Global COE Hi-Stat Discussion Paper Series 237

Research Unit for Statisticaland Empirical Analysis in Social Sciences (Hi-Stat)

Bundling Decision in Procurement Auctions with
Risk-Averse Suppliers

Hi-Stat Discussion Paper

Takeshi Nishimura

July 2012

Hi-Stat
Institute of Economic Research
Hitotsubashi University
2-1 Naka, Kunitatchi Tokyo, 186-8601 Japan
http://gcoe.ier.hit-u.ac.jp

Bundling Decision in Procurement Auctions with
Risk-Averse Suppliers
Takeshi Nishimura∗†
May 29, 2012

Abstract
In a two-stage procurement model, we compare two contracting schemes: bundlingand unbundling. They differ in whether two sequential tasks of investment and service provision are bundled or not in the auction. We show that while unbundlingcauses underinvestment in cost reduction and bundling causes the ex post inefficiency of trade, each scheme imposes some forms of risk on the suppliers. The
comparative statics results show that as the investment is more costly and/or acommon cost component is more risky, bundling becomes less attractive to boththe buyer and society.
∗

Hitotsubashi University; ed091004@g.hit-u.ac.jp.
I am grateful to my advisor Akira Okada for his invaluable guidance and encouragement. I would liketo thank Reiko Aoki, Ron Harstad, Hideshi Itoh, Takashi Kunimoto, Daisuke Oyama, Hamid Sabourian,
Hannu Salonen for their thoughtful comments and suggestions. The support of the Global COE program
and the Research Unit for Statistical and Empirical Analysis in Social Sciences at Hitotsubashi University
is gratefully acknowledged.
†

1

1

Introduction

Governments delegate the provision of a wide range of public infrastructures to the private
sectors. These infrastructures consist of physical facilities (e.g. transportation, school,
hospital) and public services (e.g. road maintenance, education, medical service). The
government, which is responsible for the use of tax revenues, must find the efficient way
of procurement.
An essential aspect of the infrastructure provision is that the process has a multi-stage
feature. A public facility is designed, built, and operated for the purpose of delivering
the associated service to people. The sequential process inevitably involves some issues
concerning efficiency. The private company’s investment in the design specifications or
the building embodying innovative ideas can reduce costs in the subsequent stages. Thus,
the public authority should induce the company to invest efficiently. On the other hand,the exact amount of cost is estimated only at the later stage, depending on exogenous
factors such as the availability of resources or technological changes, and thus is uncertain
at the outset. Then, for each task, the public authority must find a company which canperform the task at the least cost. Furthermore, the risk of construction or operation costshould be shared between the public authority and private company in the most efficientway.
As a way of procuring public infrastructures, public-private partnerships (PPPs) havebecome increasingly popular (Yescombe, 2007; OECD, 2008). One distinctive feature of
PPPs is that a private party organized by some companies is responsible for performing
many tasks such as design, construction, and operation under the long-term contract. On
the other hand, in a way of traditional procurement, these sequential tasks are separated.
In the case of construction, the scheme of design-build corresponds to PPPs, and that2

of design-bid-build corresponds to traditional procurement. Moreover, regardless of thescheme, the public procurement is often conducted through competitive bidding under
the law (e.g. the Federal Acquisition Rules in the United States).
Our aim is to discover the factors affecting the optimal choice between bundling andunbundling. The two schemes differ in whether two sequential tasks of investment andservice provision are bundled or not in the auction. We consider a two-stage model.
A risk-neutral buyer must procure one unit of service (construction or operation of the
facility) from one of risk-averse suppliers. Each supplier’s production cost is determinedby three elements: the (cost-reducing) investment, his private parameter and a common
parameter. After the investment is made, each supplier’s private parameter is privately
known to him, and the common parameter is commonly known to all suppliers. It shouldbe emphasized that no supplier initially knows these parameter values. Thus, there arerisks of production costs. Each auction is held in a first-price format. A winner is awarded
a fixed-price contract.
The main results show that the buyer and the society face the tradeoff which involves
three factors: the ex post efficiency of trade, investment incentives, and risk sharing.
First, unbundling allows the buyer to select the most efficient supplier, whereas bundlingdoes not. Second, unbundling causes underinvestment relative to bundling. Thus, as
the investment is more costly, bundling becomes relatively less attractive. Finally, eachscheme imposes some forms of risk on the suppliers. The scheme of bundling exposes awinner to the risk of production cost. Although one may expect that the suppliers undertake less risks under unbundling, the scheme entails the risk associated with competition.
On the other hand, the competition under unbundling can transfer the risk of common
cost from the risk-averse suppliers to the risk-neutral buyer. Therefore, as the common

3cost is more risky, bundling becomes relatively less attractive.
A number of articles have addressed the issues of bundling decision in the context
of PPPs. The main question is how each scheme of bundling and unbundling (possiblywith an ownership structure of the facility) affects the suppliers’ incentives for various
kinds of investments. The literature can be classified into two categories according to
the approach. The first one is the incomplete contract approach. In this strand, boththe investment and resulting outcome (e.g. production cost) are assumed to be unverifiable. Hart (2003), who develops a leading model, shows that bundling provides stronger
incentives for the cost-reducing investments than unbundling, at the expense of quality.
In the experiment, Hoppe et al. (2011) find support for the theoretical prediction. For
other studies, see Bennett and Iossa (2006), and Chen and Chiu (2010).
The second one is the complete contract approach. There are few studies which
address risk-sharing issues, apart from Martimort and Pouyet (2008), and Li and Yu(2011); see also Schmitz (2005), Maskin and Tirole (2008), and Hoppe and Schmitz(2010). Martimort and Pouyet (2008) consider the quality-enhancing investment which
reduces or increases the production cost, in an environment where contracts contingent
on the production cost (and the quality level of the facility) are feasible. They show thatthe optimal bundling decision depends on the investment externality on the production
cost. In contrast, Li and Yu (2011) explicitly consider an auction model. They examine
how the bundling decision is affected by the externality of the first task and the intensityof competition (i.e. the number of suppliers). In the models of Martimort and Pouyet(2008) and Li and Yu (2011), the risks are shared between the buyer and supplier by
writing performance contracts, as in standard moral-hazard models.
Although our model is close to that of Li and Yu (2011), there are two significant

4differences. First, we examine how the risks are shared in each scheme when the production cost is unverifiable. The point is that without relying on cost-sharing contracts, therisk of common cost (not private cost) can be effectively transferred to the buyer throughcompetition under unbundling. Notice that all suppliers’ (estimated) production costs
are equally affected by the common parameter. Second, we introduce two kinds of risks
of production cost (i.e. private and common parameters). In the model of Li and Yu(2011), there is no common cost parameter. With these parameters, it becomes clear
what kind of risks in public procurement encourage the choice of each scheme. This has
important policy implications. We will discuss the issue in Section 6.
The remainder of the paper is organized as follows. Section 2 presents the model.
Sections 3 and 4 characterize the equilibrium under each scheme. Section 5 gives thecomparative statics results which show when each scheme becomes more attractive to thebuyer and society than the other. Section 6 concludes with a summary and discussion.
All proofs are in the Appendix.

