ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E23-3

Pay for Performance in Procurement
byJuan-José Ganuza
Fernando Gómez
Makoto Hanazono

May 2023

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITYPay for Performance in Procurement
Juan-José Ganuzaa , Fernando Gómezb and Makoto Hanazonoc
January 2023

Abstract
This paper sheds light on the limitations of “pay-for-performance”in bidding markets with ex post moral hazard. High-powered incentive schemes may lead to worse
allocations. We present our analysis in a procurement setting in which potential contractors differ both in their costs and levels of wealth and are protected by limited
liability. We show first that competitive mechanisms adversely select undercapitalized
firms. Second, more powerful incentive mechanisms induce less desirable allocations:the selected contractor is likely to be at the same time less solvent and less efficient.
As a result, low-powered incentives in procurement may be optimal. This is consistent with the fact, that despite the large social welfare losses arising from inadequate
performance by contractors (delays, cost overruns, deficient quality of output) in procurement, powerful pay-for-performance mechanisms appear to be rare in reality.
Keywords: Procurement; Incentives; Pay for Performance and Limited Liability.

Corresponding author : juanjo.ganuza@upf.edu
aUniversitat Pompeu Fabra and Barcelona School of Economics,b Universitat Pompeu Fabra
School of Law, c Nagoya University.
We are grateful for the comments to Miguel Abito, Francesco Decarolis, Shingo Ishiguro, Steven
Tadelis, participants at The 38th Annual Conference of the European Association of Law and Economics (EALE), Simposio de Analisis Economico 2021, Workshop in Organizational Economics at
Summer Forum 2021, International Industrial Organization Conference, Asian Meeting of Econometric Society 2022, EARIE 2022, and participants at seminars at Universitat Pompeu Fabra,
Kyoto University, Nagoya University, and Waseda University. The usual disclaimers apply. Fernando Gomez acknowledges the financial support of the Spanish Ministry of Science and Innovation
under project DER 2017-82673-R. Juan-Jose Ganuza gratefully acknowledges the support of theBarcelona School of Economics Research, the government of Catalonia, and the Spanish Ministryof Science and Innovation through the project PID2020-115044GB-I00. Juan-Jose Ganuza is also
grateful for the support and the hospitality during his visit at the Economic Research Center,
Nagoya University (as a visiting fellow from June 2022 to August 2022). Makoto Hanazono acknowledges the support of Japan Society for the Promotion of Science, grant number 17KK0067.

“One of the main lessons from working on incentive problems for 25 years is, that withinfirms, high-powered financial incentives can be very dysfunctional...typically, it is best to
avoid high-powered incentives and sometimes not use pay-for-performance at all.” Bengt
Holmstrom, Nobel Prize Lecture (2016).

1

Introduction

Linking payoffs to outcomes (pay-for-performance) provides one of the most powerful economic tools to generate incentives. However, as Holmstrom in his Nobel Prize Lecture(quoted in the opening of the paper) pointed out, it is not difficult in reality to find situations
in which pay-for-performance incentive schemes work poorly. Common concerns expressed
about pay-for-performance refer to issues of multitasking issues and risk imposed on riskaverse agents. In this paper we explore a different shortcoming affecting pay-for-performancemechanisms: they may lead to worse allocations in settings involving competitive bidding
and ex post moral hazard when firms are protected by limited liability.
In this paper we identify a novel trade-off between the provision of incentives and theefficiency of the allocation in bidding markets. The driving force for the inefficiency in the
allocation lies in firms with lower capitalization (who will be bankrupt if they fail to perform
properly and thus will face lower penalties) exerting lower effort and expecting larger net
profits from being awarded the contract. The higher the risk (the larger the incentives), thehigher the competitive advantage of less capitalized firms and thus, the worse the allocation
will turn out to be.
When all bidding firms are fully solvent, a competitive mechanism leads to the selectionof the most efficient firm for the contract or project. This is not true in a more realistic
setting, since undercapitalized firms enjoy an advantage in the bidding contest. Incentive
mechanisms using penalties (and/or bonuses) affect the actions by the contractor in two
ways: first, by directly altering the incentives of the contractor to exert effort; second, byaffecting the incentives of firms to bid, and in this way influencing the outcome of theselection process. The higher the incentives (the penalty for mis-performance or the bonus
for good performance) the larger the likelihood of a firm with poor capitalization and higher
costs of construction to win the selection process.
This trade-off between incentives and efficiency allocation may provide relevant insights
into a number of competitive settings, especially in the design and functioning of procurement contracts. In particular, it may explain why the use of pay-for-performance mechanisms in procurement contracts seems to be of modest proportions compared to the sizeof projects and the repercussions of inadequate performance by contractors. This is a puzzling observation, given the importance of procurement in the functioning of economies,
1both in developed and developing countries,1 and the correspondingly large social welfare
losses arising from poor performance in procurement contracts (delays, deficient quality ofoutput).2
The implementation of a procurement project would typically depend on the terms of the
procurement contract, the contractor’s effort, and also unforeseen contingencies (weather
and other external conditions, shocks to the supply of input and human resources, random catastrophic events) that may arise during the contract’s performance. To tackle thecontractor’s moral hazard problem in such settings, one would expect to find widespreadand significant reliance on explicit incentives, given the magnitudes at stake when contractor’s output is poor in terms of timing and/or quality. In reality, however, one does
not observe high-powered explicit performance incentives in procurement. The most commonly used contractual format appears to be a fixed price contract stipulating price, quality/performance/timing specifications, with some financial penalties if these goals are not
met.3
Penalties, moreover, seem to be low compared to the size of the negative effects from delays or inferior quality. Lewis and Bajari (2011) discuss this apparent anomaly in real-world
procurement contracts, providing an illustrative example regarding highways in California:
they estimate that the social cost from delay was $1.75 million per day, while the daily
penalty for delay was merely $40, 000. Along similar lines, Lewis and Bajari (2014) use dataon Minnesota highway construction contracts and show that the penalty system at work
provided little incentives to reduce delays, and that switching to a more powerful incentive
system would increase welfare with only a small impact on contractors’ costs.4
We show there may be good efficiency reasons for the reluctance of procurement agencies to resort to explicit powerful incentive schemes (penalties and/or bonuses5 ) to improvethe contractors’ level of effort. The implications from our analysis may also add another1

Bosio et al. (2022) report a figure of 12% of global GDP as the share claimed by government procurementin 2018. They also report significantly larger shares for rich countries such as Switzerland, the Netherlands
or Singapore.
2
Lewis and Bajari (2011, 2014) provide evidence of the large social cost of the delays in highway construction in California and Minnesota. Decarolis et al. (2016) document sizable gains from improving performancein a case study of procurement in the electricity sector.
3
The financial penalty is typically determined and exacted ex post—when the delay or defective performance is observed by the sponsor—but it may take two forms: first, a retention (retainage) from thecontract price that the sponsor withholds until the project output is delivered and evaluated; second, afine that the sponsor has to claim from the contractor. The use of retainage is explored experimentally byWalker et al. (2020).
4Effective penalties are even lower. Penalties are imperfectly enforceable since contractors may disputethem in Courts. This is especially detrimental for incentives when the Court system is inefficient, as it isshown by Coviello et al. (2018) with Italian procurement data.
5
We will consider also incentive schemes involving positive payments or bonuses to the contractor, although they appear to be extremely rare, despite its prevalence in other settings (employment and managerial relationships). See Bigoni et al. (2014)2

explanatory factor to the empirical studies that find poorer outcomes in procurement contracts in less developed countries compared to wealthier ones. This is commonly attributed
to poor institutional quality and lower public sector capabilities in the first group of countries. Perhaps also the fact that firms who are candidates for the winning the contract havelower assets may be a contributing factor, given that this is expected to result in lower
contractors’ effort and worse allocations.
We present our analysis in a procurement setting in which a project with ex post moral
hazard has to be allocated among firms that differ in construction costs and assets (capitalization). Final quality of project outcome depends on the contractor’s effort level andexogenous shocks. The contractor is protected by limited liability—the contractor cannot
pay penalties beyond its total asset level—and incentives to exert effort depend on the
incentive (the size of the penalty) and its asset size.
We show three fundamental results. First, competitive mechanisms adversely select undercapitalized firms for undertaking procurement contracts with ex post moral hazard. Second, and counterintuitively, that more powerful incentive mechanisms lead to worse allocations: the winning firm is likely to be less solvent and less efficient in carrying out the
project. Increasing the size of the penalty that the sponsor imposes upon the contractor
for mis-performance worsens the allocation by granting an additional advantage to firms
with less assets. This may result in awarding the contract to a less efficient firm in termsof performance costs. An important implication of this result is that low powered schemes
may dominate more powerful incentive mechanisms that would be optimal absent the riskof default by the contractor. Finally, we extend the basic analysis to consider both penalties
for failing to provide proper performance and bonuses for achieving the goals set out by thesponsor and show that the allocation does not depend on the format of the explicit incentive(penalty or bonus), but solely on its size.6
We also extend our fundamental results to a setting in which the contractor’s performance is a continuous variable and show them to hold as well in this more general setup.
The tools and techniques developed for addressing this challenging extension of the binary
baseline model open the scope of the present paper to apply our results in other competitiveenvironments.
There is an extensive literature dealing with procurement and moral hazard, but previous
contributions have analyzed other dimensions of the problem. Seminal papers by Holt (1979,1980) and McAfee and McMillan (1986) study settings in which contractor’s effort reduces
construction costs and the principal (sponsor) chooses between fixed-price and cost-sharing
6

Bigoni et al. (2014) experimentally explore the use of penalties versus bonuses in a procurement setting
and provide behavioral arguments for which the sponsors have the preference for penalties. First, penalties
allow the sponsor to use exploitative offers taking advantage of contractors’ naı̈veté. Second, loss aversion on
the part of sponsors, who are thereby led to offer less generous contracts with penalties than with bonuses.

