In: Economics
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?
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