2

The model

Consider a buyer who must procure one unit of service from one of n suppliers. The
buyer is risk-neutral. Each supplier i ∈ N ≡ {1, ..., n} is risk-averse, and has a CARA
utility function u(π) = 1 − exp(−rπ), where π ∈ R is a profit from trade and r > 0 is his
coefficient of absolute risk aversion.
A supplier i’s (production) cost of service provision is given by c(a, θi , ω), wherea ∈ R+ is the investment level, θi ∈ [θ, θ̄] is the supplier i’s private parameter, andω ∈ [ω, ω̄] is a common parameter, which is identical across suppliers. We call the lattertwo variables the risk of production cost. We assume that (θ1 , ..., θn , ω) are independent
5

random variables, and the cumulative distribution functions of θi and ω are respectively
given by F and G, with F 0 = f > 0.1 For convenience, let θ(n) ≡ min{θ1 , ..., θn },F(n) (θ) ≡ 1 − (1 − F (θ))n and f(n) (θ) ≡ n(1 − F (θ))n−1 f (θ) denote the lowest cost
parameter, the cumulative distribution function and the probability density function of
θ(n) , respectively. A supplier i incurs a cost ψ(a) if he invests a. The buyer’s valuation
for the service is v > 0. Thus, all suppliers are ex ante symmetric. We make the following
assumptions.
Assumption 1. c : R+ × [θ, θ̄] × [ω, ω̄] → R+ is twice continuously differentiable in a,∂c∂2c< 0, ∂a2 ≥ 0, continuous and increasing in θi .
∂a∂c∂cAssumption 2. If c(a, θi , ω) > c(a, θi0 , ω 0 ), then − ∂a(a, θi , ω) ≥ − ∂a(a, θi0 , ω 0 ).
2

Assumption 3. ψ : R+ → R+ is twice continuously differentiable, ddaψ2 > 0, dψ(0) ≤ 0 andda(a) = ∞.
lima→∞ dψdaAssumption 1 says that the investment reduces each supplier’s production cost. Assumption 2, which means that a supplier with higher production cost can enjoy a greater
benefit of ex ante investment in a weak sense, will play a key role in the analysis toyield clear-cut results. For instance, Assumptions 1 and 2 are satisfied by c(a, θi , ω) =(θi + ω)/c̄(a) or c(a, θi , ω) = θi + ω − c̄(a) with an appropriate function c̄(·). Assumption
3 allows for the possibility that the investment cost is decreasing when the investment
levels are low. This occurs if, for example, ψ(a) = ψ̄(a) − a is a total cost of investment;
the term −a means that the investment also reduces the total cost of the task.
There are two feasible contracting schemes: bundling and unbundling. Under bundling,
the buyer bundles the two sequential tasks of investment and service provision, and awards
a contract for both tasks to a single supplier via an auction. Under unbundling, the buyer
1

We say that f > 0 if f (θi ) > 0 for all θi . We use the same notation for other functions.

6

separates those tasks, and sequentially awards a contract for each task via an auction.
Each auction is held in a first-price sealed-bid format; a supplier j ∈ N wins only if his
bid pj ∈ R+ is the lowest among (p1 , ..., pn ).2 The winner j must perform each task in
exchange for a fixed payment pj .
The game proceeds as follows. At date 0, the buyer chooses either bundling or unbundling. At date 1, each supplier i ∈ N (simultaneously and independently) submits abid p1i ∈ R+ in the first-stage auction. At date 2, a winner j in the first-stage auction
chooses an investment level a ∈ R+ . At date 3, (θ1 , ..., θn , ω) are realized; in the schemeof bundling, the game then ends. At date 4, only in the scheme of unbundling, eachsupplier i ∈ N submits a bid p2i ∈ R+ in the second-stage auction; the game then ends.
The player payoffs are defined as follows. When the game ends at date 3 underbundling, a winner j with his bid p1j obtains u(p1j − ψ(a) − c(a, θj , ω)), the other suppliers
obtain u(0) = 0, and the buyer obtains v − p1j . When the game ends at date 4 under
unbundling, a winner j in the first-stage auction obtains u(p1j − ψ(a) − πj ), the othersuppliers obtain u(πi ), and the buyer obtains v − (p1j + p2k ); (p1j , p2k ) are winning prices
in the auctions, and πi is the supplier i(∈ N )’s profit in the second-stage auction, where
πi = p2i − c(a, θi , ω) if i wins (i = k) and πi = 0 if i loses (i 6= k).
The information structure is as follows. The buyer’s choice of scheme and the identityof the winner in the first-stage auction become common knowledge among all players.
The realized values of θi and c(a, θi , ω) become the supplier i’s private information, andthat of ω becomes common knowledge among all suppliers. No supplier can observe theother suppliers’ decisions. However, the winner’s investment level a is commonly known
to suppliers at date 3 because the assumption ∂c/∂a < 0 implies that each supplier2

We assume that if there is a tie, then each supplier submitting the lowest bid wins with equal
probability. All results hold for any other tie-breaking rule.

7

exactly knows the level a by his production cost and realized cost parameters.
Each player’s (pure) strategy is defined in a standard way. The buyer’s strategy isa choice of contracting scheme. Each supplier i’s strategy is a triple (p1i , ai , p2i ), where
p1i : {bundling, unbundling} → R+ is a bidding strategy in the first-stage auction, ai :{bundling, unbundling}×R+ → R+ is a choice of investment level conditional on winning,and p2i : R+ × N × R+ × [θ, θ̄] × [ω, ω̄] → R+ (3 p2i (p1i , j, a, θi , ω) where j is a winner inthe first-stage auction) is a bidding strategy in the second-stage auction. In the following
sections, we explore the (pure strategy) perfect Bayesian equilibrium of the game. Since(θ1 , ..., θn , ω) are independent, no supplier updates his belief about the other suppliers’
types in equilibrium; we also assume that this is the case in any off-equilibrium path.
We assume that the investment level a and the realized value of ω are unverifiable. If
the buyer can initially offer a contract in which prices are contingent on the investment
level a and the realized values of (θ1 , ..., θn , ω), then the efficient outcome is realized: (i)The buyer induces an arbitrary supplier i to choose the efficient investment level ã, which
minimizes expected total cost ψ(a) + E[c(a, θ(n) , ω)].3 (ii) Given the realized values of(θ1 , ..., θn , ω), the buyer pays c(ã, θ(n) , ω) to a supplier with the lowest private parameter
θ(n) , who provides the service. The buyer then obtains the first-best (expected) utilityv − {ψ(ã) + E[c(ã, θ(n) , ω)]}.

3

Bundling

In this section, we characterize the equilibrium in the subgame after the buyer has chosen
bundling. We apply backward induction.
By investing a, a winner i in the first-stage auction obtains the expected utility
3

In this paper, E[·] represents the expectation operator of random variables.