3contracts. These papers mainly analyze the trade-off between the cost of transferring risks
to contractors versus providing incentives to reduce construction costs. A second generationof papers, such as McAfee and McMillan (1987) and Laffont and Tirole (1987), characterize
the optimal contract under risk neutrality and unlimited liability. We take as given thecontract format and the procurement mechanism (the second-price auction), but consider
the probability of default and we add the firm’s financial condition as another source of firms’
private information and heterogeneity. Recently, Chakraborty et al. (2021) characterize theoptimal contract in a competitive procurement model with adverse selection and moral
hazard, in which the private information refers to the cost of effort, similar to McAfee andMcMillan (1986). They find that the “truth-telling” constraint for the less efficient type(not to lie but to shirk ex post) matters and the associated incentive cost increases withcompetition. This creates a new trade-off between efficiency in the allocation and incentives,
so that it may be optimal to limit competition or distort the allocation. We look at the
problem from a different perspective and our approach is driven by different forces, obtaining
a result along the same lines: more powerful incentives in procurement are likely to lead toless efficient allocations.
Closely related to our work are the papers dealing with a procurement settings withcost uncertainty and limited liability (Waehrer (1995), Zheng (2001), Calveras, Ganuza andHauk (2004), Engel and Wambach (2006), Board (2007), Chillemi and Mezzetti (2010) and
Burguet, Ganuza and Hauk (2012)).What this literature shows is that in the presence of cost
uncertainty, the contractors’ limited assets pave the way to the likelihood of contractors’default, that limited liability cuts off the potential downside losses of the winning bidder,thus inducing more aggressive bids. This effect is larger for the less solvent bidders, enhancing their chances of winning the contest for the project.7 This literature, however, does notdeal with ex post moral hazard problems in procurement, and does not consider in such asetting the cost and the asset dimensions jointly, a factor that we deem very relevant in thereal-world procurement context.
The negative effect of the probability of default over incentives has been analyzed also
in the law and economics literature. The so-called judgement proof problem, introduced by
Shavell (1986) in a tort environment, is that when injurers are insolvent, first-best behavior
in terms of accident prevention cannot generally be attained through liability rules. In a
companion paper (Ganuza and Gomez (2020)), we analyze specifically a procurement problem in a tort setting where incentives are given by liability rules, show that the negligence
rule dominates strict liability, and argue that negligence standards lower than the first best
7

Manelli and Vincent (1995) and Lopomo et al. (2022), analyze a similar adverse selection problem andshow that standard competitive mechanisms work poorly when low-cost bidders are also linked with lower
quality outcomes.

4ones may be desirable. The structure of the paper is as follows. In Section 2 we present thebasic features of the model. We solve it in Section 3. In Section 4 we show our fundamental
result that more powerful pay-for-performance schemes lead to a worse allocation. In Section 5 we analyze the extensions dealing with penalties and bonuses, with negative markups
and with a continuous performance variable. Section 6 briefly concludes. All proofs arerelegated to the Appendix.

2

Model

We analyze a procurement setting in which a risk-neutral sponsor procures an indivisible
project for which he is willing to pay V . We assume V large enough so as to make thepossibility of not contracting unattractive for the sponsor. N firms compete for the indivisible contract. Firms differ both in their cost of undertaking the project and in their initialfinancial status. Let ci ≥ 0 and wi ≥ 0 denote, respectively, the cost of undertaking the
project and the value of the assets of potential contractor i = 1, 2, ..., N . Both ci and wi arecontractor i’s private information. The contract is awarded using a second-price auction.
Firms submit their bids and the winning firm is the one with the lowest bid, and the priceis set equal to the second lowest bid. Ties are resolved using a lottery.
Denote the auction price by P, and the cost and the assets of the winning firm by c andw, respectively. To simplify the presentation, we ignore the subindex i when we refer to thewinning firm. As we will focus on the weakly dominant strategy equilibrium of the second
price auction, we do not need to specify how firms’ types (ci , wi ) are distributed.
There is a performance/quality measure of the project contract θ (timeliness, quality,etc.) that we initially assume to be binary θ ∈ {θ, θ}. The contractor may invest in effort(for simplicity we assume it to be non-monetary) to improve performance.8 Let x be thecontractor’s monetary equivalent of the performance effort, and p(x) be the probability ofpoor or inadequate performance, where p(x) is decreasing and convex in x.
The sponsor provides incentives to increase effort using a pay-for-performance mechanismb(θ). We first consider a penalty. In case of bad performance θ, the contractor has to pay amonetary sum b, and 0 otherwise. Then, b(θ) = −b and b(θ) = 0. Contractors have limited
liability, i.e., the losses to contractor i from the penalty cannot be larger than the firm’s
total wealth, which at that point is P − ci + wi . Therefore, contractor i will shut down anddiscontinue the project if performance is of poor quality and the penalty, b, exceeds his total
wealth.
We now summarize the timing of the model:
8

In Appendix B, we show that our main results hold when effort is monetary.

51. Nature chooses the cost ci and the financial assets wi of each firm i.
2. A second-price auction takes place.
3. The winning firm receives price P, carries out the project incurring cost c, and exerts
the level of effort x. Poor performance takes place or not, according to probabilityp(x). The contractor pays the penalty if P −c+w −b > 0. Otherwise, the firm declares
bankruptcy and exits.9
We want to analyze how this competitive procurement setting works and what are theimplications for the design of pay-for-performance mechanisms. For most of this analysis
we do not need to specify the sponsor’s preferences. However, for some of our results wewill assume that the sponsor’s preferences are as follows: V − P + θ − b(θ)

Us = 

no default
V + θ − c + w − K otherwise.

This function describes the ex-post sponsor’s preferences that are determined by whether
or not the winning firm goes bankrupt. In case the winning firm completes the project, thesponsor’s welfare depends on the price paid, the kind of performance of the project θ, andthe payment/penalty resulting from the incentive mechanism −b(θ). Bankruptcy may arise
when poor performance is observed and the winning firm is unable to pay the penalty. In
this case, the sponsor takes the remaining assets of the firm P − c + w and incurs somedefault cost K.10
As a benchmark, we occasionally refer to total welfare, that may depend on the cost ofundertaking the project by the winning firm, the project’s degree of performance, the costof effort, and the bankruptcy costs, V +θ−x−c

W =

3

non-default
V + θ − x − c − K otherwise.

Solving the Model

We characterize the equilibrium by backwards induction. We start with the performancestage.
9In many real-world procurement processes, payments and costs may occur over a period of time. As
the firm can borrow funds or defer payments, this should not have an important impact over the firm’s
bankruptcy decision.
10

Typically, the bankruptcy of the winning firm will be costly for the sponsor for several reasons: it may
give rise to litigation costs and delays and will entail costs to complete the project, either by deploying her
own means, or by paying a new contractor to complete the unfinished project.

6

3.1

Performance StageIn this stage, the winning bidder faces an effective monetary penalty in case of poor performance. Due to limited liability, her penalty losses, z, are bounded by her available assets,z = min{P − c + w, b}. Given z, the winning firm chooses a level of effort x∗ (z) which
minimizes her expected total cost at the performance stage,x∗ (z) ∈ arg min p(x)z + x,
where z is the effective penalty faced by the contractor whenever poor performance occurs.
Let γ(z) be the expected private cost of poor performance for the contractor,
γ(z) = p(x∗ (z))z + x∗ (z).

(1)

Intuitively, x∗ (z) and γ(z) are increasing in z. For further results, some additional characterization of γ(z) is useful. In particular, γ(0) = 0, γ(z) < b, γ ′ (z) = p(x∗ (z)) < 1,11 andγ ′′ (z) ≤ 0.
3.2

Bidding StageWe focus on the weakly dominant strategy equilibrium of the second-price auction. In our
procurement setting, this implies that firms bid according to their total cost of undertaking
the project. Hence, the equilibrium bid of firm i is the minimum price Pi∗ for which firm i
is willing to accept the project. We start by characterizing the net expected profits of thecontractor
πN (P, c, w, b) = P − c − γ(min{P − c + w, b})
As 0 ≤ γ ′ (z) ≤ 1, this net expected profits, πN are increasing in P − c and weakly
decreasing in w and b. Then, wealthier firms with higher asset levels have lower net expected
profits for a given markup P − c.
Then, the equilibrium bid is defined by:
E{πN (Pi∗ , ci , wi , b)} = 0,(2)

In other words, the equilibrium bid of firm i is the price for which her net expected profits
are zero, in case firm i wins the project.
If the firm has no risk of default, the price that makes the net expected profits zero isequal to the total cost, construction cost and performance cost, Pi∗ = ci +γ(b). Equivalently,
11

∗

∂γ ∂x
∂γ
∂γ
∗
Notice that γ ′ (z) = ∂γ∂z + ∂x∗ ∂z = ∂z = p(x (z)) since by the envelope theorem ∂x∗ = 0.

7

γ(x + w)

γ(x + wi′ )γ(x + wi )g ∗ (wi )g ∗ (wi′ )xFigure 1 : The equilibrium markup equation x∗ = γ(x∗ + w)
if we take the markup between the contract price and the cost as the relevant variable,Pi∗ − ci , the non-default markup equals the expected penalty plus the expected costs ofeffort summarized in γ(b).
If the firm defaults in case of poor performance, the price/bid that makes zero net expected profits is given by the expression Pi∗ = ci + γ(Pi∗ − ci + wi ). Notice that the priceis defined as a fixed point since it also affects the expected total wealth at the performancestage. We rewrite this expression in terms of a default markup function, g ∗ (wi ) = Pi∗ − ci ,where g ∗ (wi ) is implicitly defined by g ∗ (wi ) = γ(g ∗ (wi ) + wi ).
Lemma 1. The default markup function, g ∗ (wi ), is uniquely defined for each wi . It is
increasing in wi , and g ∗ (0) = 0.
The proof of Lemma 1 is relegated to the Appendix, albeit the underlying intuition may
be grasped from Figure 1: the default markup equation x∗ = γ(x∗ +w) has a unique solution,
since γ(0 + w) > 0 and 0 < γ ′ (x) < 1 and γ(x∗ + w) is also increasing in w.

The default markup function is increasing in w, since, intuitively, in case of default,b
wealthier firms face larger losses. There exists a cut-off level w(b)= b − γ(b) such that the∗bdefault markup function is equal to the no default markup, g (b − γ(b)) = γ(b). If w = w(b)bthe firm is indifferent between making default or paying the penalty. For w > w(b),firmsstrictly prefer to pay the penalty and they bid according to the no default markup, γ(b).
Then, summarizing, the equilibrium bidding markup is γ(b)

g(b, wi ) = bif wi > w(b)= b − γ(b)∗g (wi ) otherwise.

