Tsukuba Economics Working Papers
No. 2009-013

Procurement Auctions with Pre-award Subcontractingby
Jun Nakabayashi

October 2009This version: June 2011

UNIVERSITY OF TSUKUBA
Department of Economics
1-1-1 Tennodai
Tsukuba, Ibaraki 305-8571
JAPANProcurement Auctions with Pre-award Subcontracting
Jun NAKABAYASHI∗
First Draft: October, 2009This Version: June, 2011

Abstract
To be the lowest bidders in procurement auctions, contractors commonly solicit subcontract bids at the bid preparation stage. In this research, we establish a model of a
two-stage procurement auction to analyze such pre-award upstream competitions included
in procurement auctions. Our main findings include the following: i) first-price upstream
auctions dominate second-price upstream auctions in terms of allocative eﬃciency andprocurement cost; ii) neither a first- nor a second-price auction in upstream competitions
implements an optimal outcome; and iii) the ex post negotiation between the prime andsubcontractors improves the performance of second-price auctions in upstream competitions. We show that the optimal outcome is implemented with the use of the negotiatedsecond-price auctions in upstream competitions.
Key words: procurement auctions, subcontracting, optimal mechanism
JEL classification: D44, H57, L42

1

Introduction

Subcontracting and outsourcing are common business practices in procurement markets. In ahighway construction project, for instance, the winning bidder may subcontract road marking
or signal work to specialty firms. In addition, the contractor may purchase raw materials or
equipment from other sources, which can also be considered subcontracting in the broader
sense. For prime contractors, it is not unusual that the bulk of the cost of a large construction
project that covers a wide range of work consists of subcontract payments.
To obtain qualified subcontracts at fair prices, prime contractors (PCs) commonly ask
subcontractors (SCs) to bid irrevocable price quotes (subcontract bids) prior to submitting abid in the procurement auction (Clough and sears (1994), Dyer and Kagel (1996), Marechal
and Morand (2003), Grosskopf and Medina (2007)). This practice also satisfies a PC’s need to∗Doctoral Program in Economics, Graduate School of Humanities and Social Sciences, University of Tsukuba
Tennodai 1-1-1, Tsukuba, Ibaraki 305-8571, Japan. E-mail: nakabayashi.jun.gn@u.tsukuba.ac.jp. I am
grateful to Howard P. Marvel for his guidance. I also thank P. J. Healy, Kazumi Hori, John Kagel, Georgia
Kosmopoulou, Dan Levin, Matt Lewis, Naoki Watanabe, and, in particular, Lixin Ye, for their very helpful
suggestions and comments. All remaining errors are my own.

1obtain the cost estimate for bidding.1 In this sense, PCs are not only bidders in the downstream
procurement auction but also auctioneers in the upstream subcontract auction.
The objective of this paper is to analyze such pre-award upstream auctions included in the
procurement auction. We established a model of a two-stage procurement auction in which a
procurement buyer auctions oﬀ a project for bidders (PCs). In the first stage, each PC solicits
subcontract bids from a set of SCs, where the PC is assumed to know the distribution but
not the values of the SCs’ costs to complete the subcontractable work. If the PC selects an
SC, the PC makes a pre-award subcontract agreement with the SC that specifies the amountof the subcontract payment from PC to the winner SC. In the second stage, the PC bids inthe procurement auction with private value assumption, given that his cost is the sum of thesubcontract payment plus his own cost to complete the remaining non-subcontractable workfor himself.
We demonstrate that the use of second-price auctions in upstream competitions results
likely in ex post ineﬃcient allocation and a higher procurement cost than the use of first-priceauctions. However, the defects of second-price upstream auctions are resolved by introducing
an ex post negotiation between the winner PC and sc on the subcontract payment contingent
on the outcome in the downstream auction. We show that, with the ex post negotiation,
second-price auctions in upstream competitions implement an optimal outcome in procurement
auctions with subcontracting.
The closest works related to this research are Hansen (1988) and Wambach (2009), whichshow the failure of the revenue equivalence in upstream auctions assuming that the outcomeof the downstream market is reduced into a downward-sloping demand curve. Due to modeling the downstream auction, in this study, non-trivial extensions are made to their results,
including optimal design and eﬃciency analysis in multi-layered procurement.
The remaining part of this paper is organized as follows. Section 2 describes the modelof procurement auctions with pre-award subcontracting. Section 3 examines the equilibrium
bidding behavior in upstream auctions. Section 4 analyzes the equilibrium in the downstream
auction. Section 5 provides a general model in which PCs are heterogeneous, and Section 6 isthe conclusion.

2

The model

Consider a procurement auction in which a procurement buyer solicits bids from N risk-neutral
prime contractors (PCs), each denoted by i ∈ {1, . . . , N }, to purchase a project. The value ofthe project to the procurement buyer is equal to V . The procurement buyer sets a reservation
price r in the procurement auction so that any bid above r is rejected.
1
Furthermore, some procurement buyers require PCs to submit a proposed subcontracting plan that must
be approved by the contracting oﬃcer prior to bidding. For instance, the state of Oregon requires bidders inpublic projects to submit a list of first-tier subcontractors and their subcontract bids if the amount of the bid
is greater than five percent of the total project bid or $15,000 (ORS 279C.370).

