Long Term E cient Contracting with Private Information

Transcription

1 Long Term E cient Contracting with Private Information Tracy R.Lewis Alan Schwartz y March 2010 Abstract This paper explores the possibility for e cient long term contracts for traders with changing and privately known incentives for exchange. We suggest a negociation process enabling the parties to adapt the default rules of exchange to changes in their preferences for trade. The selection of control rights and default options is delegated to the traders themselves who are collectively best informed as to what investment and exchanges are e cient. The paper demonstrates how contracts with exible and endogenous default options are tailored to maximize the gains from exchange. Applications of our ndings for contract theory and the design of commercial contracts are discussed. 1 Introduction. Contracts are intended to support e cient relationship investment and expost exchange. Of the numerous obstacles there are to e cient trade, none are thought to be more troublesome than trading with privately informed parties with changing preferences for exchange. 1 The incentives for making relation enhancing investments are reduced when the motives of traders are unknown and their actions can not be observed. Contract theory predicts that the costs of monitoring investments and bargaining for exchange may Fuqua School of Business, Duke University y Yale Law School 1 See Williamson (1979), Akerlof (1979) and Stiglitz (2002) 1

2 grow so large as to preclude e cient contracting among privately informed parties. This can arise even in environments most favorable to contracting where agreements are complete and enforceable. 2 The contract models that make this important prediction all share this feature. In the absence of an agreement the contract is governed by a rigid default rule or property right that typically cedes control of the tangible assets to one party or another. For example Coase s (1960) theory of bargaining, which was the rst and is arguably the most in uential contacting model, asserts that either the buyer or the seller have the exclusive right to property as a default position from which the parties can renegotiate. 3 This paper proposes an alternative process for assigning control and default options. We explore what possibilities there are for long term e cient contracting with private information when default rules are endogenously determined. We develop a model of complete contracting that enables the parties to negotiate the default rule and rights as part of the contract. The default rule is renegotiated at the end of each period as the parties receive new information about their preferences for continuing or terminating the agreement. For instance, this process allows for default rights to adjust in favor of the buyer or the seller in response to new requests or demands the parties may report. Control rights, like commodities and services, may be exchanged to the mutual bene t of parties. This model is then employed to predict when long term contracts can support e cient relation speci c investment and expost exchange in the most di cult environments where private parties are continually receiving private information about their preferences and options for contracting in the future. Our analysis reveals new insights on the role of legal entitlements and control rights as well as the bene ts of delegating decisions to privately informed agents in the design of long term contracts. 1.1 Recent Developments Before summarizing our ndings and the outline for this paper, we provide a brief overview of recent developments in the theory of contracting with private information to put our analysis in perspective. Modern contract 2 See Akerlof ( 1979) and Myerson and Satterthwaite (1983) 3 See Merges (1996) for another view on the importance of strong property rights in contracting 2

3 theory begins with Coases (1960) celebrated analysis of "The Problem of Social Cost", that has come to be known as the, The Coase Theorem: To achieve e cient exchange, one need only assign unambiguous rights of control to one individual and allow the parties to bargain. The Coase Theorem is virtually a tautology as it follows that rational self-interested parties will exchange to their mutual bene t when left unobstructed. Furthermore it may fail to hold when parties are asymmetrically informed 4. Nonetheless the Theorem it provides a useful guide to e cient contracting when there is complete information, or when only one of the party s is asymmetrically informed. A short while after the Coase Theorem came the important discovery by Vickrey(1961), Clark (1971 ) and Groves (1973) extending the theory of e cient contracting to the private information setting. VCG Mechanisms: E cient trade among privately informed traders is only possible if each trader pays or receives the value lost or gained by the other traders in the exchange. For instance this requires that in bilateral exchange, the buyer pay the seller s cost of supply and that the seller receive payment equal to the buyers value. In e ect each party to the exchange becomes the residual claimant of the net surplus created by the trade. Following on the heels of VCG were the disappointing disclosures of d Aspremont and Varet (1979) and Myerson and Satterthwaite (1983) that the VCG mechanism could not be implemented by many standard form contracts. Impossibility of E cient Exchange with Budget Balancing and Voluntary Participation. Private contracting requires the parties consent and voluntary participation. It also requires a balance of transfers between the parties so the contract is self nancing. Neither voluntary participation or budget balance is ensured under VCG when property rights are vested with one trader. The payments that owners demand to reveal their value of trade exceeds the potential gains from trade. E cient exchange requires subsidization from a third party, like the government, which is typically not possible in private agreements. 4 See Farrell (1987). 3