8bThen, the equilibrium bid is Pi∗ = ci + g(b, wi ). Notice that firms wealthier than w(b)do
not plan to default and that this solvency threshold is increasing in the size of the penalty
b.12 In words, the larger the penalty, the larger the set of firms that may make default.
The next proposition summarizes the characterization of the bidding equilibrium, provides direct comparative statics, and a useful feature of the equilibrium markup function,g(b, wi ).

Proposition 2. The equilibrium bid in a second-price auction is Pi∗ = ci + g(b, wi ), whichis increasing in ci and weakly increasing in b and wi . Moreover, g(b, wi ) (and consequentlyPi∗ ) is supermodular in (b, wi ).
The most important feature of this bidding equilibrium is that the equilibrium bid (probability of winning) is increasing (decreasing) in the level of assets wi of the firm. In otherwords, financially weaker contractors are more likely to win the project.
It is important to notice that Proposition 2 is not an artifact of the second-price auction.
For example, under perfect information on firms’ types, the first-price auction and thesecond-price auction are revenue equivalent in this setting. In other words, they induce thesame prices and allocations.13 The equivalence between both mechanisms does not extend
to private information settings. However, in the Appendix we generalize Proposition 2 byconsidering general incentive compatible mechanisms in which only the winning firm receives
a transfer (as it is the case in the first-price auction). We show that incentive compatibility
in this general class of mechanisms requires to adversely selects undercapitalized firms.
In words, less capitalized firms face lower penalty expected costs, which gives them anadvantage in competitive allocation mechanisms.14 We can also illustrate Proposition 2
with the following algebraic example.
3.3

Example√Consider that p(x) = 1 − x and w ∈ [0, 1]. The contractor chooses a level of care x∗ (z)which minimizes her expected total cost given the penalty losses z,√
z2x)z + x. → x∗ (z) =4∗
Given x (z), the expected total cost of the penalty isx∗ (z) ∈ arg min(1 −γ(z) = p(x∗ (z))z + x∗ (z) = z −

(3)

z2
.
4

This is also implied by 0 < γ ′ (x) < 1.
A proof of this statement can be found in Appendix C.
14
The intuition is as follows: In terms of direct mechanisms, a firm can always mimic the behavior of a firmwith a higher level of assets and the same cost. Then, incentive compatibility implies that the probabilityof winning must be non-increasing in the level of assets.
12

13

9

2

The equilibrium markup if the firm is solvent is γ(b) = b − b4 , otherwise it is given byg ∗ (wi ) = γ(g ∗ (wi ) + wi ). Theng ∗ (wi ) = γ(g ∗ (wi ) + wi ) = g ∗ (wi ) + wi −(g ∗ (wi ) + wi )2
4√∗ (w )i
This implies that g ∗ (wi ) = −wi + 4wi . Notice that g ∗ (wi ) is increasing, g ∂w
= −1+ √1wi ≥i
2

2

b = b − b4 = γ(b) which implies that firms with wealth
0. If wb = b − γ(b) = b4 then g ∗ (w)w > wb will not default and will pay the penalty.
The equilibrium bid is c + b − b2

Pi∗ =


i

2

4

ci − w i +√4wi

if wi > wb = b − γ(b) = b4
otherwise.

Figure 2 plots the equilibrium markup Pi∗ − ci and different values of firm wealth, wi ,and shows that undercapitalized firms bid more aggressively.
Pi − ci
γ(b)

ŵ(b)

wi

Figure 2 : The bidding equilibrium for b

4

Increasing incentives, worsening performance

The effects of increasing penalties on the outcome of the procurement process are counterintuitive. The following proposition presents one of our core results: more powerful incentive
contracts may lead to worse allocations.
Proposition 3. Let (w, c) be the type of the winner under penalty b. Under any higher
penalty, b′ > b, the winner (w′ , c′ ) will be weakly less solvent and have weakly higher costs,
i.e, w′ ≤ w and c′ ≥ c.
10

Figure 3 shows the bidding equilibrium when the penalty increases from b to b′ , andillustrates the intuition of Proposition 3

Pi − ciγ(b′ )
γ(b)ŵ(b)

ŵ(b′ )

wi

Figure 3 : The bidding equilibrium when the penalty increases from b to b′ ,If we increase the penalty from b to b′ , bidders weakly increase their bids. Proposition 2states that g(b, wi ) and the equilibrium bid are supermodular in (b, wi ). which means that
wealthier bidders will increase their bids more than bidders with lower assets. This implies
that the winner under b, either will remain the winner under b′ , or will be replaced by afirm with lower assets (since firms with lower assets than the original winner are now morecompetitive in relative terms). The second part of Proposition 3 is less intuitive at first
blush: the winning firm is not only weakly less solvent, but it may also have higher costs.
This is because if one firm wins the project under b′ , but loses under b < b′ , the firm musthave higher costs than the winner under b, otherwise it would also have won under b. In
sum, a higher penalty may have the effect of inducing the selection of both a less solvent
and a less efficient winner.
4.1

The optimality of low powered incentive contracts.

Proposition 3 shows that more powerful incentive contracts may lead to awarding the contract to a less efficient firm. In this subsection, we complement this result by showing that
increasing incentives may lead to worse overall performance and, consequently, that a low
powered incentive contract may be optimal in this setting. In order to do so, we analyze
a particular case of the parametric example of Section 3.3 by considering two bidders withtypes (w1 , c1 ) and (w2 , c2 ) equal to (1, 0) and (0, c) respectively. We assume that the firstbidder always pays the penalty (b < 1) in case of poor performance, has zero marginal cost11

2

and she bids her opportunity cost for facing the penalty, γ(b) = b − b4 . The opportunity
cost of the second bidder is just her cost c because she is fully insolvent, and she does notplan to pay the penalty in the case of poor performance. As γ(b) is increasing, there exists√
a cut-off b = 2(1 − 1 − c) such that γ(b) = c. Then, if b ≤ b the first bidder wins at price2c. In case of b > b the second bidder wins at price b − b4 . Then, this numerical example
illustrates Proposition 3: if we increase the penalty from b < b to b′ > b, the winner changes,
and the new winner under b′ is less solvent and has higher costs.
Now we move to analyze equilibrium effort levels and the performance of the project. If
2b < b, the first bidder wins, and will exert the level of effort x∗1 (b) = b4 . If b > b the second2(b− b −c)24
. Notice that x∗1 (b) > x∗2 (b) = 0.
firm wins, and it will exert a level of effort x∗2 (b) =4Then, by increasing the penalty from b < b to b′ > b, the exerted level of effort may decrease.
This highlights an unobserved additional cost of increasing incentives: the choice of effort(and performance) may be poorer under more powerful incentives schemes. Corollary 4summarizes this insight.

Corollary 4. Effort exerted by the winning bidder (and expected performance) is non monotonic in the penalty b.
The next figure illustrates Corollary 4 for a specific value of the cost c = .64 which leads
to b = 0, 8.
x∗

b̄

b

Figure 4 : Effort exerted in Equilibrium.

When the penalty is low, the winning bidder is fully solvent. In this case, a higher penalty
translates into higher exerted effort. This happens until the penalty reaches b, when firm 2
becomes the winning firm and the exerted effort drops to zero. However, for penalties higher
than b the effort exerted by the winning firm (firm 2) depends on wealth at the performance12

stage, which is increasing in the procurement price. Then, as the procurement price weakly
increases in the penalty, exerted effort may also increase. Along this dimension, as we willshow in Figure 4, a higher price may not be necessarily bad news for the sponsor, since ittranslates into higher effort and performance and a lower probability of default.
As we have discussed above, there are important private and social costs associated withbankruptcy. Then, it is important to assess how incentives (penalties in our binary case)affect the likelihood of bankruptcy of the competitively selected supplier. We will see thatthe non-monotonicity of exerted effort with respect to the penalty (incentives) extends tothe probability of bankruptcy. The probability of default depends on the identity of thewinner, as well as on the equilibrium effort level. As long as b < b, the first firm wins
and there will be no bankruptcy. When b > b the second firm wins and bankruptcy arises,
whenever there is poor performance (since penalty is lower than the current wealth of thefirm b > γ(b) − c). However, conditioning of the second firm being the winner, the larger
the penalty, the higher the price and the lower the probability of default.
Corollary 5. The probability of bankruptcy is non monotonic in the penalty.
The next figure illustrates Corollary 4 for an specific value of the cost c = .64 which leads
to b = 0, 8.
PDPD = 1b̄

b

Figure 5 : The probability of bankruptcy in equilibrium

Bankruptcy arises when there is poor performance of the project and the firm is unable
to pay the entire amount of the penalty. Then, in our example, when the penalty is low,b < b̄ the winner is a fully solvent firm and the probability of default is zero. For larger
penalties, b > b̄, firm 1 is replaced by firm 2, who has no assets and goes bankrupt in thecase of poor performance. Then, for this range of parameters (b > b̄), the higher the penalty,
13the higher the price and the wealth of the winning firm, which translates into higher effort
and lower probability of poor performance and bankruptcy.
All these results convey the general message that increasing incentives (penalty) may
have adverse effects on performance. As a consequence of such effect, we can make thestatement that when there is a default risk it may be optimal for the sponsor to set a
penalty lower than the penalty that provides incentives to make the efficient level of effort
without a default risk, x∗ ∈ arg max{(1 − p(x))θ + p(x)θ − x}
The first-best effort level without default risk is obtained by equalizing the performancevariable and the monetary incentives, b(θ) = θ. Consider that in our binary setting, θ = 0,and θ < 0. The first-best level effort can be achieved with a penalty bE = b(θ) = θ < 0 that
we denote as the efficient penalty. However, when there is a risk of default by the winningfirm, this efficient penalty may be far from optimal.
Proposition 6. Under default risk affecting the winning firm, the efficient penalty may be
dominated by lower powered incentives schemes (lower penalties).
As a simple proof of the previous statement, we focus on our example and start withthe simple observation that bE may be higher or lower than b. Consider the specificationof the sponsor preferences we introduced in the description of the model. If bE < b, theoptimal penalty in our simple example would be trivially bE since there is no conflict between
allocation and incentives. If bE > b, this conflict arises, and indeed, b dominates bE along
several dimensions: i) better allocation, Wb = (1, 0) and WbE = (0, c), ii) lower priceEγ(b) = c < γ(bE ), and iii) lower probability of default PD (b) = 0 < PD (bE ) = 1 − (γ(b 2)−c) .
Given that, if the default cost K is sufficiently large, a penalty b lower than the efficient onebE may be optimal.
In general, however, it is difficult to characterize the optimal penalty in this multidimensional setting. In the previous example, we have assumed that the solvent firm has acost advantage. If both firms have the same costs, the insolvent firm would win under any
amount of the penalty and defaults whenever there is poor performance. In such a case, it
may be optimal to increase the penalty even beyond bE in order to increase the price, thewealth of the contractor and the incentives to exert effort. However, in such a situation it
may not be optimal to auction the project in the first place.15 The auction is not selecting
the best contractors and the reduction in contractor’s rents produced by the auction may
be counterproductive in terms of performance. In other words, competition may not help
and other mechanisms (such as lotteries or negotiations) may be superior to auctions.16
15

