Question

3. (i) A monopolist faces the following demand and total cost
functions: Q1 = 65 -1/2P, TC = Q^{2} + 10Q + 50

(a) Calculate the profit maximizing output and price of the monopolist. Calculate the resulting profit. (12 points)

(b) Suppose the government imposes an excise tax of $30 on the production and sale of the product. Calculate the resulting optimal profit maximizing output and price for the monopolist. Also determine the level of profit. (12 points)

(c) If the government’s objective is to generate the maximum possible tax revenue from the monopolist, what excise tax rate should the government impose on the monopolist? Calculate the resulting optimal output, and price of the monopolist as well as government’s tax revenue. (16 points)

(ii) Two firms produce differentiated products and set prices to
maximize their individual profits. Demand functions for the firms
are given by Q1 = 64 – 4P_{1} + 2P_{2} &
Q_{2} = 50 – 5P_{2} + P_{1} where P1, P2,
Q1, Q2, refer to prices and outputs of firms 1 and 2 respectively.
Firm 1’s marginal cost is $5 while firm 2’s marginal cost is $4.
Each firm has a fixed cost of $50.

Assuming that the two firms decide on prices independently and simultaneously, calculate the best response function of each firm in terms of prices. Calculate the resulting equilibrium price quantity combination for each firm. Illustrate your answer with a suitable graph. Also calculate optimal profits of each firm.

Answer #1

3 (i)

(a) Q1 = 65-(1/2)P

therefore P= 2(65- Q1)

= 130 - Q1

MR = 130 - 2 Q

similarly we have

TC =Q^{2} +10Q +50

MC = 2Q + 10

MR = MC

130 - 2Q = 2Q +10

4Q = 130-10

Q = 120/4

Q=30

P = 130 - Q

= 130 -30

=$100

resulting profit = TR - TC

and TR = P*Q

= (130 - Q)*Q

= (130 - 30)*30

=$3000

and TC = Q^{2} + 10Q + 50

=(30)^{2} + 10*30 + 50

= 900 + 300 + 50

=$ 1250

resulting profit = TR -TC

= 3000 - 1250

=$ 1750

(b)

Suppose the government imposes an excise tax of $30 on the production and sale of the product.

then the resulting optimal profit maximizing output and price for the monopolist is given by

P = 130 - 30

= $100

sorry i won't able to answer further

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