2

We assume that the work to complete the project consists of two components, the subcontractable and the non-subcontractable work, and the share of the two components is given andidentical for all PCs. Prior to submitting a bid in the procurement auction, each PC solicits
subcontract bids from n risk-neutral subcontractors (SCs), denoted by {SCi,1 , . . . , SCi,n }.
Prior to being solicited by a PC, each SC draws his cost t to perform the subcontract. The
cost t has a commonly known atomless distribution Ft with support [t, t̄]. The cost of PC to¯complete the subcontractable work for himself is infinitely large, and PCs always subcontract
so that reserve prices in upstream auctions are never below t̄. The cost of the PC to complete
the remaining non-subcontractable work is normalized to be equal to zero.2
The entire game consists of two stages. In the first stage, N upstream auctions occur, andthe downstream auction takes place in the second stage. In the first stage, SC (i, j) obtains
his cost ti,j and submits a subcontract bid si,j to PC i in an upstream auction i. An SC will
be selected as a winner SC and make a subcontract agreement with PC i, which specifies theamount of a conditional subcontract payment ci from PC i to the SC.3 In the second stage,
PC i submits a bid bi in the downstream auction with reservation price r given that his cost
is equal to ci .
Throughout this paper, we assume private values. The cost of SC (i, j) is known only tohim, and the cost of PC i is known only to PC i. For the regularity condition, Ft (·) satisfies
log-concavity i.e., ft� (t)/ft (t) is non-increasing.
An optimal outcome in the double-layered procurement can be characterized by considering the situation in which each PC vertically integrates n SCs and bids for the procurement contract. Define wi ≡ min{ti,1 , . . . , ti,n }. The cumulative distribution function of wi is
given by Fw (w) = 1 − [1 − Ft (w)]n . Then, due to Riley and Samuelson (1981) and Myerson(1981), an optimal outcome is identified such that the procurement contract is allocated to
the lowest-cost integrated firm whose cost is lower than the reservation price r∗ such thatr∗ = V − Fw (r∗ )/fw (r∗ ). Taking the optimal outcome as a benchmark, we discuss how it canbe implemented through layered procurement auctions.

3

Equilibrium in upstream auctions

If second-price auctions are used in upstream competitions, the subcontract bid only determines the SC’s winning probability but not the payoﬀ conditional on winning.4 Hence, as in
the case of standard procurement auctions, submitting s = ti,j is a dominant strategy for the
SC in the second-price subcontract auction.5
2

The discussion in which the assumption is relaxed is delivered in Section 5.
The payment ci is conditional because it is paid if the PC i indeed wins in the downstream auction.
4
In this setting, revenue equivalence holds between an English (ascending) auction and a Vickery (secondprice sealed-bid) auction.
5
Let B denote the lowest competing bid in the upstream auction. By bidding ti,j , the bidder will win if
ti,j < B and not if ti,j > B. Suppose, however, that he bids an amount ti,j < z. If ti,j < z ≤ B, then he stillwins in the upstream auction, and his profit is still ti,j − B if the PC wins in the downstream competition.
However, if ti,j < B < z, then he loses, whereas, if he had bid ti,j , he would have made a positive profit33

To examine the case of first-price auctions used in upstream competitions, we restrict ourattention to a symmetric increasing equilibrium in which all SCs follow an increasing strategy
σ(·) and all PCs follow an increasing strategy β(·). Let ti,(j:n) be the cost of the jth lowest-cost
SC solicited by PC i. The cost of PC i is then given by ci = σ(ti,(1:n) ). To obtain σ, consider
the situation in which all SCs other than SC (i, j) follow σ and that in which all PCs follow β.
If SC (i, j), j = 1, . . . , n, bids s to PC i, SC (i, j) wins the upstream auction with probability:
P (s) = [1 − Ft (σ −1 (s))]n−1 .
Having his cost ci equal to s, PC i wins in the downstream auction if and only if ci is lower
than that of all the N − 1 rivals and reservation price r in the downstream auction. Hence,
provided that SC (i, j) wins in upstream auction i, the conditional probability that PC i winsin the downstream competition is given by6
Q (s|N, r) = [1 − Ft (σ −1 (s))]n(N −1) 1{s≤r} .
SCs receive positive payoﬀs with probability P (·)Q(·). Hence, the objective function is givenby��n(N −1)π(ti,j |N, r) = max(s − ti,j ) 1 − Ft (σ −1 (s)).
sTaking the derivative with respect to s, imposing a symmetric condition, and solving thediﬀerential equation yield
σ(ti,j |N, r) = ti,j +

� r �
ti,j

1 − Ft (t̂)
1 − Ft (ti,j )�n(N −1)

dt̂.

(1)

Let t(1:N )(j:n) be the cost of the jth lowest-cost SC solicited by the winner PC. Then, theexpected subcontract price is equal to E[σ(t(1:N )(1:n) |N, r)]. On the other hand, the expectedsubcontract price is equal to E[t(1:N )(2:n) ] when second-price auctions are used in upstream
competitions. Hence, the following lemma is established regarding the expected winning bid
in the upstream auction, which extends the results of Hansen (1988) and Wambach (2009) tothe model of multi-stage auctions.
Lemma 1. Subcontract prices are lower when first-price auctions are used than when secondprice auctions are used in upstream competitions, namely,E[σ(t(1:N )(1:n) |N, r)] ≤ E[t(1:N )(2:n) ].
conditional on the PC’s success in the downstream competition. Thus, bidding less than ti,j can never increase
his profit but, in some occasions, may in fact decrease it. A similar argument shows that it is not profitable tobid less than ti,j .
6
The decreasing function Q(s|·) is a reminder of the decreasing function q(·) in Hansen (1988) representing
the downstream demand schedule.

