Optimal auctions with endogenous budgets

Transcription

1 Optimal auctions with endogenous budgets Brian Baisa and Stanisla Rabinoich September 14, 2015 Abstract We study the benchmark independent priate alue auction setting when bidders hae endogenously determined budgets. Before bidding, a bidder decides how much money she will borrow. Bidders incur a cost to borrowing. We show that bidders are indifferent between participating in a first-price, second-price and all-pay auction. The all-pay auction gies higher reenue than the first-price auction, which gies higher reenue than the second price auction. In addition, when the distribution of alues satisfies the monotone hazard rate condition, the reenue maximizing auction is implemented by an all-pay auction with a suitably chosen resere price. Keywords: Optimal auction; All-pay auction; Budget Constraints; Liquidity. JEL classification: D44; D47; D82 bbaisa@amherst.edu; Amherst College, Department of Economics. srabinoich@amherst.edu; Amherst College and Uniersity of Pittsburgh, Department of Economics. We thank Justin Burkett, Jun Ishii, and Mallesh Pai for helpful comments. 1

2 1 Introduction The seminal papers on auction design assume that a bidder s ability to pay for a good exceeds her willingness to pay; preferences are quasilinear. 1 Yet, in many well-studied auction markets, this restriction does not hold - bidders face binding budget constraints. Authors hae argued that the presence of budgets limits the applicability of auction theory in real-world settings. 2 In response, a literature deeloped that analyzed how the presence of budgets changes the auction design problem. For example, Che and Gale (1996, 1998) compare standard auctions with budgets, and Laffont and Robert (1996), Che and Gale (2000), and Pai and Vohra (2014) construct reenue-maximizing auctions when bidders hae budgets. The aboe literature assumes that budgets are exogenously determined. 3 In practice, howeer, bidders can choose the amount of resources deoted towards bidding in the auction. Bidding in the auction requires liquid resources, which can be obtained by borrowing or dierting resources away from alternatie profitable inestments. does not borrow from a bank, she still incurs an opportunity cost of funds. Thus, een if a bidder This raises the question: what is the optimal selling mechanism when buyers endogenously make such liquidity choices? This question is important not only for the auctions literature, but also for the growing literature in monetary economics that models liquidity choices for the purpose of decentralized trade. 4 In this paper, we study auction design with endogenously determined budgets. We consider an auction for an indiisible good, where bidders hae independent priate alues, and endogenously determine their budgets after obsering their priate information. Borrowing is costly, and a bidder incurs a cost of borrowing whether or not her bid wins. We show that bidders are indifferent between competing in the first-price, second-price and all-pay auctions. 5 Howeer, the auctions are not reenue-equialent. The all-pay auction has the highest expected reenue, and the second-price auction has the lowest expected reenue. This reenue ranking of the standard auctions is the same as in the exogenous 1 See Myerson (1981) and Riley and Samuelson (1981). 2 See Rothkopf (2007). 3 Two exceptions are Burkett (2015a, 2015b), where budgets are endogenously determined in a principalagent relationship, and Rhodes-Kropf and Viswanathan (2005), who consider a ariety of forms of endogenous financing in a first-price auction for a risky asset. 4 Our paper s enironment, in which agents choose their money holdings before engaging in trade, is reminiscent of the monetary models in Lagos and Wright (2005) and Rocheteau and Wright (2005). See Lagos, Rocheteau and Wright (2015) for a recent surey. In this literature, the pricing mechanism is crucial for the determination of real balances, output, and efficiency. A recent strand of the literature, such as Galenianos and Kircher (2008), has introduced auctions into monetary models. Howeer, this literature assumes that goods are sold using a second-price auction, without examining whether this auction format is optimal. 5 This is similar to the benchmark quasilinear setting and in exogenous budgets case. (See Che and Gale (1996, 1998)). 2

