Question

# 1. Suppose a monopoly with a cost function of C(Q) = 0.5Q2 + 10Q + 392...

1. Suppose a monopoly with a cost function of C(Q) = 0.5Q2 + 10Q + 392 faces a market demand of P = 500 – 17Q.

A) What is its profit maximizing level of output, what price will it charge, and what will its profits be?

B) If we imposed a MC price ceiling on the monopoly, how much would we need to subsidize them in order for them to remain in business?

C) How much would the firm be willing to spend to prevent the MC price regulation from occurring?

C(Q) = 0.5Q2 + 10Q + 392

Marginal cost (MC) = dC(Q)/dQ = Q + 10

(A) Profit is maximized when Marginal revenue (MR) equals MC.

Total revenue (TR) = 500Q - 17Q2

MR = dTR/dQ = 500 - 34Q

Equating with MC,

500 - 34Q = Q + 10

35Q = 490

Q = 14

P = 500 - (17 x 14) = 500 - 238 = 262

TR = P x Q = 262 x 14 = 3,668

TC = (0.5 x 14 x 14) + (10 x 14) + 392 = 98 + 140 + 392 = 630

Profit = TR - TC = 3,668 - 630 = 3,038

(B) When Price is equated to MC,

500 - 17Q = Q + 10

18Q = 490

Q = 27.22

P = MC = 27.22 + 10 = 37.22

Subsidy per unit = Difference in price = 262 - 37.22 = 224.78

(C) When P = MC,

TR = 37.22 x 27.22 = 1,013.13

TC = (0.5 x 27.22 x 27.22) + (10 x 27.22) + 392 = 370.46 + 272.2 + 392 = 1,034.66

Required spending = Difference in profit = 3,038 - 1,034.66 = 2,003.34

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