4

Proof. Let t(2:N )(j:n) be the cost of the jth lowest-cost SC solicited by the lowest rival PCof the winner in the downstream auction. Equation (1) suggests that, for any t(1:N )(1:n) , thesubcontract price in the first-price upstream auction��n(N −1)−1
ft (t̂) 1 − Ft (t̂)σ(t(1:N )(1:n) |N, r) =t̂n(N − 1) ��n(N −1) dt̂,t(1:N )(1:n)
1 − Ft (t(1:N )(1:n) )� r= E[min{t(1:N )(2:n) , t(2:N )(1:n) , r}|t(1:N )(1:n) ]never exceeds the conditional expectation of the subcontract price in the second-price upstream
auction, E[t(1:N )(2:n) |t(1:N )(1:n) ].
Three observations are particularly noteworthy. First, if first-price auctions are used in
upstream auctions, each SC bids as if all other SCs, including the ones bidding for other PCs,
were also his rival. Thus, the expected subcontract price declines as N rises or r falls. The
equivalence of the expected subcontract price holds only when N = 1 and r ≥ t̄.7
Second, PCs using a first-price auction for the upstream competition have stronger bargaining power against SCs as the number of rival PCs increases or the reservation price falls in thedownstream auction.8 This suggests that first-price auctions in upstream competitions allow
PCs to share the risk with SCs of uncertainty in the downstream market, while second-priceupstream auctions do not.
Finally, the mechanism of the upstream auctions influences the allocative eﬃciency. In thesymmetric equilibrium under the first-price upstream competitions, the lowest-cost SC in the
upstream market always receives the subcontract. However, if second-price auctions are usedin upstream competitions, the lowest-cost SC misses the subcontract if the cost of his lowest
rival SC is higher than the cost of the second-lowest-cost SC in another upstream auction.
Hence, the following theorem is established regarding allocative eﬃciency.
Theorem 1. The use of a second-price auction in upstream competitions likely leads to an
ineﬃcient allocation.
As a simple example, suppose that N = 2, n = 2, and m = 0 (no common SC). If t1,(1:n)
is the lowest-cost SC, possible realizations of SC’s costs are
R1 : t1,(1:n) < t1,(2:n) < t2,(1:n) < t2,(2:n) ,R2 : t1,(1:n) < t2,(1:n) < t1,(2:n) < t2,(2:n) ,R3 : t1,(1:n) < t2,(1:n) < t2,(2:n) < t1,(2:n) .
7

If the winner PC solicits SCs after the downstream competition, the outcome is equivalent to the casewith N = 1 and r ≥ r̄, in which the equivalence in the expected subcontract price holds. Hence, pre-award
subcontracting with the use of first-price auctions dominates the post-award subcontracting conducted withany mechanism from the viewpoint of lowering the subcontract price.
8
This result holds even if some SCs bid to multiple PCs. As shown in Appendix A, the SC’s bidding strategy
is identical regardless of whether the SC bids exclusively to a single PC or bids simultaneously to multiple PCs.
Thus, although the multiple-bidding SC does not care which PC wins, the SC is induced to bid a lower priceby the aggressive bids of exclusively bidding SCs.

5

Since ts are i.i.d. samples, each of these realizations, {R1 , R2 , R3 }, occurs with probability 1/3.
Thus, if second-price auctions are used in all upstream competitions, an ineﬃcient outcome
results with probability 1/3.
When R3 occurs, the most eﬃcient SC may beneficially negotiate with the PC to undercut
the subcontract payment slightly below t2,(2:n) . The oﬀer expands the chance of obtaining
a positive payoﬀ for the lowest SC but never reduces the PC’s payoﬀ. Thus, the use of asecond-price auction with negotiation weakly dominates the use of the standard second-priceauction for both the PC and the winning SC.
In fact, the ex post negotiation between PCs and SCs resolves the disadvantage of thesecond-price subcontract auction in revenue and eﬃciency. Suppose that the downstream
contest takes place with a second-price auction. Let b−i be the lowest bid of i’s rival PCs in thedownstream competition. After the negotiation, the winning PC’s subcontract payment will bemin{ti,(2:n) , b−i , r}. Thus, a weakly dominant strategy of PC i in the downstream auction is tosubmit bi = ti,(1:n) .9 Therefore, i’s payoﬀ when winning is min{b−i , r} − ti,(2:n) if min{b−i , r} −ti,(2:n) > 0 or zero otherwise. Because the amount of the subcontract payment depends on thelowest-rival bid in the downstream competition, the mechanism of the downstream competition
must be a second-price auction.
Two observations were then made. First, the negotiated second-price subcontract auction
always results in an ex post eﬃcient allocation, since the PC receiving the oﬀer from thelowest-cost SC always wins in the downstream competition. Second, if all PCs use a secondprice auction with ex post negotiation, the subcontract price equals the second-lowest cost of
all n × N SCs. Hence, in the winning PC’s upstream auction, the expected subcontract priceequivalent holds between a first-price and the negotiated second-price auctions.
Since all the benefit from the negotiation belongs to SCs, the negotiated second-priceupstream auction may need regulation that restricts the freedom of PCs to choose the mechanism in the upstream market. It follows that cost-minimizing procurement buyers should notentirely delegate subcontracting to PCs.

4

The downstream auction

Let Fc (·) be the distribution of the PC’s cost. If a second-price auction is used in upstream
auctions, thenFc (c|SP) = 1 − n(n − 1)Ft (c)ft (c)[1 − Ft (c)]n−2 .
9Since PC i receives a positive payoﬀ if and only if b−i > ti,(2:n) , any bid b ∈ [ti,(1:n) , ti,(2:n) ] creates thesame expected payoﬀ for PC i.

6

If first-price auctions are used in upstream competitions and all SCs follow the symmetricequilibrium σ, the PC’s cost distribution is given by
Fc (σ(t|N, r)|N, r, FP) = 1 − nft (t)[1 − Ft (t)]n−1 .

