Contract, renegotiation, and hold up: Results on the technology of trade and investment

Watson, Joel; Buzard, Kristy Article Contract, renegotiation, and hold up: Results on the technology of trade and investment Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Watson, Joel; Buzard, Kristy (2012) : Contract, renegotiation, and hold up: Results on the technology of trade and investment, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 7, Iss. 2, pp. 283-322, https://doi.org/10.3982/TE818 This Version is available at: https://hdl.handle.net/10419/150172 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Theoretical Economics 7 (2012), 283–322 1555-7561/20120283 Contract, renegotiation, and holdup: Results on the technology of trade and investment Kristy Buzard Department of Economics, University of California, San Diego Joel Watson Department of Economics, University of California, San Diego This paper examines a class of contractual relationships with specific investment, a nondurable trading opportunity, and renegotiation. Trade actions are modeled as individual and trade-action-based option contracts (“nonforcing contracts”) are explored. The paper introduces the distinction between divided and unified investment and trade actions, and it shows the key role this distinction plays in determining whether efficient investment and trade can be achieved. Under a nonforcing dual-option contract, the party without the trade action is made the residual claimant with regard to the investment action, which induces efficient investment in the divided case. The unified case is more problematic: here, efficiency is typically not attainable, but the dual-option contract is still optimal in a wide class of settings. More generally, the paper shows that, with ex post renegotiation, constraining parties to use “forcing contracts” implies a strict reduction in the set of implementable value functions. Keywords. Contract, renegotiation, holdup, forcing contracts, nonforcing contracts, specific investment, technology of trade, mechanism design. JEL classification. C70, D23. The holdup problem arises in situations in which contracting parties can renegotiate their contract between the time they make unverifiable relation-specific investments and the time at which they can trade.1The severity of the holdup problem depends critically on the productive technology and on the timing of renegotiation opportunities. This paper contributes to the literature by examining how the nature of the “trade Kristy Buzard: [email protected] Joel Watson: [email protected] The authors thank the following people for their insightful comments: the anonymous referees, Nageeb Ali, Jeff Ely, Bob Evans, David Miller, Ben Polak, Larry Samuelson, Joel Sobel, and seminar participants at Columbia, Florida International, UCSD, USC, SWET, and Yale. Part of the analysis reported here was completed while Watson was a visitor at the Cowles Foundation, Yale, and he is grateful for the support from Cowles. 1Che and Sákovics (2008) provide a short overview of the holdup problem, which was first described by Klein et al. (1978), and Williamson (1975,1979). Analysis was provided by Grout (1984), Grossman and Hart (1986), and Hart and Moore (1988). Copyright ©2012 Kristy Buzard and Joel Watson. Licensed under the Creative Commons Attribution- NonCommercial License 3.0. Available at http://econtheory.org. DOI: 10.3982/TE818 284 Buzard and Watson Theoretical Economics 7 (2012) action” in a contractual relationship influences the prospects for achieving an efficient outcome. We introduce a new distinction—whether the party who invests also is the one who consummates trade—that plays an important role in determining the outcome of the contractual relationship. So that we can describe our modeling exercise more precisely, consider an example in which contracting parties Al and Zoe interact as follows. First Al and Zoe meet and write a contract that has an externally enforced element. Then one of them makes a private investment choice, which influences the state of the relationship. The state is commonly observed by the contracting parties, but is not verifiable to the external enforcer. Al and Zoe then send individual public messages to the external enforcer. After this, they have an opportunity to renegotiate their contract: this is called ex post renegotiation because it occurs after messages. Finally, the parties have a one-shot opportunity to trade and they also obtain external enforcement. Trade is verifiable to the external enforcer. Because the investment is unverifiable, the investor cannot be directly rewarded for choosing the efficient investment level. Instead, incentives hinge on how the terms of trade can be made sensitive to the investment choice. Typically a conflict arises between the parties’ joint interests prior to investment and their joint interests following investment and messages. In particular, investment incentives may be strengthened by specifying an inefficient trade action ex post in some off-equilibrium-path contingencies. But parties then would have the joint incentive to renegotiate and divide the surplus according to their bargaining power (holdup). Because parties rationally anticipate the renegotiation, the incentives to invest are distorted. The description above obviously leaves the mechanics of trade and enforcement ambiguous. In reality, the parties have individual actions that determine whether and how trade is consummated. Let us suppose that Al selects the individual trade action, which we call a. This could be a choice of whether to deliver or to install an intermediate good, for example. We then have an individual-action model, whereby Al chooses aand the external enforcer compels a transfer tas a function of aand the messages that the parties sent earlier. In contrast, a public-action model (or external-action model) combines the trade action and the monetary transfer into a single public action (at) that is assumed to be taken by the external enforcer. With this modeling approach, the contract specifies how the public action is conditioned on the parties’ messages. Although the public-action model may typically be a bit unrealistic, it is simple and lends itself to elegant mechanism-design analysis (for example, as in Maskin and Moore 1999 and Segal and Whinston 2002). Alternatively, Watson (2007) demonstrates that analysis of the individual-action model can be straightforward as well. He also shows that the public-action model is equivalent to examining individual trade actions, but constraining attention to “forcing contracts” in which the external enforcer induces a particular trade action as a function of messages sent by the parties (so the trade action is constant in the state). Watson (2007) provides an example in which the restriction to forcing contracts has strictly negative efficiency consequences. We deepen the examination of nonforcing contracts by investigating their efficacy in the context of different technologies of trade and investment. Specifically, we introduce the distinction between divided and unified investment and trade actions. In the Theoretical Economics 7 (2012) Contract, renegotiation, and holdup 285 divided case, the investment and trade actions are chosen by different parties (Al takes the trade action and Zoe makes the investment). In the unified case, the investment and trade actions are selected by the same party (Al does both). We show that the prospects for inducing efficient investment and trade are very different in the divided and unified cases. In fact, the efficient outcome can always be achieved in the divided case (assuming investment has no immediate benefits), but typically cannot be achieved in the unified case. Our analysis also highlights a simple contractual form that we call a dual-option contract. With the dual-option, Zoe sends a message that can be interpreted as a requested trade action or declaration of the state, and Al’s subsequent trade action also serves as an option. We show that a dual-option contract is optimal in a large class of contractual relationships. For instance, it can be used to make Al’s payoff constant in the state, gross of any investment costs, so that Zoe becomes the residual claimant with respect to the investment choice. This implies the efficiency result for the divided case. The dual option is also useful in the unified case, even though the efficient outcome typically cannot be achieved; specifically, we show that in a class of settings with a deterministic