How (Not) to Foster Innovations in Public Infrastructure Projects

The government wants an infrastructure-based public service to be provided. First the infrastructure has to be built, subsequently it has to be operated. Should the government bundle the building and operating tasks in a public-private partnership? Or should it choose traditional procurement, i.e. delegate the tasks to different firms? Each task entails unobservable investments to come up with innovations. It turns out that depending on the nature of the innovations, bundling may either stimulate or discourage investments. Moreover, we find that if renegotiation cannot be prevented, a public-private partnership may lead the government to deliberately opt for a technologically inferior project.


Introduction
To provide infrastructure-based public services is one of the main tasks of government. On average, public procurement accounts for around 12% of GDP in OECD countries (see OECD, 2017). Thus, the amounts at stake are certainly signi…cant. While the public procurement system should strive to achieve the best possible performance in terms of cost and service, examples of ine¢ ciencies are reported regularly in the daily news. 1 How to organize the provision of public goods and services is hence a highly important question.
In particular, public infrastructure such as highways, bridges, airports, and hospitals …rst has to be built and subsequently it has to be operated. Traditionally, when the government wanted to procure an infrastructure-based service, the two tasks of …rst building and then operating the infrastructure were separated. Speci…cally, the construction of a project was contracted out to a private company. This …rm built the project, received the agreed payment, and then the contract was completed. Afterwards, another party took charge of operating and maintaining the facility. Yet, around the early 1990s, public-private partnerships have emerged as a new organizational form and they have become increasingly popular since then. A key property of a public-private partnership is the fact that facility construction and service provision are bundled (see e.g. Hart, 2003); i.e., the tasks of …rst building and then operating the infrastructure are assigned to a single private company. 2 In the present paper, we consider a contract-theoretic model with two stages, a building stage and an operating stage. We investigate whether the two tasks of building and operating the infrastructure should be delegated to two di¤erent …rms or whether it is better to bundle these two tasks and assign them to a single 1 Infrastructure projects plagued by delays, cost overruns, environmental issues, and quality shortfalls attract much attention in the media. Recent examples include the Berlin Brandenburg Airport in Germany (cf. Hammer, 2015), the Honolulu Rail Transit Project in Hawaii (cf. Nagourney, 2016), or the Alaskan Way Viaduct replacement tunnel in Seattle (cf. Anderson, 2017). See Flyvbjerg et al. (2003) for discussions of many other examples. 2 Grimsey and Lewis (2004) and Yescombe (2007) also emphasize that in practice a de…ning characteristic of a public-private partnership is that design and construction as well as operation and maintenance of the public infrastructure are combined under one private contractor. company, a consortium. Advocates of public-private partnerships often argue that bundling fosters innovation incentives (see e.g. HM Treasury, 2012). In this context, innovations are de…ned in a broad sense as e¢ ciency gains achieved through productive investments. However, now that public-private partnerships have been in place for more than twenty years, the empirical evidence regarding the success of public-private partnerships in stimulating innovations is mixed. 3 On the one hand, there are case studies which document that public-private partnerships have indeed spurred innovations. For example, in the transportation sector, public-private partnerships were successful in substantially reducing construction time and in developing innovative solutions for congestion management by introducing time-varying tolls. 4 On the other hand, there is evidence which suggests that bundling may sti ‡e innovations. For example, in the health care sector it has been reported that consortia in charge of hospital projects faced strategic incentives not to come up with innovations facilitating the adaptability of the hospital's design, because in this way they would be able to "achieve additional income through alterations needed in the future". 5 Hence, when the same party is in charge of building and operating, innovations in the building stage might actually be discouraged, since they may reduce rents that could be obtained in the operating stage.
Our formal model provides an explanation for the empirical …nding that bundling the building and operating tasks in a public-private partnership can boost innovation incentives in some cases, while it may sti ‡e incentives to innovate under di¤erent circumstances. Speci…cally, we consider an extension of Tirole's (1999) "R&D game" to two stages. We assume that …rms in charge 3 See e.g. Leiringer (2006), Russell et al. (2006), Javed et al. (2013), Liu and Liu (2017), Himmel and Siemiatycki (2017), Saeed et al. (2018), and the recent survey by Carbonara and Pellegrino (2018) for empirical studies investigating whether or not public-private partnerships are conducive to fostering innovations. 4 For instance, it was a private consortium that took the initiative to introduce variable pricing for California's State Route 91 express lanes, which works well to eliminate tra¢ c congestion during peak periods. In France, a private …rm resolved a 30-year impasse over how to complete the missing link of the A86 Paris ring road, using a deep-bore tunnel under the Versailles palace. See Gilroy et al. (2007) and Small (2010) for further details. 5 See Barlow and Köberle-Gaiser (2009, p. 135). of building and/or operating the public infrastructure are protected by limited liability. In the building stage as well as in the operating stage, unobservable e¤ort can be exerted to come up with an innovation. In each stage, the outcome (i.e., whether or not there was a successful innovation) is veri…able. 6 Since e¤ort is a hidden action, the government can incentivize e¤ort only with the help of outcome-contingent contracts.
In particular, consider the operating stage. In the presence of uncertainty, the outcome is only a noisy signal of the chosen e¤ort level. Hence, if the government wants to induce the …rm in charge of operating to exert high e¤ort, it must leave a rent to the …rm (see e.g. La¤ont and Martimort, 2002). In case of a publicprivate partnership, the expected rents in the operating stage are taken into account by the consortium when it decides on how much e¤ort to spend in the building stage. Bundling can thus create positive or negative incentive spillover e¤ects which are absent in case of traditional procurement.
