Optimality of simple procurement auctions

International Journal of Industrial Organization 70 (2020) 102610

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International Journal of Industrial Organization journal homepage: www.elsevier.com/locate/ijio

Optimality of simple procurement auctionsR Oleksii Birulin School of Economics, Sydney University, Social Sciences A02, Sydney, NSW 2006, Australia

a r t i c l e

i n f o

Article history: Received 9 September 2019 Revised 2 March 2020 Accepted 2 March 2020 Available online 16 March 2020 JEL classification: D82 D44 H57

a b s t r a c t We consider procurement auctions for the projects where the cost of production is subject to ex-post shocks—cost overruns. The contractor may default due to these overruns, which affects the buyer’s expected cost. Here the lowest-bid auction emerges as the procurement mechanism that: (i) minimizes the expected transfers to the contractors, and (ii) requires the lowest surety bond to achieve a given probability of default. Since surety bonds are costly to post, the above makes a combination of the lowest-bid auction with the surety bond the optimal, i.e., the expected cost minimizing procurement mechanism in a wide range of parameters. © 2020 Elsevier B.V. All rights reserved.

Keywords: Procurement tenders Cost overruns Surety bonds Optimal auctions

1. Introduction Procurement of goods and services is a major component of public and private spending. In 2013 the government procurement alone represented 12.1% of GDP in OECD countries.1 Given such volumes of expenditures it is extremely important to design procurement methods to ensure the optimal, i.e., cost effective use of funds. The Agreement on Government Procurement (GPA), signed by most industrialized countries in 1996, requires “... transparent, nondiscriminatory, and competitive tendering for public procurement...” Tender rules are not suggested by the GPA, however, the procurers’ choices often converge. If the design of the procured object can be settled before the tender, the sealed lowest-bid auction (or its per unit version) is the most prevalent method of selecting the contractor.2 Despite their widespread use as a method of sales, sealed bid second-price or open ascending auctions are virtually never used for procurement, see Carpineti et al. (2006). Clearly, cost conscious procurers benefit from strong price competition that auctions induce. Nevertheless, the pervasiveness of a particular auction format in procurement practice is puzzling. Indeed, when the potential contractors privately know their production costs, running procurement tender as an auction would be optimal for the buyer. However, with risk-neutral contractors all standard auction formats would deliver the same expected cost. Risk aversion on the contractors’ R I am grateful to Indranil Chakraborty, Robert Gibbons, Vladimir Smirnov, Tom Wilkening and especially Murali Agastya, Sergei Izmalkov and Claudio Mezzetti for many helpful discussions. I thank the co-editor Harry J. Paarsch and the referee for their helpful comments and suggestions. E-mail address: [email protected] 1 OECD (2015), “Size of public procurement”, in Government at a Glance 2015, OECD Publishing, Paris. DOI: https://doi.org/10.1787/gov_ glance- 2015- 42- en 2 Soliciting quotes for construction or renovation of private dwellings or interstate freight transportation mimics the lowest-bid auction. Each contractor submits his quote privately, with the understanding that the compensation will be determined by his own quote.

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O. Birulin / International Journal of Industrial Organization 70 (2020) 102610

side may make the lowest-bid auction more appealing to the procurer then, say, the second lowest-bid auction. However, unlike in this paper, with risk-averse contractors the lowest-bid auction is not the optimal mechanism. Moreover, limited liability restricts the contractors’ potential losses from below, making their payoffs convex, not concave. A clue to the puzzle, we believe, lies in the uncertainty with respect to the production costs that the contractors face themselves. Many of the objects being procured do not exist at the time of the procurement tenders. The full costs of such objects are rarely known in advance even to the contractors and during production may well exceed the estimates available at the time of the tender.3 When cost overruns are significant the contractors may default and leave behind an unfinished project. Such default substantially affects the procurer, she may have to experience delays, search for another contractor, face litigation, etc. This equivalently applies to the conflicts (triggered by cost overruns) that lead to halts in the project execution. For expositional simplicity we focus on the extreme form of such conflicts – defaults. Asymmetric information and uncertainty, cost overruns and defaults are all ubiquitous features of the procurement process. The analysis of the optimal mechanism in such setting is challenging. The expected cost of the procurer includes the transfers to be made to the contractor(s) selected at the tender and additional payments that rectify the consequences of possible defaults. The structure of these additional payments, indeed, as we see later, can lead to revenue superiority of particular procurement mechanisms. The existing theoretical studies, however, suggest that the mechanism that minimizes the procurer’s expected cost is quite complex and cannot be implemented by an auction, see e.g. Burguet et al. (2012) and Chillemi and Mezzetti (2014). Even more puzzling is then the fact that real world procurement mechanisms are fairly simple: once the project design is determined, and there is a pool of potential contractors, the winner is by and large selected at the lowest-bid auction. After the allocation surety bonds are extensively used to “insulate” buyers from contractors’ defaults. This paper reconciles these real world practices with theory. In our setting the lowest-bid auction together with a surety bond—the compensation that the contractor pays to the buyer in case of default—emerges as the optimal procurement mechanism.4 This result is due to a novel combination of the model ingredients that arguably describe the procurement process reasonably well. Crucially, the cost of the project is subject to the ex-post shock (cost overrun) that is realized after the contractor makes a significant irrecoverable initial investment.5 Such assumption is particularly plausible in procurement context, where a serious amount of works always takes place before the cost overrun becomes apparent.6 We view the project execution as if it occurs in two distinct stages. The cost of the first stage is all that a contractor (privately) knows at the time of the project allocation. The cost of the second stage—the value of the cost overrun—is unknown at that time. It will be discovered by the contractor working on the project after the cost of the first stage is sunk. A cost overrun may lead to the contractor’s default, and subsequently, to complete the project, the buyer has to cover the cost overrun herself. The buyer’s expected cost then consists of the expected transfers to the contractors, of the further payments proportional to the expected probability of default and of the direct costs of deterrence against defaults. As we show, a combination of simple instruments: the lowest-bid auction together with the surety bond minimizes the buyer’s expected cost, i.e. such combination is the optimal procurement mechanism in our setting.7 We further provide the intuition on why this was not the case in the prior literature. In all of the earlier models of both sales and procurement where default is a concern the types used at the allocation also directly affect the default decision.8 This leads to the so called insolvency effect. In a standard auction IPV setting the bidder’s expected transfer depends only on his expected allocation. With insolvency effect the contractor’s expected transfer also depends on his probability of solvency. This is a major obstacle for the analysis of the optimal mechanism since any measure (say, a surety bond) that decreases the probability of default also increases the contractor’s expected transfer. In our model the cost overrun is realized after the type used at the allocation is sunk. The contractor’s decision to default depends on this type, but importantly, only indirectly, through the equilibrium award. As we show, the insolvency effect is then absent, and the expected transfer to the contractor is determined by his expected allocation, see Proposition 1. The expected cost equivalence for the seller, however, does not follow. The expected probability of default and hence the buyer’s expected cost depend both on the allocation and payment rules and on the surety bond level. Since the insolvency effect is absent, however, the expected transfers and the expected probability of default can be minimized by separate instruments that do not conflict with each other. We show that for a given allocation and given surety bond, the buyer’s expected cost is minimized by the “pay-what-yousay” mechanism where the contractor’s award only depends on his own private information, as in the lowest-bid auction,

