Strategic Incomplete Contracts: Theory and Experiments

Transcription

1 Strategic Incomplete Contracts: Theory and Experiments Nisvan Erkal University o Melbourne n.erkal@unimelb.edu.au Brian Roe The Ohio State University roe.30@osu.edu November 25, 2014 Steven Y. Wu Purdue University sywu@purdue.edu Abstract We develop a model o strategic contractual incompleteness that identiies conditions under which principals might omit even costlessly veriiable terms. We then use experiments to test comparative statics predictions o the model. While it is well known that veriiability imperections can limit complete contracting, researchers know less about how the degree o imperection aects endogenous incompleteness, particularly with repeat trading. In our baseline treatment with perect veriiability, subjects overwhelmingly used complete contracts to conduct trades, achieving nearly irst best outcomes. In our partial veriiability treatment with a reduced set o veriiable perormance levels, the results reversed and parties relied heavily on incomplete contracts that omitted even costlessly veriiable terms. However, the eicacy o incomplete contracts in outperorming available complete contracts depends critically on the continuation probability o repeat trading. With a small continuation probability, incomplete contracts did no better than complete contracts while exposing parties to considerable strategic uncertainty. KEYWORDS: incomplete contract, relational contract, endogenous incompleteness, inormal incentives, experimental economics Corresponding author. We thank Tom Wilkening, Tim Cason, Vai-Lam Mui, Klaus Abbink and seminar participants at Purdue University, Monash University, University o Sydney, and the North American Economic Science Association Conerence in Santa Cruz or helpul comments. Any remaining errors are ours. Sharon Raszap and Amy Beth Corman provided excellent research assistance. The authors grateully acknowledge inancial support provided by the USDA-NIFA grant award no , and Tim Cason or providing access to the Vernon Smith Experimental Laboratory at Purdue University. This research is not the result o a or-pay consulting relationship.

2 1 Introduction In this study, we investigate the question o why contracts used in practice oten appear to be strategically incomplete in that they omit important terms that can be veriied by a third-party at relatively low cost. For instance, Scott (2003) collected inormation on a large number o contracts and ound that a surprising number did not condition perormance on veriiable perormance measures that can be speciied in a contract at low cost. It is well known that contractual incompleteness can lead to hold-up problems and post contractual opportunism; hence, legal scholars and economists have been puzzled as to why real world contracts oten appear to be deliberately incomplete even when there are no explicit costs to including some veriiable terms. To address the above question, we develop a relational contracting model o optimal strategic incompleteness with testable implications and then empirically investigate these implications using economic experiments. While we are not the irst to theoretically investigate incompleteness within a relational contracting ramework, our major contributions are that (1) we identiy conditions under which contract draters may strategically omit even costlessly veriiable contractual terms and (2) we investigate empirically how people actually endogenously structure incomplete contracts. Indeed, many o our experimental subjects chose to omit easily veriiable contract terms in treatments that are theoretically avorable or incomplete contracts. Our model also allows us to identiy conditions in which the use o complete contracts dominate incomplete contracts both in terms o joint surplus and a reduction in strategic uncertainty. We ind that in treatments where complete contracts provide such avorable conditions, subjects overwhelmingly chose to use complete contracts. As a starting point or our analysis, we develop a principal-agent model that 1

3 allows contractual orm to emerge endogenously rom an optimization problem. We show that endogenous incompleteness can emerge even in the absence o explicit contracting costs, bounded rationality and costs o cognition. 1 That is, endogenous incompleteness can be strategic. The key ingredient is that there has to be some imperection in the perormance measurement technology that allows a third-party to veriy perormance. While it is well known that some imperection in perormance measurement is needed or endogenous incompleteness, we identiy precisely the degree o imperection required. To do this, we model the degree o imperection as a spectrum ranging rom an empty contractible set o perormance levels, which is the usual assumption in the relational contracting literature, to a very rich and dense set that includes contractible perormance levels that are arbitrarily close to the irst best level. Only when imperection leads to the absence o contractible perormance levels that are suiciently close (to be made precise in the theoretical section) to irst best do we get strategically incomplete contracts. Intuitively, when contractible perormance levels deviate too ar rom irst best, then the contracting parties preer not to use the veriiable perormance levels in a contract and instead preer incomplete contracts that possibly allow them to achieve non-contractible perormance levels that are closer to irst best. A caveat, however, is that the parties must be able to sel-enorce noncontractible perormance levels, so sel-enorcement (such as via repeat trading) is another important ingredient or endogenous incompleteness. Our model also generates comparative static insights about how exogenous changes in the quality o the third-party enorcement technology and the discount actor (to vary the strength o sel-enorcement) aects optimal contract structure. These com- 1 This is not to say that these actors do not contribute contractual incompleteness; our goal is simply to isolate the importance o enorcement environment in driving endogenous incompleteness. Extensions o our study might examine the incremental impact o adding these additional actors. 2