state, the dual-option contract is optimal. Our analysis utilizes mechanism-design techniques. With both the individualaction and public-action modeling approaches, analysis of the contractual problem centers on calculating the set of implementable value functions from just after the state is realized (before messages are sent). Formally, an implementable value function is the state-contingent continuation value that results in equilibrium for a given contract. We provide simple tools to calculate the “punishment values” that determine the implementable sets for the class of relationships we analyze here. We use these tools to characterize optimal contracts and to find bounds on the set of implementable value functions. In addition to the results on the divided and unified cases and dual-option contracts, we provide a general result on the comparison of forcing and nonforcing contracts, which shows that Watson’s (2007) conclusions are robust over a large class of contractual relationships. In particular, in the important setting of ex post renegotiation described above, limiting attention to forcing contracts reduces the set of state-contingent continuation values. This does not mean that a more efficient outcome can always be achieved when actions are modeled as individual (because efficiency depends on what region of the implementable value set is relevant for giving appropriate investment incentives), but it underscores the importance of modeling trade actions as individual. This is particularly salient for the setting of cross/cooperative investment (Che and Hausch 1999), where the investment by one party increases the benefit to the other party of subsequent trade. The literature has regarded cross-investment settings as especially prone to the holdup problem. Che and Hausch (1999) show that the optimal forcing contract is often “null” and leads to underinvestment. Our results establish that nonforcing contracts offer a significant improvement in efficiency, and our distinction between unified and divided investment and trade actions gives a basis for deeper analysis. In the class of trade technologies that we study here, a single player (player 1, Al above) has the trade action. Examples of real settings with this property are contractual relationships in which the seller provides a service or good that does not require 286 Buzard and Watson Theoretical Economics 7 (2012) the buyer’s involvement (such as consulting, advertising, and some types of construction). In these settings, the seller has the trade action. Other settings with unilateral trade actions are ones in which the seller is the investor, production is inherent in the seller’s investment, and trade is determined by whether the buyer installs or otherwise adopts the intermediate good; an example is specialized software. In these settings, the buyer has the trade action. We discuss the extension to multilateral trade actions in the Conclusion (Section 6). The only assumption required for our first result, on making player 2 (Zoe) the residual claimant, is that investment does not confer a direct gain for some minimal trade action (an assumption satisfied by the most prominent models in the holdup literature). The key economic assumption behind our other results is that player 1’s utility is supermodular as a function of the state and the trade action. That is, this player’s marginal value of the trade action is monotone in the state. Our other assumptions are mainly weak technical conditions that guarantee well defined maxima, nontrivial settings, and the like. We argue that these conditions are likely to hold in a wide range of applications and that they are consistent with typical assumptions in the literature. Our result about the optimality of the dual-option contract in the unified case requires some additional assumptions on the technology of investment and trade. The rest of the paper proceeds as follows. In the next section, we provide the details of the model. Section 2 presents an example that illustrates our main results. Section 3 contains our general results on optimal contracts and outcomes in the divided and unified cases. Readers interested in getting all of the basic ideas without the technical details can proceed from Section 3 straight to the Conclusion.Section 4 provides an overview of the basic tools for general analysis, which mostly restates material in Watson (2007). Section 5 contains our result on the difference in implementable sets based on variations regarding when renegotiation can occur and whether one restricts attention to forcing contracts. The Conclusion contains more discussion about the holdup problem and cross investment, as well as notes on the case of durable trading opportunities and multilateral trade actions. Most of the technical material and all of the proofs are contained in the Appendices. 1. The theoretical framework We look at the same class of contracting problems and use the same notation as in Watson (2007), except that we add a bit of structure on the trade technology to focus our analysis. In particular, we examine the case in which a single player has a trade action. Throughout the paper, we use the convention of labeling the player with the trade action as player 1, and we call the other player 2. These two players are the parties engaged in a contractual relationship with a nondurable trading opportunity and external enforcement. Their relationship has the following payoff-relevant components, occurring in the order shown. The state of the relationship θ. The state represents unverifiable events that are assumed to happen early in the relationship. The state may be determined by individual investment decisions and/or by random occurrences, depending on the Theoretical Economics 7 (2012) Contract, renegotiation, and holdup 287 Figure 1. Timeline of the contractual relationship. setting. When the state is realized, it becomes commonly known by the players; however, it cannot be verified to the external enforcer. Let denote the set of possible states. The trade action a. This is an individual action chosen by player 1 that determines whether and how the relationship is consummated. The trade action is commonly observed by the players and is verifiable to the external enforcer. Let Abe the set of feasible trade actions. The monetary transfers t=(t1t2).Heretidenotes the amount given to player ifor i=12, where a negative value represents an amount taken from this player. Transfers are compelled by the external enforcer, who is not a strategic player, but, rather, who behaves as directed by the contract of players 1 and 2.2Assume t1+t2≤0. We assume that the players’ payoffs are additive in money and are thus defined by a function u:A×→R2. In state θ, with trade action aand transfer t, the payoff vector is u(aθ)+t.DefineU(aθ)≡u1(a θ)+u2(a θ), which is the joint value of the contractual relationship in state θif trade action ais selected. We assume that, in each state θ,the joint value has a unique maximizer a∗(θ).Weletγ(θ) denote the maximal joint payoff in state θ,sowehave γ(θ) ≡U(a∗(θ) θ) =max a∈AU(aθ) (1) In addition to the payoff-relevant components of their relationship, we assume that the players can communicate with the external enforcer using public, verifiable messages. Let m=(m1m2)denote the profile of messages that the players send and let M1 and M2be the sets of feasible messages. The sets M1and M2are endogenous in the sense that they are specified by the players in their contract. Figure 1 shows the timeline of the contractual relationship. At even-numbered dates through date 6, the players make joint observations and they make individual decisions—jointly observing the state at date 2, sending verifiable messages at date 4, 2That the external enforcer’s role is limited to compelling transfers is consistent with what courts do in practice. 