3 In 9 out of 10 public infrastructure projects the cost exceed the original estimate with the average cost overrun at 28% of the winning bid (Flyvbjerg et al., 2002). A more recent EU report (2009) focusing on Rail, Road, Urban Transport, Water and Energy sectors, finds the average cost overruns to be about 21% of such bid. Bajari et al. (2014) estimates the average cost overrun to exceed 50% of the winning bid in the USA highway procurement data. 4 See footnote 12 for more details on surety bonds. 5 This distinguishes this setting from procurement models in Spulber (1990), Parlane (2003), Calveras et al. (2004), Burguet et al. (2012) and also from sales models in Waehrer (1995), Zheng (2001) or Board (2007). There all the uncertainty is realized immediately after the allocation. 6 Ashley and Workman (1986) in a survey of contractors and buyers in USA building industry report that project engineering must be 40–60% complete to establish a reasonable estimate for the cost. 7 The use of the surety bonds does not violate the contractors’ participation constraints. 8 In Chillemi and Mezzetti (2014) the cost overrun is revealed after the first stage cost is sunk, as here. The outside option equal to that cost, however, further affects the default decision, which leads to the insolvency effect.

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see Proposition 2. Conversely, for a given allocation the pay-what-you-say mechanism requires the lowest surety bond to achieve a given expected probability of default. The cost of posting the bond increases with the bond size, and it may or may not be optimal to eliminate defaults altogether. If defaults are eliminated at the optimum, the expected transfers reflect the (virtual) costs of production. These will be minimized by allocating the project efficiently, to the lowest cost contractor, as in Myerson (1981). The lowest-bid auction is an efficient pay-what-you-say mechanism, hence it requires the lowest surety bond to eliminate defaults. Therefore, in combination with the appropriate surety bond it is the optimal procurement mechanism. If some defaults are allowed in equilibrium, this conclusion stays true whenever the “virtual costs” are increasing. These virtual costs are endogenous and depend on the mechanism itself. Despite this, we show that in a wide range of parameters the combination of the lowest-bid auction and surety bond remains the optimal procurement mechanism. With only notational changes our analysis applies to the sales of experience durable goods of uncertain value, say apartments with a view. The buyer learns his full value only after using the good for an “introductory” period. After learning the full value she can either keep the good and pay the price set at the allocation stage, equivalent to completing the project in the procurement setting, or return the good to the seller for a pre-specified fee, i.e., default. The seller then resells the good with some discount. The buyer’s type has two components, the first is the consumption value she derives during the introductory period, the second is the further value that can be observed only after using the good for the introductory period. The seller prefers to avoid the return of the good, just as the buyer is averse to defaults in the procurement context. The rest of the paper is organized as follows. Section 2 reviews the literature, Section 3 sets up the model, Section 4 deals with the optimal mechanism, Section 5 provides concluding remarks. 2. Related literature The early theoretical literature on auctions largely focused on procurement, e.g., Holt’s (1980) work on the risk-averse bidders. Later the focus has shifted to sales. With more fine-tuned models mapping the results between the two contexts is not always possible. We study the optimal mechanism for procuring a complex object, like a new building or a custom made software, where the contractor can default. The allocation of corporate assets in Waehrer (1995) and Board (2007) is the closest analog in the literature on sales, but even there the underlying interaction is significantly different from ours. The buyers of corporate assets may adjust their values after the auction. Such adjustments are relatively quick, hence assuming that default may occur immediately after the allocation is plausible. The first-price auction leads to lower probability of default than the second-price auction and higher bonds increase solvency rates, consistently with Propositions 2 and 3 here. Parlane (2003) provides some early insights into procurement auctions with defaults. Burguet et al. (2012) and Chillemi and Mezzetti (2014) offer more elaborate procurement models. In the former the private information in on the contractor’s assets he stands to lose after a default, in the latter it is on their outside options. Both models incorporate the insolvency effect, the optimal mechanism is partially pooling and cannot be implemented by an auction.9 Decarolis (2014) and Decarolis (2018) empirically compare the lowest-bid and the average-bid auctions using Italian construction industry data. The switch from the average- to the lowest-bid auction substantially lowers the winning price, but does not reduce the overall cost unless the bid enforcement is effective. The USA practice requires the use of surety bonds. The default rate is below 1%, and exempting the projects from surety bonds reduces the contractors’ performance, see Giuffrida and Rovigatti (2019). These empirical findings are clearly aligned with our theoretical conclusions. The salience of the first-price auctions has been established on other grounds. They attract more small bidders and can generate higher revenues than other auction formats in Athey et al. (2011). The first-price auction is revenue superior to the second-price auction in a symmetric IPV setting with risk-averse bidders in Holt (1980) and in the asymmetric IPV setting where post auction resale is possible in Hafalir and Krishna (2008). They are less susceptible to bidder collusion than the second-price auctions in Marshall and Marx (2007). In Akbarpour and Li (2019) the first-price auction is shown to be optimal among all static mechanisms that are immune to the manipulations by the auctioneer. 3. The model A risk-neutral buyer procures an indivisible project of sufficient (publicly known) value, so that the option of not realizing the project is unattractive for her. A number, n, of risk-neutral contractors are capable of completing the project. Production occurs in two stages and the total cost of the project is unknown to the contractors. After the allocation the chosen contractor has to make an irrecoverable investment ci that represents all the capital, labor, and managerial resources necessary to complete the first stage of the project. Formally, for every contractor i his ci is a realization of the random variable distributed on [c, c¯] with c.d.f. F and associated log-concave density f. The draws are i.i.d. and ci constitutes i’s private information.10 Further c = (ci , c−i ) stands for the profile of the costs of all the contractor, and C = [c, c¯]n is the support of c. 9 Calveras et al. (2004) incorporate surety bonds into this framework and simulate the optimal surety bond numerically but focus on the second lowestbid auction for simplicity. 10 Some cost components are observable and contractible: e.g., the costs of the materials and of some labor and capital. These can be effectively managed by cost-plus contracts, as in Bajari and Tadelis (2001). However, a significant component of the costs (or cost savings) can be attributed to the unobservable managerial talent, quality of the organization and monitoring, other concurrent projects of the contractors, etc.

