Question

3. Sue has a monopoly over the production of strawberry shortcake. Her cost function is C(y) = y2 + 10y. The market demand curve for strawberry shortcake is y(p) = 200 – 2p.

   a. Find Sue’s profit-maximizing level of output and price.
   b. Calculate Sue’s profit and the consumer surplus at that price-quantity combination.

   Suppose Sue is forced to set her price equal to marginal cost; i.e., to behave competitively.
   c. Find Sue’s new price-quantity combination
   d. What is her new profit and the new consumer surplus? Compare these to what you found

   in parts a and b above. How does total welfare compare to the situation when Sue behaved as a monopolist?

Answer 1

Demand: y = 200 - 2p

2p = 200 - y

p = 100 - 0.5y

Marginal cost (MC) = dc(y)/dy = 2y + 10

(a)

A monopolist maximizes profit by equating MR and MC.

Total revenue (TR) = p x y = 100y - 0.5y²

MR = dTR/dy = 100 - y

Equating with MC,

100 - y = 2y + 10

3y = 90

y = 30

p = 100 - (0.5 x 30) = 100 - 15 = 85

(b)

(i)

TR = 85 x 30 = 2550

C(y) = 30 x 30 + 10 x 30 = 900 + 30 = 930

Profit = TR - C(y) = 2550 - 930 = 1620

(ii)

From demand function, when y = 0, p = 100 (vertical intercept of demand curve)

Consumer surplus (CS) = Area between demand curve and price = (1/2) x (100 - 85) x 30 = 15 x 15 = 225

(c)

When P = MC,

100 - 0.5y = 2y + 10

2.5y = 90

y = 36

P = 100 - (0.5 x 36) = 100 - 18 = 82

(d)

(i)

TR = 82 x 36 = 2952

C(y) = 36 x 36 + 10 x 36 = 1296 + 360 = 1656

Profit = 2952 - 1656 = 1296 (Profit is lower by (1620 - 1296) = 324)

(ii)

CS = (1/2) x (100 - 82) x 36 = 18 x 18 = 324 (CS is higher by (324 - 225) = 99).

(iii)

A monopoly will lead to social inefficiency (deadweight) loss causing loss of welfare.

Deadweight loss = (1/2) x Change in P x Change in y = (1/2) x (85 - 82) x (36 - 30) = (1/2) x 3 x 6 = 9