4 parative statics predictions orm the basis or our experimental design. Our experimental design places very ew restrictions on the contractual choice set available to subjects. The experiment is ramed as a buyer (principal)-seller (agent) relationship where the buyer contracts with a seller to purchase a unit o a good that can vary in quality (perormance variable). Our design can nest popular contracts seen in the literature such as ixed price contracts, which encompass eiciency wage, git-exchange, or repeat purchase mechanisms (e.g. Shapiro and Stiglitz (1984); Akerlo (1982); Klein and Leler (1981)); discretionary bonus contracts (e.g. Levin (2003); MacLeod and Malcomson (1989)); and illusory promises (e.g. Boot et al. (1993)). Ininitely repeated trading in the experiments is implemented using a airly standard random continuation rule. Our main treatment variation has to do with the eicacy o third-party enorcement o quality. In the perect enorcement treatment (E), a third-party (the computer) can perectly enorce quality and price. Thus, it is possible to achieve perect enorcement o contracts. In the partial enorcement treatments (PE), third-party enorcement o quality is imperect. This mimics an environment where, or example, a third-party can determine discrete dierences in quality such as whether a product is deective or not, but cannot detect ine gradations in quality. One can think o this as a technology that allows parties to contract on not deective but not on more reined quality levels that are closer to irst best. In both treatments, subjects are allowed to endogenously choose whether they want to avail themselves o third-party enorcement or whether to leave everything inormal. In addition, or the partial enorcement treatment, we examine two sub-treatments by varying the probability o continuation between 0.5 (expected two period repeated game) and 0.8 (expected ive period repeated game). This sub-treatment allows us to examine the interaction between ormal enorcement and sel-enorcement since an increase in continuation 3

5 probability increases the strength o sel-enorcement. Our main indings are as ollows. First, with no imperections in the third-party enorcement technology, subjects overwhelmingly used complete contracts to conduct transactions (81% o trades). Moreover, nearly hal the trades resulted in irst best levels o quality despite the act that we designed the experiment so that irst-best quality is an interior solution and ensured that it is not an obvious ocal point. This suggests that with suiciently strong third-party enorcement o contracts, trading outcomes can be highly eicient. This result is consistent with canonical contract theory that emphasizes the importance o complete contracts. Second, with imperections in third-party enorcement and a high continuation probability o 0.8, the results reversed. Approximately 65% o such trades involved the use o endogenously incomplete contracts that omit even costlessly variable quality levels in the contractible set. However, when sel-enorcement weakened, as captured by a treatment with a continuation probability o only 0.50, the majority o trades were still conducted using complete contracts. Thus, there is a strong interaction between the third-party enorcement technology and reputation eects in driving strategic incompleteness. These results are among the irst that we are aware o that provide empirical insight into second generation incomplete contracting models discussed by Aghion and Holden (2011). 2 We also empirically analyzed several important determinants o strategic incompleteness. For instance, the choice o contract was primarily driven by the motivation to provide incentives on the intensive margin (ensure higher quality) rather than the extensive margin (ensure seller participation). Moreover, we ound signiicant counter-party risk associated with promised discretionary bonuses that did not sat- 2 In their survey paper, Aghion and Holden (2011) reer to second generation models as those that examine how endogenous incompleteness responds to various determinants, including discount rates and reliance on relational contracts (p.194). 4

6 isy sel-enorcement constraints in incomplete contracts. Bonuses that were too large were not credible. Sellers tended to reject contracts with promised bonuses that were too good to be true and their oresight was correct on average, as principals tended to shirk on paying non-credible bonuses. On the other hand, when promised bonuses were too low to satisy agent s sel-enorcement constraints, agents shirked on quality. 2 Related Literature While the classical literature on incomplete contracts has ocused on bounded rationality/unoreseen contingencies (e.g. Hart (1995)), explicit contracting costs (e.g. Battigalli and Maggi (2002, 2008); Kvaløy and Olsen (2009)), and enorcement costs (See Tirole (1999) or an overview), recent theoretical work has also ocused on how repeat trading can alleviate ineiciencies that arise in one-shot environments (MacLeod (2007); Kvaløy and Olsen (2009)). In their recent survey article on incomplete contracts, Aghion and Holden (2011) point out that second generation models o incomplete contracts tend to ocus on relational contracting and the determinants o endogenous contractual incompleteness. MacLeod (2007) suggests that the way people actually engage in repeat trading is not well understood and may potentially involve more complex contractual mechanisms than the classic eiciency wage orm (Shapiro and Stiglitz (1984)) or repeat purchase mechanisms (Klein and Leler (1981)). While some theoretical progress has been made toward understanding these issues, there is a shortage o empirical studies to complement theoretical developments. Along these lines, our paper makes a contribution to the literature on second generation models o incomplete contracts and adds some empirical evidence to the discussion o how people structure contracts in repeat trading environments. Much o our work is motivated by the theoretical studies by Bernheim and Whin- 5