8E[u(p1i − ψ(a) − c(a, θi , ω))]. The certainty equivalent which gives the same utility to
the winner i is given by
p1i − ψ(a) − E[c(a, θi , ω)] − ρ∗ (a),(1)

where ρ∗ (a) = 1r ln E[exp(rc(a, θi , ω))] − E[c(a, θi , ω)] > 0 is his risk premium. We firstexamine the effect of the investment on the risk premium.
∗Lemma 1. dρ(a) ≤ 0 for all a.
da
The lemma states that a winner’s risk premium is nonincreasing in his investment.
This result depends on Assumption 2. With this assumption, the winner’s investment
has an effect to decrease the riskiness of production cost, and thus to decrease his risk
premium. We will use this lemma to compare the equilibrium investment level with theefficient level.
The next proposition characterizes the equilibrium under bundling. Now, the social
welfare in equilibrium is defined as the buyer’s expected utility plus the sum of each supplier’s certainty equivalent profit which gives the same utility as the equilibrium expectedutility to the supplier.
Proposition 1. Under bundling, any equilibrium is characterized as follows.
(i) At least two suppliers submit the same bid p1∗ = ψ(a∗ ) + E[c(a∗ , θi , ω)] + ρ∗ (a∗ ), andthe other suppliers submit bids higher than p1∗ . A winner i always chooses the investment
level a∗ determined by


∂c ∗
dρ∗ ∗
dψ ∗
(a ) = −E(a , θi , ω) −
(a ).
da
∂a
da
(ii) a∗ ≥ ã.

9(2)

(iii) The buyer’s utility EUB∗ , each supplier i’s expected utility EUi∗ , and the social welfare
W ∗ are respectively given by
EUB∗ = v − {ψ(a∗ ) + E[c(a∗ , θi , ω)] + ρ∗ (a∗ )} ,EUi∗ = 0,
W ∗ = v − {ψ(a∗ ) + E[c(a∗ , θi , ω)] + ρ∗ (a∗ )} .
This proposition identifies some properties of bundling. First, the equilibrium investment level a∗ is higher than the efficient level ã. As in the first-order condition (2), a
winner chooses the investment level to equate the marginal effects. Lemma 1 then implies
that with the positive effect on the risk premium, the winner has an additional incentive
to invest. Moreover, in contrast to the efficient outcome, the winner must provide theservice even if his production cost is extremely high. Assumption 2 then implies that
this contractual obligation increases the winner’s incentive. Notice that overinvestmentby the winner is efficient provided that he bears all the risk of production cost. This fact
partially supports the argument of OECD (2008) that risk must be transferred to theparty best able to carry it.
Second, the buyer’s utility is the same level as the social welfare. This is because eachsupplier has no private information in the first-stage auction, and thus he cannot earn
any rent.
Finally, the scheme of bundling involves two types of costs to both the buyer andsociety. The first one is the risk premium for the risk of production cost. Each supplieradds the risk premium to his bid. The second one is an efficiency loss from forgoing an
opportunity to switch to a more efficient supplier. This is because the buyer commits not
to switch suppliers under bundling.

10

4

UnbundlingIn this section, we characterize an equilibrium in the subgame after the buyer has chosen
unbundling. We apply backward induction.
The following lemma characterizes a symmetric equilibrium in the second-stage auction, in which all suppliers use the same bidding strategy. This result is based on Holt
(1980).
Lemma 2. Under unbundling, given the investment level a and the realized value of ω,
the following bidding strategy p2 (a, ·, ω) and the suppliers’ belief F constitute a symmetricequilibrium in the second-stage auction:1p2 (a, θi , ω) = − ln Eθ(n−1) [exp(−rc(a, θ(n−1) , ω)) | θ(n−1) > θi ].
r(3)

In the following analysis, we focus on this symmetric equilibrium in the second-stage
auction. This lemma has three implications. First, this bidding strategy is significantly
affected by the information structure of the game. We can easily show that p2 (a, θi , ω) >c(a, θi , ω). This is due to the “shading behavior” of the supplier i, who has private
information about his private parameter θi . On the other hand, a common parameter ω
is common knowledge among all suppliers, so that there is no room for shading. This
fact will become clearer in Proposition 4. Second, the equilibrium bidding strategy p2 (·)
is independent of the identity of the winner j in the first-stage auction. This is because
the winner j’s CARA utility function guarantees the absence of wealth effect, and thus
his bidding behavior is independent of p1j and his sunk investment cost ψ(a). Third, themost efficient supplier wins because the equilibrium bidding strategy p2 (·) is increasingin θi .
We now examine the effect of the investment on the bidding strategy in the second11

stage auction.
2∂cLemma 3. ∂p(a, θi , ω) ≤ ∂a(a, θi , ω) < 0 for all a, θi and ω.
∂a

The lemma states that each supplier’s bid in the second-stage auction is decreasing
in a winner’s investment in the first-stage. This is because the winner’s cost-reducing
investment reduces all suppliers’ production costs, and thus induces aggressive biddingby all of them. Moreover, the marginal effect is larger than that of cost reduction. As
is well known in first-price auction models, a supplier i with θi computes his equilibrium
bid by estimating the second-lowest production cost conditional on his parameter θibeing the lowest; this fact is shown in (3). In short, the supplier i cares about more
inefficient suppliers’ production costs. Then, the statement ∂p2 /∂a ≤ ∂c/∂a follows fromboth the assumption that c is increasing in θi and Assumption 2 because suppliers withhigher parameters than θi can enjoy greater benefits of cost-reducing investment than the
supplier i.
By investing a, a winner i in the first-stage auction obtains the expected utility E[(1−F(n−1) (θi ))u(p1i − ψ(a) + p2 (a, θi , ω) − c(a, θi , ω)) + F(n−1) (θi )u(p1i − ψ(a))]; the first termin the expectation corresponds to the case where the supplier i also wins the secondstage auction, and the second term corresponds to the case where he loses. The certainty
equivalent which gives the same utility to the winner i in the first-stage auction is givenbyp1i − ψ(a) + E[(1 − F(n−1) (θi ))(p2 (a, θi , ω) − c(a, θi , ω))] − ρ∗∗ (a),

12

(4)

whereρ∗∗ (a) =1
ln E[(1 − F(n−1) (θi )) exp(−r(p2 (a, θi , ω) − c(a, θi , ω))) + F(n−1) (θi )]r+ E[(1 − F(n−1) (θi ))(p2 (a, θi , ω) − c(a, θi , ω))]