288 Buzard and Watson Theoretical Economics 7 (2012) and selecting the trade actions at date 6. At date 8, the external enforcer compels transfers. At odd-numbered dates, the players make joint contracting decisions— establishing a contract at date 1 and possibly renegotiating it later. The contract has an externally enforced component consisting of (i) feasible message spaces M1and M2and (ii) a transfer function y:M×A→R2that specifies the transfer tas a function of the verifiable items mand a. That is, having seen mand a, the external enforcer compels transfer t=y(ma). The contract also has a self-enforced component, which specifies how the players coordinate their behavior for the times at which they take individual actions. Renegotiation of the contract amounts to replacing the original transfer function ywith some new function y, in which case yis the one submitted to the external enforcer at date 8. We initially assume—and maintain throughout Sections 2and 3—that the players can freely renegotiate at dates 3, 5, and 7. Renegotiation at date 5 is called ex post renegotiation. At date 3 it is called interim renegotiation.3 The players’ individual actions at dates 2, 4, and 6 are assumed to be consistent with sequential rationality; that is, each player maximizes his expected payoff, conditional on what occurred earlier and on what the other player does, and anticipating rational behavior in the future. The joint decisions (initial contracting and renegotiation at oddnumbered periods) are assumed to be consistent with a cooperative bargaining solution in which the players divide surplus according to fixed bargaining weights π1and π2for players 1 and 2, respectively. The bargaining weights are nonnegative, sum to 1, and are written π=(π1π2). The negotiation surplus is the difference between γ(θ) and the joint value that would result if the players fail to reach an agreement, where the disagreement point is given by an equilibrium in the continuation in which the externally enforced component of the contract has not been altered.4 The effect of the renegotiation opportunity at date 7 is to constrain transfers to be “balanced,” that is, satisfying t∈R2 0≡{t∈R2|t 1+t 2=0} Thus, we simply assume that transfers are balanced and then otherwise ignore date 7. Also, as we explain later, the opportunity for ex post renegotiation implies that there is never any renegotiation surplus at date 3, so we can ignore interaction at date 3. Much of our analysis does not depend on the details of date 2 interaction, but some of our key results concern the relation between the investment and the trade technologies, and for these we need to formally distinguish between different investment technologies. We assume that a single player makes an investment choice at date 2. This gives us two cases to consider: 3In Sections 4and 5, we provide some analysis for the setting in which renegotiation is possible at date 3 but not at date 5. 4The generalized Nash bargaining solution has this representation. The rationality conditions identify a contractual equilibrium;seeWatson (2006) for notes on the relation between “cooperative” and “noncooperative” approaches to modeling negotiation. The players obtain the joint value γ(θ) because, at the time of renegotiation, they know the state θand can select a contract that forces the action a∗(θ), as described in the next subsection. Theoretical Economics 7 (2012) Contract, renegotiation, and holdup 289 •Unified case. Player 1 has both the date 2 investment action and the date 6 trade action. •Divided case. Player 2 has the date 2 investment action, whereas player 1 has the date 6 trade action. We assume that the investment influences the state. In the deterministic subcase, one of the players directly selects θat date 2. More generally, the state may also depend on the outcome of a random variable. Public-action modeling and forcing contracts Because the trade action ais assumed to be taken by player 1, we have specified here an individual-action model. A public-action model, in contrast, abstracts by treating the trade action aas something that the external enforcer directly selects. Watson (2007) shows that specifying a public-action model is equivalent to examining the individualaction model but limiting attention to a particular class of contracts called forcing contracts, which, for any given message profile, prescribe that player 1 selects a particular trade action. More precisely, a forcing contract specifies a large transfer from player 1 to player 2 in the event that player 1 does not take his contractually prescribed action. This transfer is sufficiently large to give player 1 the incentive to select the prescribed action in every state. Thus, the induced trade action is constant in the state, conditional on the messages sent earlier. For example, holding the message profile fixed, the transfer function ˆ ydefined as follows forces player 1 to select action ˆ aand imposes the transfer ˆ t(as though the external enforcer chose these in a public-action model): Let Lbe such that L>supaθ u1(aθ) −infaθ u1(a θ).Thendefineˆ y(ˆ a) ≡ˆ tand, for every a= ˆ a,setˆ y(a) ≡ˆ t+(−LL). We use the term forcing for any transfer function that, given the message profile, induces player 1 to select the same trade action over all of the states.5We use the term nonforcing for transfer functions that induce player 1 to select different actions in at least two different states. Continuation value functions A(state-contingent) value function is a function from to R2that gives the players’ expected payoff vector from the start of a given date, as a function of the state. Such a value function represents the continuation values for a given outstanding contract and equilibrium behavior. We adopt the convention of not including any sunk investment 5One could add a public randomization device to the model for the purpose of achieving randomization over trade actions using forcing contracts. Allowing such randomization does not expand the set of implementable value functions here. 290 Buzard and Watson Theoretical Economics 7 (2012) costs from date 2 in the function uor in the representation of continuation values from later dates. The continuation values from the start of date 3 are important to calculate, because they determine the players’ incentives to invest at date 2. Thus, our chief objective is to characterize the set of implementable value functions from the start of date 3. A value function vis implementable if there is a contract that, if formed at date 1, would lead to continuation value v(θ) in state θfrom the start of date 3 for every θ∈. Related literature Much of the recent contract-theory literature focuses on public-action mechanismdesign models. For instance, Che and Hausch (1999), Hart and Moore (1999), Maskin and Moore (1999), Segal (1999), and Segal and Whinston (2002) have basically the same setup as we do except that their models treat trade actions as public (collapsing together the trade action and the enforcement phase), so they focus on forcing contracts.6In some related papers, the verbal description of the contracting environment identifies individuals who take the trade actions, but the actions are effectively modeled as public due to an implicit restriction to forcing contracts. In some cases, such as with the contribution of Edlin and Reichelstein (1996), simple forcing contracts (or breach remedies) are sufficient to achieve an efficient outcome and so the restriction does not have efficiency consequences.7 Examples of individual-action models in the literature, among others, are the articles of Hart and Moore (1988), MacLeod and Malcomson (1993), and Nöldeke and Schmidt (1995). Also relevant is the work of Myerson (1982,1991), whose mechanism-design analysis nicely distinguishes between inalienable individual and public actions (he uses the term “collective choice problem” to describe public-action models). Most closely related to our work is that of Evans (2006,2008), who emphasizes how efficient outcomes can be achieved by conditioning external enforcement on costly individual actions. Evans (2006) examines general mechanism-design problems; Evans (2008), which we discuss more in the Conclusion, examines contracting problems with specific investment and durable trading opportunities. Related as well is the work of Lyon and Rasmusen (2004), which shares the theme of Watson (2007), and the recent work of Boeckem and Schiller (2008)andEllman (2006).8 6Aghion et al. (1994) is another example. The more recent entries by Roider (2004) and Guriev (2003) have the same basic public-action structure. Demski and Sappington (1991), Nöldeke and Schmidt (1998), and Edlin and Hermalin (2000) examine models with sequential investments in a tradeable asset; in these models, as in Maskin and Tirole (1999), transferring the asset is essentially a public action. 7Stremitzer (forthcoming)elaboratesonEdlin and Reichelstein (1996) by examining the informational requirements of standard breach remedies (specifically, partially verifiable investments). 