = 40500$

3. Final Answer

The profit-maximizing output level is

9

0. At this output level, the price is $

5

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0. The monopolist's maximum profit is $40,

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0. The correct answer is b. Q = 90, P = $550, Profit = $40,

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0.