4 In an attempt to restore e cient contracting, Cramton, Gibbons and Klemperer CGK(1987) demonstrate how privately informed members may e ciently dissolve a partnership. 5 CGK: Possibility of E cient Exchange with shared ownership of assets. Partnerships with share ownership may be e ciently dissolved. Members who are uncertain whether to be a buyer or seller of assets demand less to reveal their value of trade. Consequently it is possible to restore e cient exchange along with voluntary participation and balanced transfers when ownership is properly distributed to traders. Default rights play an important role in contracting according to the theories we ve reviewed about. Yet this theory typically views default options as exogenous or determined by events prior to contracting. 6 When viewed this way, the success of the contract depends on whether the default happens to be set correctly in the beginning and whether the parties incentives to bargain over time remain aligned. In reality, however, as the traders preference for exchange change with time, their incentives to perform under the original contract may become misaligned. In this case default options must be adjusted if the parties are to continue supporting the contract. This calls for a bargaining processes in which the default rights at each stage are negotiated as part of the contract. 1.2 Summary of theoretical results We model a contract as an agreement that governs the parties behavior over the term of their relationship. In period the parties privately observes an estimate of their costs and bene ts of investment and exchange. The parties report this information to each other. Based on the reports, the agreement speci es what investments and exchanges the parties will undertake and the payment each will receive. Most important, the contract is voluntary. The agreement may be permanently voiding by any party in any period. In case of default, the contract provides for rights and rules that govern their behavior 5 See also Ayres and Talley (1995), Edlin and Reichelstein (1996), Lewis and Sappington (1989), Schweizer (2006) and Segal and Whinston (2009) for related results. 6 Some notable exceptions are Ayres (2004), Matuschek (2004) and Kuribko and Lewis (2009) 4

5 in the absence of agreement. For example a default rule may give each party ownership and control of all the property he has previously developed under the contract. In addition, the default rules themselves are renegotiated as part of the contract each period following the exchange of services and transfer of payments. Three main results emerge from our analysis of this model of contracting. First and most important, (a) Existence of E cient long term contracts. Voluntary and self nancing contracts that implement e cient investment and exchange exist These agreements are supported by exible default rules which are optimally adjusted each period in response to previous negotiations and the arrival of new private information. The unique feature of these contracts is that parties are delegated responsibility for the setting the default rights based on their private information. Incentives are provided for each party to negotiate default rules that ensure their continued support of the agreement. (b) Contract simplicity: The agreement is surprisingly simple. Each party receives two transfers. The rst transfer is a payment equal to the surplus created by the other parties to the agreement. This payment ensure each party is the residual claimant of the entire net surplus that the contract creates. Being an owner of the entire surplus endows each party with the right incentives to invest and exchange e ciently. The second transfer is an assessment equal to the additional surplus each party expects to receive from supporting the agreement. These assessments are collected to ensure the contract is self nancing. The default rules that the parties negotiates determine the transfer that each party is assessed to support the agreement. The assessments can not be too large, otherwise parties would void the agreement to seek their outside default option (c) Determination of Default Rights: The default rules are selected to maximize the assessments subject to the constraint that each party prefers to support the agreement rather than quit. The optimal default rule minimizes the aggregate value of the parties outside default options. Hence the parties with assets that are most highly valued outside the agreement are allocated the smallest control rights. This rule maximizes the incentives for parties to remain under contract. 5

6 Section 2 begins with a one period example that illustrates many of our main conclusion. Section 3 discusses implications of the example and extends the analyses to a multiperiod setting. Section 4 concludes with a formal model that summarizes and extends the results suggested in the examples. 2 An Example Suppose that a decision to replace a used machine must be made. There are two parties who care about the decision, the buyer, B who uses the machine and the seller, S; who originally sold the machine. The machine may be in either of two states of repair denoted by x: In state x = n the machine is not functioning. This means there is a probability of n = :00 that the machine can operate. In the other state, x = f; the machine is functioning which means that it operates with probability f = :50: A new machine operates with probability equal to 1:00: Preferences for replacing the machine naturally di er between B and S. S incurs a cost of c to replace the machine. This cost c varies with uniform probability between 0 and 100 depending on how busy S is servicing other customers. B derives a net bene t of (1 x )v from replacing a machine in state x: The bene t of replacement is the increased likelihood of operation, (1 x ) multiplied by v; the value of the machine s output. This value v varies with uniform probability between 0 and 100 depending on B 0 s demand for output. It is e cient to replace the machine if the bene t exceeds the cost, or (1 x ) v c: Otherwise the machine should not be replaced. The gain in surplus from e cient replacement available for B and S to divide is therefore, W x (v; c) = max f(1 x ) v; cg 2.1 An e cient replacement contract with complete information When the original machine is purchased, the parties arrange in advance for the e cient replacement of the machine in the future. They anticipate knowing the state of repair of the machine, x the replacement cost c and the value of production v at the time the machine may be replaced. However at the 6

7 time they negotiate the contract the parties only know the distribution of possible values for x; c and v: The parties want a contract that will implement the e cient replacement. They know this will require that it be individually rational for each party to abide by the contract at the time of replacement. They also realize that payments in exchange for replacement of the machine must nanced by the buyer and seller. The contract must balance the budget, whatever compensation one party receives must be paid for by the other. Of course by now we know that Coase (1960) discovered the very contract that the parties desire. Coase envisioned that one of the parties, say B would initially purchase the right to replace the machine from S for a mutually agreed upon price. Then at the time of replacement after the parties learned the machine s state of repair, x; the value of production v and the costs of replacement c, B would o er to sell his right of replacement to S for an amount c; provided it exceeded B 0 s replacement bene t,(1 x ) v. Otherwise B would demand S replace the machine at no cost. The Coase contract described above does implement the e cient replacement. B will demand the machine be replaced when his bene ts exceed the cost of replacement. Otherwise B will forego replacement if he receives a payment of c that exceeds his bene t. Moreover the contract is individually rational, as neither party is harmed by the agreement. And nally, of course the contract balances the budget, as the payments each party receives sum to zero. Coase s contract is so simple and compelling that it is easy for us loose sight of how it works. The party, in this case, B; who buys the right to replacement, also acquires the claim to the additional surplus W x (v; c) = max ((1 x ) v; c) generated by the e cient replacement. Since B is the chooser who decides whether to replace or not, he will opt for the choice that maximizes total surplus. Coase recognized that it is necessary as well as su cient for the chooser to be the residual claimant to the total surplus if he is to decide e ciently. 7