Suppose …rst that a successful innovation in the building stage increases the government's value of an innovation in the operating stage. The government will then implement a relatively large second-stage e¤ort level after a …rst-stage success, while it implements a relatively small second-stage e¤ort level after a …rst-stage failure. Thus, in case of a public-private partnership the consortium will be able to earn a larger rent in the operating stage if it was already successful in the building stage. As a consequence, it becomes cheaper for the government to provide incentives in the building stage, which gives a public-private partnership an advantage over traditional procurement. Now suppose that a successful innovation in the building stage reduces the government's additional value that can be generated by an innovation in the operating stage. A consortium may then prefer not to exert innovation e¤ort in the building stage, in order to obtain a larger rent in the operating stage. As a result, it can become very expensive for the government to induce high e¤ort in the building stage, such that traditional procurement may be preferred. Hence, in our model the pros and cons of bundling the building and operating tasks in a public-private partnership can be traced back to the same source, namely the 6 Note that these assumption are in line with Tirole's (1999, p. 745) one-shot model, which in turn is based on Aghion and Tirole's (1994) work on the management of innovation. e¤ect of expected second-stage rents on …rst-stage incentives.
When the government has full commitment power, then the second-stage e¤ort level that it implements after a …rst-stage success will be larger under a public-private partnership than under traditional procurement. In this way, the government further increases the consortium's incentives to exert e¤ort in the building stage. In contrast, after a …rst-stage failure the government implements a smaller second-stage e¤ort level under a public-private partnership than under traditional procurement, so the consortium is punished for not developing an innovation in the building stage.
Yet, practitioners emphasize that in reality the government often cannot commit not to renege on its contract with the consortium. 7 When mutually bene…cial renegotiation cannot be prevented, the government loses the possibility to punish the consortium for a …rst-stage failure by implementing a smaller second-stage e¤ort level than under traditional procurement. As a consequence, the possibility of renegotiations reduces the advantages of a public-private partnership compared to traditional procurement and may have important implications for the initial choice of a public infrastructure project. In particular, we …nd that if under a public-private partnership renegotiation cannot be ruled out, then the government may prefer to choose a technologically inferior project. The reason is that the choice of such a project may reduce the scope for renegotiations. In contrast, under a public-private partnership with full commitment as well as under traditional procurement the government would never choose a technologically inferior project.
Related literature. The theoretical literature analyzing the pros and cons of bundling tasks in public-private partnerships was initiated by Hart (2003), who applies the incomplete contracting approach. 8 Hart (2003) considers two di¤erent 7 The fact that in practice renegotiations cannot be prevented has often been emphasized in the empirical literature on public-private partnerships, see e.g. Guasch (2004) and Beuve et al. (2014). 8 See also Bös and De Fraja's (2002) earlier incomplete contracting model on bundling in the health care sector. The incomplete contracting paradigm was developed by Grossman and Hart (1986), Hart and Moore (1990), and Hart (1995). See Maskin andTirole (1999) andTirole (1999) for critical discussions of the foundations of the incomplete contracting methodology. kinds of investments which both can be made in the building stage in order to reduce costs in the operating stage. One kind of investment is desirable, while the other kind of investment is undesirable, since it leads to a strong reduction in service quality. A public-private partnership results in too much undesirable investments, while under traditional procurement there are weaker incentives to make desirable investments. In line with Hart (2003), we focus on the bundling decision and do not study the choice between public and private ownership. 9 In contrast, Bennett and Iossa (2006) explore the interaction of the bundling decision with the choice between di¤erent ownership structures. 10 However, in models combining agency problems and property rights, Iossa and Martimort (2015, p. 23) conclude that "the important issue is not who owns the asset but instead whether tasks are bundled or not."Following Hart (2003), these authors assume that e¤ort invested in the building stage has a direct external e¤ect on the costs incurred in the operating stage. In contrast, in our model an innovation in the building stage can make an innovation in the operating stage either more or less valuable for the government, so from the consortium's perspective an external e¤ect is created only if the government conditions payments in the operating stage on the outcome of the building stage. 11 Our contribution is based on agency problems due to moral hazard. In contrast, Hoppe and Schmitz (2013) discuss the costs and bene…ts of public-private partnerships in an adverse selection model, where the consortium may strategically gather information about future costs to adapt the service provision to 9 Hart (2003, p. C71) points out that he ignores ownership issues and that he takes bundling to be the key property of a public-private partnership. On the decision between public and private ownership, see Hart et al. (1997) and the subsequent literature, e.g. Besley and Ghatak (2001), King and Pitchford (2008), Schmitz (2010), De Brux andDesrieux (2014), and Hamada (2017). 10 See also Chen and Chiu (2010) for a variant of Bennett and Iossa's (2006) model. 11 In particular, the e¤ort costs and the success probability for a given e¤ort level in the second stage do not depend on what happened in the …rst stage. Thus, for a …xed second-stage incentive scheme the agent's second-stage behavior depends neither on the …rst-stage e¤ort nor on the …rst-stage outcome. Our model thus di¤ers from the sequential agency problems studied by Baliga and Sjöström (1998), Schmitz (2005), and Pi (2018). changing circumstances. 12 Recently, Martimort and Straub (2016) have also studied public-private partnerships in a two-stage moral hazard model with riskneutral …rms that are protected by limited liability. 13 However, there are important di¤erences. In particular, Martimort and Straub (2016) assume that the e¤ort level exerted in the second stage must always be larger than …rst-stage e¤ort, and they exogenously rule out second-stage payments that depend on the outcome of the …rst stage. Their focus is on the e¤ects of an uncertain productivity shock after the …rst stage. Our model is complementary to Martimort and Straub's (2016) setup, since we do not impose any intertemporal restrictions on the e¤ort levels and since history-dependent payments play a central role in our analysis. To the best of our knowledge, the di¤erent implications of publicprivate partnerships and traditional procurement for the initial choice of a public project have not yet been explored in the literature.