(2)Hence, the distribution is endogenous in N and r only when first-price auctions are used in
upstream auctions. Recall that σ(t|N, r) declines as N rises or r falls. Since the right-handside in Equation (2) is constant with respect to N and r, Fc (·|N, r) rises as N rises or r falls.
It follows that, if first-price auctions are used in upstream competitions, the distribution of
the bidders’ private information in the downstream competition shifts to the right in the senseof the first-order stochastic dominance as the number of bidders decreases or the reservation
price increases.
The results have impacts on the theoretical and empirical auction models that cruciallydepend on the assumption that the distribution of private information is independent of themechanism and the number of agents. For instance, the cost-minimizing reservation price inthe downstream auction must be a function of the number of bidders if the upstream contests
are conducted through first-price auctions. In addition, the pseudo-values estimated fromthe structural estimation of auctions must be pooled separately according to the number ofbidders to obtain the distribution. For instance, the cost distribution is lower-shifted if thenumber of bidders in the sample auction is greater than the average. Hence, the estimatedcost distribution would be biased upward unless the estimation is conducted separately fromthat using the pseudo-values obtained from the auction data with a larger number of bidders.
Despite the endogeneity, the distribution of the bidder in the downstream auction is i.i.d..
Hence, the downstream auction is revenue-equivalent if all upstream competitions use either afirst- or second-price auction. Using the result, we obtained the following theorem regarding
the procurement cost.
Theorem 2. The use of first-price auctions in upstream competitions lowers the procurement
cost more significantly than the use of second-price auctions.
Proof. Let t(2:N )(j:n) be the jth lowest cost of n SCs solicited by the second-lowest PC. The
cost of the second-lowest PC is equal to σ(t(2:N )(1:n) |N, r) if first-price auctions are usedin all upstream auctions, whereas it equals min{t(2:N )(2:n) , r} if second-price auctions areused. By Lemma 1, σ(t(2:N )(1:n) ) ≤ E[t(2:N )(2:N ) ], which entails that σ(t(2:N )(1:n) |N, r) ≤min{t(2:N )(2:N ) , r} for any r. Hence, if a second-price auction is used in the downstream auction, the use of first-price auctions in upstream competitions lowers the procurement cost moresignificantly than the use of second-price auctions. Revenue equivalence in the downstream
competition generalizes the result for any mechanism of downstream competition.
Besides, if the ex post negotiation is possible in all upstream competitions with the form of
second-price auctions, the bid of the lowest losing PC is t(2:N )(1:n) , whereas it is σ(t(2:N )(1:n) )if first-price auctions are used in the upstream market. Hence, if the downstream competition7

is restricted to a second-price auction, the ex post negotiation in a second-price subcontract
auction further reduces the procurement cost.
In fact, with the use of a reservation price in the downstream auction, the procurement
cost is minimized.
Theorem 3. Let Fw be the cumulative distribution function of wi = min{ti,1 , . . . , ti,n }. If upstream competitions take place with the use of a negotiated second-price auction, the procurement buyer implements the optimal outcome by using a second-price auction with reservation
price r∗ such that r∗ = V − Fw (r∗ )/fw (r∗ ) in the downstream competition.
Proof. The PC wins if the cost of his lowest-cost SC is t(1:N )(1:n) and t(1:N )(1:n) < r∗ . In
addition, the winner PC receives a payment equal to min{t(2:N )(1:n) , r∗ }. Thus, both allocationand payment rules are identical to the optimal outcome characterized in Section 2.
The intuitions behind these results are threefold. First, the ex post negotiation between
the PC and SC eliminates the double marginalization problem. The expected aggregate informational rents obtained by both winner PC and SC always coincide with the expected payoﬀ
of the winning integrated firm in the optimal outcome observed in Section 2. This also implies
that the negotiation realizes the optimal risk sharing between the PC and SC on uncertaintyof the downstream competition by limiting the informational rent accrued to both PC and
SC. Furthermore, the reservation price r∗ is independent of N , similar to the case in which
PCs vertically integrate SCs. Due to the double marginalization problem, the cost-minimizing
reservation price in the case of first-price auctions used in upstream competitions coincides
with r∗ only when N approaches infinity.
Second, to implement the dominant strategy equilibrium, procurement buyers may needto give an � small amount of transfer to the winner PC so that PCs bid the cost of the lowest
SC solicited in the upstream auction. For PC i, submitting any bid between ti,(1:n) and ti,(2:n)
yields the same expected profit. The transfer, if any, induces PC i to bid the lowest amountti,(1:n) , in which the winning probability is maximized. As � approaches zero, the expected
procurement cost is identical to that in the optimal outcome characterized in Section 2.
Finally, the ex post negotiation requires the downstream competition to be conducted withthe second-price auction format. Thus, if a first-price auction must be used in the downstream
competition, first-price auctions are the second-best mechanism in upstream competitions fromthe viewpoint of cost minimization. In this case, the procurement cost falls as the numberof PCs increases even if the total number of SCs in the upstream market is fixed.10 Thus,
10

Let Z ≡ n × N , and let t(j) be the jth lowest-order statistic among Z i.i.d. samples. Suppose also that thedownstream competition takes place �with a second-price auction. Then, the probability that the procurement
cost is equal to or above σ(t(j) ) is j−2
k=1 (n − k)/(Z − 1) for all j = 3, . . . , n + 1, and the probability thatthe procurement cost equals σ(t2 ) is (Z − n)/(Z − 1). Given Z, the probability is increasing in n for any
j = 3, . . . , n + 1 and decreasing in n for j = 2. Hence, the distribution of the procurement cost given n is
first-order stochastically dominated by the distribution given n̂ > n. Since the SC’s strategy and their expected
payoﬀs are unchanged, the severer competition in the downstream market extracts more rents only from PCs.

8the optimal number of PCs is equal to the number of SCs in the upstream market if thedownstream competition uses a first-price auction.

