Final answer:
The profit-maximizing price and quantity for the monopolist are $22.91 and 30.089 units, respectively. The monopolist's profits are approximately $537.76.
Step-by-step explanation:
For the profit-maximizing price and quantity for a monopolist, we need to set marginal revenue (MR) equal to marginal cost (MC). Given that the monopolist's average and marginal cost is $5, and the market demand curve is Q = 53 - P, we can set MR = MC:
Q = 53 - P
MR = MC = $5
Substituting MR = Q - (Q/53) into MR = MC:
Q - (Q/53) = $5
Simplifying the equation gives Q = 30.089.
To find the price, we can substitute Q into the market demand curve:
P = 53 - Q
P = 53 - 30.089 = $22.91
Therefore, the profit-maximizing quantity is approximately 30.089 units, and the price is $22.91.
To calculate profits, we need to subtract the total cost from the total revenue.
As the average cost is equal to marginal cost, the average cost is also $5.
The total cost is then $5 times the quantity, which is 30.089, giving a total cost of $150.445. The total revenue can be calculated as the price times the quantity, which is $22.91 times 30.089, giving a total revenue of approximately $688.205.
Finally, the profit can be obtained by subtracting the total cost from total revenue, which is approximately $688.205 - $150.445 = $537.76.
Therefore, the monopolist's profits are approximately $537.76.