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Fig. 1. Timeline.

Once the first stage cost is sunk the contractor faces ex-post risk. To complete the project he may have to incur extra cost (cost overrun). The realization of the cost overrun zi is observed only by the contractor who works on the project and only after he covers the cost of the first stage. The contractors can, of course, anticipate cost overruns, but any information on them remains “soft” until the project is allocated to, say, contractor i and he invests ci into the project. At the allocation stage contractor i know that his zi is distributed on [z, z¯] independently of ci with c.d.f. Fz with associated log-concave density fz .11 The pair (ci , zi ) can be thought of as a two-dimensional type of contractor i with the convention that zi is observed by i only after ci is sunk. This is similar to, say, Riordan and Sappington (1987) where the marginal cost is observed only after the fixed cost is sunk. Before starting on the project, the contractor deposits a surety bond Si with the help of a surety company or by borrowing the funds.12 Upon observing cost overrun zi , the contractor either covers it and completes the project or declares a default. In either case the contractor pays the interest on the bond ri Si to the surety company.13 We allow ri ’s to be different for different contractors, except for Proposition 4 and the discussion at the end of Section 4. Importantly, to avoid a complicated sequential screening problem we assume that the buyer takes the premium rates as given and ri ’s are independent of the current cost draws and not indicative of the chances of default on the current project. In case of a completion the contractor receives the award, bi and also, naturally, gets his bond Si back. In case of a default the contractor loses the bond Si to the buyer, who now faces the cost of project completion.14 She fully captures the effect of the investment that covers the first stage and has to only cover the cost overrun. From the buyer’s perspective the cost of completion is zi + α , where α ≥ 0 denotes the direct costs of default. These may include the adjustment costs of a replacement contractor, effects of delays and litigation, etc. Our results ultimately depend only on the fact that α > 0, not on its exact value. The timing of events is summarized in Fig. 1. The buyer minimizes the expected cost of the project inclusive of all the payments she may face. By the Revelation Prin ciple she decides on the allocation rule q(c ) = (q1 (c ), q2 (c ), . . . , qn (c ) ) with i qi (c ) = 1 in every realization c, the transfer rule b(c ) = (b1 (c ), b2 (c ), . . . , bn (c ) ), and the profile of the surety bonds S = (S1 , S2 , . . . , Sn ).15 As usual, qi (c) is the probability that contractor i obtains the project at cost profile c, bi (c) is the award that i receives if he completes the project and Si is the bond he will lose in case of default. Importantly, bi can depend on the types revealed at the allocation but the size of the bond Si even though it can be contractor specific is chosen ex-ante for each i. This conforms well with the procurement practice, where the surety bonds are determined prior to the project allocation. The buyer is able to commit to the mechanism she proposes, while no such restriction is placed on the contractors. They behave optimally at all stages: upon learning ci , after the allocation, and ex-post upon learning zi . We restrict attention to symmetric mechanisms where q(ci , c−i ) and b(ci , c−i ) are invariant to the permutations of the elements of c−i , and focus on the mechanisms where the transfers between the losing bidders (those with qi (c ) = 0) and the buyer are ruled out. This permits a rather large class of mechanisms, e.g., the lowest-, the second-lowest, the average-bid auctions, the modifications of these formats where certain bids can be rejected, exogenously set price floors, as well as any lotteries or combinations of auctions and lotteries where only the winner is allowed to work on the project and receive the transfers.

11 Distribution Fz may include a mass point at z ≥ 0 which corresponds to no cost overrun. Since the buyer always proceeds with the project we assume z¯ < ∞, although this is invoked formally only in Proposition 4. Independence between post auction shocks and pre auction values is standard in the related literature. 12 Surety bonds in the U.S. are regulated by the Miller Act of 1935 and similar state level legislations. They are also common in Canada and Japan. The purpose of the bond is to protect the buyer from the contractor’s defaults. Surety bonds are issued by surety companies. In case of a default the surety company either has to complete the project or cover the buyer’s cost of completion up to the size of the surety bond. For an easy exposition to the US surety bond industry see Donohue and Thomas (1996). 13 The premium rate for a given contractor is determined by his surety company. According to Giuffrida and Rovigatti (2019) in the U.S. construction industry it varies between 0.5% and 3% p.a. 14 We assume that the contractor himself loses his bond in case of a default, even though the bond was posted by the surety company. A surety company is not an insurance company. A contractor files a General Indemnity Agreement (GIA) with his surety company and indetifies the assets to be used as collateral in case of a default. This protects the surety company against any losses. 15 Since the value of zi is only privately known to the contractor working on the project, the allocation and the transfers in any deterministic incentive compatible mechanism cannot depend on zi .

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4. Optimality of simple procurement mechanisms Consider contractor i with true first stage cost ci who instead reported x and has received the project, that is qi (x, c−i ) > 0. After i invests ci , the value of the cost overrun z is realized.16 Since the investment of ci is irrecoverable, i ’s continuation payoff depends solely on the value of the shock and the terms of trade set at the allocation. If i covers z, he finishes the project and receives the transfer b(x, c−i ) ≡ bi determined by the mechanism. If i chooses not to cover z, he loses Si . Since i behaves optimally, for a given z his payoff is φ (bi , Si ) = max {bi − z, −Si }. It is convenient to also introduce its expectation

(b, S ) = Ez [φ (b, S )] = Ez [(b − z ) · I{z ≤ b + S}] − S · Ez [I{z > b + S}],

(1)

where I is the indicator function. It is easy to see that  is increasing in b and decreasing in S. Note that i’s expected payoff conditional on (bi , Si ) is (bi , Si ) − ri Si which also reflects the cost of posting the surety bond.  Now denote with Q (x ) = C qi (x, c−i )dF (c−i ) i’s expected allocation and consider his expected payoff. Since (bi , Si ) −i

depends on i’s report x, and not on the true value of ci , i’s expected payoff,

Ui (ci , x ) =



C−i

((b(x, c−i ), Si ) − ri Si ) · qi (x, c−i )dF (c−i ) − ci Q (x )

is quasi-linear and decreasing in ci . Incentive compatibility requires Ui (ci , ci ) ≥ Ui (ci , x) for every ci and x, and individual rationality requires Ui (ci , ci ) ≥ 0 for every ci . Then, by the standard Envelope Theorem argument, incentive compatibility implies that Q(ci ) is decreasing and leads to the expected payoff equivalence.17 Proposition 1. In any incentive compatible mechanism with allocation rule q(c) and marginal type payoff U (c ) the expected equilibrium payoff of i, Ui (ci ), depends only on i’s expected allocation and not on his probability of solvency,

Ui (ci ) =

 C−i

((bi (c ), Si ) − ri Si ) · qi (c )dF (c−i ) − ci Q (ci ) = U (c ) +



c ci

Q (t )dt .