The suboptimality of auctions has been discussed in the literature that analyzes procurement settings
with cost uncertainty and limited liability, see for example Burguet, Ganuza and Hauk (2012).Manelli andVincent (1995) and Lopomo et al. (2022), analyze a similar adverse selection problem and show that standard
competitive mechanisms work poorly when low-cost bidders are also linked with lower quality outcomes.
16This may partially explain why private firms often use negotiations instead of auctions. Bajari, McMil-

14

5

Extensions

5.1

Sticks and Carrots

In this extension, we do not impose any constraint over the pay-for-performance mechanisms
beside b(θ) < b(θ̄). Consider the possibility of using penalties and positive payments orbonuses, for example, b(θ) < 0 and b(θ̄) > 0. We will show that transfers and allocations
generated by the pay-for-performance mechanisms do not depend on the particular penalties
and bonuses used but only on the power of the incentives scheme b(θ̄) − b(θ). In order to dothat, we will show that this more general setting is in fact equivalent to the penalty setting
solved in the baseline model.
As in the previous model, default may only occur if poor performance is observed andP − c + w + b(θ) < 0.17 We rewrite this condition as
P − c + w + b(θ̄) < b(θ̄) − b(θ) ≡ b̃.
We reformulate the primitives as follows. Letc̃ = c − b(θ̄).
This is the bonus-adjusted cost. Then, the above condition is rewritten as
P − c̃ + w < b̃.
This is analogous to the condition in the main model with c̃ and b̃; default occurs if the net
penalty, b̃, exceeds the adjusted solvency, P − c̃ + w. Using this notation, we can rewrite
the payoff of the winning firm as
P − c − p(x) min{P − c + w, −b(θ)} − x + (1 − p(x))b(θ̄)= P − c − p(x) min{P − c + w + b(θ̄), b(θ̄) − b(θ)} − x + b(θ̄)
lan, and Tadelis (2009) look at private construction contracts in Northern California and find that 43% ofthem have been awarded through negotiations with a sole supplier, and only 18% used open competitive
tendering. In the same line Decarolis (2014) analyzes empirically a change in Italian public procurement
rules, forcing the use of first price auctions for public works. He finds that the increased use of auctions
between 2000 and 2006 results in significant discounts in winning bids, but also in significant deteriorationof performance from winning bidders. Similarly, Eun (2018) shows with Korean paving contract data that
reducing competition by bid screening and excluding abnormally low bids, lead to choose contractors withhigher paving costs but that this is compensated by better contractor’s performance (lower cost overruns).
17
To make this point we do not need to assume b(θ̄) > 0. Firms will not default in equilibrium in the highperformance state θ̄, since in such a case they will make negative profits and the participation constraint
would not be satisfied.

15

= P − c̃ − p(x) min{P − c̃ + w, b̃} − x.
Hence, the use of sticks and carrots amounts to the use of only the penalty b̃ with thebonus-adjusted cost c̃. Letz̃ = min{P − c̃ + w, b̃}.
The optimal effort is then given by x∗ (z̃), and the expected cost of effort is expressedby γ(z̃). Then, the firms’ problem and the bidding equilibrium are equivalent to the ones
characterized in the baseline penalty-only model, just replacing c and b by c̃ and b̃. The
equilibrium bid of each firm i is then P ∗ (c̃i , wi ; b̃) = c̃i + g(b̃, wi ) whereg(b̃, wi ) = γ(b̃) g ∗ (w )i

if wi ≥ b̃ − γ(b̃)
otherwise.

Following the same argument as in Proposition 2, the next corollary obtains immediately:
Corollary 7. With sticks and carrots, the equilibrium bid in a second-price auction isP ∗ (c̃i , wi ; b̃), which is increasing in c̃i and weakly increasing in b̃ and wi , where c̃i = ci − b(θ̄)
and b̃ = b(θ̄) − b(θ). g(b̃, wi ) (and consequently Pi∗ ) is supermodular in (b̃, wi ).
Now we consider two different incentive schemes b1 (·) and b2 (·) for which b̃ = b1 (θ̄) −b1 (θ) = b2 (θ̄) − b2 (θ). If we analyze the firm’s equilibrium bid P ∗ (c̃i , wi ; b̃) = c̃i + g(b̃, wi ),by construction g(b̃, wi ) has to be the same for the two incentive schemes. These bids only
differ in the adjusted cost part, c̃i = ci − b1 (θ̄) and c̃i = ci − b2 (θ̄). Then, the bids are thesame in the both incentive schemes except for a constant shift b1 (θ̄) − b2 (θ̄). This implies
that the winner and the second highest bidder are the same in both schemes, and so are thedifferences between the two highest bids, and consequently, the rents of the sponsor and thefirms are the same. To summarize, we have the following result:
Proposition 8. Two different incentive schemes b1 (·) and b2 (·) with the same power of
incentives b̃ = b1 (θ̄) − b1 (θ) = b2 (θ̄) − b2 (θ) yield the same equilibrium allocation and distribution of rents.
In the next subsection, we will illustrate Proposition 8 by computing explicitly the bidding
equilibrium for an incentive scheme that includes a bonus, and by obtaining the sameoutcomes as with just the penalty b for poor performance in our baseline setting.
5.2“Sticks and Carrots” versus Penalty-Only

In the baseline penalty setup, we have the incentive scheme b1 (θ̄) = 0 and b1 (θ) = −b.
Remember that the contractor’s moral hazard problem in such a case was to minimize
16

p(x)z + x, where z = min{P − c + w, b}, are the expected penalty losses. Now, we consider
this alternative sticks and carrots scheme with the same power of incentive b2 (θ̄) = b̄ andb2 (θ) = b such that b̄ > 0 > b and b̄ − b = b.
The contractor’s incentive problem with sticks and carrots is to minimize−(1 − p(x))b̄ + p(x)zB + x.

(4)

where zB = min{P −c+w, −b} is the amount of losses incurred by the contractor in the caseof poor performance, and −(1−p(x))b̄ is the reward the contractor gets for good performance(with a negative sign since the contractor’s problem is written as a minimization problem).
If we add and subtract b̄ from this losses expression zB , we get
zB = −b̄ + min{P − c + b̄ + w, b}
Let z̃B = min{P − c̃ + w, b} be the adjusted losses, that uses the bonus adjusted costdefined above c̃ = c − b̄. Then, if we plug zB = −b̄ + z̃B into the contractor’s objectivefunction, we obtain−b̄ + p(x)z̃B + x.

(5)

When we use the adjusted cost c̃ = c − b̄, the contractor’s incentive problem becomesalmost identical to the her problem when the sponsor only uses penalties (they only differ inconstant term b̄). Then, the contractor chooses optimal effort level functions with the same
shape in both cases, x∗ (z) (penalties) and x̃∗ (z̃B ) (sticks and carrots) such that x∗ (z ′ ) =x̃∗ (z ′ ) for all z ′ .
By the same token, if we plug these optimal efforts in the objective functions, we obtain
also two similar expected total cost functions γ(z) and γ(z̃B )− b̄, that should be equal to themarkup P −c in the equilibrium bidding function. If the firm is solvent, then z = zB = b andthe markup in the penalty case is γ(b) and in the sticks and carrots case it is just γ(b) − b̄.
Then, in this latter case, the markup (and the bid) is the same as in the penalty-only casebut is shifted down by the amount of the bonus b̄.
When the firm may default in the case of penalty-only incentive, we define the markup as afunction p−c = g ∗ (wi ), which was implicitly defined by the equation g ∗ (wi ) = γ(g ∗ (wi )+wi ).
In the case of sticks and carrots, z̃B = p − c̃ + wi , and then the equilibrium markup isp − c = γ(p − c̃ + wi ) − b̄. This is equivalent to p − c + b̄ = p − c̃ = γ(p − c̃ + wi ). Thenif we take p − c̃ = g̃(wi ), we can obtain the same implicit equation for the equilibriumg̃ ∗ (wi ) = γ(g̃ ∗ (wi ) + wi ). As they are characterized by the same equation, they should be
the same function g̃ ∗ (wi ) = g ∗ (wi ). Finally, p − c̃ = g ∗ (wi ), using that c̃ = c − b̄ we obtain17

that the equilibrium markup is p − c = g ∗ (wi ) − b̄ . As in the no default case, the markup
and bidding equilibrium are identical to the penalty-only case but they are shifted down by
the size of the bonus.
Given that the default markup function is increasing in w (as in the baseline penaltybmodel) there exists a cut-off level w̃(b)such that the default markup function is equal to∗ b
the no default markup, g (w̃(b)) − b̄ = γ(b) − b̄. As the markup equilibrium functions are
identical in the “sticks and carrots” case and the penalty-only case except that the amountbbof the bonus is discounted ex ante, the cut-off level also coincides w̃(b)= w(b)= b − γ(b).
Therefore, the equilibrium bidding markup in the ”sticks and carrots” case isg(b̄, b, wi ) = γ(b) − b̄b
if wi > w(b)= b − γ(b) g ∗ (w ) − b̄ otherwise.
iThen, the equilibrium bid is P̃i∗ = ci + g(b̄, b, wi ) = ci + g(b, wi ) − b̄, which is equal to thebaseline penalty-only case but with a downward shift in the amount of the bonus b̄.
5.3

Negative Markups and Budget Constraints at the Construction Stage.

