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Suppose a monopolist has a demand curve Q(p) = 12 - (1/3)P and has a quadratic cost curve C(Q) = Q2+ 4
Suppose a monopolist has a demand curve Q(p) = 12 - (1/3)P and has a quadratic cost curve C(Q) = Q2+ 4.
- What is the monopolist’s inverse demand function p(Q)?
- Find the monopolist’s profit-maximizing quantity and price.
- At this equilibrium in part b, what is the firm’s profit?
- In class, we have considered the effects of specific and ad valorem taxes on monopoly profit maximization and welfare. Another type of tax is a lump-sum tax, where the government could implement a $5 tax directly on the firm’s profits, instead of per unit. Does this change the profit-maximizing quantity and price for the monopolist? (Hint: Be careful – remember the monopolist always considers its shut down decision after the output decision)
- Suppose you are a government regulator and you want to set the price ceiling on this monopolist so that you maximize consumer and producer surplus (welfare). Explain the price at which you would set the price ceiling and solve for it.
Expert Solution
MC = dTC/dQ = 2Q
(a)
Q = 12 - (P/3)
3Q = 36 - P
P = 36 - 3Q
(b)
TR = P x Q = 36Q - 3Q2
MR = dTR/dQ = 36 - 6Q
Setting MR = MC,
36 - 6Q = 2Q
8Q = 36
Q = 4.5
P = 36 - 3 x 4.5 = 36 - 13.5 = 22.5
(c)
TR = 22.5 x 4.5 = 101.25
TC = 4.5 x 4.5 + 4 = 20.25 + 4 = 24.25
Profit = TR - TC = 101.25 - 24.25 = 77
(d)
Profit (Z) = TR - TC
Before the tax, Z = 36Q - 3Q2 - Q2 - 4 = 36Q - 4Q2 - 4
Profit is maximized when dZ/dQ = 0
dZ/dQ = 36 - 8Q = 0
Q = 4.5 and P = 22.5
After the tax, Z = 36Q - 3Q2 - Q2 - 4 - 5 = 36Q - 4Q2 - 9
Profit is maximized when dZ/dQ = 0
dZ/dQ = 36 - 8Q = 0
Q = 4.5 and P = 22.5
So, the profit tax doesn't change price and quantity.
(e)
Welfare is maximized when P = MC.
36 - 3Q = 2Q
5Q = 36
Q = 7.2
P = 2 x 7.2 = 14.4 (Required ceiling price)
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