Question

Suppose we have a monopolist that faces an inverse demand function and total cost function of

Pd= 200−2Qd

C(Qs)=Qs^2 + 8Qs + 50

Note that the last part of this question asks you to graph much of your answers to the first parts of this question.

(a) Find the profit maximizing level of output (Q) and the corresponding price charged (P).

(b) Find the Socially optimal level of output and price. (hint: this is the case when we assume perfect competition)

(c) Graph the Demand, Marginal Revenue, and Marginal Cost curves. Find the Deadweight Loss, loss to Consumer Surplus and gain to Producer Surplus due to monopolistic power.

DWL =

Loss to CS =

Gain to PS =

Answer 1

Pd = 200 - 2Qd

C(Qs) = Qs² + 8Qs + 50

Marginal cost (MC) = dC(Qs)/dQs = 2Qs + 8

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

Total revenue (TR) = Pd x Qd = 200Qd - 2Qd²

MR = dTR/dQd = 200 - 4Qd

Equating MR & MC (Qd = Qs = Q),

200 - 4Q = 2Q + 8

6Q = 192

Q = 32

P = 200 - (2 x 32) = 200 - 64 = 136

(b) In social optimal, P = MC

200 - 2Q = 2Q + 8

4Q = 192

Q = 48

P = 200 - (2 x 48) = 200 - 96 = 104

(c)

From demand function,

When Q = 0, P = 200 (Vertical intercept) & when P = 0, Q = 200/2 = 100 (Horizontal intercept)

From MR function,

When Q = 0, P = 200 (Vertical intercept) & when P = 0, Q = 200/4 = 50 (Horizontal intercept)

From MC function,

When Q = 0, MC = 8 (Vertical intercept)

In following graph, D, MR & MC are the demand, MR and MC curves. Monopoly equilibrium is at point E where MR intersects MC with price PM (= 136) and quantity QM (= 32). Socially optimal outcome is at point F where D intersects MC with price PC (= 104) and quantity QC (= 48).

When Q = 32, MR = MC = (2 x 32) + 8 = 64 + 8 = 72 (Coordinate of point C)

CS = Area between demand curve and price

CS in monopoly = Area AEPM = (1/2) x (200 - 136) x 32 = 16 x 64 = 1,024

CS in socially optimal outcome = Area AFPC = (1/2) x (200 - 104) x 48 = 24 x 96 = 2,304

Loss in CS = 2,304 - 1,024 = 1,280

PS = Area between supply curve and price

PS in monopoly = Area BCEPM = (1/2) x 32 x [(136 - 72) + (136 - 8)] = 16 x (64 + 128) = 16 x 192 = 3,072

PS in socially optimal outcome = Area BFPC = (1/2) x (104 - 8) x 48 = 24 x 96 = 2,304

Gain in PS = 3,072 - 2,304 = 768

DWL = Area CEF = (1/2) x (136 - 72) x (48 - 32) = (1/2) x 64 x 16 = 512