In the previous analysis we are not taking into account that when using bonuses the net
wealth at the construction stage may be negative, i.e., g(b̄, b, wi ) + w < 0. In this section we
consider that firms are constrained to finance ex-ante the project with their own resources
and the markup granted by the contract. Then, this is equivalent to imposing a budget
constraint at the construction stage. Namely,
P − c + w ≥ 0,
When the bidder follows the bidding strategy as above, this condition is violated if
Pi∗ − ci + wi = g(b, wi ) − b̄ + wi < 0.
As g(b, 0) = 0, then, for wi = 0 the left hand side is negative. By the previous analysis weknow that g(b, wi ) + wi is increasing and it should be positive for high values of wi . Then,
there is some level of wealth w(b̄, b) > 0 such that contractors with wi ≥ w(b̄, b) may bid
according to the bidding equilibrium with its own resources. For bidders with wi ≤ w(b̄, b),each bid is restricted to
P i = ci − w i .
We will observe that this may destroy the equivalence between bonuses, and more generally “sticks and carrots” schemes, and the exclusive use of penalties. We come back to18

our numerical example. When the incentive system is penalty b for poor performance, thisbudget constraint is always satisfied, since markups are non-negative, as the level of wealthis. Consider now a system with the same degree of power in incentives, but based only on
a bonus for good performance, (b̄ = b, b = 0). From the previous section we know that thebidding equilibrium requires to shift down the bidding function downwards in the size ofthe bonus, that is, in other words, to give a big discount in anticipation of future rents. In
such a situation the budget constraint may play an important role. In particular, the equi√
librium markup may be constrained by the wealth if −wi + 4wi − b < −wi , or equivalently,2
wi < wb = b4 . If firms are constrained to finance ex-ante the project with their own wealthand the markup, the equilibrium bid would be c − b2

Pi∗ =


i

4

ci − w i2

if wi ≥ wb = b4
otherwise.

Notice that the equilibrium bidding function is weakly higher and more importantly,
weakly decreasing in the level of wealth. The winner would be weakly wealthier, and thenthe equilibrium effort and project performance is better in this case than when b is purely
a penalty.
5.4

Continuous performance

In this section, we extend our baseline model for considering a continuous performancevariable, θ ∈ [θ, θ]. Let x be the contractor’s monetary equivalent of the performance effort,
with the probability distribution of θ; F (θ|x), where x orders the distribution in the firstorder stochastic dominance sense, so that greater effort increases the probability of obtaining
better performance θ, if x ≥ x′ , then F (θ|x) ≤ F (θ|x′ ) for all θ ∈ [θ, θ].
The sponsor provides incentives to increase effort using a pay for performance mechanismb(θ), where b(θ) is increasing and maybe positive or negative. Besides making the performance variable a continuous one, the model remains identical to the binary case. We startwith the analysis of the performance stage.
As before, the contractor’s losses are bounded by her wealth. If P − c + w + b(θ) < 0,then the winning firm goes bankrupt and loses P − c + w. As b(θ) is increasing, it may existb = −(P − c + w). Given this definition, if θ < θ,b the winning firma cut-off θb such that b(θ)
goes bankrupt. Taking into account b(θ) and her current level of wealth, the winning firmchooses a level of care xC minimizing her loss function at the performance stage18
18

For consistency with the penalty baseline model, we write the optimization problem of the contractor asa minimization problem of her expected loss at the performance stage. However, depending on the incentive
scheme b(θ), the loss function can be negative (the contractor will have positive profits at the performancestage).

19b ∈ arg minx∗C (θ)

Zθb

Zθ

θ

θb

(P − c + w)f (θ|x)dθ −b(θ)f (θ|x)dθ + x

Let x∗C (θ) be the effort undertaken by the fully solvent firms, i.e θb = θ. We will show thatas in the baseline case, the contractor’s effort is lower as the probability of default increases.
b is decreasing in θ.
b
Lemma 9. x∗C (θ)
b be the expected loss of the contractor at the performance stage in the continuous
Let γC (θ)
case:
Zθb

Zθ

θ

θb

b
(P − c + w)f (θ|x∗C (θ))dθ−bbb(θ)f (θ|x∗C (θ))dθ+ x∗C (θ)

As in the baseline case, this loss function is decreasing in the probability of default andthen, it is indirectly increasing in the current wealth of the contractor at the performancestage.
b is decreasing in θb and then, increasing in P − c and w.
Lemma 10. γC (θ)

The net expected profits of the contractor arebπN (P, c, w, b(θ)) = P − c − γC (θ(P− c + w))

Now we turn to the analysis of the bidding stage. As in the initial section, we focus
on the weakly dominant strategy equilibrium of the second-price auction. Therefore, theequilibrium bid of each firm i is the minimum price Pi∗ for which firm i is willing to accept
the project, defined by:
E{πN (Pi∗ , ci , wi , b(θ))} = 0,
(6)
If the firm has no risk of default, the price that makes the net expected profits zero isPi∗ = ci + γC (θ). Then, the equilibrium markup, Pi∗ − ci , is equal to the expected loss inthe performance stage γC (θ) (notice that γC that can be positive or negative, depending onb(θ)).
If a firm may default for a poor performance realization, the price that makes the netb ∗ − c + w )).
expected profits equal to zero is given by the expression Pi∗ = ci + γC (θ(P
i
i
i
∗
∗
∗
Let gC (wi ) be the markup in such a case, where gC (wi ) is implicitly defined by gC (w) =b ∗ (w) + w)).
γC (θ(gC
Lemma 11. gC∗ (w) is uniquely defined for each w, it is increasing in w, and gC∗ (0) = −b(θ).
20Similarly to the binary penalty case, the default markup function is increasing in w sincein the case of default, wealthier firms have larger losses (or lower expected profits if γC isbnegative). There exists a cut-off level w(b(θ))= −b(θ) − γC (θ) such that the default markup∗
function is equal to no default markup, gC (−b(θ) − γC (θ)) = γC (θ).19 Then, the equilibrium
bidding markup isgC (b(θ), w) =


 γ

C (θ)
 g ∗ (w )iCif wi ≥ −b(θ) − γC (θ)
otherwise.

Then, the equilibrium bid is Pi∗ = ci + gC (b(θ), wi ). Proposition 12 summarizes thecharacterization of the bidding equilibrium, and provides direct comparative statics.
Proposition 12. The equilibrium bid in a second-price auction is Pi∗ = ci +gC (b, wi ), whichis increasing in ci and weakly increasing in wi .
Proposition 12 expresses the same message for the continuous case as Proposition 2 inthe penalty binary baseline model: competitive mechanisms select undercapitalized firms.
We can also show that increasing incentives at the performance stage by means of a more
powerful pay-for-performance scheme may lead to awarding the contract to a less efficientand less solvent firm, and to worse overall performance. Consequently, a low powered
incentive contract may also be optimal in the continuous case.
Take an arbitrary pay for performance mechanism b(θ), where b(θ) is increasing for allθ ∈ [θ, θ]. Now, we consider a more powerful pay-for-performance scheme, b̃(θ) = (1 +α)(b(θ) − b(θ)) + b(θ), with α > 0. Notice that b̃(θ) is an upward rotation of b(θ), and it isa more powerful pay-for-performance scheme because fully solvent firms undertake higher
effort under b̃(θ) than under b(θ). Let x̃∗C (θ) be the optimal effort exerted by fully solvent
firms under b̃(θ). Then, we can show that x̃∗C (θ) ≥ x∗C (θ). This is because the difference
between the loss functions under b̃(θ) and b(θ) of fully solvent contractors at the performance
stage
Zθ

γ̃C (θ) − γC (θ) = − α[b(θ) − b(θ)]f (θ|x)dθ ≤ 0
θ

is increasing (as since b(θ) is increasing and x order the distribution F (θ|x) in the firstorder stochastic dominance sense). In other words, these loss functions have decreasing
differences and applying the results of Milgrom and Shannon (1994), we can conclude thatx̃∗C (θ) ≥ x∗C (θ).
19∗∗
For checking this, assume that gC(w)
b = γC (θ). Then, if we replace gC(w)
b by γC (θ) in the equation∗∗bbgC (w)
b = γC (θ(gC (w)b + w))
b we obtain γC (θ) = γC (θ(γC (θ) + w)).
b
For this equality be true, it has to beb C (θ) + w)
that wb is such that θ = θ(γb = b−1 (−(γC (θ) + w)
b that implies that w
b = −b(θ) − γC (θ).