>0

(5)

is his risk premium.
The next proposition characterizes an equilibrium under unbundling.
Proposition 2. Under unbundling, there exists an equilibrium characterized as follows.
(i) At least two suppliers submit the same bid p1∗∗ = ψ(a∗∗ ) in the first-stage auction, andthe other suppliers submit bids higher than p1∗∗ . A winner i always chooses the investment
level a∗∗ which satisfiesE ∂π(a∗∗ , θi , ω)(1 − F(n−1) (θi )) exp(−rπ(a∗∗ , θi , ω))dψ ∗∗∂a(a ) ≥,daE[(1 − F(n−1) (θi )) exp(−rπ(a∗∗ , θi , ω)) + F(n−1) (θi )](6)

where π(a, θi , ω) = p2 (a, θi , ω) − c(a, θi , ω). All suppliers follow the bidding strategy p2 (·)
in the second-stage auction defined in Lemma 2.
(ii) ã > a∗∗ .
(iii) The buyer’s expected utility EUB∗∗ , each supplier i’s expected utility EUi∗∗ , and thesocial welfare W ∗∗ are respectively given by
EUB∗∗ = v − ψ(a∗∗ ) + E[c(a∗∗ , θ(n) , ω)] + nE[(1 − F(n−1) (θi ))π(a∗∗ , θi , ω)] ,EUi∗∗ = u(E[(1 − F(n−1) (θi ))π(a∗∗ , θi , ω)] − ρ∗∗ (a∗∗ )),W ∗∗ = v − ψ(a∗∗ ) + E[c(a∗∗ , θ(n) , ω)] + nρ∗∗ (a∗∗ ) .
This proposition identifies some properties of unbundling. First, the equilibrium investment level a∗∗ is lower that the efficient level ã. Lemma 3 implies that the right-handside of (6) cannot be positive because the marginal effect of price reduction dominates13that of cost reduction. Thus, the winner has no incentive to invest unless the investment
has a considerable cost-reduction effect in the first stage (i.e. dψ(a)/da is sufficiently
negative for some investment levels).
Second, in contrast to bundling, the buyer’s utility is less than the social welfare. Thisis because each supplier acquires private information about his private cost parameterin the second-stage auction, and thus he can earn information rents. Notice that even
if a supplier loses in the first-stage auction, he can participate in the second-stage auction. Hence, each supplier’s reservation utility in the first-stage auction is endogenously
determined by the expected utility in the second-stage auction.
Finally, the scheme of unbundling involves two types of costs to the buyer and society.
The first one is the risk premium. Each supplier bears a fraction of the risk of production
cost because he wins with positive probability in the second-stage auction. Moreover, eachsupplier must bear the risk associated with competition; the participation in the secondstage auction entails the risk of auction outcome (i.e. winning or losing, and winning
price). The buyer must pay the premium indirectly because the expected information
rent E[(1 − F(n−1) (θi ))π(a∗∗ , θi , ω)] is greater than the risk premium ρ∗∗ (a∗∗ ). The second
one is an efficiency loss from underinvestment. Actually, a winner in the first-stage auction
does not internalize the positive externality on the other suppliers.

5

Bundling versus unbundlingThis section presents the main results. We compare the performance of bundling withthat of unbundling, by analyzing each equilibrium in Sections 3 and 4. We say thatbundling (unbundling) is socially desirable if W ∗ > W ∗∗ (W ∗ < W ∗∗ ). It is important toemphasize that the buyer and the society face the tradeoff which involves three factors:
14the ex post efficiency of trade, investment incentives, and risk sharing.
First, we consider the issue of ex post efficiency of trade. The contractual flexibility of
unbundling allows the buyer to select the most efficient supplier, whereas the contractual
rigidity of bundling does not. The latter is a harmful effect of the buyer’s commitment
not to switch suppliers under bundling. Thus, with respect to the ex post efficiency of
trade, unbundling is superior to bundling. However, notice that the buyer is obliged to
pay the information rents to the suppliers, and thus cannot extract the full surplus under
unbundling.
Second, we consider the issue of investment incentives.
(0) = 0, then a∗∗ = 0.
Corollary 1. a∗ ≥ ã > a∗∗ . If dψdaThis corollary is an immediate consequence of Propositions 1 and 2. In the scheme ofbundling, a winner is responsible for providing the service in any state. Consequently, thecontractual obligation affords strong investment incentives to the winner. On the other
hand, in the scheme of unbundling, a winner in the first-stage auction has little incentive
to invest. This is because the investment induces the suppliers to bid aggressively, andthus intensifies price competition in the second-stage auction. Without the cost-reducing
effect in the first stage, the winner makes no investment.
The next proposition provides the comparative statics with respect to the investment
cost.
Proposition 3. Assume that ψ(a) = (k + 1)ψ̄(a), where k ≥ 0 is a parameter of theinvestment cost, and the function ψ satisfies Assumption 3 with ψ̄(0) = dψ̄(0)/da = 0.
Then, there exist thresholds k and k̄ such that (i) the buyer chooses bundling in equilibrium
for all k < k̄, and unbundling for all k > k̄, (ii) the socially desirable scheme is bundlingfor all k < k, and unbundling for all k > k, and (iii) k ≤ k̄.
15

First-best utility
EUB∗ = W ∗
W ∗∗
EUB∗∗
Infimum of EUB∗
0

k

k

k̄

Figure 1: Illustration of Proposition 3.
The intuition of Proposition 3 is simple. In the scheme of unbundling, a winner in thefirst-stage auction has no incentive to invest because of the assumption dψ̄(0)/da = 0.
Both the buyer’s utility and social welfare under unbundling are thus independent of
whether the investment is costly or not. On the other hand, in the scheme of bundling, awinner chooses the positive investment level. Therefore, as the investment is more costly(i.e. k becomes greater), the scheme of bundling becomes less attractive to both thebuyer and society. Figure 1 illustrates this result.
Finally, we examine the issue of risk sharing. As analyzed in Sections 3 and 4, eachsupplier must bear some forms of risk in either scheme. Under bundling, a winner undertakes all the risk of production cost, which entails the risk premium ρ∗ (a∗ ). Under
unbundling, the suppliers bear the risk associated with competition in the second-stage
auction (and a fraction of the risk of production cost), which entails the risk premianρ∗∗ (a∗∗ ). Unfortunately, we cannot say which one is greater than the other. However,2
we can obtain a noteworthy result by specifying the production cost function as follows.
16

Suppose that a common cost parameter is additively separable from the other terms,
that is, c(a, θi , ω) = c̄(a, θi ) + ω. Notice that the equilibrium bidding strategy in thesecond-stage auction is then given by1
p2 (a, θi , ω) = − ln Eθ(n−1) [exp(−rc̄(a, θ(n−1) )) | θ(n−1) > θi ] + ω.
r
The cost of common component ω is thus compensated by the buyer through competition
regardless of its realized value. Then, in the scheme of unbundling, the risk of common
cost component is effectively transferred from the suppliers to the buyer. With thisspecification, the next proposition provides the comparative statics with respect to theriskiness of common cost component.
Proposition 4. Assume that c(a, θi , ω) = c̄(a, θi ) + ω, and an increase in β ∈ [0, β̂] isa mean-preserving spread of G(ω; β). Then, there exist thresholds β and β̄ such that (i)the buyer chooses bundling in equilibrium for all β < β̄, and unbundling for all β > β̄,(ii) the socially desirable scheme is bundling for all β < β, and unbundling for all β > β,and (iii) β ≤ β̄.
Proposition 4 establishes the main result. As explained above, in the scheme of unbundling, no supplier undertakes the risk of common cost component, which is transferredto the risk-neutral buyer. Both the buyer’s utility and social welfare under unbundlingare thus independent of the riskiness of common cost component. On the other hand, inthe scheme of bundling, a winner must undertake all the risk of production cost. Therefore, as the common cost component is more risky in the sense of Rothschild and Stiglitz(1970) (i.e. β becomes greater), the scheme of bundling becomes less attractive to boththe buyer and society. Figure 2 illustrates this result.

