4. You are the manager of a monopoly and your demand and cost functions are give

Question

4. You are the manager of a monopoly and your demand and cost functions are given by P = 200 – Q and C(Q) = 2,000 + 3Q^2, respectively. I answered a & b, just need help with the others.

a. What price-quantity combination maximizes your firm’s profits?
Profit-maximizing condition
MR = MC

TR = P * Q
TR = (200 -2Q)*Q
TR = 200Q – 2Q^2

MR = 200 – 4Q
MC = 6Q

MR = MC
200 – 4Q = 6Q
10Q = 200
Q = 20 units

P = 200 – 2Q
P = 200 – 2 * 20
P = 200 – 40
P = $160

b. Calculate the maximum profits.
p=TR-TC
= 160 * 20 – (2,000 + 3 * 20^2)
= 3,200 – 3,200
= 0

c. Is demand elastic, inelastic, or unit elastic at the profit-maximizing price-quantity combination?
d. What price-quantity combination maximizes revenue?
e. Calculate the maximum revenues.
f. Is demand elastic, inelastic, or unit elastic at the revenue-maximizing price-quantity combination?

Explanation / Answer

a) you have given demand eqn. as P = 200 - Q, but taken as 200 - 2Q in the solution. let us correct it. R = (200 - Q)*Q = 200Q - Q^2 MR = dR/dQ = 200 -2Q MC = 6Q (correct in your cal.) so, for max. profit, MR = MC i.e. 200 - 2Q = 6Q => Q = 200/8 = 25 P = 200 -Q = 200 - 25 = 175 ($) b) P(max) = R - TC = 175*25 - (2000 + 3*25^2) = 500 ($) c) AT max. profit, demand is unit elastic i.e. delta Q /unit change in P = -1 d) R = 200 Q - Q^2 for R max, dR/dQ = MR = 200 - 2Q = 0 so, Q = 100 and P = 100 e) demand elasticity = delta Q/ unit change in P = -1 i.e. unit elastic

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