8 2.2 Contracting between a Privately Informed Buyer and Seller Contracting for e cient exchange is much more di cult when the buyer and seller are privately informed about their values. Suppose B and S privately observe their bene ts and costs before negotiating for the replacement of a machine in state x = n that is known to be non functioning. Unfortunately Coasean bargaining does not work for this setting because there is no individual with complete information to propose and enforce e cient exchange. Instead a contract is needed for delegating the replacement decision to the privately informed buyer and seller. Fortunately such a contract exists that can be implemented by the following two stage process: The explanation of the contract is simpler if we start with second stage of the agreement. Agreement to Choose and Compensate: The agreement begins once B and S observe their valuations. Based on their observation B reports a bene t ~v 2 [0; 100] and S reports a cost ~c 2 [0; 100] : The party reporting the higher value becomes the "chooser" who decides on replacement. The party reporting the lower value is compensated with a payment equal to the chooser s derived surplus. The payments for B and S are therefore p n B (~v; ~c) = 0 if ~v ~c ~c if ~v < ~c ; p n S (~v; ~c) = 0 if ~c ~v ~v if ~c < ~v The e ect of the Agreement is to make each party the residual claimant of the total exchange surplus. Consequence, each party reveals his actual replacement valuation to maximize total surplus To illustrate suppose B 0 s valuation is 50: If B reports ~v = 50 he either receives 50 (if he is the chooser) or ~c (if S is the chooser) whichever amount is larger. In this case B 0 s expected surplus becomes E ~c max(50; ~c) which is exactly the net expected surplus from exchange. Consequently B is best served by truthfully reporting his valuation to maximize total surplus. By the same argument, S also reports truthfully to maximize his surplus. Hence in equilibrium the Agreement implements e cient exchange by delegating the decision to replace to the privately informed buyer and seller. The Choose and Compensation Agreement implements e cient exchange, 8

9 but it does so at a signi cant cost. The parties only reveal their private information if each is the residual claimant to the entire surplus. The parties together must receive payments equal to the expected exchange surplus of E v;c W n (v; c) = 66:67: Somehow these payments must be nanced for the budget to balance. This brings us to the initial Optional Contracting stage of the contract. Optional Contracting: Setting Default Rights Before learning their valuations, the parties agree to default control rights. The default rights are a pair of probabilities ( n B; n S) that sum to one, specifying the likelihood B or S controls the replacement decision, in the default state where there is no agreement. The control rights are strong property rights when either n B = 1 or n S = 1: Otherwise when n B and n B lie strictly between 0 and 1 the rights may be interpreted as probabilistic liability rules. 7 Optional Contracting: Determining Deposits Payments: B and S are then required to make respective deposits of d B ( n B) and d S ( n S) to ensure their participation in the subsequent stage of the contract. The deposits are collected to nance the transfer payments the parties receive from the Choose and Compensate part of the contract. Participation in the agreement is however voluntary. The deposits are returned if either party terminates the agreement to pursue his outside default option. The optional control rights, ( n B; n S) ; and the deposits d B ( n B) and d S ( n S) are crucial in ensuring the participation of B and S in the Agreement. To make the contract self nancing, the parties combined deposits must cover or exceed the expected sum of transfer payments, equal to E v;c W n (v; c) : The amount each party is willing to deposit depends on their control rights. The deposits can not exceed the minimum payments that B and S expect to gain from the contract in excess of their default option, which is calculated to be, d n B ( n B) = min v w (E ~c W n (v w ; ~c) n B (1 n ) v w ) d n S ( n S) = min c w (E ~v W n (~v; c w ) n Sc w ) 7 See Calabreshi and Melamed (1972) 9

10 Strong Default Property Rights To illustrate, imagine for instance, that B is awarded the exclusive default right to choose replacement, as we assumed before in the Coase contract. This would mean that n B = 1 and n S = 0: In this case v w = 100 is the valuation that minimizes B 0 s contract surplus compared to his outside option. The largest deposit that B is willing to make turns out to be d n B (1) = E ~c W n (100; ~c) 1x (100) = E ~c max (100; ~c) 100 = 0 The fact that B has such strong control rights in default explains why d B (1) = 0, as he is unwilling to deposit any positive amount to contract with S: In contrast S; whose minimizing cost occurs at c w = 0 is willing to deposit d n S (0) = E ~v W n (~v; 0) 0 = E ~v max (~v; 0) = 50 The relatively large deposit of d n S (0) = 50 re ects S0 s desire to negotiate away from her outside default option where she has no control. The buyer and seller s combined deposits are d n B (1) + dn S (0) = 50: This sum does cover the expected transfer payments of 66:67 that must be nanced. The budget is not balanced and hence it is not possible to implement the e cient replacement if B is given sole control of the replacement in the default. If S is awarded sole control in default instead of B; we reach the same conclusion, e cient replacement can not be implemented. This illustrates the Myerson and Satterthwaite (1983) nding, noted in the Introduction, that e cient exchange between privately informed agents with strong property rights can not sustained. Deposit Maximizing Default Rights An alternative to either assigning property rights or otherwise, arbitrary rights, is to select default rights that maximize total deposits 8. For this example deposits are maximized by assigning equal control rights where 8 Schweizer (2006) also suggests this method for selecting property right assignments. 10