8Also related are some studies of delegation in principal–agent settings with asymmetric information, where implementable outcomes depend on whether it is the principal or the agent who has the productive action. As Beaudry and Poitevin (1995) show, ex post renegotiation imposes less of a constraint in the case of “indirect revelation” (where the agent has the productive action). Thus, if it is possible to transfer “ownership” of the productive action to the agent, the threat of ex post renegotiation provides one reason for doing this. Theoretical Economics 7 (2012) Contract, renegotiation, and holdup 297 We begin with some intuition regarding the conditions for implementation. Suppose that γis increasing. Our objective is to implement a value function v1that rises with γ, so that player 1’s return on investment closely matches the joint return. The technical conditions for implementation imply upper bounds on the difference v1(θ) −v1(θ) for θ>θ . Observe that there are multiple such conditions involving each state. For example, for three states θ,θ,andθ with θ>θ >θ , there are three conditions: v1(θ) −v1(θ)≤ρ,v1(θ)−v1(θ)≤ρ,andv1(θ) −v1(θ)≤ρ for some numbers ρ,ρ, and ρ. We can call the first and second conditions local,orinside, conditions, whereas the last one is an outside condition. Note that by summing the inside conditions, we obtain a second bound on the difference v1(θ) −v1(θ); this bound is ρ+ρ. It turns out that, for a wide class of trade technologies, the outside condition is tighter than is the sum of the inside conditions; that is, ρ <ρ+ρ.Thismeansthat implementability cannot be characterized by the local conditions alone, and some of these must hold with slack to ensure that the outside conditions are satisfied. As a result, it is not possible to implement a value function v1that rises smoothly and steeply. This is not such a big problem in the divided case, where we want v1to be constant, but recall that in the unified case we want v1to rise with γ. We find that the best way to give player 1 the incentive to invest is to implement a value function with some discrete jumps. We establish conditions under which a dual-option contract optimally performs in this way, as shown in the example. We make several assumptions to structure the analysis. The first gives a set of mild technical restrictions that hold in most applications. We maintain this assumption throughout the rest of the paper. Assumption 2. (a) The sets Aand are compact subsets of Rand contain at least two elements, and u1(·θ) and u2(·θ) are continuous functions of afor every θ∈.Define a≡minA,anda≡max A,θ≡min ,andθ≡max.(b)U(·θ) is strictly quasiconcave for every θ∈. (c) Player 1’s bargaining weight is positive: π1>0. The next assumption is the main economic restriction that we impose hereinafter: that player 1’s payoff is supermodular in the state and trade action. Assumption 3. The function u1is supermodular, meaning that u1(aθ) −u1(aθ) ≥ u1(aθ)−u1(aθ)whenever a≥aand θ≥θ. With this assumption, player 1’s marginal value of increasing his trade action rises weakly with the state. In other words, higher trade actions are weakly more attractive to him as the state increases. An implication is that, for any transfers specified as a function of the trade action, player 1’s preferences satisfy the single-crossing property and he weakly prefers higher actions in higher states. This monotone structure helps us to characterize incentives at date 6. Many interesting applications studied in the literature satisfy these assumptions. For instance, consider a buyer/seller relationship in which ais the number of units of an intermediate good to be transferred from the seller to the buyer. The buyer’s benefit 298 Buzard and Watson Theoretical Economics 7 (2012) of obtaining aunits in state θis B(a θ). The seller’s cost of production and delivery is d(aθ),andweletC(aθ) =−d(aθ). Suppose, as one typically does, that Bis increasing and concave in aand that dis increasing and convex in a.Ifais the buyer’s action (he selects how many units to install, for example), then the buyer is player 1 and so we have u1≡Band u2≡C. If the seller chooses a(she decides how many units to deliver, say), then the seller is player 1 and so we have u1≡Cand u2≡B.Ineithercase,Assumption 2 is satisfied. Assumption 3 adds the weak supermodularity requirement on the payoff of the player who selects a. Our example from the previous section satisfies Assumptions 2 and 3. Note that, in a given application, if u1is submodular, then one can redefine the trade action to be −aand then Assumption 3 is satisfied. Also note that Assumption 3 is trivially satisfied in the case of pure cross investment in which u1does not depend on θ. Our final assumption, which is needed only for the results in this subsection, pertains to the relative supermodularity and investment returns for u1and u2. Assumption 4. (a) The expression π2u1−π1u2is supermodular (π2u1is relatively more supermodular than π1u2). (b) For all θ θwith θ>θ ,π1[u2(aθ) −u2(aθ)]≥ π2[u1(aθ) −u1(aθ)]. Assumption 4 is clearly restrictive, limiting the class of trade technologies that we evaluate here, but it facilitates the identification of an optimal contract. Part (a) of the assumption contributes to a tight characterization of optimal punishments in the general mechanism-design exercise. A sufficient condition for Assumption 4(a)isthatu2 is submodular. Appendix B gives an alternative assumption—on the joint value at extreme trade actions—that can be used in place of Assumption 4(a). Assumption 4(b) requires that the cross-investment component is weakly larger than the own-investment component at the highest trade action. It is easy to check that our example satisfies Assumption 4. These assumptions give us the following general result about implementation using the dual-option contract introduced in the example. Theorem 2. Consider any contractual relationship that satisfies Assumptions 2–4.For any number β, define value function vβby vβ 1(θ) ≡u1(aθ) for θ<β u1(aθ) +π1R(aθ) for θ≥β and vβ 2≡γ−vβ 1for all θ∈.Thenvβis implemented by a dual-option contract in which (i) at date 4, player 2 sends a report ˆ θof the state; and (ii) if the report is at least β,then player 1 is forced to select a=aat date 6, and otherwise player 1 is forced to choose between a∗(ˆ θ) and a. The proof of Theorem 2 is provided in Appendix B, along with the proofs of Propositions 2and 3below. Note that we have not used Assumption 1 here. Theoretical Economics 7 (2012) Contract, renegotiation, and holdup 299 We next show that value function vβachieves the best possible investment incentives in the deterministic case where player 1’s investment choice is to directly select θ. Thus, in this setting of unified investment and trade actions, the dual-option contract is optimal. Recall that we normalize so that the cost of investment is θ,andthusplayer1 selects θto maximize v1(θ) −θ. The efficient choice θ∗maximizes γ(θ) −θ. Let us say that investment θis supported by a contract if there is an implementable value function vsuch that θsolves maxθ∈v 1(θ) −θ.CallθBthe best achievable investment level if it maximizes γ(θ)−θamong all supportable θ. Proposition 2. Under Assumptions 2–4, and in the deterministic and unified case in which player 1 has both the investment and trade actions, the best achievable investment level θBis supported by value function vθBas defined in Theorem 2 (that is, setting β≡θB). Our next result gives conditions under which the efficient investment level can be supported; that is, when the best achievable investment level θBcoincides with the efficient investment level θ∗. Proposition 3. Suppose Assumptions 2–4hold. The efficient level of investment θ∗is supported in the deterministic and unified case if and only if u1(aθ∗)+π1R(aθ∗)−θ∗≥u1(aθ) −θ(2) for all θ<θ ∗. The condition from Proposition 3 ensures that we can induce a large enough discontinuity in the value function at the efficient state so that player 1 maximizes his gain net of investment cost at θ∗. If this condition fails, efficiency cannot be attained in the unified case. To summarize the results of this section, we have conditions under which player 2 can be made the residual claimant with respect to the investment action, which solves the contracting problem (yielding efficient investment and trade) in the divided case. We learn that it is generally not possible to make player 1 the residual claimant, given that player 1 has the trade action. As a result, the efficient outcome is typically not attainable in the case of unified investment and trade actions. However, for the class of trade technologies that satisfy Assumptions 2–4and for the case of a deterministic state, we are able to characterize an optimal contract and provide conditions under which the efficient outcome can be achieved. The results in this section are most pronounced when applied to settings of significant cross investment, where the optimal forcing contract is null and leads to an inefficient outcome. In the divided case, the nonforcing dual-option contract induces efficient investment and trade. In the unified case, a dual-option contract can outperform the null contract and sometimes induces the efficient outcome. We continue this theme in Section 5 by simply asking whether, in general, nonforcing contracts implement a wider range of value functions than do forcing contracts. 