(2)

Proposition 1 formalizes one of our main findings—the setting presented in Section 3 is free from the insolvency effect. The type component used at the allocation stage is sunk before the second component of the contractor’s type—the value of the cost overrun is realized. This second component affects the decision to default directly. Such decision also depends on the first component of the contractor’s type, but only indirectly, through his report and equilibrium award. Due to this the contractor’s expected equilibrium payoff is fully determined by his expected allocation and the marginal type payoff. In the rest of the literature the contractor’s decision to default depends on the type revealed at the allocation stage directly.18 This leads to the insolvency effect, i.e., the integral in the right-hand-side of (2) is over the product of the expected allocation and the probability of solvency, see e.g., Eq. (3) in Burguet et al. (2012). Then any measures that decrease the default probability always have a by-product on the contractors’ equilibrium payoffs. This is a major impediment for the analysis of the optimal mechanism that does not arise in our setting. Now consider the buyer’s expected cost conditional on the above (bi , Si ). If contractor i finishes the project, that is when z ≤ bi + Si , the buyer pays him bi . Otherwise the buyer collects Si from i and spends z + α to finish the project after i’s default. Ultimately, for a given z, the buyer’s cost is max {bi , z − Si } + α · I{z > bi + Si } = z + φ (bi , Si ) + α · I{z > bi + Si }. Then, the buyer’s expected cost conditional on (bi , Si ) is

C (bi , Si ) = Ez + (bi , Si ) + α Pr [z > (bi + Si )], where (b, S) is given by (1). Note that C(bi , Si ) by itself contains an expectation over the shocks, but is conditional on the variables set at the allocation stage and prior, that is (bi , Si ). Introduce EC(S ), the ex-ante expected cost of the buyer. Since  i qi (c ) = 1, EC (S ) contains a constant term Ez + α . The variable part of the expected cost, EVC(S ) =

 i

C

(bi (c ), Si ) · qi (c )dF (c ) − α

 i

C

Fz (bi (c ) + Si ) · qi (c )dF (c ).

(3)

The objective of the buyer is to minimize EVC(S ) subject to the IC constraint, summarized as (2) above, IR constraint, i.e.,  U (c ) ≥ 0, and feasibility, i.e., qi ≥ 0 for every i with i qi (c ) = 1. Without cost overruns EVC(S ) would be just the sum of the expected transfers to the contractors. These are the first term in (3). The second term is essentially the ex-ante expected probability of solvency. Note that the buyer is always, i.e., for any value of α > 0, averse to the contractors’ defaults. Since the insolvency effect is absent here, given the allocation rule, the buyer always prefers to lower the default probability.19 16

We further do not index z since only one contractor will observe his second type component. The equilibrium award bi certainly also depends on S and on r, but we keep this dependence implicit when this creates no confusion. 18 In Waehrer (1995) and Board (2007) the ex-post value of the item depends on the signal used at the allocation stage. In Parlane (2003) this applies to the ex-post cost. In Calveras et al. (2004) and Burguet et al. (2012) the assets that the contractor loses in case of default is his type. In Chillemi and Mezzetti (2014) the type simultaneously determines the contractor’s first stage cost and the outside option in case of default. 19 The insovency effect in other settings creates a complicated trade-off between the informational rents conceded to the contractors and the expected losses due to their defaults. Depending on the relative magnitude of the two effects, the buyer may end up default averse or default loving. 17

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The buyer controls the allocation rule q(c) and the payment rule b(c). In addition, she can choose the surety bond level  S. Using (2), setting U (c ) = 0, restricting i qi (c ) = 1 for every c, and performing the change in the order of integration as in Myerson (1981), the variable part of the buyer’s expected cost can be rewritten as EVC(S ) =



 i

C



ci +

F ( ci ) + ri Si − α Fz (bi (c ) + Si ) f ( ci )

· qi (c )dF (c ).

(4)

The ri Si in (4) is the “cost” of posting the bond that contractor i will pass on to the buyer if he gets the project. Note that these costs are independent of the contractors’ types used at the allocation. This makes the surety bonds “costly” without bringing in the insolvency effect into the model. The absence of this effect implies that the probability of default and the expected transfers can be minimized by separate   F ( ci ) instruments that do not conflict with each other. In particular, the expected transfers, the ) · qi (c )dF (c ) term C ( ci + i

f ( ci )

in (4), are invariant to the choice of the payment rule b(c) or the surety bonds S. The cost of posting the bond, ri Si , is itself invariant to choice of the payment   rule. Altering the payment rule the buyer can therefore “selectively target” the expected probability of solvency, the C Fz (bi (c ) + Si ) · qi (c )dF (c ) term in (4). This probability, of course, can be also controlled by i

the choice of S, however, setting S “large enough” is not innocuous due to the cost of posting the bond. Thus, lowering the probability of default by controlling the payment rule is not a vacuous exercise. We start with an example that provides new intuition on this. Example 1. Let S = 0, and set z = 0 and z¯ = ∞ for simplicity. With log-concave Fz the lowest-bid auction generates higher expected probability of solvency and lower expected cost than the second lowest-bid auction. b b b From (1) (b) = bFz (0 ) + 0 (b − z )dFz = bFz (b) − 0 zdFz = 0 Fz (x )dx. In the second lowest-bid auction a contractor with cost c expects zero profit if the award is determined by his own bid, hence bids b(2) (c) such that (b(2 ) (c ) ) = c. In the lowest-bid auction the same contractor bids b(1) (c) such that (b(1 ) (c ) ) = E[Y 1 |Y 1 > c], where Y1 is the lowest cost of the “others”.20 The contractor with the lowest cost c1 wins in either auction format and (upon completion) receives b(1) (c1 ) after (2) the lowest-bid auction. The second the  lowest cost c2 determines  winner’s award in the second lowest-bid auction, b (c2 ). Consider random variables t 1 =  b(1 ) (c1 ) and t 2 =  b(2 ) (c2 ) , transformed (via ) equilibrium awards in the lowest and second lowest-bid auctions. By the standard argument (see i.e. Krishna, 2002) the c.d.f. of t2 is a mean preserving spread of the c.d.f. of t1 .  Since b = −1 (t ), for a given t the probability of the winner’s solvency (t ) = Fz −1 (t ) . The expected probability of solvency is given by (t) integrated with the distribution (of t1 or t2 correspondingly) and therefore depends on the auction format. Since  (b) = Fz (b),  (t ) =

fz (−1 (t ) )  (b)