Finally, it should be noted that in practice policy makers may be tempted to favor public-private partnerships for the wrong reasons, since they are often not included in the …scal balance sheets. 14 From an economic perspective, publicprivate partnerships should be given the same treatment in budgetary accounting as traditional procurement, so the choice between the organizational forms should be based on e¢ ciency considerations (see Hart, 2003, p. C75). Hence, in the present contribution we abstract from …nancing issues and instead focus on the di¤erent incentive structures that prevail in public-private partnerships and traditional procurement.
Organization of the paper. The remainder of the paper is organized as follows. In Section 2, we introduce the model. The case of traditional procurement is analyzed in Section 3, while the organizational form of a public-private partnership is investigated in Section 4. In Section 5, we compare the two modes of 12 The role of adverse selection in the context of public-private partnerships has recently also been studied by Buso (2018). 13 For dynamic moral hazard problems in settings not speci…c to procurement, see e.g. Rogerson (1985), Ohlendorf and Schmitz (2012), Kräkel and Schöttner (2016), and Schöttner (2017).
Yet, these papers do not study the di¤erences between bundling and unbundling.
14 See e.g. Vining and Boardman (2008, p. 153), Martimort (2015, p. 29), andBuso et al. (2017). provision. In Section 6, we analyze a scenario where renegotiations cannot be ruled out and we explore the implications for project choice. Concluding remarks follow in Section 7. All formal proofs have been relegated to the Appendix.

The model
Suppose the government (the principal) wants two sequential tasks to be performed in order to provide a public good or service. First an infrastructure has to be designed and built (stage 1), subsequently it has to be maintained and operated (stage 2). Before the …rst stage begins, the government has the choice between two di¤erent governance structures, traditional procurement (TP) and a public-private partnership (PPP). In the case of traditional procurement, the government contracts with one agent (the builder) in charge of stage 1 and with another agent (the operator) in charge of stage 2. In the case of a public-private partnership, the two tasks are bundled; i.e., the government contracts with only one agent (a consortium) that is in charge of both stages. We assume that all parties are risk-neutral, the agents are protected by limited liability, and their reservation utilities are zero. 15 In the …rst stage, the agent in charge of designing and building the infrastructure can choose an unobservable e¤ort level E 2 f0; 1g in order to come up with innovative ideas to improve the social value of the infrastructure. The veri…able outcome of the building stage is a success (x = 1) with probability pE and a failure (x = 0) otherwise, where 0 < p < 1. Let the agent's disutility of e¤ort be given by E, where > 0. 16 In the second stage, the agent in charge of operating and maintaining the infrastructure exerts unobservable e¤ort e 2 [0; 1], incurring a disutility of e¤ort given by 1 2 e 2 . The second-stage e¤ort aims at innovations to further increase the 15 Similar assumptions are often made in the related literature, see e.g. the recent work by Martimort and Straub (2016). 16 We model …rst-stage e¤ort in this simple way as a binary decision, so that if there were only the building stage the government would not have to leave a rent to the agent in order to motivate him to exert high e¤ort. This assumption simpli…es the exposition, since it allows us to clarify the e¤ects that anticipated second-stage rents have on the behavior in the …rst stage.
social value of providing the public good or service. The veri…able outcome of the operating stage is a success (y = 1) with probability e and a failure (y = 0) otherwise. Note that the e¤ort level e(x) chosen in the second stage can depend on the outcome x of the …rst stage.
The social bene…ts generated by the public good or service are given by The bene…ts are net of the monetary and veri…able costs of building and operating the infrastructure, which are always reimbursed by the government. We assume that a successful innovation always increases the social value. Speci…cally, B 1 > B 0 > 0, so the bene…ts are larger when an innovative infrastructure was built in the …rst stage. Similarly, b 0 > 0 and b 1 > 0, so a second-stage innovation always increases the bene…ts from service provision.
Note that the magnitude of the increase may depend on whether or not there was an innovation in the building stage. Moreover, we make the technical assumptions that b 0 1 and b 1 1; this normalization allows us to follow the usual convention that e¤ort e can be directly interpreted as a success probability. 17 Furthermore, in order to focus the analysis on the economically most interesting case, we assume throughout that > 1 2 pb 2 1 , i.e. the …rst-stage e¤ort costs are su¢ ciently large. 18 For simplicity, we assume throughout that there is no discounting. 19 The sequence of events is illustrated in Figure 1. At the outset, the government chooses the organizational mode (traditional procurement or a publicprivate partnership). In the building stage, the agent who is in charge can exert unobservable e¤ort E. Since the outcome is veri…able, it is feasible to contractually specify a payment T that the government must make to the agent whenever there was a success (x = 1). In the operating stage, the agent who is in charge can exert unobservable e¤ort e. The outcome is again veri…able, so it is possible to contractually specify a payment t x that has to be made from the government to the agent whenever there was a success (y = 1). Note that the amount to be paid for a second-stage innovation can depend on whether or not there was a …rst-stage innovation. Under traditional procurement, the government o¤ers a contract (specifying T ) to the builder at the outset, while it o¤ers a contract (specifying t x ) to the operator at the beginning of the second stage. Under a public-private partnership, the government o¤ers a contract (T; t 0 ; t 1 ) to the consortium at the outset. 20 We will study the case in which the government has full commitment power as well as the case in which it cannot commit not to renege on the contract at the beginning of the operating stage.