11 ( n B = :5; n S = :5) : In this instance B and S deposit d n B (:5) = dn S (:5) = 37:50 for a total of 75:00: 9 The deposits easily cover the expected sum of transfer payments E v;c W n (v; c) = 66:67 so that the contract is self nancing. To ensure the payments are balanced for all realizations of (v; c) ; requires one to adjust the transfer payments. This adjustment, which we describe in section 4 is easy to implement. 2.3 Contracting with Private Information Under different Conditions To complete the example let s derive the e cient contract under di erent conditions when the machine to be replaced is known to be functioning. The contracting process will remain the same, of course, however the terms of the contract will be modi ed to account for the di erence in operating status of the machine. When the machine is functioning the bene t to B from replacement is just 5v; since the old machine functions 50 percent of the time. The buyer s production value v is still the same ranging with equal probability between 0 and 100. S 0 s cost of replacement, is still c; and it also ranges between 0 and 100 with equal probability. The transfer payments under the Choose and Compensate Agreement are now modi ed to account for the diminished bene t B derives from replacement, p f B (~v; ~c) = 0 if.:5~v ~c ~c if 5~v < ~c ; p f S (~v; ~c) = 0 if ~c 5~v ~v if ~c < 5~v These are the payments ensuring each of the parties is a residual claimant of the entire replacement surplus. The expected sum of these transfers is now, E v;c max [v; c] = 54:16 The transfers are smaller because the bene ts from replacement are decreased compared to the case where the old machine is non functioning. 9 For example S s deposit is calculated as The calculation for B is the same. d n S (:5) = E v max (v; 50) :25 = 37:50 11

12 In order to nance these transfers we select control rights f B ; f S maximize the sum of the deposits, d f B + df S, that are collected. The depositmaximizing control rights are calculated to be fb = :66; fs = :33 and the corresponding deposits turn out to be d f S (:33) = 25:00 and df B (:66) = 33:33: The sum of these deposits exceed the expected transfers, with d f S (:33) + df B (:67) E v;cw f (v; c) = 58:33 54:16 > 0: Hence, as with the non functioning machine, this contract is self nancing and therefore capable of implementing e cient replacement. Comparing the control rights for the two cases x = f and n reveals that f B > n B: The buyer is allocated greater control when the old machine is functioning as compared to when it is not. B 0 s replacement demand is greater for a non functioning machine. Hence B 0 s control rights vary inversely with his demand for replacement. We show in section 4, this is a general property of optimal control right allocations. The rationale is that controls are chosen to minimize the parties outside options to create the greatest incentives for them to continue contracting. 3 Implications Coase was, and still is, right. To contract e ciently requires that traders have clearly de ned legal rights and that they be allowed to bargain, unobstructed, to their mutual bene t. What does require some reconsideration, however, is the way in which contracts are implemented when traders are privately about their preference for exchange 3.1 Coasean Contracting with Private Information We argue through example that two modi cations to the Coase contract are required to contract e ciently. The rst arises because the information required to exchange e ciently resides separately with the privately informed traders. There is no individual who can make e cient choices alone, as in Coase s world. Instead, decisions must be collectively delegated to the di erent privately informed parties. However, as in Coase s world, e cient decision making requires that those to 12

13 who choose must become the residual claimants of all the surplus they create. This is easily accomplished through transfers between the parties. Each trader is paid the surplus that the other traders generate through their exchange. So for example, the seller who supplies the machine is paid the bene t that the buyer receives when the old machine is replaced. But how is this bene t determined if the buyer is privately informed about his valuation? Fortunately there is a relatively straightforward process for gathering this information. As we explained in our example of the Choose and Compensate agreement, the buyer and seller report their willingness to pay in order to decide. The party bidding the highest amount, in this case the buyer, selects his preferred option, to replace the machine. The loosing party, the seller, is paid the buyer s bene t as compensation. All the information needed to implement these payments is provided by the buyer and seller. The decision to exchange, is in e ect, delegated to the privately informed buyer and seller to make. Given the elegance and simplicity of this process, it s probably no surprise that agreements like these are frequently used in bilateral contracting, market allocations and as default options in the absence of consensual agreement. Perhaps the best known of these mechanisms is the second price auction, that Vickrey(1961) rst noticed. In this case the winner pays the second highest bid as compensation for the reduction in surplus he imposes on the other bidders. Most liability rules are also based on some type of choose and compensate agreement that we introduced in the example. Infact when contracting and bargaining are viewed through the lens of option theory as in Ayres (2004), there are countless examples of formal and informal allocation processes that are implemented in one form or another as a choose and compensate agreement. What makes residual claimancy so attractive is that each party acts ef- ciently because he owns the total surplus. But therein lies the problem there is only one surplus to distribute among a multitude of claimants. It is this imbalance between the promised transfer payments and the surplus available to fund these payments that calls for the second and more di cult modi cation of the Coase contract. In order to nance the transfers, traders must deposit su cient funds in advance of bargaining with each other. Since contracting is voluntary, each trader will only deposit the amount he expects to earn in excess of what he would receive from his outside option. Of course the value of the outside option is determined by the control rights of each party in the absence of 13

