A monopolist with total cost function c(Q) 10 + Q2 faces a market with two disti
ID: 1148450 • Letter: A
Question
A monopolist with total cost function c(Q) 10 + Q2 faces a market with two distinct consumer groups with the first group having a demand function of q1(P) 80 -5P1 and the second having a demand function of q2(P) = 40-2P2. 1- The monopolist can now engage in third degree price discrimination, setting a different price for each market segment. Calculate the monopolist's optimal prices charged to each market segment, the quantities consumed by each market segment, and the monopolist's profits. Would the monopolist rather engage in this price discrimination or would they rather set a uniform price?Explanation / Answer
The cost function is C = 10 + 0.5Q^2. Hence marginal cost = Q = Q1 + Q2.
Market 1
Inverse demand is P1 = 16 - 0.2Q1
Marginal revenue MR1 = 16 - 0.4Q1.
MC = Q1 + Q2
Equate MR1 = MC
16 - 0.4Q1 = Q1 + Q2
This gives us the equation 1.4Q1 + Q2 = 16.
In market 2,
Inverse demand function is P2 = 20 - 0.5Q2
MR2 = 20 - Q2
MC = Q1 + Q2
MR2 = MC
20 - Q2 = Q1 + Q2
This gives the second equation Q1 + 2Q2 = 20.
Solve 1.4Q1 + Q2 = 16 and Q1 + 2Q2 = 20 to get Q1 = 20/3 and Q2 = 20/3
Prices in two markets are P1 = 16 - 0.2*20/3 = 44/3 and P2 = 20 - 0.5*20/3 = 50/3.
Profits in two markets is arrived at
Profit = TR1 + TR2 - TC = (20/3)*(44/3) + (20/3)*(50/3) - 10 - (1/2)*(20/3 + 20/3)^2 = $110.
When it charges a single price, the market demand is Q1 + Q2 = Q
Q = 80 - 5P + 40 - 2P
Q = 120 - 7P or P = 120/7 - (1/7)Q
Marginal revenue is MR = 120/7 - (2/7)Q
Profit maximizing quantity is
MR = MC
120/7 - (2/7)Q = Q
This gives Q = 40/3 and a price of P = 120/7 - (1/7)*(40/3) = 320/21. Profit = (40/3)*(320/21) - 10 - (1/2)*(40/3)^2 = 104.3
Observe that profit is increased with price discrimination from $104.3 to $110. Hence price discrimination is better.
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