5

The noise term in PC’s costs

In this section, we briefly investigate the case in which PCs have heterogeneity in the production cost to complete the non-subcontractable work. Let θi be the cost of PC i, i = 1, . . . , N tocomplete the work. Then, PC i’s total cost ci is the sum of θi plus the subcontract payment.
Let Fθ (·) be the commonly known atomless distribution of θ with support [θ, θ̄] and let fθ be¯
its density. As for regularity condition, fθ is log-concave.
The SC’s dominant strategy is still to bid his cost if a second-price auction is used in
the upstream competition. On the other hand, suppose that upstream competitions use first�
�n−1
price auctions. Then, if all other SCs follow σ, P (s) = 1 − Ft (σ −1 (s))
1{s≤t̄} is theprobability that SC (i, j) will win in upstream auction i when his bid is equal to s. In addition,
Q(s|N, r, σ −i ) is the conditional probability that PC i wins in the downstream auction giventhat SC (i, j) wins in upstream auction i, where σ −i is the strategy profile of n × (N − 1) SCs
who bid for PCs except i. Thus, SC (i, j)’s objective function in equilibrium is given as��n−1π(ti,j |N, r, σ −i ) = max(s − ti,j ) 1 − Ft (σ −1 (s|·)|·)
Q(s|N, r, σ −i ),s(3)

To obtain Q(·), we first examine the probability that the PC i’s total cost ci is lower
than another PC’s cost, provided that the PC i’s cost for non-subcontractable work equals
θi and the selected subcontract bid equals s. The convolution theorem gives the cumulative
distribution function of ci by
1 − Fc (s + θi |σ) =

� t̄
t

¯

nft (t) [1 − Ft (t)]n−1 [1 − Fθ (s + θi − σ (t|·))]dt.

Since the number of PCs equals N , the probability that PC i will win in the downstream
auction is equal to−i

Q(s|N, r, σ ) =

� r−s
θ

¯

[1 − Fc (s + θi |σ)]N −1 fθ (θi )dθi ,(4)and the derivative is given by�−i

Q (s|N, r, σ ) = −

� r−s
θ

¯

[1 − Fc (s + θi |σ)]N −1 fθ� (θi )dθi(5)if N > 1 and Q� = 0 if N = 1, implying that Q is decreasing in s.
To identify a symmetric increasing equilibrium of the SC’s strategy σ, the derivative of (3)9is taken with respect to s. Suppressing subscripts and replacing σ −1 (s|·) = t yield
1
ft (t)1Q� (σ(t|N, r)|N, r, σ −i )− (n − 1)=−.
σ(t|N, r) − t
1 − Ft (t) σ � (t|N, r)
Q(σ(t|N, r)|N, r, σ −i )(6)This equation holds for any t̂ ∈ [t, t̄] in equilibrium. Thus, the integral is taken from t through¯t̄, and the integration by parts is used on the right-hand side to obtainσ(t|N, r) = t +� t̄

n−1 Q(σ(t̂|·)|N, r, σ −i )dt̂t [1 − Ft (t̂)].
[1 − Ft (t)]n−1 Q(σ(t|·)|N, r, σ −i )(7)Solving this for σ(t|·) gives the SC’s bidding function. Although not a closed form, many
insights are drawn from (7). First, for any non-increasing function Q(·), σ is strictly increasing.
Second, Lemma 1 still holds: the optimal bidding strategy is to bid the next lowest costconditional both on the SC’s winning in the upstream auction bid and on the PC’s winningin the downstream auction.11 Finally, the SC’s bidding strategy σ is a function of the numberof PCs and the reservation price in the downstream auction. The hazard function of Q isincreasing in N and decreasing in the reservation price r (see Appendix B). Furthermore, theSC’s two bidding functions with diﬀerent numbers of bidders or diﬀerent reservation prices
never cross each other for any t < t̄ (see Appendix C). Hence, Lemma 1 holds even if θ is arandom variable (see Appendix D).
Theorem 1 can hold when θ is random. For simplicity, we assume that N = 2, n = 2, andm = 0. Suppose that σ(ti,(1:n) ) and θ have the same distribution, namely, Fθ (θ) = 1 − [1 −Ft (σ −1 (θ))]2 . Then, the probability that both σ(t1,(1:n) ) < σ(t2,(1:n) ) and σ(t1,(1:n) ) + θ1 >σ(t2,(1:n) ) + θ2 occur is 0.25,12 implying that an ineﬃcient allocation occurs with probability
0.25 in first-price upstream auctions. In contrast, if a second-price auction is used in upstream
auctions, the probability of an ineﬃcient allocation occurring is strictly more than 0.33. Hence,
the advantage of the first-price auction in terms of allocative eﬃciency is maintained even when
unobserved heterogeneity exists in the PC’s cost to complete the non-subcontractable work.
11

Equation (7) is rearranged as� t̄σ(t|N, r) =tt̂(n − 1)ft (t̂)[1 − Ft (t̂)]n−2 Q(σ(t̂|·)|N, r, σ −i )dt̂
[1 − Ft (t)]n−1 Q(σ(t|·)|N, r, σ −i )��� t̄t̂[1 − Ft (t̂)]n−1 −Q� (σ(t̂|·)|N, r, σ −i ) dt̂
t+.
[1 − Ft (t)]n−1 Q(σ(t|·)|N, r, σ −i )The first term on the right-hand side is the expected cost of the lowest rival SC in the upstream auction
conditional on SC’s winning and PC i’s winning. The second term is the conditional expected cost of PC i’slowest rival PCs.
12
Let µ = t1,(1:n) − t2,(1:n) , and let ξ = θ1 − θ2 . Let Fξ (·) be the cumulative distribution function for theserandom variables. By construction, both µ and ξ have a mean of zero, and Fξ is symmetric so that Fξ (0) = 0.5.
For any µ ≥ 0, the probability that µ + ξ ≤ 0 is Fξ (−µ). Hence, if µ ≥ 0, then the probability that µ + ξ ≤ 0 is
P≡

� θ̄

Fξ (−θ)fξ (−θ)dθ =

0

� 0
θ

Fξ (θ)fξ (θ)dθ.

