A firm sells a single good to a group of customers. Each customer either buys zero or exactly one unit of the good; the good cannot be divided or resold. However it can be delivered as either a high-quality or a low-quality good. The quality is characterised by a non-negative number q; the cost of producing one unit of good at quality q is C(q) where C is an increasing and strictly convex function. The taste of customer h is Th - the marginal willingness to pay for quality. Utility for h is
Uh (q, x) Thq + x
where Th is a positive taste parameter and x is the quantity of consumed of all other goods.
1. If F is the fee required as payment for the good write down the budget constraint for the individual customer.
2. If there are two types of customer show that the single-crossing condition is satisfied and establish the conditions for a full-information solution.
3. Show that the second-best solution must satisfy the no-distortion-at-the-top principle (page 343).
4. Derive the second-best optimum.(Mussa and Rosen 1978)