_The Optimality of Being Efficient_ _4_ Treasury auctions, where resale is especially important. Bidding behavior is significantly affected by resale. Agastya and Daripa (1998) also focus on Treasury auctions, emphasizing the interaction of the futures market, the auction, and resale. Haile (1998), using a reduced-form representation of the resale market, characterizes equilibrium bidding behavior in standard single-good auctions. Haile (1997) examines resale in a setting where bidders acquire additional information after the auction. Haile (1996) empirically tests the model using U.S. Forest Service timber data. Horstmann and LaCasse (1997) consider resale by the seller to a potentially different set of bidders. Our paper is most closely related to several recent studies of optimal auctions with multiple goods. For example, Krishna and Perry (1997) find conditions under which the Vickrey auction is optimal among efficient mechanisms. Armstrong (1997) is also interested in when an optimal auction is efficient. He shows that with two goods and two types, the optimal auction is always efficient (this result, however, does not generalize to three types). Avery and Hendershott (1998) analyze optimal bundling in a multiple objects setting.

Our paper is organized as follows. In section 2, we establish the seller's general incentive to misassign goods and we identify settings where the optimal auction is efficient. In section 3, we solve two variations on the optimal auction, which recognize the possibility of resale. Section 4 proves that perfect resale destroys the seller's incentive to misassign goods. Section _5_ establishes tiit, with perfect resale, any misassignment of goods results in strictly lower seller revenues than the best efficient assignment. In section 6, we show that the Vickrey auction is not distorted by the possibility of resale.

# **2 THE INCENTIVE TO MISASSIGN THE GOOD**

There are two ways an optimal auction can fail to be efficient: (1) the seller can withhold some quantity; and (2) the seller can award quantity to a bidder with a lower marginal value instead of a bidder with a higher marginal value. Myerson (1981) demonstrates both inefficiencies in deriving the optimal auction in an independent private value auction for a single good. We begin by examining the incentive to misassign goods in a multiple object setting.

# **_2.1 Identical objects model_**

For most of the paper, we consider a model with multiple identical objects or close substitutes. The seller has a quantity 1 of a divisible good to sell to n bidders. The seller's valuation for the good equals zero. Each bidder i can consume any quantity qi E [O,Xi], where _Xi_ E (0,1]. We can interpret _qi_ as bidder i's share of the total quantity being auctioned, and Xi as i's capacity or quantity restriction. Let _q = (q l, -,qn), Q = {q I qie_ [O,Xi] and _X-i qi_ S 1}, and Q _=_ {q _I qiE [0,Xi]_ and _Xi qi =_ 1)}. Then _Q_ is the set of all feasible assignments and Q is the set of all feasible assignments in which the seller sells the entire quantity available. Bidder i has a diminishing marginal value, which may depend on all the bidders' private information. Let _ti E Ti_ be bidder i's type, t = _(tl,...,tn), t-i_ = _t\ti,_ and ni = min {ti _I ti E Ti}._ The bidders' types are drawn independently from the distribution functions Fi with full support on _Ti._ A bidder's type is private information; whereas, the value functions, capacities, and distributions of types are common knowledge. The bidders are risk-neutral. A bidder i with marginal value _vi(t,qi)_ who receives quantity qi E [O,Xi] and pays x for it has a payoff **v** :' _(t,y)dy_ - _x._