Question

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?

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