17

First-best utility
EUB∗ = W ∗
W ∗∗
EUB∗∗

0

β

β

β̄

β̂

Figure 2: Illustration of Proposition 4.

6

Concluding remarks

We have explored what factors affect the buyer’s optimal choice between bundling andunbundling. The buyer (and the society) faces the tradeoff which involves three factors:
the ex post efficiency of trade, investment incentives, and risk sharing. The comparative
statics results show that as the investment is more costly and/or a common cost component is more risky, bundling becomes relatively less attractive to both the buyer andsociety. The interesting effect of unbundling on risk sharing is that although the common
cost parameter is unverifiable, this information is revealed through competition, so thatthe risk can be transferred from the suppliers to the buyer. This effect improves risk
sharing.
It is now possible to answer the question what kind of risks encourage the choiceof each scheme; Yescombe (2007) classifies project risks into some categories (p. 246).
3First, the increase in the riskiness of common
cost parameter will encourage the choice18

of unbundling. We can give the following examples of the risk of common cost: theuncertainty about a project scope, the uncertainty about ground conditions, and rapid
technological changes. Yescombe (2007) argues that those projects where technology
is changing rapidly are not suitable for PPPs (p. 27). Actually, the United Kingdom
abandoned the uses of PPPs for IT projects. Second, the increase in the riskiness ofprivate cost parameter may or may not encourage the choice of unbundling. We can givethe following examples of the risk of private cost: the uncertainty about the availability of
a company’s human and material resources. As we have shown, the scheme of unbundlingentails the risk associated with competition, which is affected by the riskiness of private
parameter. Thus, the effect of the riskiness on the public authority’s choice is ambiguous.
We make some final remarks. First, we discuss the assumption that a winner in thefirst stage auction can participate in the second stage auction under unbundling. Thissetting is different from other studies on PPPs. However, even if the winner is excluded
from the second stage auction, all our results are essentially unchanged. In this case, it
is clear that the winner has no incentive to invest unless the cost-reducing effect in thefirst stage is positive.
Second, we discuss the issue of the buyer’s commitment. The crucial difference between bundling and unbundling is the buyer’s commitment. In particular, there is room
for Pareto improvement under bundling. Suppose now that under bundling, the buyer canswitch suppliers at date 4 by the first-price auction. Then, the equilibrium of bundlingis equivalent to that of unbundling. There may be another scenario. Suppose that underbundling, the buyer allows a winner in the first-stage auction to hire a subcontractor in
the second stage. When the winner selects the subcontractor by the first-price auction,
the winner will undertake all the risk of common cost. Thus, our main result (Proposition

194) will still hold.
Finally, we examine how cost-sharing contracts affect the result. Suppose now thatthe production cost becomes verifiable after the service is provided. Then, some costsharing contracts are feasible. Under bundling, the buyer faces a standard tradeoff inmoral hazard models. If the buyer compensates all the production cost, then a winnermakes no investment. If the winner bears all the production cost, then he bears all therisk of production cost. Thus, the buyer optimally chooses the intermediate sharing levelso that the winner always bears a fraction of common cost. On the other hand, in thescheme of unbundling, the risk of common cost can be transferred to the buyer, as in
Proposition 4. We leave this analysis for future research.

References
J. Bennett and E. Iossa. Building and managing facilities for public services. Journal of
Public Economics, 90(10-11):2143–2160, 2006.
B.R. Chen and Y.S. Chiu. Public-private partnerships: Task interdependence and contractibility. International Journal of Industrial Organization, 28(6):591–603, 2010.
O. Hart. Incomplete contracts and public ownership: Remarks, and an application to
public-private partnerships. Economic Journal, 113(C69-C76):69–76, 2003.
C. Holt. Competitive bidding for contracts under alternative auction procedures. Journalof Political Economy, 88(3):433–445, 1980.
E.I. Hoppe, D.J. Kusterer, and P.W. Schmitz. Public-private partnerships versus traditional procurement: An experimental investigation. Journal of Economic Behavior &Organization, 2011.
20

E.I. Hoppe and P.W. Schmitz. Public-private partnerships versus traditional procurement: Innovation incentives and information gathering. CEPR Discussion Papers,
2010.
V. Krishna. Auction theory. Academic press, 2009.
S. Li and J. Yu. Bundling decisions in procurement auctions. 2011.
D. Martimort and J. Pouyet. To build or not to build: Normative and positive theories
of public-private partnerships. International Journal of Industrial Organization, 26(2):393–411, 2008.
E. Maskin and J. Tirole. Public-private partnerships and government spending limits.
International Journal of Industrial Organization, 26(2):412–420, 2008.
OECD. Public-private partnerships – In pursuit of risk sharing and value for money.
2008.
M. Rothschild and J.E. Stiglitz. Increasing risk: I. A definition. Journal of Economic
Theory, 2(3):225–243, 1970.
P.W. Schmitz. Allocating Control in Agency Problems with Limited Liability and Sequential Hidden Actions. RAND Journal of Economics, 36(2):318–336, 2005.
E.R. Yescombe.

Public-private partnerships:

Butterworth-Heinemann, 2007.

21

principles of policy and finance.

Appendix
Proof of Lemma 1. Differentiating ρ∗ with respect to a yields
 ∂c




(a,
θ
,
ω)
·
exp(rc(a,
θ
,
ω))
∂c
i
i
∂a
−E
(a, θi , ω)E[exp(rc(a, θi , ω))]∂a∂c
Cov − ∂a(a, θi , ω), exp(rc(a, θi , ω))=−E[exp(rc(a, θi , ω))]

E
dρ∗
(a) =
da

≤ 0;
the second equality follows from the formula for the covariance, and the inequality follows
from the fact that the covariance between two positively covarying variables is nonnegative, together with Assumption 2.
Proof of Proposition 1. (i) A winner i chooses an investment level to maximize his expected utility E[u(p1i − ψ(a) − c(a, θi , ω))], which is equal to the utility level u(p1i − ψ(a) −E[c(a, θi , ω)] − ρ∗ (a)) from the certainty equivalent (1). Note that the expected utility
E[u(p1i − ψ(a) − c(a, θi , ω))] is strictly concave in a because the second derivative is givenby E[u00 (π) · (∂π/∂a)2 + u0 (π) · ∂ 2 π/∂a2 ] < 0, where π = p1i − ψ(a) − c(a, θi , ω); thus,u(p1i − ψ(a) − E[c(a, θi , ω)] − ρ∗ (a)) is also strictly concave in a. Hence, the equilibrium
investment level a∗ is uniquely determined by the first-order condition


dψ ∗
∂c ∗
dρ∗ ∗
(a ) = −E(a , θi , ω) −
(a ).
da
∂a
da
If a supplier i who submits a bid p1i wins the first-stage auction, then he can obtain
the expected utility E[u(p1i −ψ(a∗ )−c(a∗ , θi , ω))] in equilibrium. The certainty equivalentis p1i − ψ(a∗ ) − E[c(a∗ , θi , ω)] − ρ∗ (a∗ ). The first-stage auction is equivalent to the gameof Bertrand competition among symmetric suppliers. Hence, in any equilibrium, there
exist at least two suppliers who submit the same bid p1∗ = ψ(a∗ ) + E[c(a∗ , θ, ω)] + ρ∗ (a∗ ),22

which makes each supplier indifferent between winning and losing, and the other suppliers
submit bids higher than p1∗ .
(ii) First, the necessary and sufficient first-order condition for the efficient investment
level ã = arg min{ψ(a) + E[c(a, θ(n) , ω)]} is given by


