E cient Multi-unit Auctions for Normal Goods

I study e cient multi-unit auction design when bidders have private values, multiunit demands, and non-quasilinear preferences. Without quasilinearity, the Vickrey auction loses its desired incentive and e ciency properties. Instead of assuming that bidders have quasilinear preferences, I assume that bidders have positive wealth effects. This nests cases where bidders are risk averse, face financial constraints, or have budgets. With two bidders, I show that there is a mechanism that retains the desirable properties of the Vickrey auction if and only if bidders have single dimensional types. If bidders have multi-dimensional types, there is no mechanism that satisfies (1) individual rationality, (2) dominant strategy incentive compatibility, (3) ex-post Pareto e ciency, and (4) weak budget balance. When there are more than two bidders, I show that there is no mechanism with desirable incentive and e ciency properties, even if bidders have single dimensional types.