7 ston (1998), Baker et al. (1994), and Boot et al. (1993). The theory o strategic ambiguity described by Bernheim and Whinston (1998) suggests that, when there are imperections in third-party enorcement o contracts, greater incompleteness may enhance surplus by providing greater discretionary latitude to use inormal incentives. Our study can be seen as an experimental test o strategic ambiguity which is arguably one o the most important theories o incomplete contracting. Similarly, Baker et al. (1994) suggest that i a principal has access to both objective and subjective perormance measures, imperections in the objective measure can cause trading parties to rely on inormal contracts that rely on the subjective measure. While our model does not include two perormance measures, our model does show that imperections over a single perormance measure can lead to similar results. Boot et al. (1993) emphasize the point that, while incomplete contracts that leave parties with more ex post discretion can increase perormance in some situations, discretion also increases the scope or abuse. However, concern or uture reputation may discipline the abuse o discretion so there is interplay between incompleteness and reputation. Our study shows that the use o incomplete contracts, which leaves more ex post discretion, increases with the discount actor. This suggests that, when the uture is more important, incomplete contracts are more likely to emerge. One contribution o our research is that we capture the remarkable insights o Bernheim and Whinston (1998), Boot et al. (1993) and Baker et al. (1994) within a single principal-agent ramework that is suiciently parsimonious to acilitate experimental implementation and empirical testing. 3 Our model precisely identiies important thresholds and interactions between third-party and sel-enorcement that drive strategic incompleness. Finally, our model yields clean comparative statics predictions concerning how vari- 3 Baker et al. (1994) also use a airly parsimonious principal-agent ramework though implementing two perormance measures in the laboratory can be potentially too complicated or many experimental subjects. 6

8 ations in the caliber o the third-party enorcement technology and the strength o sel-enorcement induce strategic incompleteness. Our paper is also related to the work o Kvaløy and Olsen (2009)(KO) who examine endogenous incompleteness within a relational contracting ramework. However, our paper diers rom KO s paper in that we consider contracting parties motivation to omit even costlessly veriiable terms in a contract whereas KO examine the case where a principal can make costly ex ante preparations to improve the probability o third-party veriication. Thus, the two papers examine dierent determinants o endogenous incompleteness in a complementary way with our speciic ocus being why contracting parties seemingly omit even terms that are costless to veriy. There is also an experimental economics literature on incomplete contracts, the most notable being the novel work o Ernst Fehr and colleagues (e.g. Fehr et al. (1997); Fehr and Falk (1999); Fehr and Gächter (2000); Brown et al. (2004); Fehr et al. (2007) among others). However, our objectives are undamentally dierent rom the objectives o these papers as we use standard methodology to study a long-standing question in the contract theory literature concerning why contracts are incomplete in the irst place, whereas these papers examine how recent behavioral theories can alleviate ineiciencies arising rom contractual incompleteness. Nonetheless, we view our paper as complementary to these papers in providing insights into the nature o trading under incomplete contracts. Our paper also has some overlap with the work o Sloo and Sonnemans (2011)(SS) who study the interaction between explicit and relational incentives. While we ocus on enorcement conditions that make contracts endogenously complete or incomplete in the irst place, they ocus on how sub-optimal explicit incentives can support relational contracts or trigger social preerences. Both papers ind that weaker explicit contracts can support stronger relational contracts. The consistency in results holds despite substantial design dierences between the SS 7

9 experiments and ours, which suggests that the inding that weaker explicit incentives support inormal incentives might be airly robust. 4 3 Theoretical Framework 3.1 Model setup We assume that a principal contracts with an agent to produce one unit o a good or which quality is important. The typical assumption in the relational contracting literature is that the key perormance variable (i.e. quality in our case) is mutually observable to the contracting parties but not to outsiders (Bernheim and Whinston, 1998; Levin, 2003; Dixit, 2007; MacLeod, 2007). This precludes the use o ormal contracts that are third-party enorced. However, we relax this assumption by introducing a third-party enorcement technology that varies in precision and is exogenous to the principal and agent. By varying the caliber o the third-party enorcement technology, we can nest both ormal contracting and relational contracting. To keep the analysis rom getting overly complex, we abstract away rom asymmetric inormation problems. Thus, our environment is similar to MacLeod and Malcomson (1989) where the key barrier to contracting is related to enorcement problems. Beore describing the enorcement technology in detail, we irst emphasize an assumption that is important or deining the contractible set or a given enorcement technology. Note that under a contractual agreement, each party has obligations to take actions consistent with the agreement. We can speciy the principal and agent s 4 For example, Sloo and Sonnemans use a reduced orm contracting environment where they impose a restricted set o contracts (trust games) on subjects. Our experimental design is based on a ully speciied principal-agent model that imposes very ew restrictions on contractual orm; i.e. contract structure emerges endogenously subject to incentive compatibility constraints, participation constraints, and enorcement limits. In addition, in their design, it is the agent who makes the contract choice whereas in our model, the principal designs the contract. 8

10 actions generically as a i A i where i = A, P. Assumption 1. There is an agency conict in that or a A > a A and a P > a P, where a A, a A A A and a P, a P A P, we have: π P (a P, a A ) > π P (a P, a A ); π P (a P, a A ) < π P (a P, a A ); π A (a P, a A ) < π A (a P, a A ); and π A (a P, a A ) > π A (a P, a A ) In words, ceteris paribus, the principal s payo is increasing in the agent s action and decreasing in her own actions. The opposite is true or the agent. This assumption easily holds in our model where the agent s action/obligation is to deliver quality level q Q where capital Q reers to the quality level speciied in the contract and q reers to the actual quality delivered. The principal s action/obligation is to make a total payment w = p + b W where w is total actual payment and W is the total obligated payment under the contract. The total payment w can consist o a salary or base price p and a bonus payment b. We use capitalized P and B to denote the contractually speciied promised payments. The parties payo unctions are assumed to be π P = r(q) p b and π A = p + b c(q) where r(q) and c(q) are dierentiable unctions such that r (q) > 0, r (q) 0, c (q) > 0 and c (q) 0, q [q, ˇq] R +. Here, the principal preers higher quality all else equal and the agent always preers higher payment, all else equal. The reservation payos or the principal and agent are π and u, respectively. We make the additional assumption that there exists some minimal quality threshold q (q, ˇq) such that r(q h ) c(q h ) u + π > r(q l ) c(q l ) or q l [q, q) and q h [q, ˇq]. This implies that some minimum quality level must be produced to generate positive surplus. Our enorcement technology is based on the veriiability constraints o Bernheim and Whinston (1998) although our speciication extends their ramework to accommodate continous action spaces and to deine a contractible set o actions. Deinition 1. The enorcement technology is an ordered partition o A i R, 9