21

The next step is to prove that when comparing the allocation under b̃(θ) and b(θ), we canalso state that increasing incentives may lead to worse allocations. As b̃(θ) > b(θ) for all θ ∈(θ, θ], the expected losses at the performance stage for the fully solvent firms are lower for thismore powerful incentive scheme, γ̃C (θ) < γC (θ). This implies that the default cut-off levelb b̃(θ)) = −b(θ) − γ̃C (θ) > w(b(θ))
bis higher when higher incentives are in place, w(= −b(θ) −bb b̃(θ))]γC (θ). Then, it is possible that a contractor with a level of wealth, w ∈ [w(b(θ)),
w(
is fully solvent under b(θ) but not under b̃(θ). In such a case, this contractor would haveunder b̃(θ) a competitive advantage over wealthier fully solvent contractors, and it may be
the winning firm under b̃(θ) even if it has higher costs and it would not have been thewinning firm under b(θ).
Despite the fact that a more powerful incentive scheme leads to higher effort from fully
solvent contractors, x̃∗C (θ) ≥ x∗C (θ), it may also induce lower exerted effort from a winningbidder and thus, poorer performance. Firstly, because, as we show above, under a more
powerful incentive scheme, a solvent winner may be replaced by a contractor who has a riskof default and may have lower incentives to exert effort. But even if the winner does notchange, it may have less incentives to exert effort under a more powerful incentive scheme
since it may default for a larger set of performance realizations.
Lemma 13. Consider a contractor with a level of wealth such that it defaults for θ < θbbb
under b(θ) and for θ < θ̃ under b̃(θ) respectively. Then, θb < θ̃.
The intuition for this result is as follows. Under a more powerful incentive scheme,
contractors will obtain higher earnings in case of high performance realizations. As in thebidding equilibrium, contractors make zero profits with their bids. This would imply higher
losses in the poor performance realization case, which leads contractors to default for alarger set of performance realizations. This result implies that the effort exerted by thewinning bidder is not necessarily higher under a more powerful incentive scheme, as westate in Corollary 4 in the binary case.
Corollary 14. Effort exerted by the winning bidder (and expected performance) is nonmonotonic in the power of the incentive scheme.
The intuition for the lack of monotonicity may be easily grasped by analyzing how thepayoffs (and incentives) of the winning bidders depend on the performance realizations underb the firm makes default and loses its wealth under b(θ) andb(θ) and under b̃(θ). If θ < θ,bb θ̃],
under b̃(θ). If θ ∈ [θ,then the firm makes default under b̃(θ) but its payoff increasesbon the performance realization under b(θ). Finally, if θ > θ̃ firm does not make default,
payoffs are increasing in the performance realization but at a higher rate under b̃(θ) thanunder b(θ). Then, these pay-off functions do not satisfy a single-crossing condition and the22effort exerted under b(θ) can be higher or lower than the effort exerted under b̃(θ). We
have shown above that the effort exerted by fully solvent firms is greater under the more
powerful incentive scheme. However, we can think of examples where firms that may default
may exert lower effort under a more powerful incentive scheme. Take as, in Lemma 13, abbb and under b̃(θ) if θ < θ̃,
contractor that may default under b(θ) if θ < θ,
where θb < θ̃.
Consider that θ ∈ [θ, θ] = [0, 1] and there exist only two types of effort, x = c leading
to a uniform performance distribution function over the interval [0, 1] and x = c + ϵ thatb a constantleads to a three steps distribution function, zero density over the interval [0, θ],b θ̃] and a density of 1 over the interval [θ̃, 1]. The
density equal to b̃ θ over the interval [θ,
b̃

b

b

θ−θb

b to the interval [θ,b θ̃], andeffect of high effort is just to transfer all the measure on [0, θ]
then it generates a distribution that dominates the one generated by low effort according
to FOSD. It is clear that the contractor will choose high effort, x = c + ϵ, under b(θ), sincebb θ̃]b′ (θ) > 0 on [θ,but low effort, x = c, under b̃(θ) since the improvement of the performance
distribution will have no effect on their profit (given that it will default if the performancebrealization is below θ̃).
For a more general result of the adverse effect of incentives on the allocation, in the line of
Proposition 3, we need to introduce an auxiliary lemma stating that the markup equilibrium
function has increasing differences between incentives and wealth:bb ∗ ), weakly increases as the incentive scheme
Lemma 15. If the probability of default, F (θ|xC
becomes more powerful for all w, the equilibrium markups have increasing differences: forw > w′gC (b̃(θ), w) − gC (b(θ), w) > gC (b̃(θ), w′ ) − gC (b(θ), w′ ).
b ∗ ) weakly increases is triviallyIf we compare b(θ) and b̃(θ), the condition that F (θ|xCbbb b̃(θ))], itsatisfied if w > w(b̃(θ)) since it is zero in both cases. For w ∈ [w(b(θ)),
w(
also holds generally, since the probability of default is zero for b(θ) and positive for b̃(θ).
bb
For w < w(b(θ)),
as θb < θ̃ if the effort is lower under b̃(θ), the condition automaticallyholds. Otherwise it may hold if the effect on the default probability of a higher performancethreshold under b̃(θ) compensates the also higher performance effort.
Finally, we can state in the continuous case that more powerful schemes may lead theselection of both a less solvent and a less efficient winner.
b ∗ ) weakly increases as the incentive scheme becomes
Proposition 16. Suppose that F (θ|xC
more powerful. Let (w, c) be the type of the winner under b(θ), and (w̃, c̃) under a more
powerful incentive scheme b̃(θ). Then, the winner (w̃, c̃) will be weakly less solvent and have
weakly higher costs, i.e, w̃ ≤ w, and c̃ ≥ c.

23

Lemma 15 states that wealthier winners decrease (weakly) less of their bids when there
is upward rotation of the incentive scheme (which gives bidders more expected rents). As
in the binary case, this implies that the winner under b(θ), either will remain the winnerunder b̃(θ) or will be replaced by a firm with lower assets, since wealthier firms are now
less competitive in relative terms. Following the same argument as in Proposition 3, thewinning firm is not only weakly less solvent, but it may also have higher costs. The newwinner under b̃(θ) must have higher costs than the winner under b(θ) since otherwise it
would also have won under b(θ).
It is important to notice that we have focused on a particular way to increase the incentives, namely a linear upward rotation of the initial incentive scheme, b̃(θ) = (1 + α)(b(θ) −b(θ))+b(θ). However, our results are more general in two dimensions. Firstly, following similar arguments than in the subsection 5.1 and 5.2, it can be shown that rents and allocation
generated by b̃(θ) are invariant with respect to a parallel shift of b̃(θ) (adding or subtracting
a constant). Then, in particular our results holds for a pure rotation of the incentive scheme,or when the more powerful incentive scheme implies a reduction in performance pay for allstates. Secondly, our results also hold when α depend on θ, as long as α(θ)′ ≥ 0 for all θ.

6Conclusions

The trade-off between providing incentives and achieving efficient allocations affects bidding
markets generally. Perhaps most ostensibly, procurement seems to be particularly vulnerable to this. If not for reasons other than sheer magnitude, procurement is of key importancefor economies of all kinds and sizes. Understanding what are the effects of rules and practices governing procurement processes is likely to prove relevant for ensuring value fromthe enormous amounts of public money invested in the provision of infrastructure, goods
and services for citizens. Eventually, better understanding may lead to improvements inthe design of current arrangements and institutions at play in this sector. Our analysis, we
believe, sheds light on widespread features of procurement contracts. In particular, we explain why selection mechanisms based upon competitive bidding may favor undercapitalized
firms for undertaking projects and how, when moral hazard is present in the execution of thecontract, negatively affect the quality of outcomes. More importantly, we show that more
powerful incentives may be counterproductive, leading to worse allocations (chosen contractors will have higher costs and lower assets than their rivals) and poorer performance of thecontract. This constitutes an approach to explain the puzzling observation that, despite theimportant negative effects from inadequate performance in procurement projects, sponsors
seem to deploy only mild penalties on non-performing contractors. One way to interpret
our results is that the ability of sponsors to use powerful pay-for-performance mechanism is24

limited by the distribution of assets of potential contractors. In a number of recent empiricalstudies of procurement, examining the role of granting discretion to public officials in chargeof organizing the procurement process and making decisions, it has been found that granting
enhanced ability to public buyers results in selected contractors being financially larger andless risky, more experienced as partners of that buyer, and leading to better procurement
auctions. This is the case in Coviello et al. (2018) and Coviello et al. (2022). In this spirit, ina recent study linked to the World Bank (Bosio et al. (2022)), analyzing procurement laws,
practices and outcomes across a large dataset of countries, a clear negative relationshipbetween timely outcomes in procurement and income level of countries is identified. In that
paper, the driving force of the model and the empirical predictions is that in lower income
countries, the capabilities of public sector sponsors are also more limited. Discretion, thusis largely exploited in these countries by public officials to choose less efficient contractors
who offer benefits to the awarding individuals, whereas in high income countries with solid
public sector capabilities, discretion is fruitfully used to improve contractor’s selection and,with it, procurement outcomes (in line with Coviello et al. (2018), Coviello et al. (2022), or
Decarolis et al. (2020)). In light of our findings, one could add a different explanation for theobserved pattern. In lower income countries, the financial standing of the pool of potential
contractors is worse than in higher income ones. Therefore, in the former set of countries,
the ability of sponsors to use more ambitious incentives mechanisms is also more limited
than in the latter. Surely, looking at one side of the interaction (the sponsor, its capabilities, institutional constraints and framework) is important for understanding procurement
outcomes, but keeping an eye on the other side (the pool of contracting candidates) is, we
believe, also relevant. In the end, the outcomes will result from the incentives faced by bothsides in the interaction.
Acknowledgement: We are grateful for the comments to Miguel Abito, Francesco Decarolis, Shingo Ishiguro, Steven Tadelis, participants at The 38th Annual Conference of
the European Association of Law and Economics (EALE), Simposio de Analisis Economico
2021, Workshop in Organizational Economics at Summer Forum 2021, International Industrial Organization Conference, Asian Meeting of Econometric Society 2022, EARIE 2022,
and participants at seminars at Universitat Pompeu Fabra, Hitotsubashi University, Kyoto
University, Nagoya University, and Waseda University. The usual disclaimers apply. Fernando Gomez acknowledges the financial support of the Spanish Ministry of Science andInnovation under project DER 2017-82673-R. Juan-Jose Ganuza gratefully acknowledges the
support of the Barcelona School of Economics Research, the government of Catalonia, andthe Spanish Ministry of Science and Innovation through the project PID2020-115044GBI00. Juan-Jose Ganuza is also grateful for the support and the hospitality during his visit
25

at the Economic Research Center, Nagoya University (as a visiting fellow from June 2022
to August 2022). Makoto Hanazono acknowledges the support of Japan Society for the
Promotion of Science, grant number 17KK0067.