∂c
dψ
(ã) = −E(ã, θ(n) , ω) .
da
∂a

(7)

(0) ≤ 0 that ã > 0.
It follows from the assumption dψdaSecond, the necessary and sufficient first-order condition for the equilibrium investment level a∗ under bundling is given by (2) in Proposition 1. The right-hand side of (2)satisfies the following inequalities:





∂c ∗
∂c ∗
∂c ∗
dρ∗ ∗
−E
(a , θi , ω) −(a ) ≥ −E(a , θi , ω) ≥ −E
(a , θ(n) , ω) ;
∂a
da
∂a
∂a


(8)

the first inequality follows from Lemma 1, and the second inequality follows from the factthat the distribution function F of θi first-order stochastically dominates that of θ(n) ,together with Assumptions 1 and 2 (see, for example, Appendix B of Krishna (2009)). It
follows from (2), (7), (8) and the convexity of ψ that a∗ ≥ ã.
(iii) The buyer’s equilibrium utility is given by
EUB∗ = v − p1∗ = v − {ψ(a∗ ) + E[c(a∗ , θ, ω)] + ρ∗ (a∗ )} .
Since each supplier i who submits p1∗ wins with equal probability under the tie-breaking
rule, his equilibrium expected utility is given by
EUi∗ =1m−1u(p1∗ − ψ(a∗ ) − E[c(a∗ , θi , ω)] − ρ∗ (a∗ )) +u(0) = 0,mmwhere m ∈ {2, ..., n} is the number of suppliers who submit p1∗ . Each supplier who
submits a higher bid than p1∗ receives zero payoff. Finally, the social welfare is given by
W ∗ = EUB∗ + n · u−1 (EUi∗ ) = v − {ψ(a∗ ) + E[c(a∗ , θ, ω)] + ρ∗ (a∗ )} .
23

Proof of Lemma 2. We must show that for all a and ω, a supplier i with θ cannot gain bydeviating from a bid p2 (a, θ, ω) when the other suppliers follow the strategy p2 (·). Notethat p2 (·) is increasing and continuous in θ.
First, we can assume without loss of generality that no supplier submits a bid p 6∈
[p2 (a, θ, ω), p2 (a, θ̄, ω)]. If a supplier i bids p > p2 (a, θ̄, ω), then he loses with probability
one; by bidding p2 (a, θ̄, ω), he can obtain the same utility. If a supplier i bids p <p2 (a, θ, ω), then he wins with probability one; by bidding p2 (a, θ, ω), he can win withprobability one and obtain higher utility.
Second, we show that it is optimal for the supplier i with θ to bid p = p2 (a, θ, ω).
There are two cases to consider: The supplier i wins or loses in the first-stage auction.
If the supplier i is different from the winner in the first-stage auction, then his expectedutility from bidding p2 (a, θ̂, ω) is given by(1 − F(n−1) (θ̂))u(p2 (a, θ̂, ω) − c(a, θ, ω)) + F(n−1) (θ̂)u(0)"Z θ̄= (1 − F(n−1) (θ̂)) 1 − exp(rc(a, θ, ω))exp(−rc(a, s, ω))θ̂Z θ̄= (1 − F(n−1) (θ̂))[1 − exp(−r(c(a, s, ω) − c(a, θ, ω)))]θ̂

#f(n−1) (s)1 − F(n−1) (θ̂)
f(n−1) (s)1 − F(n−1) (θ̂)

ds

ds

Z θ̄
=

u(c(a, s, ω) − c(a, θ, ω))f(n−1) (s)ds;
θ̂

the first equality is obtained by substituting p2 (a, θ̂, ω), and the other equalities follow
from simple calculations. Thus, the difference between the expected utility from bidding
p2 (a, θ, ω) and that from bidding p2 (a, θ̂, ω) 6= p2 (a, θ, ω) is given byZ θ̂
u(c(a, s, ω) − c(a, θ, ω))f(n−1) (s)ds > 0;θ

the inequality follows from the assumption that c(a, θi , ω) is increasing in θi .
24

If the supplier i is the winner with a winning price p1i in the first-stage auction, thenhis expected utility from bidding p2 (a, θ̂, ω) is given by(1 − F(n−1) (θ̂))u(p1i − ψ(a) + p2 (a, θ̂, ω) − c(a, θ, ω)) + F(n−1) (θ̂)u(p1i − ψ(a))
no= 1 − exp(−r(p1i − ψ(a))) (1 − F(n−1) (θ̂)) exp(−r(p2 (a, θ̂, ω) − c(a, θ, ω))) + F(n−1) (θ̂)
nhio
= 1 + exp(−r(p1i − ψ(a))) (1 − F(n−1) (θ̂)) 1 − exp(−r(p2 (a, θ̂, ω) − c(a, θ, ω))) − 1(Z)θ̄= 1 + exp(−r(p1i − ψ(a)))
u(c(a, s, ω) − c(a, θ, ω))f(n−1) (s)ds − 1 ;θ̂the third equality is obtained in the same way as above, and the other equalities follow
from simple calculations. Thus, the difference between the expected utility from bidding
p2 (a, θ, ω) and that from bidding p2 (a, θ̂, ω) 6= p2 (a, θ, ω) is given byexp(−r(p1i − ψ(a)))

(Z
θ̂

)
u(c(a, s, ω) − c(a, θ, ω))f(n−1) (s)ds

> 0.

θ

Therefore, it is optimal for the supplier i with θ to bid p = p2 (a, θ, ω), no matter
whether he wins or loses in the first-stage auction.
Proof of Lemma 3. Differentiating p2 with respect to a yields ∂cEθ(n−1) ∂a(a, θ(n−1) , ω) · exp(−rc(a, θ(n−1) , ω)) | θ(n−1) > θi∂p2(a, θi , ω) =∂aEθ(n−1) exp(−rc(a, θ(n−1) , ω)) | θ(n−1) > θi∂c(a, θi , ω) · Eθ(n−1) exp(−rc(a, θ(n−1) , ω)) | θ(n−1) > θi∂a≤Eθ(n−1) exp(−rc(a, θ(n−1) , ω)) | θ(n−1) > θi=∂c(a, θi , ω) < 0;∂a

the first inequality follows from both the assumption that c is increasing in θi and Assumption 2.
Proof of Proposition 2. We show that the subgame has the equilibrium characterized in
the proposition, by backward induction.