¯Integration by parts yields P = [Fξ ]0θ − P. Since [Fξ ]0θ = 0.5, P = 0.25.
¯¯

10Theorem 2 holds since ci = θi + σ(ti,(1:n) ) is an i.i.d. random variable and the revenue
equivalence holds in the downstream auction. As for Theorem 3, define the optimal allocation
as the one in which integrated firm i wins if his total cost ci is the lowest and does not exceedr∗ such that r∗ = V − Fc (r∗ )/fc (r∗ ), where Fc (·) is the cumulative distribution function ofci . Suppose that the downstream competition takes place with a second-price auction and the
upstream competitions are on the basis of a second-price auction with the ex post negotiation.
Then, the PC will bid ci = θi + ti,(1:n) and receive min{Bi , r} − (ti(2:n) + θi ) if it is positive andzero otherwise. As long as θi is known and verifiable by the winner SC, the SC beneficiallyaccepts the payment scheme. Thus, the optimal outcome is implemented.

6Conclusion

In this study, we have explored the optimal mechanism, eﬃciency, and equivalence in the
procurement cost of a procurement auction with pre-award subcontracting. Motivated by thefact that goods and services are typically produced by a team of firms (main firms and subfirms)
in many industries, we constructed a Bayesian game in which the lower-tiered subfirms andsuppliers are non-negligible players who also possess private information. Then, we foundthat, although the aggressive bids of SCs help their PC win if first-price auctions are used in
an upstream auction, the double marginalization problem never vanishes. The results showthat the use of neither a first- nor a second-price auction in upstream competitions fails to
implement the optimal outcome.
The proposed optimal mechanism in upstream competitions suggests that transactions
between PCs and SCs in practice may be quite complicated, especially in private projects
in which the PC is not selected through the first-price sealed-bid auction. Moreover, thecost-minimizing project owner must take care not only of the mechanism of the downstream
market but also of that used in the upstream market, since the auctioneer in the upstream
competition, a PC, may not be better oﬀ by using the optimal mechanism.
Although our framework may shed new light on some practical questions of subcontracting,there remain many unanswered questions. For instance, the incentive for subcontracting may
not stem from cost reduction. Marechal and Morand (2003) point out that subcontracting
can reduce the risk of potential change orders.13 Given the sheer volume of procurement,
it is clear that more serious research and evaluation are needed to investigate the eﬀect of
subcontracting.

Appendix A
Suppose that there are two types of SCs, exclusive and common. The exclusive SC submits
a subcontract bid to a particular PC, whereas the common SC submits subcontract bids to
13

The eﬀect of such ex post changes on procurement contracts is thoroughly analyzed in Bajari and Tadelis(2001).

11

all PCs. Let m ≤ n denote the number of common SCs solicited by each PC. In the firststage, SC (l, j) with l ∈ {0, i} obtains his cost tl,j and submits a subcontract bid sl,j to PC
i in upstream auction i, and SC (0, j) submits N bids to all PCs. Which SC is common or
exclusive is known to PCs. All other settings are the same as those presented in Section 2.
Then, consider a situation in which all SCs other than SC (l, j) follow σ and in which all PCs
follow β. If a common SC (0, j), j = 1, . . . , m, bids s to PC i, SC (0, j) wins in the upstream
auction with probability P (s), as shown in Section 2. PC i wins in the downstream competition
unless any rival PC accepts a subcontract bid below σ(t0,j ). This happens with probability[1 − Ft (σ −1 (s))](n−m)(N −1) 1{s≤r} . Since all PCs bid the same amount in this occasion, PC i
receives the contract with probability 1/N . Hence, the conditional probability that PC i winsisQc (s|N, r) =1
[1 − Ft (σ −1 (s))](n−m)(N −1) 1{s≤r} .
N

The common SC receives a positive payoﬀ with probability N P (·)Qc (·). Hence, the objectivefunction is identical for both types of SCs.

Appendix B
From (4) and (5),� r−s
[1 − Fc (s + θi )]N −1 fθ� (θi )dθiQ� (s|N, r, σ −i )θ−= �¯r−s,Q(s|N, r, σ −i )[1 − Fc (s + θi )]N −1 fθ (θi )dθiθ¯f � (r−s)(8)f � (θ)for any N > 1. Let k0 be a positive real number such that fθθ (r−s) = k0 . Since fθθ (θ) is nonincreasing in θ, there exists a non-negative and non-increasing function k(θ) such that, for allθ ∈ [θ, r − s], fθ� (θ)/fθ (θ) − k0 = k(θ), or equivalently fθ� (θ) = [k0 + k(θ)]fθ (θ). Substituting¯
out f � (θ) in (8) gives� r−s[1 − Fc (s + θi )]N −1 k(θi )fθ (θi )dθiQ� (s|N, r, σ −i )
θ
−
=
k
+
�¯r−s
0
Q(s|N, r, σ −i )[1 − Fc (s + θi )]N −1fθ (θi )dθiθ
�¯r−s
= k0 +g(θi |N )k(θi )dθi ,
θ

¯

12

(9)

� r−swhere g(θ|N, r) ≡ [1 − Fc (s + θ)]N −1 fθ (θ)/ θ [1 − Fc (s + θi )]N −1 fθ (θi )dθi . Define G(θ|N, r) =�θ¯θ g(θ̂|·)dθ̂. Then, for any N > 1,¯
G(θ|N + 1) − G(θ|N )�θ�θN
N −1 f (θ̂)dθ̂θθ [1 − Fc (s + θ̂)] fθ (θ̂)dθ̂θ [1 − Fc (s + θ̂)]= � r−s−�¯¯
r−s[1 − Fc (s + θi )]N fθ (θi )dθi
[1 − Fc (s + θi )]N −1 fθ (θi )dθi
θ
θ
¯ ��
¯
� r−s
θ
1
N
=
[1 − Fc (s + θ̂)] fθ (θ̂)dθ̂[1 − Fc (s + θi )]N −1 fθ (θi )dθi
∆
θ
θ
¯
¯
�
� θ
� r−s
− [1 − Fc (s + θ̂)]N −1 fθ (θ̂)dθ̂[1 − Fc (s + θi )]N fθ (θi )dθi
=