11 E i = {e i1,..., e ini } where each subset o the partition is let closed, the mesh o the partition decreases with N i, all actions are ordered rom lowest to highest both within and across subsets, and a third-party cannot distinguish any two actions that belong to the same subset e in but can distinguish two actions that belong to dierent subsets o the partition, e ik and e il, where k l. The assumption that the mesh o the partition is decreasing in N i is required to guarantee that the enorcement technology improves as the number o subintervals in the partition increases. For example, i the mesh did not decrease as N i tends to ininity and the irst best action is in the mesh, then increasing the number o subintervals would not impact the ability to contract or irst best. Assumption 1 implies, ceteris paribus, party i tries to minimize her actions in A i but party j preers higher actions in A i ; i.e., the principal s payo is monotonically increasing in a A A A while the agent s payo is monotonically decreasing in a A A A. Similarly, the principal s payo is monotonically decreasing in a P A P while the agent s payo is monotonically increasing in a P A P. This implies that each party will seek to minimize his own actions while desiring that the counter-party chooses higher actions. Thus, a contract is needed to coordinate desirable actions. However, only certain actions are contractible; i.e. third-party enorceable. Since each party wants to minimize his own actions, only the smallest action in each subset o the partition is contractible. This allows us to deine the contractible set. Deinition 2. The contractible set or i = A, P is deined as E i = {e i1, e i2,..., e ini } where e in is the inimum o e in n = 1, 2,..., N i. In other words, only the smallest action in each subset o the partition can be third-party enorced. Improvements in the enorcement technology can be modeled via iner partitions o E i so that an increase in N i implies an increase in the number 10

12 o contractible actions in the contractible set. To illustrate both the enorcement technology and contractible set, consider a simple example where q A A = [0, 10] and the enorcement technology is E A = {e A1, e A2, e A3 } = {[0, 3), [3, 7), [7, 10)}, which has an associated contractible set o E A = {0, 3, 7}. An example o a reinement o this enorcement technology would be E A = {e A1, e A2, e A3, e A4, e A5, e A6 } = {[0, 2), [2, 4), [4, 6), [6, 8), [8, 10), {10}} with the contractible set E A = {0, 2, 4, 6, 8, 10}. Here, E A is the superior enorcement technology as it allows the parties to ormally contract over a greater range o Q. Thus, iner partitions are synonymous with better enorcement technologies. While the contractible set o actions is countable, as the enorcement technology improves, the contractible set E A will approximate A A. We say approximate rather than converge because E A can only be made countably ininite whereas A A is a subset o real numbers. Thereore A A will always have greater cardinality. Nonetheless, E A is dense in A A since every point in A A is either in E A or can be made arbitrarily close to a member o E A i the partition is made ine enough. Lemma 1. Consider a principal-agent relationship that is constrained by the enorcement technology and contractible set speciied in Deinitions 1 and 2. Let q A A denote the irst-best level o quality; i.e. q = argmax{r(q) c(q) π u}. Then or every ɛ > 0, there exists a partition o A A with N A > N ɛ that has an associated con- tractible set E Aɛ that contains a contractible quality level q ɛ such that q ɛ q < ɛ. Proo: See Appendix A or all proos. In other words, in the limit, as the partition gets ine enough, it is possible to contract or irst best quality level. Thus, the enorcement apparatus developed in this section can nest both the complete absence o third-party enorcement (when N A = 1) and perect third-party enorcement (when N A ). 11

13 3.2 Conceptualizing Contractual Incompleteness Under a complete contract, a complete state-contingent plan governs perormance. Thereore, all perormance obligations o both parties are ully speciied or all contingencies in the initial contract. Moreover, the contract is third-party enorced by a court and suicient penalties exist to deter both parties rom shirking on their obligations. This implies that no party has ex post discretionary latitude to deviate rom the initial contract. One can view the presence o ex post discretionary latitude to deviate as being synonymous with an incomplete contract. Our enorcement technology machinery gives us a precise way o speciying both complete and incomplete contracts. To illustrate, recall our earlier example where A A = [0, 10] and E A = {e A1, e A2, e A3 } = {[0, 3), [3, 7), [7, 10)}, which has an associated contractible set o E A = {0, 3, 7}. Here, despite the enorcement technology being imperect, it is still possible to write a complete contract. The complete contract can either speciy state-contingent prices P 0, P 3, and P 7 to be paid under each contractible quality realization, or the principal can speciy, or example, Q = 7 in exchange or a ixed payment P 7 (we will reer to this as the simple contract ). In the ormer case, a third-party enorces the contingent payments P 0, P 3, and P 7 whereas in the simple contract, a third-party enorces Q = 7 and P 7 directly. Note that in either case, all variables are third-party enorceable since they are either in the contractible set or depend only on variables in the contractible set. I the contingent payments P 0, P 3, and P 7 are chosen in an incentive compatible manner to implement q = 7, then the two types o contracts are outcome equivalent in that they both implement q = 7. However, given the simplicity o the simple contract, we will ocus on simple contracts going orward. The important point is that a simple contract is a complete contract because it can credibly tie down all obligations so that neither party has ex 12