14 an agreement. If these rights are mis-speci ed, the contract for e cient exchange can not be implemented. For instance, in the foregoing example we illustrate that strong property rights, that gives one of the traders exclusive control of the assets in default, will typically not work. Strong rights endow the parties with asymmetric bargaining power. The trader who controls the property is not eager to bargain and therefore unwilling to post a deposit to ensure his participation in the exchange. Consequently the contract fails because the transfers between traders required to make them residual claimants can not be internally nanced. 10 This is the cause for what Myerson and Satterthwaite (1983) observe as an ine ciency in bilateral exchange. The solution, as we show by example, is to select control rights that maximize the deposits collected. The optimal control rights are simple to characterize: they are the controls that minimize the traders outside option value. The intuition is that traders with a low outside options are most eager to bargain and are therefore willing to post the largest deposits. These deposits are typically su cient to cover the transfers between the parties, so the contract can be internally nanced and is therefore feasible 11. In principle the setting of control rights to maximize deposits is straightforward. All that is required is for the traders to know the distributions of each other s values, and to anticipate the types of exchanges that they will eventually negotiate. This is su cient information to predict the expected gains from exchange for each party and therefore to select the default controls that will maximize the traders deposits. It seems plausible to ful ll these information requirements in short term contracts during which the private information of the traders does not change. In long term contracts the initial common information about the traders preferences for exchange and the types of exchanges that may arise is certain change with the passage of time. The likelihood that traders acquire additional private information with time, may further constrain the set of e cient contracts that one can implement. 10 Strong property rights are however e ective when the traders have interdependent valuations for the exhange. In that case the amount the parties are willing to deposit is maximized by giving one of the traders exclusvie rights of control See Jehiel and Pauzner( 2006) and Fieseler,Kittsteiner and Moldovanu (2003) 11 Our s is not the rst demonstration that properly set control rights can support ef- cient contractual agreements. Aside from CGK(1987)) who demonstrate this result for equal share partnerships, see Segal and Whinston (2009), and Schweizer (2006). 14

15 3.2 Boundaries of E cient Contracting with Private Information To our knowledge the models and examples demonstrating e cient contracting with private information, all consider short term contracts in which the common information of the parties about their distribution of values and the types of exchanges that arise does not change. 12 For instance in CGK (1987) the distribution of values of the partners is assumed to be known and xed at the time the partnership is dissolved. All of these model rely on common information about the traders exchanges and preferences to determine default control rights and transfer payment that will support e cient exchange. When the contract last for several periods, the assumption that the parties future trade opportunities and distribution of their preferences for exchange is common knowledge becomes untenable. In this case we propose an augmented contract that solicits updated information from the traders each period to set default rights and transfer payments for future exchanges. 3.3 Augmented Contracts with Endogenous Default Controls: Imagine that in our example of machine replacement that the type x = n; f of the machine to be replaced is initially unknown. Moreover assume that B may make an unobserved investment to a ect the reliability of the machine in future use and that only B will privately observe the repair state of when it is replaced. While these are realistic possibilities to model, they all serve to complicate the contract designer s life in two ways. First, we must now rely on B to provide information on the type of machine to be replaced. This information is required to set the terms including default rights and transfer payments for e cient exchange. Second, we rely on B to invest e ciently to until replacement. B 0 s investment incentives are, of course, be a ected by the replacement agreements. We address these complications with an augmented contracting process that is fully explained by the formal model presented section 4, The process relies on information provided by B of his observation of the machine type 12 One exception is Kuribko and Lewis (2009), who describe an e cient multiperiod contract where the traders receive new and presistent private information in each of the period of the contract. 15

16 to design e cient exchange. While we have no guarantee that B reports accurate information we can provide incentives for B to disclose truthfully. Working backwards, we begin with the augmented contract just prior to period 1; after B has invested and learns the working state of the used machine. Realizing this,b is o ered a menu of choices denoted by 8 9 >< >= M = >: hqn ; M n ( n ; p n ; d n )i q f ; M f f ; p f d f {z } {z } >; non functional option functional option B may select the non functional or the functional machine replacement option. When B chooses option x = n or f; he receives payment qb x and the subsequent replacement of the machine is governed by the agreement M x that speci es the contract terms ( x ; p x ; d x ) that we have previously derived in the example. To induce B to select the e cient agreement, requires that he be the residual claimant of the replacement surplus. Suppose U B (M x ) is the expected surplus B expects from the replacement of a type x machine under agreement M x. Then B is o ered a payment of qb x that satis es, q x B + U B (M x ) = E v;c W x (v; c) The payment to B must be nanced by S; so that qs x = surplus from B 0 s selection is equals qx B and S0 s net q x S + U S (M x ) = U B (M x ) E v;c W x (v; c) + U S (M x ) = 0 At this stage B is the sole claimant to all future expected surplus from exchange. Rolling the contract back to the beginning, the parties anticipate B will be the sole claimant to surplus at the time of replacement. Consequently B has the incentive to invest e ciently to maintain the machine because he owns the stream of bene ts the machine produces and the surplus created after the machine is replaced. Therefore it is e cient for B to purchase the exclusive rights from S to maintain and contract for the eventual replacement 16

