How to Determine Third-Degree Price Discrimination in Managerial Economics

In order to determine the profit-maximizing price and quantity for each group of customers by using third-degree price discrimination, you must satisfy the following condition:

where MR_A is group A’s marginal revenue for the last unit it buys, MR_B is group B’s marginal revenue for the last unit it buys, and MC is marginal cost for the last unit you produce.

Assume that the following equations describe the demand for your firm’s product for two different groups of customers. Group A’s demand is

where P_A is the price in dollars charged to group A customers, and q_A is the quantity sold to group A customers. Group B’s demand is

where P_B is the price in dollars charged to group B customers, and q_B is the quantity sold to group B customers.

Your firm’s total cost and marginal cost equations are

where TC is your total cost in dollars, MC is marginal cost in dollars, and q is the total quantity of the good your firm produces.

Using this information and the following steps, you can determine the price to charge each group of consumers, how much to sell each group of consumers, and how much to produce:

1. Determine the marginal revenue for group A customers.

   First, multiply the demand equation by q_A to determine total revenue, and then take the derivative of total revenue with respect to q_A to determine marginal revenue.

   

2. Determine the marginal revenue for group B customers.

   First, determine total revenue by multiplying P_B by q_B, and then take the derivative of total revenue with respect to q_B to determine marginal revenue.

   

3. Set MR_A = MC.

   

4. Substitute q_A + q_B for q.

   The total quantity of output you produce and sell, q, is sold to customers in either group A or group B.

   

5. Solve the equation in Step 4 for q_B.

   

6. Set MR_A equal to MR_B.

   

7. Substitute q_B = 950 – 5q_A in the equation in Step 6.

   

8. Solve for q_A.

   

9. Substitute q_A = 150 in group A’s demand equation to determine the price, P_A, to charge customers in group A.

   

10. Substitute q_A = 150 in the equation from Step 5 to determine q_B.

    

11. Substitute q_B = 200 in group B’s demand equation to determine the price, P_B, to charge customers in group B.

    

12. Determine the total quantity of output that your firm produces.

    

Your firm produces 350 units of the good. It sells 150 units to customers in group A at a price of $70 per unit and 200 units to customers in group B at a price of $60 per unit.