Equations for the maximum profit

Question

Problem: An amusement park, whose customer set is made up of two markets, adults and children, has developed demand schedules as follows:

                                    Quantity

Price ($)              Adults           Children

5                                  15                    20

6                                  14                    18

7                                  13                    16

8                                  12                    14

9                                  11                    12

10                                10                    10

11                                  9                      8

12                                  8                      6

13                                  7                      4

14                                  6                      2

The marginal operating cost of each unit of quantity is $5.  (Hint: Since marginal cost is a constant, so is average variable cost. Ignore fixed cost.) The owners of the amusement park wish to maximize profits.

A. Calculate the price, quantity, and profit if

(1) The amusement park charges a different price in each market.

(2) The amusement park charges the same price in the two markets combined.

(3) Explain the difference in the profit realized under the two situations.

b. (Mathematical solution) The demand schedules presented in problem 2 can be expressed in equation form as follows where subscription A refers to the adult market, subscript C to the market for children, and subscript T to the two markets combined):

Q_A = 20 – 1P_A
Q_C = 20 – 2P_C
Q_T = 50 – 3P_T

Solve these equations for the maximum profit that the amusement park will attain when it charges different prices in the two markets and when it charges a single price for the combined market.