17 of the machine. Assume W : is the maximized value of the machine that is e ciently maintained and replaced. Then provided B and S negotiate to split the excess surplus from e cient investment and exchange, B will pay S an amount qb 0 that is equal to q 0 B = W E c c 2 Summarizing, the augmented contract works by delegating the choice of control rights and contract terms to the privately informed parties. The missing common information on the types of exchanges that may arise in the future and the distribution of value for exchange is supplied by the contracting parties. Their incentives are to report this information truthfully since they are residual claimants to the surplus that results. In cases where a one party has all the information required to decide, he acquires the sole claim on future surplus by buying out the other party. For instance, the buyer purchases the sole rights to maintain and direct replacement of the machine because he alone is informed about the e cient replacement decision. Most important, though, augmented contracting can substitute for common information that is needed to structure contracts that implement e cient investment and exchange Model of Supply Contracting This section formally models the supply contracting process. The supply contract consists of two stages. In the rst stage the buyer B and the supplier, S contract for the purchase and investment in machine maintenance. In the second stage B and S contract for the servicing and possible replacement of the machine in the future. As with many analyses of long term contracts, it s most instructive to start with the second stage contract and work backwards: 13 By comparison,other dynamic models including Bergemann and Valimaki (2006) and Athey and Segal (2007) analyze e cient VCG types of contracts, where however the property and default rights are xed. These models nd, that although there is e cient investment and exchange of services, the contract are not necessarily individually rational or budget balanced. The inability of parties to adjust control through time with the arrival of new information precludes these contract from be implemented as voluntary budget balance mechanisms. 17

18 4.1 Part 1: The Service and Maintenance Agreement Suppose that a decision to replace a used machine must be made. There are two parties who care about the decision, the buyer, B who purchased and uses the machine and the seller, S; who originally sold the machine and would replace it if required to do so. The current reliability of the machine is denoted by 2 [0; 1] : is the probability the machine is up and running at any time. The only way to restore the reliability of the current machine is to replace it with a new one. The buyer and seller preference s for replacing the machine naturally differ. The seller incurs a cost of c to replace the machine. The cost c varies with uniform probability between 0 and 100 depending on how busy S is servicing other buyers. The buyer derives a bene t from having a machine of type replaced that equals (1 )v; which is the increased probability the machine operates after being replaced, multiplied by v the value of the materials produced by the machine. The value v varies with uniform probability between 0 and 100 depending on the buyers need for materials. The e cient replacement rule is readily derived. Let (v; c) be the e cient decision,where 1 if (1 ) v c (v; c) = 0 if (1 ) v < c It is only e cient to replace the machine if the bene t exceeds the cost, or (1 x ) v c; : otherwise the machine should not be replaced. The total surplus from replacement for the parties to divide is may therefore be written as The Contract W (v; c) = (v; v) (1 ) v + (1 (v; v)) c = max f(1 ) v; cg In order to derive the e cient contract we make some assumptions about the sequence at which information is acquired and contracted upon by the parties which is summarized in Figure 1. Suppose that initially the parties 18

19 observe that the machine reliability is < 1: 14. At this time all that is known about c and v is that each is independently and uniformly distributed over the [0; 100] interval. Following this in period 0 the parties agree to a contract, denoted by M = ( ; (~v; ~c) ; p (~v; ~c) ; d ( )) The contract has the following provisions: Default Option: Beginning of period 1; B privately observes the realization of v and S privately observes the realization of cost, c: At that time either party may choose to default. In case of default the contract is governed by control rights = ( B ; S ) where B can have the machine replaced with probability B ; and S can not replace the machine with probability S = 1 B :15 Note, M is designed to be individually rational, so that in equilibrium both traders prefer to contract rather than default. Nonetheless, the rights are important in determining the parties bargaining position in subsequent stages of the contract. Payment of Deposit: If the parties decide to continue they make deposits d B ( ) and d S ( ) : These deposits ensure the parties participation in the subsequent agreement. Replacement Decision: Based on their observation B reports valuation ~v 2 [0; 100] and S reports a cost ~c 2 [0; 100] : The reports may or may not truthful since they can not be veri ed. Based on the reports, the machine replacement occurs according to rule (~v; ~c) : Following this B and S exchange payments p B (~v; ~c) and p S (~v; ~c) and the contract ends. 14 If = 1 there is no reason to replace the machine. 15 Note when B = 1 or S = 1; this corresponds to Coasian property rights where one party has exclusive control rights. 19