f z (b ) . When Fz is Fz (b) c.d.f. of t2 is a mean

=

log-concave, its reverse hazard rate σz (b) = fz (b)/Fz (b) is

decreasing, hence (t) is concave. Since the preserving spread of the c.d.f. of t1 , and (t) is concave, the expected probability of solvency is higher after the lowest-bid auction. The expected cost ranking in Example 1, albeit similar to Holt’s (1980) classic revenue ranking with risk averse bidders, is driven by different principles. In Holt’s full commitment IPV setting risk aversion makes contractors bid more aggressively, that is bid lower, in the lowest-bid auction, whereas their bidding behavior in the second lowest-bid auction is unaffected. Since the buyer minimizes the expected cost he prefers the lowest-bid auction.21 Here, introducing the possibility of default (setting S = 0 is without loss of generality) changes the bidding behavior both in the lowest and in the second lowest-bid auctions. The bidding is more aggressive in either auction format, but the expected payoff of the winning contractor is the same in both auctions, recall (2), hence this by itself does not affect the expected cost. The difference in the expected cost stems from the difference in the probability of default, which is higher after the second lowest-bid auction. Note that in the lowest-bid auction, conditional on the information of the winner the award is non-stochastic, while in the second lowest-bid auction with the same conditioning it remains random. The next Proposition builds on this insight and generalizes the above ranking to arbitrary incentive compatible allocation rules. Proposition 2. Fix S and consider incentive compatible mechanisms with the same allocation rule and the same marginal type payoff. Among those, the “pay-what-you-say” mechanism, where the winner’s award does not depend on the types of the others, i.e., bi (c ) = b(ci ), minimizes the expected probability of default.22 All proofs are in Appendix. Burguet et al. (2012) provide a similar result. The expected continuation payoff (b, S) is convex in b, hence reducing the randomness in the payment rule leads to a higher expected award and lower expected probability of default. The default occurs when the realized award falls below a certain threshold. The value of the threshold is random, given by z in Burguet et al. (2012) and by z − S here. Intuitively, when the threshold is lower than the average award, the probability of default is higher when the award itself is more random. When the threshold is higher than the Since  is strictly increasing, the above equations uniquely define the equilibrium bidding strategies b(1) (c) and b(2) (c) in the two auction formats. Matthews (1983) derives the optimal mechanism when the bidders have CARA utilities, Maskin and Riley (1984) provide a more general characterization of the optimal mechanism. 22 Similarly an additional exogenous randomness in the payment rule also hurts the buyer. 20

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Fig. 2. Default frontiers for the lowest-bid auction L and some efficient auction A for surety bonds S and S > S.

average award, the probability of default is higher when the award itself is less random. Log-concavity of Fz ensures that the threshold is high “not too often”. The “default frontiers” for the lowest-bid auction L (solid lines) and some other efficient mechanism A (dotted lines) for two surety bond levels S and S > S are schematically depicted on Fig. 2 for the case where the initial cost draws and the cost overruns are uniformly distributed (hence independent of each other). The region, say, above the curve bL (c, S ) + S = z corresponds to the type profiles (c, z) where a default will occur when the project is allocated at the lowest-bid auction and the surety bond is S. Other regions are defined similarly. Despite the fact that the curves bL (c, S ) + S = z and bA (c, S ) + S = z intersect, the area above bL (c, S ) + S = z curve is smaller than the area above bA (c, S ) + S = z curve. This reflects Proposition 2, since the lowest-bid auction delivers a lower ex-ante expected probability of default. A higher surety bond (or a higher surety bond premium) pushes both of these curves higher (perhaps at different rates), i.e., reduces the expected probability of default for either mechanism. However, by Proposition 2 this does not change the relationship between the areas above the corresponding curves. More generally, Proposition 2 also shows that the same applies to the pay-what-you-say mechanism and any other mechanism A with the same allocation rule and the same marginal type payoff. In view of Proposition 2 to minimize (3) the buyer should restrict her attention to pay-what-you-say mechanisms. The following Proposition 3 further states that in our setting the equilibrium award in any incentive compatible pay-what-yousay mechanism is determined in the manner similar to that in the lowest-bid auction. To determine the equilibrium bid in any pay-what-you-say mechanism a contractor calculates a conditional expectation of the cost of the “next highest” contractor. Instead of the exogenously given distribution of that cost, here the distribution is endogenous, and its decumulative distribution function matches the reduced form allocation rule Q(c).23 From (2) follows that in any pay-what-you-say mechanism with U (c¯ ) = 0,

(b(ci ), Si ) − ri Si = ci +

1 Q ( ci )



c ci

Q (t )dt,

(5)

for every ci with Q(ci ) > 0. Recall that Q(c) is decreasing with 0 ≤ Q(c) ≤ 1, introduce increasing function P (c ) ≡ 1 − Q (c ) and consider random variable x such that Pr [x < c] = P (c ) and Pr [x ≥ c] = Q (c ). Discontinuity points of Q(c) (if any) correspond to mass points in P(c). If Q (c¯ ) > 0, the c.d.f. of x involves a mass point of value Q (c¯ ) at c¯, and if Q(c) < 1 the c.d.f. of x involves a mass point of value 1 − Q (c ) located below c. The following Proposition shows that in a pay-what-you-say mechanism the allocation rule Q(c) determines the winner’s award (b(ci ), Si ) − ri Si in the manner similar to that in the lowest-bid auction. Proposition 3. In any pay-what-you-say mechanism, given Q(c) and S, (i) the equilibrium award b(c) solves

(b(c ), Si ) − ri Si = E[x|x ≥ c], where Pr [x ≥ c] = Q (c ).

(6)

(ii) c ≤ (b(c ), Si ) − ri Si ≤ c¯ for any c, (iii) b decreases in n for every c, (iv) b is increasing in c, (v) b increases in Si and in ri for every c, and (vi) the expected probability of default is decreasing in Si and in ri .24 Note that (6) generalizes the equilibrium bidding strategy in the lowest-bid auction, where Q ∗ (c ) = (1 − F (c ) )n−1 , to an arbitrary incentive compatible allocation rule. (ii ) of Proposition 3 implies that any contractor assigned to the project prefers to post the bond and make the investment that covers the first stage. (iii ) is a known comparative statics with full 23

This applies also to the mechanisms where the allocation is inefficient and includes randomization. The proof follows from the integration by parts when Q is continuously differentiable. The proof in the Appendix relies on Fubini’s theorem and does not require differentiability. By convention E[x|x ≥ c] = c for c such that Q (c ) = 0. 24