1
∆

��

θ

θ
θ

¯

θ

¯

[1 − Fc (s + θ̂)]N fθ (θ̂)dθ̂
−

� θ
θ

¯

[1 − Fc (s + θ̂)]

� r−s
θ

N −1

¯

[1 − Fc (s + θi )]N −1 fθ (θi )dθifθ (θ̂)dθ̂

� r−s
θ

N

[1 − Fc (s + θi )] fθ (θi )dθi

�

,

� r−s

� r−s
where ∆ = θ [1 − Fc (s + θi )]N fθ (θi )dθi · θ [1 − Fc (s + θi ]N +1 fθ (θi )dθi > 0. Applying the¯Mean Value ¯Theorem yields
1
=
∆

�

�

(1 − Fc (θ− )) − (1 − Fc (θ+ ))
×

� θ
θ

¯

�

� r−s
θ

[x − y][1 − Fc (s + θ̂)]N −1 fθ (θ̂)dθ̂

� r−s
θ

[1 − Fc (s + θ̃)]N −1 fθ (θ̃)dθ̃�,where θ− ∈ [θ, θ] and θ+ ∈ [θ, r − s]. Since Fc is strictly increasing, the whole terms are strictly¯
positive. Hence, given s, G(θ|N + 1) is first-order stochastically dominated by G(θ|N ). Sincek(θi ) is non-increasing, −Q� (s|N, r, σ −i )/Q(s|N, r, σ −i ) is strictly increasing in N .
� (s|N,r,σ −i )Finally, we show that − QQ(s|N,r,σ−i ) is weakly decreasing in r. Let r < r̃. From Equation(9), we obtain
G(θ|N, r̃) − G(θ|N, r)�θ�θ
N −1 f (θ̂)dθ̂
N −1 f (θ̂)dθ̂θθθ [1 − Fc (s + θ̂)]θ [1 − Fc (s + θ̂)]= � r̃−s
−
� r−s
¯
¯
[1 − Fc (s + θ̃)]N −1 fθ (θ̃)dθ̃[1 − Fc (s + θ̃)]N −1 fθ (θ̃)dθ̃
θ
θ
¯
¯
≤ 0.
It follows that G(θ|N, r̃) first-order stochastically dominates G(θ|N, r) and that equality holds

13

if and only if Fc (r) = 1. Hence, from Equation (9), we obtain��Q� (s|N, r̃)Q� (s|N, r)−− −Q(s|N, r̃)Q(s|N, r)� r̃−s� r−s=g(θi |N, r̃)k(θi )dθi −g(θi |N, r)k(θi )dθi
θ

=

�¯r−s
θ

θ

¯
� r̃−s[g(θi |N, r̃) − g(θi |N, r)] k(θi )dθi +g(θi |N, r̃)k(θi )dθir−s

¯ � r−s� r̃−s−+= k(θ )[g(θi |N, r̃) − g(θi |N, r)] dθi + k(θ )g(θi |N, r̃)dθi
θ

r−s

¯

≤ 0,

where θ− < θ+ . The third equality is obtained by the Mean Value Theorem. Furthermore,� r̃−s� r−sg(·|N, r̃) = θ g(·|N, r) = 1, or equivalently
θ
¯
¯
� r−s
� r̃−s[g(θ|N, r̃) − g(θ|N, r)] dθ +g(θ|N, r̃)dθ = 0.
θ

θ

¯

¯

Together with the fact that k(·) is non-increasing, we obtain the last inequality.

Appendix C
Suppose there exists t such that σ(t|m̃, ·) = σ(t|m, ·) = ξ(t). Then, by Equation (7),
� t̄
t

�

[1 − Ft (t̂)]

n−1 Q(ξ (t̂)|m̃, r, σ

−i )

dt̂ =
Q(ξ (t)|m̃, r, σ −i )� t̄t

[1 − Ft (t̂)]n−1

Q(ξ (t̂)|m, r, σ −i )dt̂.
Q(ξ (t)|m, r, σ −i )−i(s|m,r,σ )Since − QQ(s|m,r,σ−i ) is increasing in m,��Q(s|m̃, r, σ −i ) Q� (s|m̃, r, σ −i ) Q� (s|m, r, σ −i )0>−Q(s|m, r, σ −i ) Q(s|m̃, r, σ −i )Q(s|m, r, σ −i )Q� (s|m̃, r, σ −i )Q(s|m, r, σ −i ) − Q� (s|m, r, σ −i )Q(s|m̃, r, σ −i )d Q(s|m̃, r, σ −i )==,ds Q(s|m, r, σ −i )[Q(s|m, r, σ −i )]2
for any m̃ > m ≥ 1. It follows that, for any ŝ > s,Q(ŝ|m̃, r, σ −i )Q(s|m̃, r, σ −i )Q(ŝ|m̃, r, σ −i )Q(ŝ|m, r, σ −i )<⇔<.
Q(ŝ|m, r, σ −i )Q(s|m, r, σ −i )Q(s|m̃, r, σ −i )Q(s|m, r, σ −i )14

Replacing ŝ = ξ(t̂) and ξ = σ(t) and multiplying by [1 − Ft (t̂)]n−1 on both sides will yield−i
Q(ξ (t̂)|m̃, r, σ −i )n−1 Q(ξ (t̂)|m, r, σ )
<
[1
−
F
(t̂)]
t
Q(ξ (t)|m̃, r, σ −i )Q(ξ (t)|m, r, σ −i )
� t̄
�
t̄
−i
Q(ξ (t̂)|m̃, r, σ −i )n−1 Q(ξ (t̂)|m, r, σ )⇔[1 − Ft (t̂)]n−1
d
t̂
<
[1
−
F
(t̂)]
dt̂.
t
Q(ξ (t)|m̃, r, σ −i )Q(ξ (t)|m, r, σ −i )tt