20 Parties observe reliability Negotiation of contract B learns v S leans c Parties consider default option B and S continue and make deposits B reports v S reports c Contract is implemented t=0 t= Response of B and S Timing of Replacement Contract Each party selects a report that to maximize his expected surplus, given his information and the expected report of the other party. B chooses ~v to maximize expected utility U B (v) which is U B (v) max E ~c (p B (~v; ~c) + ~v (~v; ~c) (1 ) v) d B (1) B 0 s surplus consists of his expected transfer payment plus the expected surplus from replacement minus his deposit. S similarly selects a report ~c to maximize expected surplus U S (c) which is U S (c) max E ~v (p S (~v; ~c) + (1 ~c (~v; ~c)) ~c) d S (2) Desired Properties of the Contract We would like the contract to be individually rational, budget balanced and that it be expost e cient. Speci cally this requires the following: We require M to be incentive compatible, so that B and S truthfully report their valuations ~c = c; ~v = v for all c; v 2 [0; 100] (IC) 20

21 The actual replacement rule (~v; ~c) will be e cient provided there is truthful reporting of valuations. M should be individually rational. This means both parties prefer the contract to the default, which requires U B (v) B (1 ) v 0 (IIR) U S (v) S c 0 M should balance the budget so that p B (v; c) + p S (v; c) d B ( ) d S ( ) = 0 (BB) Derivation of the Contract Incentive Compatible Payments: To construct an incentive compatible contract that implements e cient exchange it is necessary that each party become the residual claimant of the entire exchange surplus. Notice, the parties individual contribution to surplus W B (v; c) and W S (v; c) are given by W B (v; c) = (v; c) (1 ) v (3) W S (v; c) = (1 (v; c))c (4) To induce truthtelling it is su cient for the parties expected utilities to take the form U B U S (v) = max E ~c (W B (~v; ~c) + W S (~v; ~c)) ~v d B ( ) (5) (c) = max E ~v (W B (~v; ~c) + W S (~v; ~c)) ~c d S ( ) (6) Substituting for U B (v) and U S (c) into (1) and (2) and solving for the transfers, we obtain the payment that induce truthful reporting are 21

22 U B (v) = E ~cw (v; ~c) d B B (1 ) v (IIR) p S (~v; ~c) = W B (v; c) + W S (~v; ~c) (1 (v; c))c = W B (~v; ~c) (7) p x B (~v; ~c) = W B (v; c) + W S (~v; ~c) (~v; ~c) (1 ) v = W S (~v; ~c) (8) Each party is paid the contribution to surplus provided by their trading partner. For example B receives a payment equal to the surplus generated by S. Hence each party becomes a claimant for the total surplus from trade. Deposits Required for Individually Rationality and Budget Balance To ensure both parties agree to contract requires that the deposit payments are not so large as to induce a party to default. The parties will not default provided their expected surplus from contracting exceeds their surplus from default, or U S (v) = E ~cw (v; ~c) d S S c This implies that the deposits cannot exceed d B ( ) and d S ( ) which are the largest allowable deposits for which the contract is individually rational, where d B ( ) = min v w E ~c W (v w ; ~c) B (1 ) v w (9) d S ( ) = min c w E ~v W (~v; c w ) S c w (10) Note that the minimizing valuations v w and c w are de ned by B = E c (v w ; c) S = E v (1 (v; c w )) 22

23 Hence substituting for the control rights into (9) and (10) we obtain: d B ( ) = E ~c (W (v w ; ~c) (v w ; c) (1 ) v w ) = E ~c p B (v w; ~c) (11) d S ( ) = E ~v (W (~v; c w ) S c w) = E ~v p S (~v; c w) (12) The deposits d B ( ) and d S ( ) represent the amount each party expects to be paid under the agreement, evaluated at their initial default bargaining position To balance the budget we want the largest deposits possible that ensure the contract is individually rational. De ne as the deposit maximizing control rights. is are determined as the solution to, = arg max ((d B ( ) + d S ( ))) (13) s:t: B + x S = 1 Necessary and su cient conditions for the maximization are d d B (E c W (v w ; c) B () (1 ) v w + E v W (v; c w ) (1 B ) c w) = 0 Given the envelop theorem this is equivalent to minimizing the sum of the parties outside options, which requires that (1 ) v w + c w = 0 Summarizing, the optimal default rights that maximize the sum of deposits are characterized by B = E c (v w; c) = prob (c (1 ) v w ) S = E v (1 (v; c w )) = prob (v c w = (1 )) It s worthwhile pausing here to interpret these conditions. First the control rights are selected to minimize the value of the parties s outside option. This is intuitive. To limit the payments, the parties are o ered an outside option that is least attractive, so that they are willing to post higher fees to 23

24 participate in the contract. Second the controls adjust to the parties willingness to trade. For instance, compare B 0 s control when = :5 and when = 0: When the machine is nonfunctional, B is more eager to trade for a replacement than when the product is relatively functional. B default control rights for the two cases are :5 B = :66 and 0 B = :33: Hence, when B 0 s demand for a replacement is greater his control rights are smaller. This is done to place B on the "right side of the market" where he will be most eager to trade. In other words, the control rights are set to maximize the parties willingness to exchange Verifying Payments are Balanced Budget It remains for us to verify that the maximum deposits characterized in (9) (10) are large enough to cover the transfers payments so that E v;c (p B (v; c) + p S (v; c)) d B d S 0 From (7) (8) and (11) (12) note that we calculate that the sum of expected payments as follows. E v;c (p B (v; c) + p S (v; c)) d B d S = (14) E v;c W (v; c) (E ~c W (v w ; ~c) B (1 ) v w ) (E ~v W (~v; c w ) S c w) For the purposes of this calculation assume v; c 2 0; k and v is distributed by F (v) and c is distributed by G (c) with strictly positive densities in the 0; k interval. Then we can write Ev;c W x (v; c) as E c;v W (v; c) = Z k 0 E v W (v; c) dg (c) (15) = E v W k; v = = Z k 0 Z k 0 Z k 0 de v W (v; c) G (c) dc dc de v W (v; c) (1 G (c)) dc + E v W (0; v) dc c F 1 Z k (1 G (c)) dc + (1 F (v)) (1 ) dv 0 24