14 post discretion to deviate. Hence, imperections in the enorcement technology does not preclude the possibility o writing complete contracts, though imperections do limit the set o available complete contracts. What does an incomplete contract look like? There is no unique incomplete contract, but we will provide one example or illustration purposes. Suppose a contract speciies Q = 10, a ixed payment P 10 and a bonus B i q = 10 is realized. Because Q = 10 is not in the contractible set, it ollows that the agent has ex post discretion to deviate rom Q = 10 without legal consequence. Additionally, because B is contingent on q = 10, B is a discretionary bonus that is not contractible. Thereore, the principal can shirk on the bonus even i the agent perorms. The important point is that both parties have ex post discretion to deviate rom the initial agreement. The traditional incomplete contracts literature has recognized the damaging eects o ex post discretionary latitude through various examples, case studies and ormal models. The crux o the issue can be illustrated with the above incomplete contract. Note that the principal would oer a contract that just satisies the agent s participation and incentive compatibility constraints. The participation constraint is π A = P 10 + B c(10) = u, where u is the agent s reservation utility. The incentive compatibility condition is B = c(10) c(0). B can be substituted into the participation constraint to get P 10 = u + c(0). Since this is a sequential game where the principal is the last mover, a rational principal would not pay an unenorceable bonus. Thereore, the agent would only receive P 10 c(10) = u + c(0) c(10) < u by delivering q = 10. A rational, orward looking agent would then only choose q = 0 to ensure a payo o u. Backward inducting one more stage, the principal expects to earn only π P = r(0) P 10 = r(0) u c(0) < π and oers no contract. Even i there is no clear last mover or certain actors are only contractible ex post, the parties may still have to engage in bargaining which might also create ineiciencies. 13

15 We now introduce more notation that will be used requently or the remainder o the paper. We denote π N A and u N A to be the payos obtained rom the best complete contract or an enorcement technology o partition size N A. The best complete contract is the one that yields the highest joint surplus under the enorcement technology. We denote Q N A as the associated best contracted level o quality, and q N A to be the best actual quality delivered, but under a complete contract, Q N A = q N A. To illustrate, suppose surplus, S(q) = r(q) c(q) π u, is monotonically increasing in quality over A A = [0, 10]. 5 Then or the contractible set E A = {0, 3, 7}, the complete contract that speciies Q = 7 in exchange or a ixed payment P 7 is the complete contract that yields the highest surplus. No other complete contract dominates it since a complete contract or Q > 7 is not easible and a complete contract or Q < 7 does not yield as high o surplus. Thereore, or N A = 3, Q 3 = q3 = 7. More generally, the eiciency o complete contracts improves with the enorcement technology. For ine enough partitions, a complete contract can implement quality levels that are close to irst best. Lemma 2. Let S N A = π N A + u N A u π = r(q N A surplus generated by the best complete contract. Then ) c(q N A ) u π denote the lim N A SN A = S(q ). 3.3 Optimal Contracting In this section, we will show how the optimal contract structure can be made endogenous to the enorcement technology. Our oundational model is a principal-agent model o relational contracting with symmetric inormation. The enorcement technology acts as an exogenous constraint to the contract design problem. We can 5 The assumption o monotonicity is made only to keep this example simple. Monotonicity need not be true in general. For example, or a continuous quality space, there could be an optimal level o quality. Going beyond that level may reduce joint surplus as marginal returns may be less than the marginal costs. 14

16 exogenously vary the quality o the enorcement technology to generate comparative statics predictions about the types o optimal contracts used by the principal. Thus, the incomplete contractual orm can be endogenized. Repeat trading acilitates the use o incomplete contracts because, as noted by Boot et al. (1993), reputation concerns can discipline the abuse o ex post discretion that exists under an incomplete contract. Let the agent s action set (quality choice) be denoted by A A = [q, ˇq] R + where the enorcement technology, E A, (see Deinition 1) is a partition o A A. To economize on notation, we do not speciy the principal s action set because the agent s quality choice is the only determinant o eiciency and the principal s actions are only contingent on agent s quality (i.e. the discretionary bonus). We also deine a binary variable α {0, 1} where α equals 1 i u N A + π N A u + π and 0 otherwise. That is, α = 1 i the best complete contract delivers joint proits that exceed joint proits rom the parties reservation payos. The timeline within a stage-game ollows the typical textbook principal-agent sequence: 1. The principal oers a contract, which is an agreement over a price/bonus/quality triplicate, (P, B, Q). 2. The agent decides whether to accept or reject. I rejected, the principal and agent deault to the best ormal contract i u N A + π N A u + π (i.e. α = 1) and to their outside options, which yield payos u and π, otherwise (i.e. α = 0). 3. I accepted, the agent chooses actual quality q. 4. The principal observes q and chooses actual bonus b. The promised ixed payment, P, is also made. 6 6 Recall that the promised ixed payment is always third party enorceable because it is not contingent on quality. Thereore, the principal has no ex post discretion to deviate rom P. 15