3 budget case of Che and Gale (1996), but the intuition is distinct. In Che and Gale (1996), the all-pay auction yields higher reenues than first or second price because the budget constraint is less likely to bind in the all-pay. In our model, the all-pay auction yields higher reenues because it economizes on the bidders borrowing costs. Turning to the design of reenue-maximizing auctions, we show that the optimal auction can be implemented by an all-pay auction with a suitably chosen resere price. 6 This differs from an optimal auction with exogenous budgets: the latter would not necessarily sell the good to the highest-alue bidder. 7 The reason for the difference is that, with exogenous budgets, high-alue bidders are unable to express high demand for the good. With endogenous budgets, high-alue bidders are able to reeal that they hae a higher demand by borrowing more money to place higher bids. While placing higher bids comes at a cost, the auctioneer can minimize these costs by using an all-pay payment scheme. 2 Model 2.1 Enironment A seller has one unit of an indiisible good, which she has no alue for. There are N 2 risk-neutral potential buyers. A buyer s preferences are described by a single dimensional ariable, her aluation, which is independently and identically distributed across bidders with a density f that has full support oer [, ] R +. Thus, f has an associated distribution function F which is continuous and strictly increasing oer [, ], with F () =0and F () =1. Abidderdeterminesherbudgetafterfindingoutheralue,butbeforeplacingherbid and obsering competing bids. If a bidder borrows b from the bank, she repays b + c(b), where c(b) is continuous, differentiable, strictly increasing, and weakly conex. 8 Thus, if bidder i with aluation i wins the good, pays p to the auctioneer and has borrowed b, where b p, herutilityis i p c(b). Abiddercannotplaceabidthatexceedstheamountofmoneythatshehasborrowed. 6 This requires that bidders hae monotone irtual alues, as in Myerson (1981). 7 For example, in Laffont and Robert s (1996) optimal auction, all bidders with alues aboe some threshold win with equal probability. 8 When determining the optimal auction, we assume that c is linear. 3

4 2.2 Mechanisms By the reelation principle, we limit attention to direct reelation mechanisms. Gien the profile of reported types =( 1,..., N ),thedirectreelationmechanismstatesabidder s probability of winning Q i (), expectedtransfert i (), and borrowing b i ( i ). The amount a bidder borrows is independent of other bidders reported types, because a bidder decides her budget before bidding. Since borrowing is costly, bidders do not borrow more money than they need to place their bid; hence b i () = max i T i (). Feasibilityrequires NX Q i () apple 1. i=1 Denote by q i () =E i (Q i (, i )) bidder i s interim probability of winning when reporting type. Similarly, t i () =E i (T i (, i )) denotes the interim expected payment made by bidder i. Therefore, the expected utility of bidder i, ifhertruetypeis i and she reports type, is U i (, i )=q i () i t i () c(b i ()). The direct reelation mechanism is (interim) incentie-compatible if U i ( i, i ) U i (, i ) 8 2 [, ], i=1,...,n. 3 Standard Auctions Incentie compatibility implies that q i () and t i () are weakly increasing. Thus, both functions are differentiable almost eerywhere along [, ]. Pick any point where U i (, i ) is differentiable with respect to at i. The necessary first order condition for incentie compatibility i (, i = q 0 i() i t 0 i() c 0 (t i ())t 0 i() =0. Therefore, the total deriatie of U i ( i, i ) with respect to i is du i ( i, i ) d i = q i ( i ). Since U i is differentiable almost eerywhere, this implies U i ( i, i )=U i (,)+ ˆ i q i (s)ds. (1) 4

5 Thus, the standard Myersonian approach can be used to characterize bidder i s interim expected utility. Equation (1) implies that bidder i is indifferent between any two mechanisms that gie the same interim probability of winning, and gie the same expected utility to the low type. In particular, bidders are indifferent between participating in the first-price, second-price, and all-pay auction. We use equation (1) to establish a reenue ranking of the three standard auctions. We consider symmetric Bayes Nash Equilibria. In all three mechanisms, equilibrium bid functions are strictly increasing in reported alues. Proposition 1. (Reenue ranking of standard auctions) The all-pay auction has strictly greater expected reenue than the first price auction. The first price auction has strictly greater expected reenue than the second price auction. Proof. Since bids are strictly increasing in each auction, q() =F () N 1,andU(,)=0. Let b f ( i ), b s ( i ),andb a ( i ) be the symmetric equilibrium bid functions in the first price auction, the second price auction, and the all-pay auction, respectiely. First, we show that b f ( i ) >b a ( i ) 8 i 2 (, ). Equation (1) implies that and therefore F ( i ) N 1 i b f ( i ) c(b f ( i )) = F ( i ) N 1 i b a ( i ) c(b a ( i )), F ( i ) N 1 b f ( i )+c(b f ( i )) = b a ( i )+c(b a ( i )). Since F ( i ) N 1 2 (0, 1) 8 i 2 (, ), it follows that b f ( i ) >b a ( i ) 8 i 2 (, ). But then, c(b f ( i )) >c(b a ( i )), andsotheaboeequalityimmediatelyimpliesb a ( i ) >F( i ) N 1 b f ( i ) for all i 2 (, ). Thus, bidder i makes a greater expected payment in the all pay auction ersus the first price auction conditional on being type i. The proof that the first price has higher reenue than the second price is similar. In the first price auction, a bidder s utility when she is type i is F ( i ) N 1 i b f ( i ) c(b f ( i )). In the second price auction, a bidder s utility when she is type i is F ( i ) N 1 i E(max b s ( j ) i j6=i max j ) j6=i Equation (1) implies that these utilities must be equal, and so c(b s ( i )). 5