25

(i) Lemma 2 implies that p2 (·) can be an equilibrium bidding strategy in the secondstage auction.
A winner i in the first-stage auction chooses an investment level to maximize his
expected utility
E[(1 − F(n−1) (θi ))u(p1i − ψ(a) + p2 (a, θi , ω) − c(a, θi , ω)) + F(n−1) (θi )u(p1i − ψ(a))],which is equal to the utility levelu(p1i − ψ(a) + E[(1 − F(n−1) (θi ))(p2 (a, θi , ω) − c(a, θi , ω))] − ρ∗∗ (a))
from the certainty equivalent (4). Differentiating the certainty equivalent (4) yields


 2
∂c
dρ∗∗
dψ
∂p
(a, θi , ω) −(a, θi , ω) −
(a) −
(a).
E (1 − F(n−1) (θi ))
∂a
∂a
da
da
If this first-order derivative is greater than zero at a = 0, then there is an interior solutiona∗∗ determined by (6) with equality. Otherwise, either a corner solution a∗∗ = 0 or aninterior solution exists.
If a supplier i who submits a bid p1i wins the first-stage auction, then he can obtain the expected utility E[(1 − F(n−1) (θi ))u(p1i − ψ(a∗∗ ) + p2 (a∗∗ , θi , ω) − c(a∗∗ , θi , ω)) +F(n−1) (θi )u(p1i − ψ(a∗∗ ))] in equilibrium. The certainty equivalent is p1i − ψ(a∗∗ ) + E[(1 −F(n−1) (θi ))(p2 (a∗∗ , θi , ω) − c(a∗∗ , θi , ω))] − ρ∗∗ (a∗∗ ). The first-stage auction is equivalent to
the game of Bertrand competition among symmetric suppliers. However, note that even
if the supplier i loses the first-stage auction, he can obtain the expected utility
E[(1 − F(n−1) (θi ))u(p2 (a∗∗ , θi , ω) − c(a∗∗ , θi , ω)) + F(n−1) (θi )u(0)]in the second-stage auction; the certainty equivalent is given byE[(1 − F(n−1) (θi ))(p2 (a∗∗ , θi , ω) − c(a∗∗ , θi , ω))] − ρ∗∗ (a∗∗ ).
26

Hence, in any equilibrium, there exist at least two suppliers who submit the same bid
p1∗∗ = ψ(a∗∗ ), which makes each supplier indifferent between winning and losing, and theother suppliers submit bids higher than p1∗∗ .
(ii) We show that ã > a∗∗ . If there exists a corner solution a∗∗ = 0, then ã > a∗∗ = 0.
If there exists an interior solution a∗∗ > 0, then a∗∗ is determined by (6) with equality in
Proposition 2. The first-order condition can be rewritten as
h 2

i
∂p
∂c
E
(a, θ, ω) − ∂a (a, θ, ω) (1 − F(n−1) (θ)) exp(−rπ(a, θ, ω))
∂a
dψ


(a) =
da
E (1 − F(n−1) (θ)) exp(−rπ(a, θ, ω)) + F(n−1) (θ)
≤ 0;

(9)

the inequality follows from Lemma 3. It then follows from (7) and (9) with the convexityof ψ that ã > a∗∗ .
(iii) Since in equilibrium the supplier with θ(n) wins the second-stage auction, thebuyer’s expected utility is EUB∗∗ = v − p1∗∗ − E[p2 (a∗∗ , θ(n) , ω)]. It follows from (i)that p1∗∗ = ψ(a∗∗ ). The expected payment in the second-stage auction is given by27

Eω [Eθ(n) [p2 (a∗∗ , θ(n) , ω)]]. Now, Eθ(n) [p2 (a∗∗ , θ(n) , ω)] can be rewritten as
Eθ(n) [p2 (a∗∗ , θ(n) , ω)]1∗∗= Eθ(n) − ln Eθ(n−1) [exp(−rc(a , θ(n−1) , ω)) | θ(n−1) > θ(n) ]
r
Z θ̄
∗∗
c(a∗∗ , s, ω)f(n) (s)ds= Eθ(n) [c(a , θ(n) , ω)] −
θ

Z θ̄
+

1
− ln Eθ(n−1) [exp(−rc(a∗∗ , θ(n−1) , ω)) | θ(n−1) > s]f(n) (s)dsrθ= Eθ(n) [c(a∗∗ , θ(n) , ω)]Z θ̄1− ln Eθ(n−1) [exp(−r(c(a∗∗ , θ(n−1) , ω) − c(a∗∗ , s, ω))) | θ(n−1) > s]f(n) (s)ds+rθ= Eθ(n) [c(a∗∗ , θ(n) , ω)]Z θ̄1ln Eθ(n−1) [exp(−r(c(a∗∗ , θ(n−1) , ω) − c(a∗∗ , s, ω))) | θ(n−1) > s]n(1 − F(n−1) (s))f (s)ds−rθ= Eθ(n) [c(a∗∗ , θ(n) , ω)] + nEθi [(1 − F(n−1) (θi ))(p2 (a∗∗ , θi , ω) − c(a∗∗ , θi , ω))];the first and fifth equalities follow from the definition of p2 (·) in Lemma 2, and the fourthequality follows from the definition of f(n) . Thus, EUB∗∗ is given by
EUB∗∗ = v − ψ(a∗∗ ) + E[c(a∗∗ , θ(n) , ω)]+ nE[(1 − F(n−1) (θi ))(p2 (a∗∗ , θi , ω) − c(a∗∗ , θi , ω))] .
Since each supplier i who submits p1∗∗ wins the first-stage auction with equal probability under the tie-breaking rule, his equilibrium expected utility is given by
EUi∗∗ =1u(p1∗∗ − ψ(a∗∗ ) + E[(1 − F(n−1) (θi ))(p2 (a∗∗ , θi , ω) − c(a∗∗ , θi , ω))] − ρ∗∗ (a∗∗ ))mm−1+u(E[(1 − F(n−1) (θi ))(p2 (a∗∗ , θi , ω) − c(a∗∗ , θi , ω))] − ρ∗∗ (a∗∗ ))m= u(E[(1 − F(n−1) (θi ))(p2 (a∗∗ , θi , ω) − c(a∗∗ , θi , ω))] − ρ∗∗ (a∗∗ )),where m ∈ {2, ..., n} is the number of suppliers who submit p1∗∗ . Each supplier who
submits a higher bid than p1∗∗ also obtains the same level of expected utility.
28

Finally, the social welfare is given by
W ∗∗ = EUB∗ + n · u−1 (EUi∗∗ ) = v − ψ(a∗∗ ) + E[c(a∗∗ , θ(n) , ω)] + nρ∗∗ (a∗∗ ) .