17 A relational contract is an ininite repetition o the above contracting stage-game so that in each period t and or each history up to t, the relational contract describes a sequence (1)-(4). Moreover, the relational contract is sel-enorcing i it describes a subgame perect equilibrium o the ininitely repeated game. In addition, Levin (2003) and Halac (2012) have shown that, with symmetric inormation, the optimal contract is stationary in that the same (optimal) contract is oered in every t. 7 Letting δ be the discount actor and multiplying the payos by 1 δ to express them as per-period averages, the principal s contract design problem is: max(1 δ) [r(q) P B] + δv [C] s.t. (1) Q,P,B (1 δ) [r(q) P B] + δv [C] απ N A + (1 α)π (2) (1 δ) [P + B c(q)] + δu [C] αu N A + (1 α)u (3) (1 δ) [r(q) P B] + δv [C] (1 δ) [r(q) P ] + δ [ απ N A + (1 α)π ] (4) (1 δ) [P + B c(q)] + δu [C] (1 δ) [ P c(q) ] + δ [ αu N A + (1 α)u ] (5) Constraints 2 and 3 are the participation constraints and 4 and 5 are the dynamic incentive compatibility or sel-enorcement constraints. To understand the expressions V (C) and U(C), let Γ denote the set o easible contracts, which can be partitioned as C F = Γ and C F =. Then either (P, B, Q) C or F, where the C contracts 7 Nonstationary contracts arise primarily in the context o private inormation where one has to model relational dynamics due to the revelation o the private inormation over time (e.g. see Halac (2012) or Yang (2013)). It is important to point out that nearly all experiments involve some dynamics simply because subjects need to learn how to play the game. Hence, researchers typically treat predictions arising rom stationary symmetric inormation games as theoretical benchmarks that subjects should converge to ater suicient rounds o learning. The actual dynamics that lead to the convergence is typically not o theoretical interest and early period departures rom theoretical benchmarks is treated as noise that can be reduced i subjects are given adequate time to gain experience. Thus, to test the comparative statics predictions generated rom our model, every eort was made to ensure that our experimental design provides subjects with suicient rounds o play. 16

18 are relational contracts that satisy contraints 2-5, and F denotes ormal (i.e. complete) contracts that satisy participation constraints. Thus, V (C) and U(C) are the low payos or the principal and agent, respectively, rom the optimal selenorcing relational contract (P, B, Q) C. I either party shirks on the relational contract or the principal oers a relational contract that ails to satisy constraints 2-5, then the parties respond by reusing to engage in relational contracting in the uture; they will either rely on complete contracts or select their outside options. 8 Due to stationarity, the same contract is oered every t, problem 1 becomes essentially a static optimization problem. Proposition 1. The solution to optimization problem 1 yields an optimal stationary contract that requests some Q Q where Q is a request or the irst best quality level. The associated payment scheme is W ( Q) = P + B( Q) such that: 1. αun A +(1 α)u+c( Q) δ {r( Q) απ N A 1 δ 1 δ (1 α)π} P αu N A + (1 α)u + c(q) 2. c( Q) c(q) B( Q) δ {r( Q) c( Q) απ N A 1 δ (1 α)π αu N A (1 α)u} 3. P + B( Q) c( Q) = αu N A + (1 α)u In words, under the optimal contract, the principal will contract or a quality level that is less than or equal to the irst best quality level. Moreover, the discretionary bonus must simultaneously satisy both the agent s and principal s sel-enorcement constraints. Finally, the base price P and the discretionary bonus must be simultaneously chosen to orce the agent s participation constraint to bind with equality. For a more intuitive look at sel-enorcement, we can also solve condition (2) in 8 Another possibility is that they might restructure the relational contract so that the breaching party receives only his/her reservation payo. This possibility makes relational contracts renegotiation proo (Levin, 2003) 17

19 Proposition 1 or δ which yields: δ δ(q) = c(q) c(q) r(q) c(q) α [ ] (6) π N A + u N A (1 α) [π + u] = c(q) c(q) r(q) c(q) α [ r(q N A ) c(q N A ) ] (1 α) [π + u] The above critical value δ(q) is the threshold or the incomplete contract to be selenorcing. Notice that the threshold depends on Q; thereore, i the principal wants to implement a higher Q, this puts pressure on sel-enorcement. Consequently, the need or sel-enorcement can limit the quality level that can be implemented. Finally, the threshold also depends on the payos u N A and π N A, which in turn, depend on the eicacy o the enorcement technology. This suggests a non-trivial interaction between sel-enorcement and third-party enorcement. To see the interaction, suppose that Q N A is the enorceable quality level that yields the highest joint surplus among all contractible quality levels. Thus, a complete contract (Q N A, P N A ) yields payos π = P N A c(q N A ) and u N A = P N A c(q N A ). These payos can be substituted in the middle term o 6 to get the last term o 6. One can now conduct comparative statics analysis by letting N A in the enorcement technology increase, which would make E A a more reined partition o A A. This in turn, increases the joint proit r(q N A ) c(q N A ) which weakly raises the threshold or selenorcement 6. 9 In short, an improvement in the enorcement technology should cause some relational contracts to unravel and be replaced by complete contracts. The ollowing proposition provides speciic conditions under which the parties endogenously use an incomplete contract over a complete contract. 9 We say weakly because i α = 0, then the threshold does not change until joint surplus under the complete contracts exceeds joint surplus rom the outside options, triggering α = 1. 18