PPP or TP
Effort E∈{0,1} Building stage Operating stage The …rst-best benchmark. Consider for a moment a …rst-best world in which the e¤ort decisions are veri…able. The …rst-best e¤ort level in the operating stage maximizes eb x 1 2 e 2 . Hence, the marginal e¤ort costs must be equal to the marginal bene…t, e F B (x) = b x . In the building stage, it is …rst-best to choose high e¤ort (E = 1) whenever The left-hand side is the expected total bene…t net of e¤ort costs given high e¤ort in the building stage, while the right-hand side is the corresponding expression given low e¤ort in the building stage. Hence, there is a cuto¤ value If the e¤ort levels were veri…able, the government would implement the …rstbest e¤ort choices with a simple forcing contract that would in each stage reimburse the agent in charge for his e¤ort costs. Thus, the government would be indi¤erent with regard to the bundling decision. Yet, in the remainder of the paper we assume that the e¤ort choices are hidden actions. As a consequence, when we will …nd that one of the two organizational forms is strictly preferred by the government, then this result must be due to incentive considerations only.

Traditional procurement
We now investigate the incentive structure under traditional procurement. Consider …rst the operating stage, so the outcome of the building stage x 2 f0; 1g has already been realized. In the operating stage, given the contractually spec-i…ed reward t x 1 for a second-stage innovation, 21 the operator maximizes his expected payo¤ et x 1 2 e 2 . Thus, the operator chooses e(x) = t x . Anticipating the operator's e¤ort choice, at the beginning of the second stage the government sets the reward t x in order to maximize its expected payo¤ . Thus, the government will specify the payment t T P x = 1 2 b x . Observe that the operator's expected rent 1 2 t 2 x = 1 8 b 2 x is increasing in the 21 It is straightforward to verify that the government will never o¤er a reward larger than one, since the additional bene…t generated by a second-stage innovation b x is smaller than one. additional bene…t generated by a second-stage innovation. Moreover, note that the government's second-stage payo¤ is 1 4 b 2 x . Next, consider the building stage. Given that the reward T was contractually speci…ed for a …rst-stage innovation, the builder will choose high e¤ort (E = 1) whenever the incentive compatibility constraint pT 0 is satis…ed. Hence, the government sets T T P = =p if it wants to induce high e¤ort in the …rst stage, It is optimal for the government to implement high e¤ort in the …rst stage whenever i.e., whenever the expected social bene…ts net of the payments to the agents are larger in the case of high …rst-stage e¤ort than in the case of low …rst-stage e¤ort. Rewriting the condition we …nd that the government implements E = 1 whenever T P , where The preceding discussion can thus be summarized as follows.
Proposition 1 Consider traditional procurement.
(i) If T P , it is optimal for the government to set t T P 0 = 1 2 b 0 , t T P 1 = 1 2 b 1 , and T T P = =p. Then the builder will choose E T P = 1 and the operator will choose e T P (1) = 1 2 b 1 , e T P (0) = 1 2 b 0 . (ii) If > T P , it is optimal for the government to set t T P 0 = 1 2 b 0 , t T P 1 = 1 2 b 1 , and T T P = 0. Then the builder will choose E T P = 0 and the operator will choose e T P (0) = 1 2 b 0 .
Note that when the government implements low e¤ort in the building stage, there will be no …rst-stage success, so on the equilibrium path the payment t 1 is irrelevant if > T P . Furthermore, observe that T P is smaller than F B whenever b 0 < b 1 . Hence, the following result holds.
(i) In the building stage, (ii) In the operating stage, e T P (x) < e F B (x) for x 2 f0; 1g.
Compared to the benchmark case in which e¤orts are veri…able, unobservability of e¤orts leads to a smaller e¤ort level in the second stage, since in this way the expected rent that must be left to the operator is reduced. In the building stage, there may be e¤ort cost parameters such that high e¤ort would be chosen when e¤orts were veri…able, while only low e¤ort is induced when e¤orts are hidden actions. This happens when b 0 < b 1 , because in this case the second-stage rent is larger following a …rst-stage success, so from the government's perspective the value of a …rst-stage innovation is reduced. In contrast, if b 0 > b 1 , there are cost parameters such that low e¤ort would be preferred when e¤ort was veri…able, while high e¤ort is induced when e¤ort is unobservable. The reason is that in this case a larger second-stage rent must be paid following a …rst-stage failure, which from the government's perspective further increases the attractiveness of a …rst-stage success.

Public-private partnership
Let us now analyze the incentive structure in case of a public-private partnership, assuming that the government can commit not to renege on the contractually speci…ed payments. Suppose that the payments t 0 1 and t 1 1 have been contractually agreed upon. 22 In the second stage, following the …rst-stage outcome x 2 f0; 1g, the consortium chooses the e¤ort level e that maximizes its expected payo¤ et x 1 2 e 2 . Thus, the consortium will exert e¤ort e(x) = t x . Observe that the consortium's expected second-stage rent is 1 2 t 2 x . Applying backward induction, we can now study the consortium's e¤ort decision in the …rst stage. Given that the payment T was speci…ed in the contract, the consortium prefers to exert high e¤ort (E = 1) whenever 22 It is again straightforward to verify that the government will never o¤er payments larger than one.
i.e., whenever the consortium's expected payo¤ over the whole life of the project is larger if it exerts high instead of low e¤ort in the building stage. This incentive compatibility constraint can be rewritten as Anticipating the consortium's behavior, at the outset the government o¤ers a contract (T; t 0 ; t 1 ) that maximizes the expected social bene…ts net of the payments made to the consortium, In order to characterize the solution to the government's problem, let us de…ne a threshold level of the …rst-stage e¤ort costs, Then the solution under a public-private partnership can be summarized as follows.
Proposition 2 Consider a public-private partnership and suppose the government has full commitment power.
(i) If P P P , it is optimal for the government to set t P P P 0 = 1 p 2 p b 0 , t P P P 1 = b 1 , and T P P P = =p 1 2 b 2 1 + 1 2 ( 1 p 2 p b 0 ) 2 . Then the consortium will choose E P P P = 1 in the building stage and e P P P (1) = b 1 , e P P P (0) = 1 p 2 p b 0 in the operating stage. (ii) If > P P P , it is optimal for the government to set t P P P 0 = 1 2 b 0 , t P P P 1 = 1 2 b 1 , and T P P P = 0. Then the consortium will choose E P P P = 0 in the building stage and e P P P (0) = 1 2 b 0 in the operating stage.
Proof. See the Appendix.
When the same agent is in charge of both stages, the government can make use of incentive spillovers from the second to the …rst stage. In the building stage, the consortium's e¤ort decision will not only depend on the payment T for a …rst-stage success, but also on the expected rents that it may get in the second stage. Suppose the government wants to implement high …rst-stage e¤ort.
The government can indirectly reward the consortium for a …rst-stage success by implementing a relatively large second-stage e¤ort (and thus a large rent) following x = 1, while it can punish the consortium for a …rst-stage failure by implementing a relatively small second-stage e¤ort (and thus a small rent) following x = 0. Observe that according to Proposition 2(i), following a …rst-stage success the government implements the …rst-best e¤ort level in the operating stage. While the expected rent could be further increased by specifying an even larger second-stage e¤ort level, this would be an ine¢ cient way to reward the consortium; i.e., it would be cheaper for the government to increase the direct reward T for a …rst-stage success.
Furthermore, note that P P P > F B , so there are …rst-stage e¤ort cost parameters such that in the building stage low e¤ort would be chosen when e¤orts were veri…able, while high e¤ort is chosen when they are unobservable.
Intuitively, since the government must leave a rent to the consortium in order to induce second-stage e¤ort, it would like to extract this rent from the consortium by an up-front payment. Yet, since negative payments are ruled out due to limited liability, utility may instead be transferred from the consortium to the government in an ine¢ cient way only, namely by implementing an ine¢ ciently large …rst-stage e¤ort level.
These …ndings are summarized in the following result.
Corollary 2 Consider a public-private partnership and suppose the government has full commitment power.
(i) In the building stage, E P P P E F B .
(ii) In the operating stage, e P P P (1) = e F B (1) and e P P P (0) < e F B (0) if P P P , while e P P P (0) < e F B (0) if > P P P .

Public-private partnership versus traditional procurement
We can now analyze the government's choice between the two organizational modes. Propositions 1 and 2 immediately reveal that if the government implements low e¤ort in the building stage, the second-stage e¤ort level does not depend on the organizational form; i.e., in this case the government is indi¤erent between traditional procurement and a public-private partnership. However, when the government wants to implement high e¤ort in the building stage, the two modes of provision lead to di¤erent agency costs.
Speci…cally, suppose that b 1 is larger than b 0 , so under traditional procurement higher second-stage rents are earned by the operator after a …rst-stage success than after a …rst-stage failure. In this case it is clearly better to bundle the two tasks, because then the payment that is necessary to directly reward a …rst-stage success can be reduced. The reason is that in the building stage, the consortium already has an indirect incentive to exert e¤ort, since a …rst-stage success leads to a larger rent in the second-stage.
In contrast, if b 1 is smaller than b 0 , then under traditional procurement a larger second-stage rent is earned after a …rst-stage failure. At …rst glance, one might guess that in this case bundling would be undesirable, since the consortium would have an indirect incentive not to exert e¤ort in the building stage, in order to avoid a …rst-stage success. However, this intuition is correct only if b 1 is much smaller than b 0 . Otherwise, a public-private partnership can still outperform traditional procurement. To see this, observe that if the government implements high e¤ort in the building stage, then compared to traditional procurement under a public-private partnership the second-stage e¤ort is larger in case of a …rststage success, e P P P (1) > e T P (1), while it is smaller in case of a …rst-stage failure, e P P P (0) < e T P (0). Hence, when the same agent is in charge of both stages, the government can commit to second-stage e¤ort levels that are di¤erent from the ones implemented under traditional procurement in order to indirectly reward a …rst-stage success and punish a …rst-stage failure.
Let us now take a closer look at the government's expected payo¤s under the two organizational modes. Under traditional procurement, Proposition 1 implies that high e¤ort (E = 1) is implemented in the building stage whenever T P .