[1 − Ft (t̂)]n−1

Thus, we obtain a contradiction.
Next, suppose by contradiction that there exists t such that σ(t|N, r̃) = σ(t|N, r) = ξ(t).
Then, by Equation (7),
� t̄
t

�

[1 − Ft (t̂)]

n−1 Q(ξ (t̂)|N, r̃, σ

−i )

dt̂ =
Q(ξ (t)|N, r̃, σ −i )� t̄t

[1 − Ft (t̂)]n−1

Q(ξ (t̂)|N, r, σ −i )dt̂.
Q(ξ (t)|N, r, σ −i )−i(s|N,r,σ )Since − QQ(s|N,r,σ−i ) is decreasing in r,

Q(s|N, r̃, σ −i )0<
Q(s|N, r, σ −i )�Q� (s|N, r̃, σ −i ) Q� (s|N, r, σ −i )−Q(s|N, r̃, σ −i )Q(s|N, r, σ −i )�=d Q(s|N, r̃, σ −i )ds Q(s|N, r, σ −i )for any r < r̃. It follows that, for any ŝ > s,Q(s|N, r̃, σ −i )Q(ŝ|N, r̃, σ −i )Q(ŝ|N, r, σ −i )Q(ŝ|N, r̃, σ −i )<⇔<.
Q(s|N, r, σ −i )Q(ŝ|N, r, σ −i )Q(s|N, r, σ −i )Q(s|N, r̃, σ −i )Replacing ŝ = ξ(t̂) and ξ = σ(t) and multiplying by [1 − Ft (t̂)]n−1 on both sides will yield−i
Q(ξ (t̂)|N, r, σ −i )n−1 Q(ξ (t̂)|N, r̃, σ )
<
[1
−
F
(t̂)]
t
Q(ξ (t)|N, r, σ −i )Q(ξ (t)|N, r̃, σ −i )
� t̄
�
t̄
−i
Q(ξ (t̂)|N, r, σ −i )n−1 Q(ξ (t̂)|N, r̃, σ )⇔[1 − Ft (t̂)]n−1
d
t̂
<
[1
−
F
(t̂)]
dt̂.
t
Q(ξ (t)|N, r, σ −i )Q(ξ (t)|N, r̃, σ −i )tt

[1 − Ft (t̂)]n−1

Thus, we have reached a contradiction.

Appendix D
Since σ(t|m) and σ(t|m̃) never cross each other at t < t̄ for any m̃ > m ≥ 1, σ(t|m) must
be monotone in m, the number of bidders, even if m is a real number instead of an integer.
Suppose by contradiction that σ(t|m) is increasing in m ≥ 1. Then, replacing N with a real
number m in (6) yields
1
ft (t)1Q� (σ(t|m, r)|m, r, σ −i )− (n − 1)=−.
σ(t|m, r) − t
1 − Ft (t) σ � (t|m, r)Q(σ(t|m, r)|m, r, σ −i )The right-hand side is positive if m > 1 and vanishes if m = 1. Therefore, if the numberof bidders increases to m̃ > 1, then σ � (t|m̃, ·) > σ � (t|m, ·) must hold for all t ∈ [t, t̄). Thus,
¯
15

� t̄

�
� (t̂|m̃, ·)dt̂ = t̄ − σ(t|m̃, ·) > t̄ − σ(t|m, ·) = t̄ σ � (t̂|m̃, ·)dt̂, implying σ(t|m̃, ·) < σ(t|m, ·). Aσttcontradiction is reached.
Similarly, σ(t|N, r) is strictly increasing in r. Letting r− ≤ t + θ < t̄ + θ ≤ r+ , suppose by¯ ¯¯contradiction that there exists t < t̄ such that σ(t|N, r+ ) < σ(t|N, r− ). Since an SC has nochance to obtain a job if the reservation price in the downstream auction is equal to r− , his
strategy is σ(t|·, r− ) = t for all t, whereas if r = r+ , then an SC seeks a positive bid marginwhen his cost is strictly smaller than t̄, i.e., σ(t|·, r+ ) > t for any t < t̄. A contradiction isreached. Thus, σ(t|N, r) is strictly increasing in r.

References
Bajari, Patrick and Steven Tadelis, “Incentives versus Transaction Costs: A Theory ofProcurement Contracts,” RAND Journal of Economics, Autumn 2001, 32 (3), 387–407.
Clough, Richard Hudson and Clenn A sears, Construction contracting, John Wiley &Sons. Inc., 1994.
Dyer, Douglas and John H. Kagel, “Bidding in Common Value Auctions: How theCommercial Construction Industry Corrects for the Winner’s Curse,” Management Science,
October 1996, 42 (10), 1463–1475.
Grosskopf, Ofer and Barak Medina, “Rationalizing Drennan: On Irrevocable Oﬀers, Bid
Shopping and Binding Range,” Review of Law & Economics, 2007, 3 (2), 4.
Hansen, Robert G., “Auctions with Endogenous Quantity,” RAND Journal of Economics,
Spring 1988, 19 (1), 44–58.
Marechal, Francois and Pierre-Henri Morand, “Pre vs. post-award subcontracting plans
in procurement bidding,” Economics Letters, October 2003, 81 (1), 23–30.
Myerson, Roger B, “Optimal auction design,” Mathematics of Operations Research, 1981,
6 (1), 58–73.
Riley, John G and William F Samuelson, “Optimal Auctions,” American Economic
Review, June 1981, 71 (3), 381–92.
Wambach, Achim, “How to subcontract?,” Economic Letters, November 2009, 105 (2),152–155.

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