20 Proposition 2. Let Q be the irst best quality request such that Q arg max{s(q)}. Q I there exists Q such that S(Q ) S( Q) > S(Q N A ) and δ δ( Q), then an incomplete contract that implements Q is preerred over the best complete contract. Proposition 2 states that i there is enough imperection in the enorcement technology, which allows or the existence o some sel-enorcing level o Q that yields joint surplus that is greater than what is obtainable under the best complete contract, then the parties will eschew a complete contract and instead relational contract or Q. The above model can also be used to generate some comparative statics predictions that will be useul or organizing our experimental design and data analyses. Speciically, we ocus on how exogenous changes in sel-enorcement ( δ) and thirdparty enorcement ( N A ) aects contracting outcomes. Corollary 1. The ollowing comparative statics ollow rom the above model: 1. (Exogenous change in enorcement technology): Consider the enorcement technology E A = {e A1, e A2,..., e ANA } and let Q Q = { Q : S(Q ) S( Q) > S(Q N A )}. As N A, then δ( Q) 1 or any Q Q and all incomplete contracts are endogenously replaced with complete contracts. 2. (Exogenous change in δ): Suppose that Q is such that S( Q) > S(Q N A ) and δ δ( Q). Then an exogenous decrease in δ has the ollowing eects: (a) I δ δ( Q) continues to hold, then the principal continues to contract or Q using an incomplete contract. (b) I δ < δ( Q), then the principal contracts or a lower ˆQ where δ = δ( ˆQ) using an incomplete contract i S( ˆQ) > S(Q N A ). (c) I δ < δ( Q), then the principal switches to a complete contract that implements Q N A i there exists no ˆQ such that S( ˆQ) > S(Q N A ) 19

21 The key testable implications o the above corollary are that an improvement in the enorcement technology will increase the use o complete contracts over relational contracts. Moreover, a decrease in δ can lead to a decrease in the quality level that the principal contracts or and/or the decreased use o relational contracts. 4 Experimental Design Our experimental design is based on the above contracting model though we did have to impose speciic parameters and unctional orms and these are described in detail below. However, the unctional orms and parameters were chosen to all within the general assumptions imposed in the theoretical model so that results can be interpreted with minimum loss o generality. In addition, our principal-agent relationship was ramed as a buyer (principal) - seller (agent) relationship or the experiments. Note that a crucial aspect o our design is that we must allow buyers to endogenously choose contractual orm subject to exogenously imposed limits on the enorcement technology. To achieve this, we irst speciy sellers action space as q A A = {1, 2,..., 15} where our sellers can only choose a natural number between 1 and 15. While we can easily incorporate continuous values, we chose natural numbers to keep the action space simple or subjects. This allowed us to deine two enorcement technologies which represents our major treatment variation: 1. Technology E: This is the perect enorcement technology E E A = {e A1, e A2,..., e A15 } = {{1}, {2},..., {15}}. This technology allows a third-party to enorce every single quality level in A A so that the contractible set is E E A = A A. Here N A = Technology PE: This is the partial enorcement technology E P E A = {e A1, e A2 } = 20

22 {{1, 2, 3, 4}, {5, 6,..., 15}}. The contractible set is E P E A = {1, 5}. Here N A = 2. One can interpret Technology E as a technology that provides perect grading o quality whereas Technology PE allows a third-party only to arbitrate on whether the product was deective or not (i.e. below or above 5). Thus, even under PE the parties can write a complete contract that conditions on whether the product is deective. To understand how we allow our buyers to endogenously structure complete contracts under both E and PE, we now describe the contracting stage-game. Within each stage-game, each buyer can (but is not required to) oer a contract, which is a triplicate (P, B, Q) where P {0, 1,..., 200} is a ixed price, B {0, 1,..., 200} is a discretionary bonus, and Q {1, 2,..., 15} is the buyer s requested quality level. A crucial point to note is that we allowed buyers to speciy simple complete contracts by speciying Q E A along with a ixed price P and then clicking a binding option on the computer screen. When binding is checked, neither party has ex post discretionary latitude to deviate as the computer enorces both P and Q. A discretionary bonus B is redundant since it plays no incentive role as the seller has no discretion to choose q Q. 10 Apart rom these restrictions, we imposed no other structure on the contracts. Thus, our subjects can endogenously speciy complete contracts as speciied above, as well as a range o incomplete contracts seen in the literature including git-exchange/eiciency wage contracts (P > 0, B = 0), discretionary bonus contracts (P > 0, B > 0), and illusory promises (P = 0, B > 0). Speciying an incomplete contract only requires the buyer to check the discretionary 10 We thank Tim Cason or pointing out that it is possible to create binding contracts that only prevent downward discretion which might allow subjects to express reciprocal behavior by delivering quality and/or bonuses beyond the contractually speciied levels. We conducted some downward binding but upward discretionary sessions. Though there were some changes in numeric results, the qualitatively results and central conclusions did not change. Moreover, because our paper is not about social preerences and we did not integrate reciprocal behavior in our model, our theoretical predictions under the assumption o pure sel-interest would not change. Hence, in order to maintain clarity and conserve space, we did not include the results o these additional sessions in this paper. 21

