Suppose your firm sells only x and y individually what

Question

Your firm sells two goods, X and Y. There are three customers who have different valuations for your products, but you cannot determine which customer is which, and thus cannot charge a different price for each customer. Each customer wants to buy at most one unit of X and one unit of Y. Your firm's cost of producing each good is zero. The valuations for each consumer are as follows:

Customer                           Product X                          Product Y

1                                          $80                                 $50

2                                          $60                                 $130

3                                          $30                                 $150

a) Suppose your firm sells only X and Y individually. What prices, P_x and P_y, should your firm charge to maximize profits, and how much profit do your earn? (note: costs are zero, so profits and revenues are the same thing.)

b) Now suppose your firm only sells X and Y in a bundle, with one unit of each. What bundle price P_B should your firm charge, and how much profit do your earn?

c) Instead, you decide to sell X and Y in a bundle for price P_B, and you also sell good X individually for price P_x, but you do not sell Y individually. Find the values of P_B and P_x that will maximize your profit.