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Homework answers / question archive / A monopoly’s long-run total cost function is given by ?(?) = 4?, − 8? + 200 and the market demand curve (inverse demand curve) is given by ?(?) = 124 − 4?
A monopoly’s long-run total cost function is given by ?(?) = 4?, − 8? + 200 and the
market demand curve (inverse demand curve) is given by ?(?) = 124 − 4?.
a. Find the profit-maximizing quantity of output for the monopoly and the price the
monopolist will set.
b. Calculate the monopolist’s profits (?m) and consumer’s surplus (??m) .
c. What quantity of output would be produced if the monopolist acted like a perfect
competitor?
d. Calculate profits (?pc) and consumer surplus (??pc) corresponding to the competitive
equilibrium.
e. Find the deadweight loss due to the monopoly.
C = 4q2 - 8q + 200
MC = dC/dq = 8q - 8
(a)
P = 124 - 4q
TR = P x q = 124q - 4q2
MR = dTR/dq = 124 - 8q
Setting MR = MC,
124 - 8q = 8q - 8
16q = 132
q = 8.25
P = 124 - 4 x 8.25 = 91
(b)
(i)
TR = P x q = 91 x 8.25 = 750.75
C = 4 x 8.25 x 8.25 - 8 x 4.25 + 200 = 438.25
Profit = TR - C = 750.75 - 438.25 = 312.5
(ii)
From demand: When q = 0, P = 124
Consumer surplus (CS) = area between demand curve and price = (1/2) x (124 - 91) x 8.25 = 136.125
(c)
In perfect competition, P = MC
124 - 4q = 8q - 8
12q = 132
q = 11
P = 124 - 4 x 11 = 80
(d)
(i)
TR = 80 x 11 = 880
C = 4 x 11 x 11 - 8 x 11 + 200 = 596
Profit = 880 - 596 = 284
(ii)
CS = (1/2) x (124 - 80) x 11 = 5.5 x 44 = 242
(e)
DWL = (1/2) x Change in P x Change in q = (1/2) x (91 - 80) x (11 - 8.25) = (1/2) x 11 x 2.75 = 15.125
