(public good)* Consider an economy with I consumers with “quasi-linear” utility functions, where t…

(public good)* Consider an economy with I consumers with “quasi-linear” utility functions,

where ti is consumer i's income, x is a public decision (for instance, the quantity of a public good), Vi(x, 0;) is consumer i's gross surplus for decision x. and k is a utility parameter. The monetary cost of decision x is C(x).

The socially efficient decision is

Assume 0) that the maximand in this program is strictly concave and (ii) that for all θ I, θi, and ,

Condition ii says that the optimal decision is responsive to the utility parameter of each consumer. (Condition i is satisfied if x belongs to ℝ, Vi is strictly concave in x, and C is strictly convex in x. Furthermore, if 0; belongs to an interval of ℝ, Vi and C are twice differentiable,   or 0, and x* is an interior solution, then x* is strictly increasing or strictly decreasing in θi, so that condition (ii) is satisfied as well.)

Now consider the following “demand-revelation game”: Consumers are asked to announce their utility parameters simultaneously. A pure strategy for consumer i is thus an announcement i of his parameter (i, may differ from the true parameter i). The realized decision is the optimal one for the announced parameters x*(i, …,i), and consumer i receives a transfer from a “social planner” equal to

when Ki is a constant.

Show that telling the truth is dominant, in that any report A # Of is strictly dominated by the truthful report i = θi.

Because each player has a dominant strategy, it does not matter whether he knows the other players utility parameters. Hence, even if the players do not know one another's payoffs (see chapter 6), it is still rational for them to tell the truth. This property of the dominant-strategy demand-revelation mechanism (called the Groves mechanism) makes it particularly interesting in a situation in which a consumer's utility parameter is known only to that consumer.