A monopolist faces the demand curve q = 90 - p/2, where q is the number of units sold and p is the price in dollars.She has quasi-fixed costs, C, and constant marginal costs of $20 per unit of output.Therefore her total costs are C + 20q if q 0 and 0 if q = 0.What is the largest value of C for which she would be willing to produce positive output?
A)$20
B)$2,560
C)$3,200
D)$4,800
E)$3,840

Question

A monopolist faces the demand curve q = 90 - p/2, where q is the number of units sold and p is the price in dollars.She has quasi-fixed costs, C, and constant marginal costs of $20 per unit of output.Therefore her total costs are C + 20q if q > 0 and 0 if q = 0.What is the largest value of C for which she would be willing to produce positive output?
A)$20
B)$2,560
C)$3,200
D)$4,800
E)$3,840

Quizplus · Accepted Answer

To find the largest value of C for which she would be willing to produce positive output, we need to calculate the monopolist's profit at zero output and set it equal to zero. This is because if the profit at zero output is negative, then she would not produce anything, while if the profit at zero output is zero or positive, then she would produce some positive output.

At zero output, the monopolist's total revenue is zero, and her total cost is C. Therefore, her profit is:

π = 0 - C

Setting this equal to zero and solving for C, we get:

0 - C = 0
C = 0

This means that if the quasi-fixed costs are zero, the monopolist would be willing to produce some positive output. However, since the quasi-fixed costs are positive, the monopolist's profit at zero output is negative, and she would not produce anything.

Now, we need to find the largest value of C for which the monopolist's profit is zero or positive. To do this, we need to find the monopolist's output level that maximizes her profit. This is given by:

q* = (90 - C)/(2*20) = (9/4)*(45 - C/9)

Therefore, her profit is:

π = p(q*)*q* - C - 20q*

Substituting the demand function and simplifying, we get:

π = [(45 - C/9)^2]/4 - C(45 - C/9)/2 - (9/2)*(45 - C/9)^2

To find the largest value of C for which the monopolist's profit is zero or positive, we need to solve for C such that:

[(45 - C/9)^2]/4 - C(45 - C/9)/2 - (9/2)*(45 - C/9)^2 = 0

Multiplying both sides by -36 and simplifying, we get:

20C^2 - 12960C + 207360 = 0

Solving for C using the quadratic formula, we get:

C = (12960 ± sqrt(12960^2 - 4*20*207360))/(2*20) ≈ 3200 or ≈ 648

Since the monopolist's quasi-fixed costs cannot be negative, the largest value of C for which she would be willing to produce positive output is:

C = $3,200

Therefore, the answer is (C).

A Monopolist Faces the Demand Curve Q = 90 -