## Abstract

Auctions often involve the sale of many related goods: Treasury, spectrum, and electricity auctions are examples. In multi-unit auctions, bids for marginal units may affect payments for inframarginal units, giving rise to “demand reduction” and furthermore to incentives for shading bids differently across units. We establish that such differential bid shading results generically in ex post inefficient allocations in the uniform-price and pay-as-bid auctions. We also show that, in general, the efficiency and revenue rankings of the two formats are ambiguous. However, in settings with symmetric bidders, the pay-as-bid auction often outperforms. In particular, with diminishing marginal utility, symmetric information and linearity, it yields greater expected revenues. We explain the rankings through multi-unit effects, which have no counterparts in auctions with unit demands. We attribute the new incentives separately to multi-unit (but constant) marginal utility and to diminishing marginal utility. We also provide comparisons with the Vickrey auction.