300 Buzard and Watson Theoretical Economics 7 (2012) 4. Implementable value functions This section summarizes how to calculate implementable value functions in general. Much of the analysis here repeats material in Watson (2007), so we keep this text brief and ask the reader to see Watson (2007) for more details. The culmination of the basic analysis here are some simple characterization results from Watson (2007), which we build on in the subsequent section. In previous sections we assumed that the players can freely renegotiate at dates 3 and 5, but now we also consider the case in which renegotiation is possible at date 3 only (the interim phase). We let VEPF be the set of implementable value functions from date 3 for the case of ex post renegotiation and with the restriction to forcing contracts. We let VEP be the corresponding set for the case of ex post renegotiation and no contractual restrictions. Further, we let VIbe the set of implementable value functions for the case in which renegotiation can occur only at date 3.15 We can characterize the implementable value functions by backward induction, starting with date 6 where player 1 selects the trade action. State-contingent values from date 6 To calculate the value functions that are supported from date 6, we can ignore the payoff-irrelevant messages sent earlier (or equivalently, fix a message profile from date 4) and simply write the externally enforced transfer function as ˆ y:A→R2.That is, ˆ ygives the monetary transfer as a function of player 1’s trade action. Given the state θ,ˆ ydefines a trading game in which player 1 selects an action a∈A and the payoff vector is then u(aθ) +ˆ y(a). Focusing on pure strategies, we let ˆ a(θ) denote the action chosen by player 1 in state θ. This specification is rational for player 1 if, for every θ∈,ˆ amaximizes u1(aθ) +ˆ y1(a) by choice of a. The state-contingent payoff vector from date 6 is then given by the outcome function w:→R2defined by w(θ) ≡u(ˆ a(θ)θ) +ˆ y(ˆ a(θ)) (3) Let Wdenote the set of supportable outcome functions. That is, w∈Wif and only if there are functions ˆ yand ˆ asuch that ˆ ais rational for player 1 and, for every θ∈, (3) holds. Furthermore, let WFbe the subset of outcomes that can be supported using forcing contracts. It is easy to see that w∈WFif and only if there is a trade action ˆ aand a transfer vector ˆ tsuch that w(θ) =u(ˆ aθ) +ˆ tfor all θ∈. We can compare individualaction and public-action models by determining whether the restriction to forcing contracts implies a significant constraint on the set of implementable value functions. State-contingent values from date 5 We next step back to date 5. If there is no opportunity for ex post renegotiation, then nothing happens at date 5 and so Wand WFare the supported state-contingent value 15In the case of only interim renegotiation, a restriction to forcing contracts does not affect the implementable set. Theoretical Economics 7 (2012) Contract, renegotiation, and holdup 301 sets from the start of date 5 as well. On the other hand, if ex post renegotiation is allowed, then at date 5 the players have an opportunity to discard their originally specified contract yand replace it with another, y. By picking a new contract y, the players are effectively choosing a new outcome function win place of the function wthat would have resulted from the original contract y. The players can freely select wfrom the set Wor the set WF, depending on whether they are restricted to forcing contracts. The players divide the renegotiation surplus according to the fixed bargaining weights π1and π2. Dividing the surplus in this way is feasible because Wand WFare closed under constant transfers. Clearly, we have γ(θ) =maxw∈WF[w1(θ) +w2(θ)]because the trade action that solves the maximization problem in (1) can be specified in a forcing contract to yield the desired outcome. The renegotiation surplus is r(wθ) ≡γ(θ) −w1(θ) −w2(θ) The bargaining solution implies that the players settle on a new outcome in which the payoff vector in state θis w(θ) +πr(wθ). We define an ex post renegotiation outcome to be the state-contingent payoff vector that results when, in every state, the players renegotiate from a fixed outcome in W. That is, a value function zis an ex post renegotiation outcome if and only if there is an outcome w∈Wsuch that z(θ) =w(θ) +πr(wθ) for every θ∈.LetZdenote the set of ex post renegotiation outcomes.16 If trade actions are treated as public (and so attention is limited to forcing contracts), then the set of ex post renegotiation outcomes contains only the value functions of the form z=w+πr(w·)with the constraint that w∈WF.Let ZFdenote the set of ex post renegotiation outcomes under forcing contracts. Although the terminology is a bit loose, we refer to functions in Zand ZF, in addition to functions in Wand WF, simply as “outcomes.” State-contingent values from dates 4 and 3 Analysis of contract selection and incentives at date 4 can be viewed as a standard mechanism-design problem. The players’ contract is equivalent to a mechanism that maps messages sent at date 4 to outcomes induced in the trade and enforcement phase (possibly renegotiated at date 5). The revelation principle applies, so we can restrict attention to direct-revelation mechanisms defined by (i) a message space M≡2and (ii) a function that maps 2to the relevant outcome set that gives the state-contingent value functions from the start of date 5. The outcome set is either W,WF,Z,orZF,depending on whether ex post renegotiation and/or nonforcing contracts are allowed. We concentrate on Nash equilibria of the mechanism in which the parties report truthfully in each state.17 Let us write ψθ1θ2for the outcome that the mechanism prescribes when player 1 reports the state to be θ1and player 2 reports the state to be θ2. Note that, in any given 16All elements of Zare efficient in every state; also, Zand Ware generally not ranked by inclusion. 17The revelation principle usually requires a public randomization device to create lotteries over outcomes (or that the outcome set is a mixture space), but it is not needed here. 302 Buzard and Watson Theoretical Economics 7 (2012) state θ(the actual state that occurred), the mechanism implies a “message game” with strategy space 2and payoffs given by ψθ1θ2(θ) for each strategy profile (θ1θ2).For truthful reporting to be a Nash equilibrium of this game, it must be that ψθθ 1(θ) ≥ψ˜ θθ 1(θ) and ψθθ 2(θ) ≥ψθ˜ θ 2(θ) for all ˜ θ∈. We proceed using standard techniques for mechanism design with transfers, following Watson (2007). The key step is observing that, for any two states θand θ, the outcome specified for the “off-diagonal” message profile (θθ)must be sufficient to simultaneously (i) dissuade player 1 from declaring the state to be θwhen the state is actually θand (ii) discourage player 2 from declaring θin state θ. Thus, we require ψθθ 1(θ) ≥ψθθ 1(θ) and ψθθ 2(θ)≥ψθθ 2(θ) Because the outcome sets are closed under constant transfers, we can choose the outcome to effectively raise or lower ψθθ 1and ψθθ 2while keeping the sum constant. Thus, a sufficient condition for these two inequalities is that the sum of the two holds. Letting ψ≡ψθθ and ψ≡ψθθ, we thus have the following necessary condition for implementing outcome ψin state θand outcome ψin state θ: There exists an outcome ˆ ψsatisfying ψ1(θ)+ψ 2(θ)≥ˆ ψ1(θ)+ˆ ψ2(θ). This condition, applied to all ordered pairs (θθ), is necessary and sufficient for implementation. The sum ˆ ψ1(θ) +ˆ ψ2(θ)is called the punishment value corresponding to the ordered pair (θ θ). The punishment value plays a central role in our analysis. Lower punishment values imply a greater set of implementable outcomes. Interim renegotiation has the effect of requiring each “on-diagonal” outcome to be efficient in the relevant state; that is, for each θ, we need ψθθ to be efficient in this state. In the case of ex post renegotiation, allowing interim renegotiation entails no further constraint because every outcome in Zis efficient in every state. It is also the case that without ex post renegotiation, Wand WFyield the same set of implementable value functions from date 3. Therefore, we have three settings to compare: unrestricted contracts with ex post renegotiation, forcing contracts (public actions) with ex post renegotiation, and forcing contracts with interim (but not ex post) renegotiation. Call a value function vefficient if v1(θ) +v2(θ) =γ(θ) for every θ∈. The following results summarize the characterization of VEP,VEPF,andVIand provide a general comparison. Result 1(Watson (2007)). Consider any value function v:→R2. •Implementation with interim renegotiation. Value function vis an element of VIif and only if vis efficient and, for every pair of states θand θ, there is an outcome ˆ w∈WFsuch that v1(θ) +v2(θ)≥ˆ w1(θ) +ˆ w2(θ). •Implementation with ex post renegotiation. Value function vis an element of VEP if and only if vis efficient and, for every pair of states θand θ, there is an outcome ˆ z∈Zsuch that v1(θ) +v2(θ)≥ˆ z1(θ) +ˆ z2(θ). Theoretical Economics 7 (2012) Contract, renegotiation, and holdup 303 •Implementation with ex post renegotiation and forcing contracts. Value function v is an element of VEPF if and only if vis efficient and, for every pair of states θand θ, there is an outcome ˆ z∈ZFsuch that v1(θ) +v2(θ)≥ˆ z1(θ) +ˆ z2(θ). Furthermore, the sets VEP,VEPF,andVIare closed under constant transfers. Result 2(Watson (2007)). The implementable sets are weakly nested in that VEPF ⊆ VEP ⊆VI. Furthermore, VEPF =VEP if and only if, for every pair of states θ θ∈and every ˆ z∈Z, there is an ex post renegotiation outcome ˜ z∈ZFsuch that ˜ z1(θ) +˜ z2(θ)≤ ˆ z1(θ) +ˆ z2(θ). Likewise, VEP =VIif and only if, for all θ θ∈and every ˆ w∈WF,there is an ex post renegotiation outcome ˆ z∈Zsuch that ˆ z1(θ) +ˆ z2(θ)≤ˆ w1(θ) +ˆ w2(θ).18 To summarize, we have thus far analyzed the players’ behavior at the various dates in the contractual relationship, leading to a simple characterization of implementable value functions from date 3. The characterization is in terms of the minimum punishment values for each pair of states, which yields a way to relate the implementable sets for the cases of interim renegotiation, ex post renegotiation, and ex post renegotiation and forcing contracts. We next turn to investigate the relation more deeply. 5. A robustness result for non-forcing contracts The example from Watson (2007) and our example in Section 2 provide illustrations of VEPF =VEP =VI. Our main objective in this section is to examine the robustness of this conclusion. We consider the wide class of contractual relationships that satisfy Assumptions 2,3,and5(which follows). Assumption 5. There exist states θ1θ2∈such that θ1>θ 2,a∗(θ1)>a,a∗(θ2)<aand either U(aθ2)<U(aθ2)or U(aθ1)>U(aθ1). This is a weak assumption that removes a knife-edge case concerning the relative joint values of the extreme trade actions in the various states. For instance, if has more than two elements and U(aθ)=U(a θ) for some θstrictly between θand θwith interior optimal actions, then Assumption 5 is satisfied. We have the following robustness result. Theorem 3. Consider any contractual relationship that satisfies Assumptions 2,3,and5. The sets of implementable value functions in the cases of unrestricted contracts with ex post renegotiation, forcing contracts with ex post renegotiation, and interim renegotiation are all distinct. That is, VEPF =VEP =VI. 18Watson’s (2005) Lemma 1 provides some of the supporting analysis (which was not explained fully in the relevant proof in Watson 2007). This lemma establishes that, for any given ordered pair of states θand θ and any supportable outcome ψ, there exists an implementable value function vfor which v1(θ) +v2(θ)= ψ1(θ)+ψ2(θ). Because the minimum punishment values exist, in each case we can let ψequal the outcome that attains the minimum. 304 Buzard and Watson Theoretical Economics 7 (2012) The analysis underlying Theorem 3 amounts to characterizing and comparing the minimum punishment values that can be supported for each of the settings of interest. Recall that the punishment value for the ordered pair (θθ)is the value ψ1(θ) +ψ2(θ), where ψis the outcome specified in the message game when player 1 reports the state to be θand player 2 reports the state to be θ. Lower punishment values serve to relax incentive conditions, so to characterize the sets of implementable value functions completely, we must find the minimum punishment values. We let PI,PEP,andPEPF denote the minimum punishment values for the settings of interim renegotiation, ex post renegotiation, and ex post renegotiation and forcing contracts, respectively: PI(θθ)≡min w∈WFw1(θ) +w2(θ) PEP(θθ)≡min ˆ z∈Zˆ z1(θ) +ˆ z2(θ) PEPF(θθ)≡min ˆ z∈ZFˆ z1(θ) +ˆ z2(θ) Our assumptions on the trade technology guarantee that these minima exist. From Result 2, we know that Theorem 3 is equivalent to saying that there exist states θ θ∈such that PI(θθ)<P EP(θθ)and there exist (possibly different) states θ θ∈such that PEP(θ θ)<P EPF(θθ).Thus,toproveTheorem 3, we examine the punishment values achieved by various contractual specifications in the different settings. We develop some elements of the proof in the remainder of this section; Appendix C contains the rest of the analysis. We focus in this section on the relation between VEPF and VEP. The analysis of the relation between VEP and VIis considerably simpler and is wholly contained in Appendix C. We establish PEP <PEPF by comparing the punishment values implied by (i) the outcome in which player 1 is forced to take the trade action that yields the lowest punishment value among forcing contracts, and (ii) a related nonforcing specification in which player 1 is given the incentive to select some action ain state θand a different action a in state θ. We derive conditions under which aand acan be arranged to strictly lower the punishment value for (θθ), relative to the best forcing case. We then find states θ1 and θ2such that the conditions must hold for at least one of the ordered pairs (θ1θ2) and (θ2θ1). To explore the possible outcomes in the cases of ex post renegotiation, consider player 1’s incentives at date 6. For any given transfer function ˆ y, necessary conditions for player 1 to select trade action ain state θand action ain state θare u1(aθ) +ˆ y1(a) ≥u1(aθ)+ˆ y1(a) (4) u1(aθ)+ˆ y1(a)≥u1(aθ)+ˆ y1(a) Transfer function ˆ ycan be specified so that player 1 is harshly punished for selecting any trade action other than aor a. Then, in every state, either aor amaximizes player 1’s payoff from date 6. Thus, we can state the following fact. Theoretical Economics 7 (2012) Contract, renegotiation, and holdup 305 Fact 1. Consider two states θ θ∈and two trade actions a a∈A. Expression (4)is necessary and sufficient for the existence of a transfer function ˆ y:A→R2 0(defined over all trade actions) such that player 1’s optimal trade action in state θis aand player 1’s optimal trade action in state θis a. Summing the inequalities of expression (4), we see that there are values ˆ y(a) ˆ y(a)∈ R2 0that satisfy (4) if and only if u1(aθ) −u1(aθ)≥u1(a θ)−u1(aθ) (5) Assumption 3 then implies the following fact. Fact 2. If θ>θ ,thena≥aimplies inequality (5). If θ<θ ,thena≤aimplies inequality (5). Note that Fact 2 gives sufficient conditions. In the case in which u1(··)is strictly supermodular (replacing weak inequalities in Assumption 3 with strict inequalities), player 1 can be given only the incentive to choose greater trade actions in higher states. For any two states θ θ∈,define E(θθ)≡{(a a)∈A×A|inequality (5)issatisfied} Also, for states θ θ∈and trade actions a a∈Awith (a a)∈E(θθ),define Y(aaθθ)≡{ˆ y:A→R2 0|condition (4)issatisfied} Condition (4), combined with the identity ˆ y1=−ˆ y2, implies the next fact. Fact 3. For any θ θ∈and a a∈A,with(a a)∈E(θθ),wehave min ˆ y∈Y(aaθθ)ˆ y1(a) +ˆ y2(a)=u1(aθ)−u1(a θ) Using the definition of the set W(recall expression (3)), any given w∈Wcan be written in terms of the trade actions and transfers that support it. We have w(θ) =u(ˆ a(θ)θ) +ˆ y(ˆ a(θ)) and w(θ)=u(ˆ a(θ) θ)+ˆ y(ˆ a(θ)) where ˆ agives player 1’s choice of trade action as a function of the state and ˆ yis the transfer function that supports w. For any state ˜ θand trade action ˜ a,defineR(˜ a ˜ θ) to be the renegotiation surplus if, without renegotiation, player 1 would select ˜ a.Thatis,R(˜ a ˜ θ) =U(a∗(˜ θ) ˜ θ) −U(˜ a ˜ θ). Combining the expressions for win the previous paragraph with Fact 1 and the definition of ex post renegotiation outcomes, we obtain Fact 4. 