25 and Z k d B = Ec W S (v w; c) = (1 G (c)) dc + B (1 ) v w (16) c w Z k d S = Ev W B (v; c w) = (1 F (v)) (1 ) dv + c w S (17) v w Substituting (15) (17) into (14) we obtain, E v;c W (v; c) E ~c W S v w; ~c) E ~v W B (~v; c w) (18) Z cw 0 = (1 ) Z k c w c 1 F (1 G (c)) dc 1 c c + F F G (c) dc < 0 Hence we see that the deposits more than cover the transfer payments. In order to insure the budget is balanced at each realization of (v; c) requires that we adjust the transfer payments in the following way. De ne new transfer payments p B (v; c) and p S (v; c) as p B (v; c) = E Cp B (v; c) E v;c (p B (v; c) + p S (v; c)) d B 2 +E vc p S (v; c) E v p S (v; c) d S p S (v; c) = E V p S (v; c) E v;c (p B (v; c) + p S (v; c)) d B 2 +E vc p B (v; c) E cp B (v; c) d S Then it is easy to verify that, p B (v; c) + p S (v; c) d B d S = 0 The sum of the newly de ned payments is zero at each realization (v; c) : 25

26 4.1.6 Summary of Service Contract: The provisions of the type Service Contract are summarized by the following M = ( ; (~v; ~c) ; p (~v; ~c) ; d ( )) where B = E c (v w ; c) ; S = (1 E c (v w ; c)) p B (v; c) = E cw S (v; c) + d B + d S ( ) E v;c W (v; c) 2 +E vc p S (v; c) E vp S (v; c) p S (v; c) = E sw S (v; c) + d B + d S ( ) E v;c W (v; c) 2 +E vc p B (v; c) E cp B (v; c) (v; c) = 1 if (1 ) v c 0 if (1 ) v < c d B = Ec (W x S (v w ; c)) d S = Ev (W x B (v; c w )) U B = E v;cw (v; c) + d S 2 U S = E v;cw (v; c) + d B 2 d B d S 4.2 Part II: Purchase and Investment The foregoing analysis describes the e cient contract, beginning at the stage where the operating state of the machine is known. Now we wish to model the contracting process that proceeds this stage where the initial purchase and investment in machine reliability are undertaken. At the time of purchase the parties bargain over a sequence of two events that are illustrated in Figure 26

27 2. First shortly after the machine is purchased, the buyer is expected to make an investment to maintain the machine in good working order. Second following his investment B uses the machine for a period of time, where upon he assesses how reliable the machine is for future use. Based on this assessment, B selects a service Agreement, M that provides for the eventual replacement of the machine if warranted. The parties negotiations of these events are governed by the contract M 0 ; q 0 ; I ; M ~ ; q (~) with the following provisions Default Options: Initially default options 0 B ; S 0 are assigned and payments (qb 0 ; q0 S ) are exchanged. Prior to each event, either party may decide to void the contract in favor of their default option. Maintenance Investment: The contract calls for B to make an e - cient investment I in reliability to maintain and improve the operating state of the machine. The buyer s actual investment is voluntary and unobservable. The operating status of the machine, described by 2 [0; 1] is the probability the machine will fail in succeeding periods. The surplus derived from a type of machine is E v;c (v + W (v; c)) where E vc v is the expected value the machine produces and E v;c W (v; c) is the additional surplus from e cient replacement of the machine. The likelihood the machine operates atleast percent of the time, is 1 F ( j I) which is increasing in I: The recommended investment I is the investment that maximizes W (I) the surplus from reliability 16 where, W (I) = max I Z 1 0 E v;c (v + W (v; c)) df ( j I) I 16 We assume I exists and is well de ned. 27

28 Selection of M Agreement: After investing, B observes the resulting reliability, ; of the machine just prior to period 1. B selects a replacement agreement and payment from the menu fm ; q B ()g, where M = M ( ; p ; ; d ( )) is the contract governing replacement and q B () is B 0 s payment. Negotiation of contract B Obtains control rights B observe reliability B reports reliability Replacement Contract Implemented B invests t= 1 t=0 t= Response of the Parties Contracting for Investment In this stage of the contract, B is the only party to take an action, (invest) or to observe information (machine reliability). Consequently the contract delegates the decision to invest and to choose the supply agreement to B: U B is B0 s replacement surplus from a type machine, which is U B = max U B ~ M ~ + q B (~) where U B M ~ ; is the surplus from the M ~ agreement, q B (~) is B 0 s payment and d B 0 is a payment depending only on the default right allocation. U S is S0 s corresponding replacement surplus which is U S = U S M ~ + q S (~) 28