In this case, the government's expected payo¤ is Under a public-private partnership, Proposition 2 implies that high e¤ort is implemented in the building stage whenever P P P . In this case, the govern-ment's expected payo¤ reads Furthermore, under both modes of provision the government's expected payo¤ is given by when low e¤ort is implemented in the building stage.  Figure 2 depicts the government's expected payo¤ depending on the …rststage e¤ort costs . In the left panel, the condition b 2 1 > 1 p 2 p b 2 0 is satis…ed. 23 This condition implies that P P P > T P and G P P P H > G T P H must hold. The government will implement high e¤ort in the building stage whenever the e¤ort costs are smaller than P P P and in this case the government strictly prefers a public-private partnership.
In the right panel, the condition b 2 1 < 1 p 2 p b 2 0 holds (i.e., a second-stage success is su¢ ciently more valuable when there was a …rst-stage failure). This condition implies P P P < T P and G P P P H < G T P H , so whenever is smaller than T P the government implements high e¤ort in the building stage and strictly prefers traditional procurement.
Taken together, the following result holds.
Proposition 3 Suppose the government has full commitment power.
(i) If b 2 1 > 1 p 2 p b 2 0 and < P P P , the government strictly prefers a public-private partnership.
(ii) If b 2 1 < 1 p 2 p b 2 0 and < T P , the government strictly prefers traditional procurement.
(iii) Otherwise, the government is indi¤erent between the two modes of provision.

ment reconsidered
So far, we have assumed that the government can commit not to renege on the contractually speci…ed payments. We now relax this assumption and explore what will happen if mutually bene…cial renegotiation at the beginning of the operating stage cannot be prevented. 24 In the case of traditional procurement the analysis remains unchanged, since two di¤erent parties are in charge of the two stages and hence the government has no reason to ex ante commit to a secondstage contract that it would want to renege on after the building stage is …nished.
However, in case of a public-private partnership, ex ante the government wants to commit to second-stage payments that a¤ect the consortium's second-stage incentives as well as its …rst-stage incentives. Once it is known whether or not there was an innovation in the building stage, the government is interested only in the consortium's incentives in the operating stage and thus the government might want to renege on the original contract.
Speci…cally, consider a public-private partnership and suppose that P P P , so that the government would implement high …rst-stage e¤ort (E = 1) if 24 Several authors have pointed out that renegotiation is an important problem in the context of public-private partnerships, see e.g. the recent contributions by Henckel and McKibbin (2017) and Ahmad et al. (2018). renegotiation could be ruled out. According to Proposition 2, under full commitment the contract speci…es t P P P 1 = b 1 , so that following a …rst-stage innovation the second-stage e¤ort level is e P P P (1) = b 1 . Yet, when there was a …rst-stage innovation, then at the beginning of the second stage the government would prefer to implement only e(1) = 1 2 b 1 in order to reduce the expected second-stage rent, as we have seen in the analysis of the operating stage under traditional procurement. However, if the government tried to renege on the agreed-upon contract by reducing the payment for a second-stage innovation, the consortium would not give in. The consortium would insist on the original contract, since otherwise its expected rent would be reduced. Hence, there is no scope for mutually bene…cial renegotiation when there was a …rst-stage success. Now suppose that there was no innovation in the building stage. According to Proposition 2, for this case the contract under full commitment speci…es t P P P 0 = 1 p 2 p b 0 . As a consequence, the consortium would choose the second-stage e¤ort level e P P P (0) = 1 p 2 p b 0 , while at the beginning of the operating stage the government would prefer to implement the e¤ort level e(0) = 1 2 b 0 , as we know from the analysis of traditional procurement. Clearly, when the government o¤ers to increase the payment for a second stage innovation to t 0 = 1 2 b 0 , the consortium will accept the o¤er, since then its expected rent will be larger. Therefore, the outcome described in Proposition 2 is no longer sustainable when mutually bene…cial renegotiation cannot be prevented.
To characterize the solution to the government's problem when renegotiation cannot be ruled out, let us de…ne a new threshold level of the …rst-stage e¤ort costs,^ P P P : Applying the renegotiation-proofness principle, we can without loss of generality focus on contracts that are not renegotiated in equilibrium. 25 We thus obtain the following result.
25 See Hart and Tirole (1988) for more on the renegotiation-proofness principle. Intuitively, the allocation that would result from renegotiation can already be speci…ed in the original contract, so there is no need to consider contracts that are renegotiated on the equilibrium path.
Proposition 4 Consider a public-private partnership and suppose that mutually bene…cial renegotiation cannot be prevented.
(i) If ^ P P P , it is optimal for the government to set t P P P 0 = 1 2 b 0 , t P P P 1 = b 1 , and T P P P = =p 1 2 b 2 1 + 1 8 b 2 0 . Then the consortium will choose E P P P = 1 in the building stage and e P P P (1) = b 1 , e P P P (0) = 1 2 b 0 in the operating stage. (ii) If >^ P P P , it is optimal for the government to set t P P P 0 = 1 2 b 0 , t P P P 1 = 1 2 b 1 , and T P P P = 0. Then the consortium will choose E P P P = 0 in the building stage and e P P P (0) = 1 2 b 0 in the operating stage.
Proof. See the Appendix.
Observe that the government still rewards the consortium for a …rst-stage success by implementing a larger second-stage e¤ort level in this case than it would do under traditional procurement. However, when renegotiation cannot be ruled out the government loses its possibility to punish the consortium for a …rst-stage failure by implementing a smaller second-stage e¤ort level than under traditional procurement. As a consequence, given that high …rst-stage e¤ort is implemented, the government's expected payo¤ is smaller when renegotiation cannot be prevented than in the case of full commitment.