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O. Birulin / International Journal of Industrial Organization 70 (2020) 102610

commitment. It extends to pay-what-you-say mechanisms since it relies on the fact that the cost draws are ordered. We later use this comparative statics to characterize the optimal surety bond in Proposition 5. (iv ) implies that for a given Q, a contractor with lower ci expects lower award b(ci ). This feature, called “abnormally low tenders” in prior literature, implies that a mechanism that allocates the project to the contractor with the lowest cost also allocates it to the contractor who is most likely to default. This trade-off is an obstacle to the study of the optimal mechanism when the allocation rule is a sole control of the buyer. The mere availability of additional instruments, however, does not solve the problem. With the insolvency effect, any measure that reduces the probability of default increases the expected transfers. Our setting is free from such conflict. Then, by raising Si the buyer can decrease the probability of i’s default without affecting the expected transfers to the contractor, see (vi ) of Proposition 3. Minimization of the expected transfers requires allocating the project efficiently, to the contractor with the lowest ci , for a given profile S. Proposition 2 suggests that among the efficient mechanisms the lowest-bid auction minimizes the probability of default for a given S. If ri = 0 for every i, it is optimal to eliminate defaults altogether. The optimal level of the bond S∗ for the lowest-bid auction can be obtained from (6). The contractor with c = c who submits the lowest bid has a particularly high risk of default. For such contractor (b(c ), S ) = E[Y 1 ], see Example 1. To eliminate defaults we need b(c ) + S∗ = z¯, and for such b(c), (b(c ), S∗ ) = z¯ − Ez − S∗ . Therefore S∗ = z¯ − Ez − E[Y 1 ]. Proposition 4. Suppose ri = 0 for every i. Then the lowest-bid auction coupled with surety bond S∗ = z¯ − Ez − E[Y 1 ] is an optimal procurement mechanism, it minimizes the expected cost (3). The lowest-bid auction also requires the lowest surety bond to achieve the minimum of the expected cost. With ri > 0 the buyer faces a new trade-off between reducing the expected probability of default and reducing the surety bond premia. Since the bond premium increases with the bond size, it may or may not be optimal to eliminate defaults. If defaults are eliminated at the optimum, the expected transfers reflect the (virtual) costs of production, and these will be minimized by allocating the project efficiently, to the lowest cost contractor. The lowest-bid auction is an efficient paywhat-you-say mechanism, hence it requires the lowest surety bond to eliminate defaults. Therefore, in combination with the appropriate surety bond it is the optimal procurement mechanism, as in Proposition 4. In general, eliminating defaults may be “too costly” due to the ri Si term in (4). We further extend Proposition 4 in two ways. In the next Section 4.1 we characterize the optimal surety bond. In the remainder of this section we argue that in a wide range of parameters, it is optimal to use the lowest-bid auction as the allocation mechanism. The optimal mechanism always allocates the project to the contractor whose “virtual cost” —the integrand of (4)—is the lowest at a given profile of the costs c. The contractors may differ not only in terms of ci but also in terms of ri Si . Suppose that ri = r > 0 for every i. By symmetry Si = S for every i as well, so that the contractors differ only due to the difference in their ci ’s. The lowest-bid auction then always allocates the project to the contractor with the lowest true cost ci . The contractor with the lowest ci always has the lowest virtual cost if and only the virtual cost is increasing in ci . This virtual cost, ϕ (c, S ) ≡ c + Ff ((cc )) − α Fz (b(c ) + S ), is the difference between the transfers, c + Ff ((cc )) , and the probability of solvency. Both of these terms are increasing in c. Whether ϕ is itself increasing in c depends on the mechanism itself (since b depends on Q and S), not only on the exogenous features of the environment.25 We nevertheless argue that in a wide range of parameters ϕ is increasing. Recall that f(c) is log-concave. Then,

  ∂ϕ (c, S ) d F (c ) ∂ b( c ) = c+ − α f z ( b( c ) + S ) , ∂c dc f (c ) ∂c

(7)

d with dc (c + Ff ((cc)) ) ≥ 1.26 To obtain an upper bound on the second term in (7) differentiate both sides of (6) with respect to c for a given allocation Q and given S. Thus,

b (b, S )

∂ b( c ) ∂ b ( c ) d E [x | x ≥ c ] = Fz (b(c ) + S ) = . ∂c ∂c dc

For a log-concave density function f(c) the corresponding decumulative function 1 − F (c ) is log-concave. Since ln Q (c ) = (n − 1 ) ln(1 − F (c ) ), the efficient allocation rule Q(c) is also a log-concave decumulative function. The left-truncated mean of the random variable sampled from such Q(c) satisfies dE[x|x ≥ c]/dc ≤ 1 for every c.27 Thus,

  d F (c ) f z ( b( c ) + S ) ∂ϕ (c, S ) ≥ c+ −α . ∂c dc f (c ) Fz (b(c ) + S )

For a log-concave Fz , the reverse hazard rate σz (x ) = fz (x )/Fz (x ) is decreasing. Many log-concave distributions have a “light tale”, and the reverse hazard rate decreases to 0 very rapidly. As we already know higher S leads to higher b. The fact that the contractors pay the interest on the bond only amplifies this effect. Indeed, the only way the contractors can recover the cost of posting the bond, rS, is via an increase in the equilibrium award b. This, of course, further raises b(c ) + S, which, in turn, decreases the reverse hazard rate σz (b(c ) + S ). Naturally σz (b(c ) + S ) ≤ σz (S ) for every c. The bond, S, without 25 26 27

Hence, we cannot rely on the usual techniques, like “ironing”, to make ϕ non-decreasing. The inequality follows from the log-concavity of f(c) and hence of F(c). Burdett (1996) contains a nice summary of results on log-concave distributions.

O. Birulin / International Journal of Industrial Organization 70 (2020) 102610

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F c c + f ((c )) ≥ 1 this makes the virtual cost ϕ (c, S) increasing in c. The efficient allocation rule is then the only one consistent with optimality. Such outcome is particularly likely when r and α are not too large. We already know that the optimal mechanism is in the pay-what-you-say class. Whenever the virtual cost ϕ is increasing, insisting on the efficient allocation is optimal. The lowest-bid auction coupled with the surety bond then emerges as the uniquely optimal procurement mechanism. We reiterate that this conclusion may well stand true when defaults are expected in equilibrium. When ri are not the same across the contractors, their virtual costs also differ due to the differences in ri Si . The formal argument for the optimality of the lowest-bid auction is then less clear cut. Notice, however, that other things equal, the virtual cost is higher for the contractor with high ri Si . Note also that either high ri or high Si increases the contractor’s bid in the lowest-bid auction, hence the buyer who simply selects the contractor with the lowest bid may not be that far from selecting the contractor with the lowest virtual cost as she should in the optimal procurement mechanism.

