Imagine a single consumer whose utility function takes the money-leftover form U(x1, x2, . . . , xk,m) = v1(x1) + V (x2, . . . , xk) + m, where m is money left over after purchases of goods 1 through k and the function v1 is strictly concave. Suppose that the consumer starts with enough money so that, after her purchases of these goods, she has a strictly positive amount of money left over. Therefore, if good 1 were sold in the standard method of a set price p1 and the consumer allowed to choose how much of the good to buy at that price, she would choose x⇤1 that solves v01 (x⇤1 ) = 1. It is also the case that, no matter how good 1 is sold, as long as the terms of sale in no way affect the consumer’s ability to purchase other goods, we can ignore the other goods in deciding how much good 1 the consumer buys.
The firm selling good 1 has constant marginal cost

c. Suppose the firm can make the following offer to the particular consumer: “If you pay a fixed fee F , you can buy as much or as little as you want at the price-per-unit p.” What are the optimal (profit-maximizing) levels of F and p for the firm to set? What is the connection with first-degree price discrimination?