The new threshold value^ P P P satis…es F B <^ P P P < P P P . High e¤ort in the building stage is now implemented for a smaller range of …rst-stage e¤ort costs compared to the case of a public-private partnership where the government has full commitment power. Yet, the impossibility to prevent renegotiation does not qualitatively change the comparison with the …rst-best benchmark where e¤orts are veri…able.
Corollary 3 Consider a public-private partnership and suppose that mutually bene…cial renegotiation cannot be prevented.
(i) In the building stage, E P P P E F B .
(ii) In the operating stage, e P P P (1) = e F B (1) and e P P P (0) < e F B (0) if ^ P P P , while e P P P (0) < e F B (0) if >^ P P P .
Let us now turn to the comparison between the two organizational modes.
When renegotiation cannot be ruled out, Proposition 4 implies that under a public-private partnership high e¤ort is implemented in the building stage whenever ^ P P P . In this case, the government's expected payo¤ readŝ which is smaller than G P P P H . Otherwise, the government's expected payo¤s remain unchanged.
Suppose now that the condition b 2 1 > 1 2 b 2 0 holds, which is always the case if a second-stage innovation is more valuable when there also was a …rst-stage innovation. Then^ P P P > T P andĜ P P P H > G T P H hold. Hence, the government implements high …rst-stage e¤ort whenever the e¤ort costs are smaller than P P P and in this case the government strictly prefers a public-private partnership. Next, suppose that the condition b 2 1 < 1 2 b 2 0 is satis…ed, so a second-stage innovation is su¢ ciently more valuable when there was no …rst-stage innovation. Then^ P P P < T P andĜ P P P H < G T P H hold. Thus, whenever is smaller than T P the government implements high …rst-stage e¤ort and strictly prefers traditional procurement.
Taken together, the parameter range where a public-private partnership is optimal is now smaller than in the case of full commitment.
Proposition 5 Suppose that mutually bene…cial renegotiation cannot be prevented.
(i) If b 2 1 > 1 2 b 2 0 and <^ P P P , the government strictly prefers a public-private partnership.
(iii) Otherwise, the government is indi¤erent between the two modes of provision.

Project choice
We now investigate implications that the impossibility to rule out renegotiation in case of a public-private partnership may have with regard to the initial choice of a public project. Suppose that at the outset, the government has the choice between two di¤erent projects I and II . In what follows, we assume that b I 0 > b II 0 , while the projects are identical otherwise. Hence, the two projects di¤er only in the value of a second-stage innovation when there was no …rst-stage success.
This value is larger in case of project I , which means that project I is the technologically superior project. 26 If the government implements low …rst-stage e¤ort (so that the organizational mode does not matter) or if the government opts for traditional procurement, it is obvious that it will never choose the technologically inferior project II . To see this formally, observe that G L and G T P H are increasing in b 0 . Moreover, in case of a public-private partnership the government always prefers project I when it has full commitment power, since also G P P P H is increasing in b 0 . Now consider a public-private partnership and suppose that renegotiation cannot be prevented. Recall that when high …rst-stage e¤ort is implemented, the government's expected payo¤ as a function of b 0 is given bŷ Observe that when the probability of a …rst-stage success given high …rst-stage e¤ort is relatively large, then the government's expected payo¤ is decreasing in Figure 3. Choice between projects I and II with b I 0 > b II 0 , when p > 2=3. 26 It should be noted that we could also assume that in addition b I 1 > b II 1 . If the di¤erence between b I 1 and b II 1 was su¢ ciently small, qualitatively similar insights would still hold by continuity.
As an illustration consider Figure 3, which depicts the government's expected payo¤ from a given project depending on the …rst-stage e¤ort costs. The green curves refer to the technologically superior project I , while the red curves refer to the technologically inferior project II . Recall that when low …rst-stage e¤ort is implemented, the government's expected payo¤ as a function of b 0 is given by In each project, high …rst-stage e¤ort is implemented when the …rst-stage e¤ort costs are su¢ ciently small, . Note that^ P P P (b I 0 ) <^ P P P (b II 0 ) must hold. When low …rst-stage e¤ort is implemented, the government prefers project I , since ). Yet, when high …rst-stage e¤ort is implemented, the government prefers project II , given that p > 2=3. Hence, there exits a threshold value such that the government chooses the technologically inferior project whenever the …rst-stage e¤ort costs are smaller than .
Proposition 6 Consider a public-private partnership and suppose that mutually bene…cial renegotiation cannot be prevented.
(i) If p < 2=3, the government chooses the technologically superior project I.
(ii) If p > 2=3, there exists a cuto¤-value 2 (^ P P P (b I 0 );^ P P P (b II 0 )) such that the government chooses project I if > , while it chooses the technologically inferior project II if < .
Proof. See the Appendix.
Intuitively, the government may prefer the technologically inferior project II because a smaller b 0 means that there is less scope for renegotiation. Recall that when renegotiation cannot be prevented the government loses its ability to punish the consortium for a …rst-stage failure by implementing a very small second-stage e¤ort. This is less of a problem when b 0 is small, since then the second-stage e¤ort (and hence the consortium's expected second-stage rent) following a …rst-stage failure will be small anyway; i.e., renegotiation has less bite. However, choosing the technologically inferior project can be optimal only if the probability p is relatively large, so the probability that b 0 will actually become relevant on the equilibrium path is relatively small.
Finally, regarding the choice between a public-private partnership and traditional procurement, it should be noted that the preceding …ndings imply that the availability of a technologically inferior project can increase the parameter range for which the government prefers a public-private partnership. In particular, the following result holds.