Proof of Proposition 3. The proof consists of four steps: We show that (a) EUB∗ is continuous and decreasing in k, (b) both EUB∗∗ and W ∗∗ are independent of k, (c) EUB∗ = W ∗for all k, limk→∞ EUB∗ < EUB∗∗ < W ∗∗ , and (d) the statements in the proposition follow
from (a)-(c).
(a) Since the equilibrium investment level a∗ under bundling depends on k, we denotea∗ = a∗ (k) and EUB∗ = EUB∗ (k). Now, the buyer’s equilibrium utility under bundling isEUB∗ (k) = v − (k + 1)ψ̄(a∗ (k)) + E[c(a∗ (k), θi , ω)] + ρ∗ (a∗ (k)) .
The assumption that both c(a, θi , ω) and ψ̄(a) are twice continuously differentiable in a
implies that a∗ (k) is differentiable, and thus continuous in k. Thus, EUB∗ is continuousin k. Using the envelope theorem, differentiating EUB∗ (k) with respect to k yields
dEUB∗(k) = −ψ̄(a∗ (k)) < 0.
dk
Hence, EUB∗ (k) is decreasing in k.
(b) It follows from Corollary 1 that a∗∗ = 0. Since the investment cost is ψ(0) =(k + 1)ψ̄(0) = 0 in equilibrium, both EUB∗∗ and W ∗∗ are independent of k.
(c) First, Proposition 1 states that EUB∗ (k) = W ∗ (k) for all k. Second, it follows fromthe first-order condition (2) that
lim a∗ (k) = 0.

k→∞

29

This implies that the infimum of EUB∗ (k) is given bylim EUB∗ (k) = v − {E[c(0, θi , ω)] + ρ∗ (0)}
k→∞


1
ln E[exp(rc(0, θi , ω))] ;=v−
r
the second equality follows from the definition of ρ∗ . The bracketed term satisfies thefollowing inequalities:
1ln E[exp(rc(0, θi , ω))] > E[c(0, θi , ω)]r> E[c(0, θ(n) , ω)]1= E − ln exp(−rc(0, θ(n) , ω))Eθ(n−1) [1 | θ(n−1) > θ(n) ]r1> E − ln Eθ(n−1) [exp(−rc(0, θ(n−1) , ω)) | θ(n−1) > θ(n) ]r= E[p2 (0, θ(n) , ω)];the first inequality follows by Jensen’s inequality, the second inequality follows from thefact that the distribution function F of θi first-order stochastically dominates that of θ(n) ,together with Assumptions 1 and 2 (see, for example, Appendix B of Krishna (2009)),
and the last equality follows from the definition of p2 (·). On the other hand, the buyer’s
equilibrium expected utility under unbundling isEUB∗∗ = v − p1∗∗ + E[p2 (a∗∗ , θ(n) , ω)]= v − (k + 1)ψ̄(a∗∗ ) + E[p2 (a∗∗ , θ(n) , ω)]= v − E[p2 (0, θ(n) , ω)];the third equality follows from ψ̄(a∗∗ ) = ψ̄(0) = 0. Therefore, we obtain thatlim EUB∗ (k) < EUB∗∗ .

k→∞

30

Third, it follows from Proposition 2 that W ∗∗ − EUB∗∗ = n · u−1 (EUi∗∗ ) > 0. We thus
obtain EUB∗∗ < W ∗∗ .
(d) There are three cases to consider.
First, suppose that W ∗∗ < EUB∗ (0). Then, it follows from (a)-(c) that there exist
thresholds k and k̄ such that the statements (i) and (ii) in the proposition hold, with
0 < k < k̄ < ∞.
Second, suppose that EUB∗∗ < EUB∗ (0) ≤ W ∗∗ . Then, it follows from (a)-(c) that
there exist thresholds k and k̄ such that the statements (i) and (ii) in the proposition
hold, with 0 = k < k̄ < ∞.
Last, suppose that EUB∗ (0) < EUB∗∗ . Then, it follows from (a)-(c) that EUB∗ (k) <EUB∗∗ and W ∗ (k) < W ∗∗ for all k. By setting k = k̄ = 0, the statements (i) and (ii) in
the proposition hold.
Proof of Proposition 4. The proof proceeds in the same way as Proposition 3, and consistsof four steps: We show that (a) EUB∗ is decreasing in β, (b) both EUB∗∗ and W ∗∗ areindependent of β, (c) EUB∗ = W ∗ for all β, EUB∗∗ < W ∗∗ and (d) the statements in the
proposition follow from (a)-(c).
(a) Since the risk premium ρ∗ (a∗ ) under bundling depends on β, we denote ρ∗ (a∗ ) =ρ∗ (a∗ ; β) and EUB∗ = EUB∗ (β). Now, the buyer’s equilibrium utility under bundling is
EUB∗ (β) = v −



Z ω̄
∗
∗
∗ ∗
ψ(a ) + E[c̄(a , θi )] +ωdG(ω; β) + ρ (a ; β) .
ωSince an increase in β represents a mean-preserving spread of G, the expected value of ω
is independent of β, but the risk premium ρ∗ (a∗ ; β) is increasing in β. Hence, EUB∗ (β) is
decreasing in β.

31

(b) First, the buyer’s equilibrium utility under unbundling is
EUB∗∗


∗∗

Z ω̄

∗∗

= v − ψ(a ) + E[c̄(a , θ(n) )] +ωdG(ω; β)ω+ nE[(1 − F(n−1) (θi ))(p (a , θi , ω) − (c̄(a , θi ) + ω))]
2



∗∗

∗∗

∗∗

∗∗

Z ω̄

= v − ψ(a ) + E[c̄(a , θ(n) )] +ωdG(ω; β)
ω





1
+ nE (1 − F(n−1) (θi )) − ln Eθ(n−1) [exp(−rc̄(a∗∗ , θ(n−1) )) | θ(n−1) > θi ] − c̄(a∗∗ , θi )r
note that under the assumption c(a, θi , ω) = c̄(a, θi ) + ω, the bidding strategy p2 (a, θi , ω)
is now given by1
p2 (a, θi , ω) = − ln Eθ(n−1) [exp(−rc̄(a, θ(n−1) )) | θ(n−1) > θi ] + ω.
r

(10)

Since an increase in β represents a mean-preserving spread of G, EUB∗∗ is independent ofβ.
Second, it follows from (5) and (10) with the assumption c(a, θi , ω) = c̄(a, θi ) + ω
that the risk premium ρ∗∗ (a∗∗ ) under unbundling is independent of β; note that a∗∗ isindependent of β because the first-order condition (9) does not include ω. Hence, W ∗∗ isalso independent of β.
(c) Proposition 1 states that EUB∗ (β) = W ∗ (β) for all β. In the same way as (a), we
can show that EUB∗∗ < W ∗∗ .
(d) There are four cases to consider.
First, suppose that W ∗∗ ≤ EUB∗ (β̂). Then, it follows from (a)-(c) that EUB∗ (β) >EUB∗∗ and W ∗ (β) ≥ W ∗∗ for all β. By setting β = β̄ = β̂, the statements (i) and (ii) in
the proposition hold.
Second, suppose that EUB∗∗ ≤ EUB∗ (β̂) < W ∗∗ . Then, it follows from (a)-(c) that
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;

there exist thresholds β and β̄ such that the statements (i) and (ii) in the proposition
hold, with 0 ≤ β < β̄ = β̂.
Third, suppose that EUB∗ (β̂) < EUB∗∗ < EUB∗ (0). Then, it follows from (a)-(c) that
there exist thresholds β and β̄ such that the statements (i) and (ii) in the proposition
hold, with 0 ≤ β < β̄ < β̂.
Last, suppose that EUB∗ (0) ≤ EUB∗∗ . Then, it follows from (a)-(c) that EUB∗ (β) ≤EUB∗∗ and W ∗ (β) < W ∗∗ for all β. By setting β = β̄ = 0, the statements (i) and (ii) in
the proposition hold.

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