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28

AppendicesA

Proofs

Proof of Lemma 1. As g ∗ (wi ) = γ(g ∗ (wi ) + wi ) for proving the result, we have to showthat the equation x∗ = γ(x∗ +wi ) has a unique solution. This follows for the features of γ(x),that it is an increasing function, it has a slope lower than one, and γ(0 + wi ) ≥ 0. Then,γ(0 + wi ) ≥ 0 and given that γ ′ (x) ≤ 1 it cross the 45 degree line only once, x∗ = γ(x∗ + wi ).
Moreover, g ∗ (wi ) is increasing in wi . Let g ∗ − γ(g ∗ + wi ) = 0 by taking the total derivative
γ ′ (g ∗ +wi )dg ∗(1 − γ ′ (g ∗ + wi ))dg ∗ − γ ′ (g ∗ + wi ))dwi = 0. Then, dw=′ (g ∗ +w ) > 0. Finally, as γ(0) = 0,1−γii
the unique solution of g ∗ (0) = γ(g ∗ (0)) is g ∗ (0) = 0. QEDProof of Proposition 2. The proof of the first part of the Proposition 2 (Pi∗ being
increasing in ci , and weakly increasing in b, and wi ) is immediate from the argumentsin the main text. Now, we need to show that g(b, wi ) is supermodular, if x′ > x theng(b′ , w) − g(b, w) is weakly increasing in w.
g(b, wi ) = γ(b)b
if wi ≥ w(b)= b − γ(b) g ∗ (w ) otherwise.
ib
As γ(b) has a slope lower than 1, w(b)= b − γ(b) is increasing. Consider b′ > b,






b ′ ).
γ(b′ ) − γ(b)if w ≥ w(bb ′ ) ≥ w ≥ w(b)bg(b′ , w) − g(b, w) = g ∗ (w) − γ(b) if w(b




b
0
if w < w(b)bb ′ )) = γ(b′ ), then
Notice that as g ∗ (w) is increasing in w, g ∗ (w(b))= γ(b) and g ∗ (w(bg(b′ , w) − g(b, w) is a continuous and weakly increasing function of w. This implies thatg(b, w) is a supermodular function. QEDProof of Proposition 3: Let (w′ , c′ ) be the wealth and cost of the winning firm underb′ , and (w, c) under b. Consider contrary to the Proposition 3 that the winner under b′ ismore solvent than the winner under b, that is, w′ > w. Given that P ∗ is increasing in c andw, and that P (w′ , c′ , b′ ) < P (w, c, b′ ), we must have c′ < c. Then, the next two conditions
have to be satisfied.
P (w′ , c′ , b′ ) = c′ + g(b′ , w′ ) < c + g(b′ , w) = P (w, c, b′ ),P (w′ , c′ , b) = c′ + g(b, w′ ) > c + g(b, w) = P (w, c, b).
29

They are equivalent tog(b′ , w′ ) − g(b′ , w) < c − c′g(b, w′ ) − g(b, w) > c − c′ .
These two conditions imply thatg(b, w′ ) − g(b, w) > g(b′ , w′ ) − g(b′ , w)or equivalently,g(b′ , w) − g(b, w) > g(b′ , w′ ) − g(b, w′ ).
This leads to a contradiction with Proposition 1 , since g(b, w) is supermodular, which
implies thatg(b′ , w) − g(b, w) ≤ g(b′ , w′ ) − g(b, w′ )Finally, consider contrary to the Proposition 3, that the winner under b′ is more efficientthan the winner under b, c′ < c. The case in which the winner under b′ is also more solvent,is discussed above. But, if the winner under b′ is more efficient and less solvent, he shouldhave won also under x, since her bid is lower than the bid of (w, c) for any penalty. QEDProof of Corollaries 4 and 5: See the explanations given in the main text as well as
the example and figures 3 and 4. QEDProof of Proposition 6 : The example provided in the main text and the discussion
after Proposition 6 is a simple proof the result. QEDProofs of Corollary 7 and Proposition 8: Follow from the arguments of the maintext. QED
Proof of Lemma 9: We can rewrite loss function of the contractor at the performancestage as

γC

Zθb
Zθ
b
b
(x, θ) = − b(θ)f (θ|x)dθ− b(θ)f (θ|x)dθ + x
θ

θb

b has
Using the results of Milgrom and Shannon (1994), it is enough to show that γC (x, θ)
increasing differences. Notice thatb∂γC (x, θ)b (θ|x)b= −b′ (θ)F
b
∂θ

30

b > 0 and that x orders the distribution in the first-order stochastic
Given that b′ (θ)θb)
dominance sense, ∂γC (x,is increasing in x, which concludes the proof. QEDb∂θb = −(P − c + w). we rewrite the loss function γ (θ)
b asProof of Lemma 10: Given b(θ)
C
followsZθbb (θ|x∗ (θ))dθ
b
b(θ)f
−

−

Zθ

C

θ

b
b
b(θ)f (θ|x∗C (θ))dθ+ x∗C (θ)θbThen by the envelope theoremdγC (θb)b (θ|xb ∗ (θ))b <0= −b′ (θ)Fdθb

Finally, as θb = b−1 (−(P − c + w)), is decreasing in P − c and w, γC is increasing in P − cand w. QEDProof of Lemma 11: The proof is very similar to Lemma 1 related to the binary penalty
case. We start by showing that gC∗ (0) = −b(θ). Intuitively, the contractor with zero wealthb ∗ (0))) =only can make zero profits when she defaults with probability one. Then, θ(g
C
∗
∗
−1
∗
b
b (−gC (0)) = θ which implies that gC (0) = −b(θ). Alternatively, γC (θ(gC (0))) = γC (θ) =−b(θ).
b ∗ (w ) + w)) has a unique solution. For
The next step is to show that gC∗ (wi ) = γC (θ(giC
the same arguments as Lemma 1, this is because γC is increasing in w, it has a slope lower
b < 1), and γ (0 + w ) > g ∗ (0). Finally, we show that g ∗ (w ) isb ∗ (θ))C
than one ( ∂γ= F (θ|x
i
C
i
C
C
∂w
∗
∗
increasing in wi , following also the same arguments in Lemma 1. Let gC − γC (gC + w) = 0
dg ∗by taking the total derivative (1 − γC′ (gC∗ + w))dgC∗ − γC′ (gC∗ + w))dw = 0. Then, dwC =′ (g ∗ +w)
γCC> 0. This is because, 0 < γC′ < 1.. QED
1−γ ′ (g ∗ +w)
C

C

Proof of Lemma 13: We know that the bidding equilibrium mark up has to lead to zeroprofit ex-ante with the optimal level of effort, we can write this condition for both incentive
schemes as follows, for b(θ):

b ∗) +−wF (θ|xZθ

(b(θ) + g ∗ (w))f (θ|x∗ )dθ − x∗ = 0

θb

for b̃(θ)∗−wF (θ̃|x̃ ) +bZθ(b̃(θ) + g̃ ∗ (w)))f (θ|x̃∗ )dθ − x̃∗ = 0.

b̃θ

31

If we replace in this later equilibrium condition, x̃∗ by x∗ , the expected ex-ante pay-off haveto be negative since we are introducing a non optimal effort. Then, it has to be true thatb ∗) +−wF (θ|x

Zθ

∗

∗

∗

(b(θ) + g (w))f (θ|x )dθ > −wF (θ̃|x ) +
b

θb

Zθ

(b̃(θ) + g̃ ∗ (w))f (θ|x∗ )dθb̃θ

Consider, contrary to the Lemma that θb ≥ θ̃. Then, the previous inequalty can be written
in such a case as follows:
b

Zθb

∗

∗

(−w − (b̃(θ) + g̃ (w)))f (θ|x )dθ >Zθ((b̃(θ) + g̃ ∗ (w) − b(θ) − g ∗ (w))f (θ|x∗ )dθ

θb

b̃θ

However, the left hand side is negative (because −w < (b̃(θ) + g̃ ∗ (w)) for θ > θ̃) and theright hand side is positive since (because (b̃(θ) + g̃ ∗ (w) > b(θ) + g ∗ (w))). Then, we have abcontradiction, which implies that θb < θ̃. This concludes the proof. QEDb

Proof of Lemma 15: Recall: γbif w ≥ w(b(θ))= −b(θ) − γC (θ)
otherwise.

 γ̃b b̃(θ)) = −b̃(θ) − γ̃C (θ)
if w ≥ w(otherwise.

gC (b(θ), w) = andgC (b̃(θ), w) = C (θ)∗
gC (w)C (θ)∗
g̃C (w)where γ̃C (θ) and g̃C∗ (w) are accordingly defined with b̃(θ).
Since b̃(θ) is a rotation of b(θ), the expected losses of performance stage for fully solvent
firms are lower under b̃(θ) than under b(θ), i.e., γ̃C (θ) < γC (θ). Noting that b̃(θ) = b(θ), web b̃(θ)) > w(b(θ)).
bhave w(
Then






b b̃(θ)),γ̃C (θ) − γC (θ) if w ≥ w(∗b b̃(θ)) ≥ w ≥ w(b(θ)),
bgC (b̃(θ), w) − gC (b(θ), w) =  g̃C (w) − γC (θ) if w(
 g̃ ∗ (w) − g ∗ (w) if w < w(b(θ)).
bCCb b̃(θ))) = γ̃C (θ),
b
Notice that as g̃C∗ (w) is increasing in w, gC∗ (w(b(θ)))= γC (θ) and g̃C∗ (w(then gC (b̃(θ), w) − gC (b(θ), w) is a continuous and weakly increasing function of w for w ≥bw(b(θ))
as seen in the binary penalty case. To complete the proof, we need to show that∗g̃C (w) − gC∗ (w) is also weakly increasing in w.

32b ∗ (w)+w)), and θ(gb ∗ (w)+w) satisfies
Recall gC∗ (w) is implicitly defined by gC∗ (w) = γC (θ(gCCb ∗ (w) + w)) = −(g ∗ (w) + w). Substituting and rearranging, we have
b(θ(g
C
C

b ∗ (θ)))bb ∗ (θ))b −gC∗ (w)(1 − F (θ|x
= wF (θ|x
C
C

Zθ

b
b
b(θ)f (θ|x∗C (θ))dθ+ x∗C (θ).

θbb ∗ (w) + w). Differentiating with respect to w, we obtain
where θb = θ(gC

gC∗′ (w) =b ∗ (θ))b
F (θ|xC.
bb ∗ (θ))1 − F (θ|xC

Under the assumption that the equilibrium probability of default weakly increases as the
incentive scheme becomes more powerful, we haveg̃C∗′ (w) ≥ gC∗′ (w)Integrating this for w > w′ and rearranging,g̃c∗ (w) − gc∗ (w) ≥ g̃c∗ (w′ ) − gc∗ (w′ ).
This concludes that the equilibrium markups have increasing differences. QEDProof of Proposition 16: As Lemma 15 has stablished the supermodularity of themarkup functions, it follows from the same arguments than the proof of Proposition 3.
QED

B

Monetary Care.