(i) If only project I is available, the government strictly prefers a public-private partnership over traditional procurement whenever b 2 1 > 1 2 (b I 0 ) 2 . (ii) If in addition the technologically inferior project II becomes available, the government chooses project II and strictly prefers a public-private partnership over traditional procurement whenever b 2 Proof. See the Appendix.

Concluding remarks
The relatively new organizational form of public-private partnerships was promoted to foster incentives to innovate, such that increased quality would be achieved at lower costs. However, after more than 20 years of experience we observe mixed evidence regarding innovation incentives within public-private partnerships. Our model explains in a uni…ed framework that compared to traditional procurement, a public-private partnership may indeed either foster or sti ‡e innovation incentives.
In contrast to earlier contributions to the literature on public-private partnerships that was initiated by Hart (2003) in our model there is a single force working in two ways. The costs and bene…ts of bundling the building and operating tasks in a public-private partnership are two sides of the same coin, since in each case they stem from the intricate e¤ects that expected rents have on the prevailing incentive structure.
Speci…cally, it turns out that if a …rst-stage innovation increases the social value of a second-stage innovation, then bundling the tasks in a public private partnership reduces the agency costs. In contrast, if a …rst-stage innovation reduces the social value of a second-stage innovation, then in a public-private partnership the consortium may face strategic reasons not to exert innovation e¤ort in the building stage, in order to extract a larger rent in the provision stage. Moreover, we …nd that the impossibility to rule out mutually bene…cial renegotiations reduces the advantages of bundling and that in this case a publicprivate partnership may even lead the government to opt for a technologically inferior project.
We hope that the insights gained by our analysis will help to spur further empirical research on the important topic of innovations in public infrastructure projects. 28 Moreover, from a contract-theoretic perspective, our model could be extended in several directions. For example, following most of the theoretical literature on public-private partnerships, we have abstracted from agency problems within the consortium and we have focused our analysis on the relationship between the government and a given consortium. Exploring the interactions of internal agency problems and of the award procedure with the incentive e¤ects identi…ed in the present paper might be interesting avenues for future research. 29 28 As has been pointed out by Iossa and Martimort (2015, p. 40), in spite of the policy relevance, still relatively little research has been carried out on public-private partnerships. In particular, the empirical literature on innovations in public-private partnerships (cf. footnote 3 above) is still scarce, so much more work needs to be done on that front. 29 Regarding agency problems within consortia, see Greco (2015) for an analysis of imperfect bundling in an incomplete contracting model based on Hart (2003) and Bennett and Iossa (2006). With regard to award procedures, see Li et al. (2015) who study the bundling of tasks in procurement auctions where the …rms have private information about their costs.
its expected stage-2 payo¤ et x 1 2 e 2 and thus exert e¤ort e(x) = t x . Recall from our analysis of traditional procurement that at the beginning of the operating stage, the government would like to set t x = 1 2 b x . Due to concavity of the government's payo¤, at the beginning of the operating stage the government would like to reduce t x when in the original contract it was larger than 1 2 b x , while the government would like to increase t x when in the original contract it was smaller than 1 2 b x . Since the consortium's expected second-stage rent 1 2 t 2 x is increasing in t x , the consortium will accept a renegotiation o¤er at the beginning of the operating stage whenever the payment is larger than in the original contract.
If the government wants to implement E = 0, it sets T P P P = 0 and t P P P 0 = 1 2 b 0 , which maximizes B 0 +t 0 (b 0 t 0 ). To satisfy the constraint T < =p 1 2 t 2 1 + 1 2 t 2 0 , under our assumption > 1 2 pb 2 1 the government can specify t P P P 1 = 1 2 b 1 . Note that the contract is renegotiation-proof.
A comparison of the government's expected payo¤s implies that the government implements E = 1 whenever so the proposition must hold.
Proof of Proposition 6. Recall that b I 0 > b II 0 implies G L (b I 0 ) > G L (b II 0 ). Hence, part (i) of the proposition immediately follows from the fact thatĜ P P P H (b I 0 ) > G P P P H (b II 0 ) when p < 2=3. Now consider part (ii) of the proposition. Recall that p > 2=3 impliesĜ P P P H (b II 0 ) >Ĝ P P P H (b I 0 ). Hence, while the government prefers project I when it implements low …rst-stage e¤ort, it prefers project II when it implements high …rst-stage e¤ort. The government thus implements high …rststage e¤ort wheneverĜ P P P H (b II 0 ) G L (b I 0 ). This condition can be rewritten as , where It is straightforward to verify that the cuto¤-value satis…es the condition P P P (b I 0 ) < <^ P P P (b II 0 ).
Proof of Corollary 4. Part (i) of the corollary follows immediately from Propositions 5 and 6. Now consider part (ii) of the corollary. We already know that the government prefers project I in case of traditional procurement. Given a public-private partnership, the government prefers project II if it wants to implement high …rst-stage e¤ort, while it prefers project I otherwise. Hence, the government chooses the technologically inferior project II and a public-private partnership ifĜ P P P H (b II 0 ) > G T P H (b I 0 ) = p[B 1 + 1 4 b 2 1 ] + (1 p)[B 0 + 1 4 (b I 0 ) 2 ] andĜ P P P H (b II 0 ) > G L (b I 0 ). The latter condition is satis…ed since by assumption < . The former condition can be rewritten as Observe that p > 2=3 implies 1 p p (b I 0 ) 2 + 3p 2 2p (b II 0 ) 2 < 1 2 (b I 0 ) 2 , so a publicprivate partnership is preferred for a larger parameter range when project II is available.