eliminating defaults entirely, may well decrease σ z (S) below the level of 1/α for every c. With

d dc

4.1. Optimal surety bonds This section characterizes the optimal level of the surety bond for a given, and not necessarily efficient, allocation rule. Refer once again to the expected cost (4). It contains the expected transfers invariant to the choice of Si ’s, the expected probability of solvency and the cost of posting the bond. Rewrite the expected probability of solvency in terms of the expected allocation Q(c). Thus, the buyer chooses Si ’s to minimize EVC(Si , ri ) = Eci



i

[(ri Si − α Fz (b(ci ) + Si ) ) · Q (ci )],

(8)

where b(ci ) is, of course, connected to Si via (6) for a given Q(c). As argued above, bi increases in Si so that the expected probability of solvency increases in Si . Since (8) is a difference of two increasing functions, the optimal Si can be in the corner, i.e., either Si = 0, or Si fully eliminates defaults as in Proposition 4. We are concerned with the cases where Si is interior and allows for some defaults. The following Proposition characterizes the optimal (interior) surety bond(s) and provides the comparative statics. Proposition 5. Fix the allocation rule (not necessarily the efficient one) of the optimal mechanism and denote the corresponding expected allocation with Q(c). The optimal procurement mechanism is the pay-what-you-say mechanism with the payment rule given by (6) and the optimal surety bond Si is given by Eci

r i

 i

1 + ri



− ασz (b(ci ) + Si ) · Q (ci ) = 0,

(9)

where σ z is the reverse hazard rate of Fz . i) A higher bond premium ri , or a lower effective cost of default α command a lower Si . ii) Consider two log-concave distributions Fz and Gz such that Fz dominates Gz in terms of the reverse hazard rate. Then the optimal Si with Fz is higher. iii) If ri = r for all i, then a higher n leads to a higher S. We further reconcile Proposition 5 with the earlier literature. Waehrer (1995) examines sales auctions with defaults. The seller’s payoff is either decreasing in the level of the bond or constant depending on what can be “recovered” after the default. When the seller recovers the winner’s value she is “default loving” and prefers a lower bond, when she recovers a lower constant value, she is indifferent to the size of the bond. In a loose parallel to this variation in the “recovery regime” lower α corresponds to higher value recovery. Our buyer, however, will always choose higher surety bond when the bankruptcy has higher social cost, i.e., α is higher. In procurement auctions with full commitment the expected cost decreases in the number of bidders, n, due to the more aggressive, i.e., lower, bidding. Here the lower informational rents can be offset by the higher expected probability of default. Hence whether n raises or lowers the expected cost is ambiguous and depends on α . Proposition 5 concentrates, instead, on the optimal size of the surety bond. The bond does not affect the informational rents. Varying the bond the buyer operates on the probability of default vs. the bond premium margin. The cost overrun is independent of n. The number of bidders then affects the optimal bond only through the competition at the allocation stage. For a given S tighter competition, i.e. higher n, leads to a lower equilibrium award in any pay-what-you-say mechanism. A lower award leads to a lower expected probability of default. To counter this negative effect the optimal bond has to increase with n, as in conclusion ii) of Proposition 5. 5. Discussion Procured objects differ in their complexity. For the “simple” ones, like stationary, most procurement methods are equivalent since the production costs are known. Extremely complex and unique projects, like new weaponry or large infrastructure, cause obvious challenges. It is difficult to identify even one qualified contractor, and his default would most likely mean the whole project cancellation. The project design cannot be resolved without the contractor’s input and will likely be adjusted during the production. The cases of scandalous cost overruns are largely from this class. When the procurer and

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O. Birulin / International Journal of Industrial Organization 70 (2020) 102610

the contractor sit as equal at the bargaining table and can blame the cost overruns on each other, the project’s budget is likely to soar. Then there is a whole range of objects of “intermediate” complexity. On the one hand, they are not too complex so that the procurer can fix the design ex-ante, and there are several potential contractors to choose from. On the other hand, the objects are not too simple, so that the production costs are unknown in advance. Optimal procurement of such objects is the theme of this paper. The optimal mechanism here is not the same as the ones proposed for unique projects. In our setting the procurer never takes the blame for the cost overruns, dominates at the allocation stage and deals with any disputes with the contractors very harshly. The inability to blame your misfortune on the others is a serious incentive instrument. This paper gives a context where this can be formalized. An important takeaway is that different classes of objects pose different challenges, and a one-size-fits-all approach to their procurement is unwise. In this paper the set of potential contractors is taken as given, however, entry into procurement tenders is an important question for future research. Typically with endogenous entry, the bidders participate in the auction if their expected payoff exceeds the entry cost. In a symmetric IPV setting the auction format is then irrelevant due to the expected payoff equivalence. With heterogeneous bidders the format matters, see e.g., Athey et al. (2011). In the setting explored in this paper the contractor may not participate in the tender not only due to his low expected payoff but also due to his inability to post the surety bond. Assume that all the contractors have the same exogenously given “chance” to clear the bond requirement, and this chance is decreasing in the size of the bond. By lowering the bond the procurer increases entry, however, when defaults are possible encouraging participation does not always benefit the procurer. Higher entry decreases the expected transfers, but at the same time it decreases the winner’s award and increases the chances of default. The optimal bond size under such scenario is an important question, but the optimal auction format is likely to be shaped by the same principles as here; the format does not affect the expected transfers, but matters for the probability of default. Interestingly, since different auction formats require different sizes of the bond to achieve the same expected cost, the auction format may also affect entry even in a completely symmetric IPV setting as the one outlined above. Another possible extension can relax independence between the initial cost draw and the cost overrun. Such problem could be amenable to recently developed dynamic mechanism design, although unlike in the traditional dynamic setting the decision after the cost overrun obviously should depend on the allocation at the first stage. The optimal (dynamic) mechanism is unlikely to be simple, since allowing correlation between the ex-post shock and the signal used at the allocation stage (to an extent) reintroduces the insolvency effect in the model. This paper also suggests further empirical work, in particular, assessing the effects of contract enforcement measures on the bidding and the project costs.28 Bajari et al. (2014) may serve as a benchmark. They study cost overruns using the data on procurement auctions for highways construction. Before the bidding all the contractors are provided with the estimates of the quantities of the inputs needed, and they bid in the per unit input prices. The estimates can differ from the actual quantities, but in Bajari et al.. the latter are assumed to be known to all the contractors ex-ante. In this paper the contractors are uncertain about the actual quantities and learn them once the project unfolds. This affects their bidding behavior as long as the procurer does not absorb all the cost overruns. Positive effects of contract enforcement measures on the bids would suggest that uncertainty (with respect to the quantities) on the contractors’ side plays an important role. Appendix. Proofs Proof of Proposition 2. Fix S and consider the probabilities of default in the mechanism with bi = bi (ci ), and any other incentive compatible mechanism. When bi ≥ z¯ − Si there is no default in the former. Then focus on the realization ci where bi < z¯ − Si . To minimize on notation introduce dGi (c ) ≡ qi (c )dF (c−i ). From (2)



C−i

(bi (c ), Si )dGi (c ) =



C−i

(bi , Si )dGi (c ).