6 F ( i ) N 1 b f ( i )+c(b f ( i )) = F ( i ) N 1 E(max b s ( j ) i j6=i max j )+c(b s ( i )). j6=i Since b s ( i ) > E(max j6=i b s ( j ) i max j6=i j ) 8 i >, the aboe equality implies b s ( i ) > b f ( i ) 8 i >. This, in turn, implies c(b s ( i )) > c(b f ( i )), and so the aboe equality immediately implies F ( i ) N 1 b f ( i ) F ( i ) N 1 E(max j6=i b s ( j ) i max j6=i j ) for all i 2 (, ). Thus, bidder i makes a higher expected payment conditional on winning in the first price auction than in the second price auction. Since both auctions assign the good to the bidder with the highest i,thisimpliesthattheexpectedtotalpaymentmadeinthe first price auction exceeds the expected total payment made in the second price auction. Although Proposition 1 mirrors results from Che and Gale (1996, 2006) with exogenous budgets, the intuition is different. With exogenous budgets, Che and Gale argue that the all-pay auction gies higher reenues because the budget constraint is less likely to bind in an all-pay auction. In our setting, the all-pay auction gies greater reenues because bidders incur lower bid preparation costs. Since the equilibrium bid distribution is higher in the first-price or second-price auction, bidders spend more money to finance their bids. It may seem that bidders should then prefer the all-pay auction to the first or second-price auctions. Howeer, equation (1) shows us that the gains in surplus generated by borrowing less are competed away in the auction enironment. Since the bidders borrow less money, yet get the same surplus, it then must be the case that the extra surplus goes to the auctioneer. 4 Reenue maximizing auction In this section, we assume that c(b) =rb: thereisafixedinterestrater>0. Wefirstshow that the optimal mechanism has an all-pay payment structure. For ease of notation, we write U i (, ) as U i (). Lemma 1. Consider any two incentie-compatible direct reelation mechanisms, where Ui () =U i () and qi () =q i () 8. In addition, suppose that b i () =t i () 8. Then, t i () t i () 8. Proof. Let q i () =qi () =q i (). Byequation(1)wehae q i () (1 + r)t i () = and so ˆ q i (s)ds = q i () t i () rb i (), 6

7 (1 + r)t i () =t i ()+rb i (). From the restriction b i () > max i T i (), it follows that b i () t i (). This implies that (1 + r)t i () =t i ()+rb i () (1 + r)t i (), andthereforet i () t i (). Thus, when considering reenue-maximizing auctions, it suffices to consider only all-pay mechanisms, i.e. mechanisms in which b i () =t i () 8. Fromequation(1), wethengetthat for any incentie compatible allocation function q i, where U() =0, U i () = ˆ q i (s)ds = q i () (1 + r)t i (). Bidder i s expected payment can therefore be written as a function of the allocation function, Thus, expected reenue is t i () = 1 q i () 1+r ˆ q i (s)ds. 1 1+r NX i=1 ˆ q i () ˆ q i (s)ds f()d. This is equialent to the problem soled by Myerson (1981) and Riley and Samuelson (1981). Using their results, we can rewrite the aboe expression in terms of irtual alues. () = equal to 1 F () f() be the irtual alue of a bidder with type. Expected reenue is then 1 1+r NX E (q i () ()). Maximizing pointwise subject to feasibility shows that, if Thus, if i=1 8 < 1 if () 0 and i > max i Q i ( i, i )=. : 0 if () < 0 or max i > i () is weakly increasing, () is weakly increasing, the optimal mechanism can be implemented by an all-pay auction with a suitably chosen resere price b. 9, where Let The resere price makes a bidder with type ( )=0, indifferent between bidding and not bidding. That is, b is such that F ( ) N 1 (1 + r)b =0. 9 If F satisfies the monotone hazard rate condition, then is weakly increasing in. 7

8 We hae thus established the following result: Proposition 2. Suppose that () is weakly increasing. The optimal mechanism can be implemented by an all-pay auction with minimum bid b. The minimum bid b satisfies where is such that b = F ( ) N 1+r 1 = 1 F ( ). f( ) The result differs from results on auctions with exogenous budgets, e.g., Laffont and Robert (1996) and Pai and Vohra (2014). Both papers suggest using a mechanism with an all-pay payment scheme, but a different winning rule. Specifically, they show that the bidder with the highest aluation does not necessarily win the object with probability 1, een if all bidders hae the same budget. Here we show that, when we endogenize the budget constraint, we get a different result, which looks closer to that of Myerson (1981). When bidders hae weakly increasing irtual alues, the optimal mechanism is a standard all-pay auction with a resere price, which allocates the good to the highest-alue bidder., 8

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