a firm with market power faces the demand function q=4000-40p. The firm\'s total
ID: 1200365 • Letter: A
Question
a firm with market power faces the demand function q=4000-40p. The firm's total cost function is TC(q)=10q+.001q^2+1000.
a. If the firm behaves as a single price monopoly, identify the firm's optional price and output level.
b.Demonstrate that the single price monopolist's profit maximizing choice of price and ouput also maximizes producer surpulus.
c. Identify the output level that would maximize total surplus.
d.Identify the output level that a perfect price discriminating monopolist would produce.
Explanation / Answer
a)
The monopolist will try to maximize her profit by producing the level of output at which MC = MR. Marginal cost is the first derivative of total cost. Hence the marginal cost MC = 10 + 0.002Q
Marginal revenue is the derivative of Total revenue. For this find the inverse demand function as P = 100 - 0.025Q
Note that total revenue is P*Q or 100Q - 0.025Q2. Thus, marginal revenue MR = 100 - 0.05Q
Hence, the profit-maximizing output level is:
100 - 0.05Q = 10 + 0.002Q
90 = 0.052Q
Q = 1730, P = 56.75
Hence, the profit maximizing level of output is 1730 units and price is $56.75 per unit.
b) Producer surplus for monopolist is its profit.
Profit can be computed as TR - TC and is maximum at Q*.
Profit = P*Q - TC
= 100Q - 0.025Q2 - 10Q - 0.001Q2 - 1000
= 90Q - 0.026Q2
Profit is maximum whenist derivative is zero.
90 - 0.052Q = 0
Q* = 1730
Hence it is shwon that MR=MC or TR-TC both approches confirm that at Q = 1730, profit is maximum.
c) Total surplus is maximum in a monopoly when its produces at MC = AR
AR is the average revenue or TR/Q which in this case, is AR = P = 100 - 0.025Q
MC = AR
10 + 0.002Q = 100 - 0.025Q
0.027Q = 90
Q* = 3333 units.
Hence, total surplus maximizing output level is 3333.
d) The perfect price discriminating monopolist produces a quantity where its AR = MC. The MR curve is the same as its demand curve it charges the reservation price. Hence a perfect price discriminating monopolist produces a quantity Q* = 3333
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