Note that for b + S < z¯, (b, S ) = (b − z )Fz (z ) +

 b+S z

Fz (x )dx − S. This implies





C−i

bi +Si

z

Fz (x )dx −



bAi (c )+Si

z

 b+S z

(b − z )dFz − S

 z¯ b+S

dFz . Integrating by parts and rearranging, (b, S ) =

 Fz (x )dx dGi (c ) = 0.

Next concentrate on the inner integrals. Partition the set of c−i into set , where bi ≥ bAi (ci , c−i ) and the complementary set C . Then the last integral is equivalent to

 

bi +Si

 bAi (c )+Si

28

Fz (x )dxdGi (c ) −



 C

bAi (c )+Si

bi +Si

Fz (x )dxdGi (c ).

Giuffrida and Rovigatti (2019) appears to be the first study of surety bonds in the procurement auctions context.

O. Birulin / International Journal of Industrial Organization 70 (2020) 102610

11

Dividing and multiplying both integrals by fz (x)

 

bi +Si

 bAi (c )+Si

Fz (x ) f f z (x ) z

Fz (b + S ) f z (b + S )

≤



(x )dxdGi (c ) −

 

bi +Si

 bAi (c )+Si

qi ( c )

C



bAi (c )+Si bi +Si

fz (x )dxdGi (c ) −





C

Fz (x ) f f z (x ) z

bAi (c )+Si bi +Si

(x )dxdGi (c )



fz (x )dxdGi (c ) .

The inequality follows since Fz /fz is increasing for log-concave Fz . Thus

 

bi +Si

 bAi (c )+Si



fz (x )dxdGi (c ) −



C−i

bi +Si z



 C

fz (x )dx −

bAi (c )+Si bi +Si



bAi (c )+Si z

fz (x )dxdGi (c ) ≥ 0, so that

 fz (x )dx dGi (c ) ≥ 0.

Since the above argument applies for any ci , Si , and qi ,

 i



C

Fz (b(ci ) + Si )dGi (c ) ≥





Fz



C

i

bAi (c ) + Si



dGi (c ),

where on the left is the expected probability of no default in the mechanism with bi (c ) = bi (ci ), and on the right is that probability for any other incentive compatible mechanism with the same allocation rule and S. Proof of Proposition 3. Extending (b(c ), S ) − rS = c for c with Q (c ) = 0 does not violate incentive compatibility. Further, c relying on Fubini’s theorem, we show that c + Q 1(c ) c Q (t )dt = E[x|x ≥ c]. Without assuming differentiability of Q we still have,

(b(c ), Si ) − ri Si = c¯ +

1 Q (c )

= c¯ −

1 Q (c )



c c



c c

(Q (t ) − Q (c ) )dt = c¯ − 

c c

1 Q (c )

I{x < t }dP (x )dt = c¯ −



c c

1 Q (c )

Pr [c ≤ x < t ]dt =



c



c

x

c

dtdP (x ),

where I is, as usual, the indicator function. The last equality is obtained by changing the order of integration. Further,

(b(c ), Si ) − ri Si = c¯ +

1 Q (c )



c¯ c

(x − c¯ )dP (x ) =

1 c¯ · Q (c¯ ) + Q (c ) Q (c )

 c

c¯

xdP (x ) ≡ E[x|x ≥ c].

(ii) clearly follows from (i). To establish (iii ) − (v ) recall that for bi + Si < z¯, (bi , Si ) − ri Si =

 b +S z

i

i

Fz (x )dx − (1 + ri )Si . For

bi + Si ≥ z¯, i finishes the project regardless of z so that (bi , Si ) − ri Si = bi − Ez − ri Si The above implies that (bi , Si ) − rS is increasing in bi , which together with the fact that E[x|x ≥ c] is increasing in c implies that bi increases in c for every Si , which gives (iv). Further note that (bi , Si ) − ri Si also decreases in Si and in ri hence bi increases in Si and in ri for every c, which gives (v). Since E[x|x ≥ c] decreases in n for every c and (bi , Si ) − ri Si depends on n only through bi , (iii) follows as well. Since b(c) increases with Si , the expected probability of default, 1 − Fz (b(c ) + Si ) decreases in Si , which gives (vi). .29

Proof of Proposition 5. Since we are focusing on the interior optimal Si it is appropriate to use the first-order condition of (8) to characterize the optimum. The optimal (interior) Si satisfies Eci









Q ( ci ) ri − α f z ( b( ci , Si ) + Si )

i

∂ b( ci , Si ) +1 ∂ Si



= 0.

(10)

The rate at which the awards increase with the bond level, ∂ b(c, S)/∂ S can be obtained from (6). Totally differentiate (6) with respect to b and Si for a given c. Since the right-hand side does not depend on Si this implies

b (b(c, Si ), Si )db = (ri − S (b(c, Si ), Si ))dSi . Since (b(c, S ), S ) =

 b(c,S )+S z

(11)

Fz (t )dt − S, it follows that

b (b(c, Si ), Si ) = Fz (b(c, Si ) + Si ) and S (b(c, Si ), Si ) = Fz (b(c, Si ) + Si ) − 1. Together with (11) those imply

∂ b( ci , Si ) ri − S (b(ci , Si ), Si ) 1 + ri +1= +1= . ∂ Si b (b(ci , Si ), Si ) Fz (b(ci , Si ) + Si ) 29

Note that

 b+S z

Fz (x )dx − S = b − Ez for b + S = z, since

z z

Fz (x )dx = z − Ez.

(12)

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O. Birulin / International Journal of Industrial Organization 70 (2020) 102610

The expression Proposition 3 b(ci ) inates Gz in terms and b(ci ) increases

(9) in the Proposition results from combining (10) and (12). Clearly ri /(1 + ri ) increases in ri . By increases in ri and in Si . Since Fz is log-concave, σ z decreases in b(ci ) + Si . These facts give i). Fz domof the reverse hazard rate implies fz (t)/Fz (t) ≥ gz (t)/Gz (t) for any t. These ratios are also decreasing in t in Si for any ci which implies (ii). To obtain (iii) note that with ri = r for all i the optimal bond satisfies

Eci [Q (ci ) · σz (b(ci ) + S )] =

1

r

αn 1 + r

,

so that the comparative statics with respect to n mimics the comparative statics with respect to α .



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