306 Buzard and Watson Theoretical Economics 7 (2012) Fact 4. Consider any two states θ θ∈and let αbe any number. There is an ex post renegotiation outcome z∈Zthat satisfies z1(θ) +z2(θ)=ρif and only if there are trade actions aa∈Aand a transfer function ˆ ysuch that (a a)∈E(θθ),ˆ y∈Y(aaθθ), and ρ=u1(aθ) +ˆ y1(a) +π1R(aθ) +u2(aθ)+ˆ y2(a)+π2R(aθ) (6) In the last line, the first three terms are w1(θ) plus player 1’s share of the renegotiation surplus in state θ, totaling z1(θ). The last three terms are w2(θ)plus player 2’s share of the renegotiation surplus in state θ, totaling z2(θ). Finding the best (minimum) punishment value for states θand θmeans minimizing ˆ z1(θ) +ˆ z2(θ)by choice of ˆ z∈Z. For now, holding fixed the trade actions aand athat player 1 is induced to select in states θand θ, let us minimize the punishment value by choice of ˆ y∈Y(aaθθ).Tothisend,wecanuseFact 3 to substitute for ˆ y1(a) +ˆ y2(a) in expression (6). This yields the punishment value for trade actions aand ain states θ and θ, respectively, written λ(aaθθ)≡u1(aθ)+π1R(a θ) +u2(aθ)+π2R(aθ) (7) Next, we consider the step of minimizing the punishment value by choice of the trade actions aand a, which gives us a useful characterization of PEP(θθ).Assumption 2(a) guarantees that λ(aaθθ)has a minimum. Fact 5. The minimum punishment value in the setting of ex post renegotiation is characterized as PEP(θθ)=min (aa)∈E(θθ)λ(a aθθ) We obtain a similar characterization of the minimal punishment value for the setting in which attention is restricted to forcing contracts. The characterization is exactly as in Fact 5 except with the additional requirement that a=abecause forcing contracts compelthesameactionineverystate. Fact 6. The minimum punishment value for the setting of forcing contracts and ex post renegotiation is characterized as PEPF(θθ)≡min a∈Aλ(aaθθ) Recall that proving Theorem 3 requires us to establish that PEP(θθ)<P EPF(θθ) for some pair of states θ θ∈.Appendix C finishes the analysis by exploring how one can depart from the optimal forcing specification in a way that strictly reduces the value λ(aaθθ). Theoretical Economics 7 (2012) Contract, renegotiation, and holdup 313 Assumption 4. (a) For all θ∈(θθ],U(aθ)≥U(a θ). Lemma 2. Under Assumptions 2,3, and either Assumption 4(a) or 4(a), for any pair of states (θθ) with θ<θ, the optimal punishment involves inducing player 1 to select a∗(θ)in state θand ain state θ. That is, PEP(θθ)=λ(a∗(θ)aθθ). Proof.From(7), the punishment value for (θθ),λ(aaθθ),isgivenby u1(aθ)+π1R(aθ)+u2(a θ) +π2R(aθ) which can be rewritten as u1(aθ)+π1U(a∗(θ)θ)−π1U(aθ)+π1u2(aθ) +π2U(a∗(θ) θ) −π2u1(aθ) The optimal punishment value is obtained by choosing aand ato minimize this objective function under the constraint that a≥a(because of supermodularity of u1). Ignoring the constant terms that do not contain aor a, and substituting π2=(1−π1)and π1u1(aθ) +π1u2(a θ) =π1U(aθ), the objective function becomes −[u1(aθ) −u1(a θ)]+π1U(aθ)−π1U(aθ) (14) The number aaffects only the last term; to minimize it (that is, maximize U(aθ)) without consideration of the constraint a≥a,itisoptimaltoseta=a∗(θ).Fromsupermodularity of u1, the negative bracketed term is minimized by choosing a=a.By Assumption 4(a), the final term is also minimized at a.Thus,λ(aaθθ) attains its lowest value when a=aand a=a∗(θ). To see that the same result holds with Assumption 4(a) in place of Assumption 4(a), observe that if the minimizing values aand asatisfy a>a then it must be that a=aand a=a∗(θ).Thata=a∗(θ)is an implication of strict quasiconcavity of U(·θ),forifa< aand a=a∗(θ), then it must be that a∗(θ)>a, but then raising ato astrictly increases U(aθ). The conclusion that a=afollows from strict quasiconcavity of U(·θ) and from supermodularity of u1, which imply that it is not optimal to set a∈(aa). So we know that either it is optimal to have a=aand a=a∗(θ),ora=ais optimal. In the latter case, Assumption 4(a) implies that a=a=ais best. This is apparent by rearranging terms to show that, by substituting a=ainto expression (14), the expression becomes π1[u2(aθ) −u2(a θ)]−π2[u1(a θ) −u1(aθ)] which is decreasing in a. This yields a contradiction because we can lower ato a∗(θ)to strictly decrease the objective function.  A note about inside and outside constraints on value functions Lemma 2 allows us to easily calculate the lowest possible punishment values for any unilateral deviation from truth-telling, and with this in hand we can begin to evaluate how 314 Buzard and Watson Theoretical Economics 7 (2012) the conditions for implementability come together to constrain the value function. For the unified case, we want to know whether we can implement a value function so that v1 increases at the same rate as does γ. One way to get at this is to examine constraints on v1(θ) −v1(θ)at the margin where θand θare very close, and then chain together these inside conditions to characterize the optimal implementable value function. Unfortunately, there are also outside conditions to examine; they give constraints on v1(θ) −v1(θ)for θand θthat are far apart. We demonstrate that the outside conditions are typically tighter than the sum of the inside conditions, so a triangle inequality fails. Thus, one cannot rely on marginal analysis to calculate bounds on implementable value functions (a cautionary note relative to Segal and Whinston 2002). Consider any three states satisfying θL<θ M<θ H, such that the optimal trade action in state θMis interior so that a∗(θM)∈(aa). Also assume that the assumptions for Lemma 2 hold. Using inequality (13), we have three necessary conditions for implementation: v1(θM)−v1(θL)≤γ(θM)−PEP(θLθM) v1(θH)−v1(θM)≤γ(θH)−PEP(θMθH) v1(θH)−v1(θL)≤γ(θH)−PEP(θLθH) (15) Summing the first two yields v1(θH)−v1(θL)≤γ(θH)−PEP(θMθH)+γ(θM)−PEP(θLθM) (16) We want to know whether (16) is a weakly tighter bound than is (15), which would mean that the inside conditions imply the outside conditions and allow implementability to be characterized by marginal analysis. So we must establish whether the following triangle inequality holds: PEP(θLθH)+γ(θM)≤PEP(θLθM)+PEP(θMθH) Expanding terms using expression (7)andLemma 2,weget u1(aθL)+u2(aθH)+π2R(a θH)+γ(θM) ≤u1(aθL)+u2(aθM)+π2R(a θM)+u1(a θM)+u2(aθH)+π2R(aθH) which simplifies to γ(θM)≤u2(aθM)+π2R(a θM)+u1(a θM) This is equivalent to γ(θM)≤U(a θM)+π2γ(θM)−π2U(a θM) which simplifies to γ(θM)≤U(a θM). This inequality, coupled with Assumption 2(b), requires that a∗(θM)=a, which contradicts what we assumed earlier. Theoretical Economics 7 (2012) Contract, renegotiation, and holdup 315 Proof of Theorem 2 Consider the following contract: In the message phase (date 4), player 2 must declare the state. Let ˆ θdenote player 2’s announcement. If ˆ θ≥β, then player 1 is forced to select aat date 6. Otherwise, player 1 is forced to choose between a∗(ˆ θ) and a.Inthiscase,if player 1 selects action a∗(ˆ θ), then the enforcer compels a transfer of ˆ t=u1(a ˆ θ) −u1(a∗(ˆ θ) ˆ θ)u1(a∗(ˆ θ) ˆ θ) −u1(a ˆ θ)(17) and if player 1 selects action a, then the transfer is t=(00). The forcing arrangement is achieved by specifying a transfer of (−τ τ) if player 1 picks any other trade action, where τis set large enough to keep him from doing so. We show that this contract implements the value function vβdefined in the text. First note that, in any state θ, if at date 4 player 2 declares the state to be ˆ θ∈[θβ) and the players do not renegotiate at date 5, then player 1 obtains at least u1(a θ) because he has the option of choosing awith no transfer. Further, if player 1 selects a∗(ˆ θ), then (from expression (17)) he gets u1(a∗(ˆ θ)θ) +u1(a ˆ θ) −u1(a∗(ˆ θ) ˆ θ) whereas he gets u1(a θ) by choosing a. The latter payoff weakly exceeds the former if and only if u1(aθ) −u1(a∗(ˆ θ)θ) ≥u1(a ˆ θ) −u1(a∗(ˆ θ) ˆ θ) From the supermodularity of u1and given that a≥a∗(ˆ θ), we know that it is rational for player 1 to choose ainthecaseof ˆ θ<θand it is rational for player 1 to choose a∗(ˆ θ) in the case of ˆ θ>θ. Player 1 is indifferent if ˆ θ=θ. We can therefore prescribe the following behavior, for any state θ. •If player 2 declares ˆ θ∈[θ β), then, absent renegotiation, player 1 chooses a∗(ˆ θ) at date 6. •For any other message (either ˆ θ≥βor ˆ θ<θ), absent renegotiation, player 1 chooses a. It is clear that, given player 1’s behavior just specified, it is optimal for player 2 to report truthfully at date 4. For instance, in a state θ<β, if player 2 reports honestly, then there is no renegotiation and she gets γ(θ) −u1(a θ). If she reports a different state, then player 1 is expected to take an ex post inefficient trade action that gives him at least u1(aθ), so player 2 fares less well. With the specified behavior for the players, in any state θ<βthere is no renegotiation and player 1 obtains the payoff u1(aθ).Inanystateθ≥β, the players renegotiate away from the anticipated action of aand player 1 gets u1(a θ) +π1R(aθ). Thus, value function vβis implemented.  