Let z = P − c + w be now the initial wealth (before the effort decision) at the construction
stage, of a firm that plans to make default in case of bad performance. This firm chooses alevel effort xM ∗ (z) which minimizes his expected total cost,xM ∗ (z) ∈ arg min p(x)z + (1 − p(x))x,
In case of a bad performance the firm looses all its wealth and otherwise, its cost islimited to the exerted effort.20 Compared to the non monetary effort case, the cost of effort
is subsidized in the case of bankruptcy. Let γ M (z) be the expected private cost of being
Notice that the problem remains regular since the second order condition is satisfied, p(x)′′ (z − x) −2p (x)) ≥ 0.
20
′

33

liable when effort is monetary.
γ M (z) = p(xM ∗ (z))z + (1 − p(xM ∗ (z))xM ∗ (z).
As in the case of non monetary effort, xM ∗ (z) and γ M (z) are increasing functions, andγ M (0) = 0, and γ M ′ (z) < 1. Compared with the non monetary case, when effort is monetaryand firms are not fully solvent, xM ∗ (z) ≥ x∗ (z) (since care is partially subsidized) and γ M (z)≤ γ(z) (since liability costs are lower). However, for the fully solvent firms, the equilibrium
level of effort is the same than in the non monetary case, and then also the expected payoffs
during the contracting stage.
The equilibrium level of effort will be x∗ (b)xM
E (x, w) = 

if P − c + w > b + x∗ (b)xM ∗ (z) otherwise.

Following the arguments of the main text, we characterize the bidding equilibrium bid ofeach firm i as the minimum price Pi∗ for which firm i is willing to accept the project. As in thenon monetary case, if the firm does not plan to make default in the case of bad performance,
the price that makes the net expected profits zero is PiM ∗ = ci + γ(b) , and the equilibrium
markup in such case is: PiM ∗ − ci = γ(b). Similarly to the main text, if the firm is not
solvent, the price that makes the net expected profits equal to zero is given by the expression
PiM ∗ = ci + γ M (PiM ∗ − ci + wi ) or, in terms of the markup, g M ∗ (wi ) = γ M (g M ∗ (wi ) + wi ),where g M ∗ (wi ) is unique and increasing in wi for the same arguments than in the nonmonetary case. It is important to notice that γ M (z) ≤ γ(z) ⇒ g M ∗ (wi ) ≤ g ∗ (wi )dM , such that the default
As in the non monetary case, there exists a cut-off level wdM) =markup function is equal to non-default markup, γ(b). Then for this cut-off g M ∗ (wddM ) = γ(b). For this later equality to be true, it is necessary that γ(b) + w
M =γ M (γ(b) + wdd
M = (1 − p(x∗ (b)))b, where w
M (b) is increasing on b. As
b + x∗ (b). Then, the cut-off level wdM (b) = b − γ(b) + x∗ (b), more types will be liable under monetary care,bw(b)= b − γ(b) ≤ w
but the equilibrium bidding markup is qualitatively the same:g M (b, wi ) = γ(b)dM (b)if wi > w
 g M ∗ (w ) otherwise.
i

In particular, g M (b, wi ) is supermodular in (b, wi ), if b′ > b then g M (b′ , w) − g M (b, w)−

34

is weakly increasing in w. Then, the proof is identical to the non-monetary case, if b′ > b,






d
M (b′ ),
γ(b′ ) − γ(b) if w ≥ wdd
M (b′ ) ≥ w ≥ w
M (b),g M (b′ , w) − g M (b, w) = g M ∗ (w) − γ(b) if w



d

M (b).
0
if w < wdd
M (b)) = γ(b) and g M ∗ (wM (b′ )) =As g M ∗ (w) is increasing in w, and by definition g M ∗ (wγ(b′ ), then g M (b′ , w) − g M (b, w) is increasing in w. This implies that g M (b, w) and the equilibrium bid, P M ∗ = c + g M (b, w), are supermodular functions, and consequently Proposition
2 and Proposition 3 hold under monetary care.

C

Revenue Equivalence between Second-Price Auction and First
Price Auction with Perfect Information.

Let P1∗ < P2∗ < ... < Pi∗ < ... < Pn∗ be the bids of the second-price auction, where we havelabeled firms according to their bids, being firm 1 the firm with the lowest bid, and firm n,
the firm with the highest bid. Hence, the outcome of the second-price auction will be thatfirm 1 wins at a price equal to P2∗ . In a first-price auction with perfect information, there
exists a Nash equilibrium that generates the same outcome. Consider the following profile ofbidding strategies (P2∗ , P2∗ + ε, P3∗ , ..., Pi∗ , ..., Pn∗ ) where ε is the minimum amount in whicha bid can be modified. This is a Nash equilibrium because no one can profit by unilaterally
deviating and it generates the same allocation and revenue than the second-price auction:
firm 1 wins at a price equal to P2∗ . Before proceeding with the proof, it is important to
notice that the bids in the second-price auction are the opportunity costs of the firm fromaccepting the contract. Meaning that firm i with a price lower than Pi∗ would make negative
profits in expectation, and with a price higher than Pi∗ will make positive profits. Then,if firm 1 deviates and bids more than P2∗ , she looses the contract (50% of probability ofloosing in the case of bidding P2∗ + ε) and gets zero instead of a positive payoff. Bidding
lower than P2∗ reduces directly her profits. If firm 2 deviates and bids P2∗ , she gets zeroprofits as with the current bid. Bidding lower than P2∗ generates negative profits for Firm 2.
Finally, making a bid higher than P2∗ + ε, does not modify the allocation and consequently,
firm 2 also obtains zero profits. For some firm other than firm 1 and 2, if she bids a lower
amount, either it does not win yet, or she gets a negative payoff in case she bids lower than
the winning bid. If she bids higher, her payoff remains at zero. So this strategy profile is aNash Equilibrium since no profitable unilateral deviations exists.

35

D

General Allocation Mechanisms.

Proposition 2 in the main text states that a second-price auction mechanism adversely selects
undercapitalized firms for the project, since for a given level of private cost in delivering the
project, the lower the wealth w, the larger the probability of winning.
This market failure does not depend on the competitive mechanism used for allocating
the project, since incentive compatibility leads to adverse selection in this incentive settingin which firms are protected by limited liability.
In order to formalize this claim, in this appendix we consider an arbitrary direct incentive
compatible mechanism Ψ(w, c), P (w, c) specifying a probability of winning Ψi and a price
Pi in the case of winning for each firm i and for every vector of assets and costs (w, c).21
We focus on the incentives of agents with the same cost, but different assets, to truthfully
reveal their types. Notice that by looking at the case of firms with identical costs anddifferent levels of wealth, we do not restrict the domain of the types we are considering
but we want to characterize a minimal necessary condition that must be satisfied for any
incentive compatible mechanism. In order to do so, by taking expectations over the othertypes (w−i , c−i ), we can redefine the mechanism according to the individual probability ofwinning
Ψi (wi ) = Ew−i , c−i {Ψ(wi , w−i , c−i )}and a distribution of prices for each wi , Pi (wi ).
Proposition 17. In any IC mechanism (Ψi (wi ), Pi (wi )), for which the distribution of prices
can be ranked in terms of First Order Stochastic Dominance, the probability of winning of
an individual firm, Ψi (wi ), is monotonically decreasing in wi , and the expected price in caseof winning, Pi (wi ), is monotonically increasing in wi .
Proof: The firm indirect utility function isb cb, wi , ci ) = Ψi (w,b cb)πi (Pi (w,b cb), wi )Πi (w,

where wb and cb refer to revealed types while wi and ci are real types. We want to analyze
the ex-ante incentives to reveal truthfully the level of assets of firms with the same cost
but different assets. For each w, Ψi (wi ) is a number, and Pi (wi ) is a distribution of prices.
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We are assuming that only the winning firm receives a transfer. This is consistent with the motivationof the paper, focused on firms with different wealth levels and protected by limited liability, and moreover
it is a common feature of most of the real world procurement processes. Alternatively, we could consider,for example, that all firms pay for participating in the auction and that the money thus raised is added tothe price paid to the winning firm, reducing or eliminating the probability of expost bankruptcy. Chillemi
and Mezzetti (2011) explores these procurement mechanisms.

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Then, we define the ex-ante profit function.
Πi (wi′ , wi ) = Ψi (wi′ )EPi (wi′ ) {πi (P, wi )}.
Where πi (P, wi ) is the profit function defined in the baseline model but to economize innotation we are omitting the dependence of profits with respect to b and ci . Consider thatwi′ > wi . The incentive compatibility constraint on the general mechanism, Ψ(w, c), P (w, c),
has to imply the following conditions over the ex-ante incentives to reveal truthfully thelevel of assets of firms with the same cost but different levels of wealth
Πi (wi , wi ) ≥ Πi (wi′ , wi )(7)

Πi (wi′ , wi′ ) ≥ Πi (wi , wi′ ).

(8)

and

Combining these two equations we obtain
Πi (wi , wi ) − Πi (wi , wi′ ) ≥ Πi (wi′ , wi ) − Πi (wi′ , wi′ )which is equivalent to
Ψi (wi )EPi (wi ) {πi (P, wi ) − πi (P, wi′ )} ≥ Ψi (wi′ )EPi (wi′ ) {πi (P, wi ) − πi (P, wi′ )}.

(9)

From Proposition 3 of the basic model, we know that πi (P, wi ) − πi (P, wi′ ) is weakly
decreasing in P. Now, assume for contradiction that Pi (wi′ ) < Pi (wi ) (in terms of FOSD)
since πi (P, wi ) − πi (P, wi′ ) is decreasing in P, this implies thatEPi (wi ) {πi (P, wi ) − πi (P, wi′ )} ≤ EPi (wi′ ) {πi (P, wi ) − πi (P, wi′ )}.

(10)

Given (10), inequality(9) can only be satisfied if Ψi (wi′ ) ≤ Ψi (wi ). But Pi (wi′ ) < Pi (wi ) andΨi (wi′ ) ≤ Ψi (wi ) violates the incentive compatibility constraint of type wi′ (inequality (8)).
Hence Pi (wi′ ) > Pi (wi ). But then Ψi (wi′ ) ≤ Ψi (wi ). Otherwise, the incentive compatibility
constraint of type wi would be violated (inequality (7)). This concludes the proof.
Then, financially weaker contractors are selected with higher probability in any incentive
compatible mechanism. In our setting, the opportunity cost of undertaking the projectis decreasing in the wealth of the contractor and then, any competitive mechanism would
adversely select undercapitalized bidders.

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