Question

Suppose that the monopolist gas producer UPEC operates in two distinct markets and charges customers in...

Suppose that the monopolist gas producer UPEC operates in two distinct markets and charges customers in each market a different price (i.e., practicing third-degree price discrimination). In addition, suppose that in producing gas, UPEC incurs a fixed cost of $10 and a variable cost of 2Q. The (separate) demand functions are given by:

 Demand for gas among Group 1: P1 = 24 – Q1

 Demand for gas among Group 2: P2 = 10 – 0.5Q2

A. Find the optimal quantities and prices for each market separately. Show your work.

b. Under this kind of price discrimination, how much in profit does UPEC earn?

Homework Answers

Answer #1

Let the total cost for both the markets be:

Total Cost = Fixed Cost + Variable Cost

TC = FC + VC = 10 + 2Q      {Q = Q1 + Q2}

Thus, Marginal Cost (MC) = 2       {First order derivative of TC with respect to Q}

a.

For Group 1

P1 = 24 – Q1

Total Revenue (TR1) = P1 * Q1 = (24 – Q1)*Q1 = 24Q1 – Q12

Marginal Revenue (MR1) = 24 – 2Q1        {First order derivative of TR(1) with respect to Q1}

The profit maximizing condition for optimal price and quantity is :

MR(1) = MC

24 – 2Q1 = 2

24 – 2 = 2Q1

Q1 = 22/2 = 11

Thus, P1 = 24 – Q1 = 24 – 11 = $13

For Group 2

P2 = 10 – 0.5Q2

Total Revenue (TR2) = P2 * Q2 = (10 – 0.5Q2)*Q2 = 10Q2 – 0.5Q22

Marginal Revenue (MR2) = 10 – Q2        {First order derivative of TR(2) with respect to Q2}

The profit maximizing condition for optimal price and quantity is:

MR(2) = MC

10 – Q2 = 2

Q2 = 10 – 2 = 8

Thus, P2 = 10 – 0.5Q2 = 10 – 0.5(8) = 10 – 4 = $6

Thus total quantity sold by the monopolist is Q = Q1 + Q2 = 11 + 8 = 19

b. Profits earned by UPEC = Total Revenue earned from both groups + Total Cost

Profit = [TR(1) + TR(2)] - TC

Profit = [(24Q1 – Q12)+ (10Q2 – 0.5Q22)] – (10 + 2Q)

Profit = 24(11) – (11)2 + 10(8) – 0.5(8)2 - [10 + 2(19)]

= 264 – 121 + 80 -32 – 10 – 38

= $143

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