316 Buzard and Watson Theoretical Economics 7 (2012) Proof of Proposition 2 Consider any two states θand θwith θ<θ. Using Lemma 2 for the pairing (θθ),which shows that the minimum punishment value for such a pair involves inducing player 1 to select a∗(θ)in state θand ain state θ,wehavethat PEP(θθ)=u1(a θ)+u2(aθ) +π2R(a θ) We use this equality to substitute for PEP in the necessary condition (13). Rearranging terms yields the following upper bound on the value difference between the two states for player 1: v1(θ) −v1(θ)≤π1R(a θ) +u1(a θ) −u1(a θ) (18) Consider any contract that supports the best achievable investment level θBand let vBbe the implemented value function. By definition of θB, we know that θBsolves player 1’s investment problem of maximizing vB 1(θ) −θ. Define β≡θBand consider the value function vβdefined in Theorem 2.Toseethat vβsupports θB(that is, θBmaximizes vβ 1(θ) −θ), first observe that for any state θ<θ B, we have vβ 1(θB)−vβ 1(θ) =u1(a θB)+π1R(a θB)−u1(a θ) so vβmeets the upper bound on player 1’s payoff difference between θand θB,as identified in inequality (18). Since vBalso must satisfy the bound (18), we conclude that vβ 1(θB)−vβ 1(θ) ≥vB 1(θB)−vB 1(θ) for all θ<θ B. This implies that the maximizer of vβ 1(θ) −θmust be no less than θB. The final step is to consider the implications of θBnot maximizing vβ 1(θ) −θ.In this case, let ˜ θ>θ Bdenote an investment that player 1 strictly prefers. We then have vβ 1(˜ θ) −˜ θ>v β 1(θB)−θB. Plugging in the implemented values of vβ 1, this is equivalent to u1(a ˜ θ) +π1R(a ˜ θ) −˜ θ>u 1(aθB)+π1R(a θB)−θB Rearranging terms, we see that this is equivalent to π1γ(˜ θ) −˜ θ−[π1γ(θB)−θB]>π 2U(aθB)−π2U(a ˜ θ) +u2(a ˜ θ) −u2(aθB) It is not difficult to verify that Assumption 4(b) implies that the expression on the right side is weakly positive and thus the expression on the left side is strictly positive. This further implies that γ( ˜ θ) > γ(θB), which means that γ(˜ θ) −˜ θ−[γ(θB)−θB]>0 contradicting that θBis the best achievable investment level. Thus, we know that θB maximizes vβ 1(θ) −θ. Theoretical Economics 7 (2012) Contract, renegotiation, and holdup 317 Proof of Proposition 3 If θ∗is supported, then Proposition 2 implies that it is supported by the value function vθ∗from Theorem 2. We then know that θ∗maximizes vθ∗ 1(θ) −θby choice of θ,which implies that inequality (2) holds for all θ<θ ∗. Thus, the condition of the proposition is necessary. Sufficiency requires not only that inequality (2) hold for all θ<θ ∗,butalso that player 1 prefer not to invest ˜ θ>θ ∗when vθ∗is the implemented value function. This follows from the argument in the final paragraph of the proof of Proposition 2, replacing θBwith θ∗and “best achievable” with “efficient.”  Appendix C: Proof of Theorem 3 In this appendix, we complete the proof of Theorem 3. We start with the comparison of VEPF and VEP and then provide the analysis for the comparison of VEP and VI. Completion of the proof that VEPF =VEP We pick up from the analysis at the end of Section 5. Consider a pair of states θ1and θ2 that satisfies Assumption 5. That is, we have θ1>θ 2and either U(aθ2)<U(aθ2)or U(aθ1)>U(aθ1).Letb1denote a solution to the forcing-contract problem min a∈Aλ(aaθ1θ2) and let b2denote a solution to the forcing-contract problem min a∈Aλ(aaθ2θ1) It is easy to show that b1≥a∗(θ1)>aand b2≤a∗(θ2)<afollow from Assumptions 2(b) and 5. We use these facts below. We demonstrate that either PEP(θ1θ2)<P EPF(θ1θ2) or PEP(θ2θ1)<PEPF(θ2θ1)or both, which implies that VEPF =VEP. Let us evaluate the minimum punishment value that corresponds to the ordered pair of states (θ1θ2). Specifically, compare the optimal forcing-contract punishment (forcing player 1 to select b1in both states) with a nonforcing specification in which player 1 is induced to select b1in state θ1and ain state θ2. This is a valid nonforcing contractual specification because, by Fact 2,θ1>θ 2and b1>aimply (b1a)∈E(θ1θ2). If VEP =VEPF, then it must be that λ(b1b1θ1θ2)≤λ(b1aθ1θ2). Applying the definition of λ,thisis u1(b1θ1)+π1R(b1θ1)+u2(b1θ2)+π2R(b1θ2) ≤u1(aθ1)+π1R(b1θ1)+u2(aθ2)+π2R(aθ2) Canceling the second term on each side and using the definition of R,wearriveat u1(b1θ1)+u2(b1θ2)−π2U(b1θ2)≤u1(aθ1)+u2(aθ2)−π2U(aθ2) 318 Buzard and Watson Theoretical Economics 7 (2012) Substituting u2(·θ2)=U(·θ2)−u1(·θ2)on both sides, we have u1(b1θ1)+U(b1θ2)−u1(b1θ2)−π2U(b1θ2) ≤u1(aθ1)+U(aθ2)−u1(aθ2)−π2U(aθ2) Finally, rearranging this expression a bit and using π1+π2=1, we conclude that λ(b1b1θ1θ2)≤λ(b1aθ1θ2)is equivalent to u1(b1θ1)−u1(aθ1)−[u1(b1θ2)−u1(aθ2)]≤π1[U(aθ2)−U(b1θ2)](23) Similarly, ordering states θ1and θ2in the opposite way, we compare the optimal forcing-contract punishment (forcing player 1 to select b2in both states) with a nonforcing specification in which player 1 is induced to select b2in state θ2and ain state θ1. Note that θ2<θ 1and b2< a imply (b2a) ∈E(θ2θ1).IfVEP =VEPF,thenitmust be that λ(b2b2θ2θ1)≤λ(b2aθ2θ1), which similar algebraic manipulation reveals to be equivalent to u1(aθ1)−u1(b2θ1)−[u1(aθ2)−u1(b2θ2)]≤π1[U(a θ1)−U(b2θ1)](24) The foregoing analysis shows that if VEPF =VEP, then expressions (23)and(24) hold. Assumption 3 then implies that the left sides of these inequalities are nonnegative, which implies U(aθ2)≥U(b1θ2)and U(a θ1)≥U(b2θ1) Using Assumption 2(b), and that b1>aand b2< a, we obtain the following fact. Fact 7. If VEPF =VEP, then U(aθ2)≥U(a θ2)and U(a θ1)≥U(aθ1). Assumption 5 and the contrapositive of Fact 7 provide the contradiction that proves VEPF =VEP. Proof that VEP =VI We next prove the claim about the relation between VIand VEP. Since forcing contracts are sufficient to construct VI, we can state the following fact. Fact 8. The minimum punishment value in the setting of interim renegotiation is characterized as PI(θθ)=min a∈Au1(aθ)+u2(aθ) Remember that, by Result 2,VI=VEP if and only if PEP(θ θ)=PI(θθ)for all θ θ∈. We can again compare the minimization problems to determine if this is the case. Theoretical Economics 7 (2012) Contract, renegotiation, and holdup 319 Take θ1and θ2as satisfying Assumption 5. Consider any solution to the minimization problem that defines PEP(θ1θ2)and denote it (b b).Thatis,(b b)solves min (aa)∈E(θ1θ2) u1(aθ1)+π1R(a θ1)+u2(aθ2)+π2R(aθ2) Then PEP(θ1θ2)=PI(θ1θ2)is equivalent to u1(bθ1)+π1R(b θ1)+u2(bθ2)+π2R(bθ2)=min a∈Au1(aθ1)+u2(aθ2) Because R(··)≥0,weseethatPEP(θ1θ2)=PI(θ1θ2)only if bsolves the minimization problem on the right side of the above equation and also R(b θ1)=R(bθ2)=0. By Assumption 2(b), R(bθ2)=0if and only if b=a∗(θ2). Combining this with the requirement that bmust minimize u1(·θ1)+u2(·θ2),wederivethat u1(a∗(θ2) θ1)+u2(a∗(θ2)θ2)≤u1(aθ1)+u2(aθ2) for all a. In particular, the following inequality must hold: u1(a∗(θ2) θ1)+u2(a∗(θ2)θ2)≤u1(aθ1)+u2(aθ2) Using the identity u2=U−u1and rearranging terms, we see that this is equivalent to u1(a∗(θ2) θ1)−u1(aθ1)−u1(a∗(θ2)θ2)−u1(aθ2) ≤U(aθ2)−U(a∗(θ2) θ2) (25) Similarly, ordering states θ1and θ2in the opposite way, it is necessary that a∗(θ1) must solve PI(θ2θ1)so that PEP(θ2θ1)=PI(θ2θ1). In particular, we must have u1(a∗(θ1) θ2)+u2(a∗(θ1)θ1)≤u1(a θ2)+u2(a θ1) This inequality is equivalent to u1(aθ1)−u1(a∗(θ1) θ1)−u1(aθ2)−u1(a∗(θ1) θ2) ≤U(aθ1)−U(a∗(θ1)θ1) (26) By Assumption 3, the left sides of expressions (25)and(26) must be nonnegative, which implies both U(aθ2)≥U(a∗(θ2) θ2)and U(a θ1)≥U(a∗(θ1) θ1).FromAssumption 2(b), we see that this is only possible if a=a∗(θ2)and a=a∗(θ1).Ifthisis the case, Assumption 2(b) also implies that U(aθ2)≥U(a θ2)and U(a θ1)≥U(aθ1). Thus we obtain our last fact. Fact 9. If VI=VEP, then U(aθ2)≥U(a θ2)and U(a θ1)≥U(aθ1). The contrapositive of Fact 9 combined with Assumption 5 provides the contradiction that proves VI=VEP. 320 Buzard and Watson Theoretical Economics 7 (2012) References Aghion, Philippe, Mathias Dewatripont, and Patrick Rey (1994), “Renegotiation design with unverifiable information.” Econometrica, 62, 257–282. 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