23 box rather than the binding box. When discretionary is checked, Q and B are not enorced by the computer. P is still guaranteed because a ixed payment can always be enorced since it is not contingent on anything that is imperectly veriiable. 11 The ollowing summarizes the sequence o events in a stage-game. 1. Proposal phase-buyer can oer a single contract with terms (P, B, Q) to seller. Contract can be made binding (quality 3rd party enorced) but in the PE treatments, the buyer can only make either Q = 1 or Q = 5 binding. The seller can accept or reject. So a participation constraint is active. 2. Quality phase-seller chooses q i Q is not binding. 3. Payment phase-i the buyer oered B, the buyer can choose actual b to pay. Note that under binding complete contracts, there are no Quality or Payment phases since neither party has ex post discretion to deviate rom the initial contract. While the stage-game is a one-shot contracting relationship, our theoretical model also allows or relational contracting, so our experimental design must permit repeat trading. We ollow the typical approach o implementing an ininitely repeated game using a random continuation rule (e.g. Bó (2005)). Speciically, we exogenously ormed buyer-seller pairs where the buyer and seller within each pairing can trade with each other or a random number o stage-games. In other words, in each period, there is δ probability that the same buyer and seller can trade with each other again the next period. This allows us to create a second treatment variation with respect to δ: treatment: δ = 0.8 implies an expected ive period repeated game In practice, even i no legal system existed, one can mimic the enorcement o P simply by having the buyer pay it upront The expected number o periods is 1 δ. 22

24 treatment: δ = 0.5 implies an expected two period repeated game. Sel-enorcement is obviously stronger in the 0.8 treatment because, in any given period, there is a larger probability that the two parties will trade in the ollowing period. We reer to the repeated game or each buyer-seller pair as a supergame. Thus, in the 0.8 treatment, the expected length o a supergame is ive periods and in the 0.5 treatment, the expected length o a supergame is two periods. Table 1: Treatments δ = 0.5 δ = 0.80 Perect enorcement (E) two sessions Partial enorcement (PE) three sessions three sessions Our treatment variations are summarized in Table 1. Note that we only ran the E treatment under δ = 0.80 because, by Corollary 1 part (1), when the enorcement technology is perect, complete contracts should be used regardless o the size o δ. Nonetheless, we ran the E treatment at δ = 0.80, which is the avorable δ level or incomplete contracting. I subjects use complete contracts under E0.80, then we can be conident that they will use complete contracts under E0.50. With respect to parameterization, the stage-game payos are π = 12q p b and u = p+b (q 2 )/2 or the buyer and seller, respectively. Our sellers were provided with Table 2 so that they could quickly get their costs or each quality level. Reservation Table 2: Seller s Cost Quality Cost payos are π = u = 15. These reservation payos are triggered i either the buyer does not oer a contract or the seller rejects a contract. First best is realized at q = 12 which yields joint payos o 72, which exceeds the joint payos o 30 i the 23

25 parties do not contract. Also note that joint proit is below the joint outside option payos o 30 i q < 3. Thus, it is risky or the parties to engage in contracting i the seller does not deliver at least q > 2. Additionally, in the PE treatments, the best contractible quality is Q 2 = 5 which yields joint payos o 47.5 which exceed joint outside option payos. In the E treatment, even irst best is in the contractible set, so Q 15 = 12 = Q. All interactions between subjects occurred via computers and subjects identiied each other through assigned ID numbers that were not associated with actual identities. Once subjects were seated at their computer terminals, our z-tree program randomly assigned hal the subjects to be buyers (principals) and the other hal to be sellers (agents). These roles were ixed or the duration o the experiment. The z-tree program also randomly assigned ID numbers to each subject. Subjects were then read instructions and answered some control questionnaires to ensure understanding. Ater this, they conducted two trial periods to get accustomed to the computerized trading platorm. Their ID numbers were suppressed during the trial periods. Once the live rounds began, each buyer was exogenously matched to a seller to play a supergame which consisted o a sequence o repeated stage-games until randomly terminated. The continuation probability ollowing each stage game was either δ = 0.8 or δ = 0.5 depending on the treatment described above. No subjects were matched or more than one supergame, so we imposed stranger matching. The experiment ended when one o two conditions occurred: (1) All possible supergame matches have been exhausted and the last pairing randomly terminates; (2) I all pairings have not been exhausted and the subjects have played at least 18 periods (across all supergames) in the δ = 0.8 treatment or at least 20 periods in the δ = 0.50 treatment, then they were in their last supergame and the experiment ended when that supergame terminated. These long period sessions were used to ensure 24

26 there was adequate opportunity or learning to take eect. We recruited either 20 or 22 subjects per-session or our 0.5 experiments and either 16 or 18 subjects per-session or the 0.8 experiments. 13 Our experiments were conducted in the Vernon Smith Experimental Economics Laboratory (VSEEL) at Purdue University. This laboratory is used only or economic experiments with an explicit no deception policy. The subject pool consisted o undergraduate students in the VSEEL subject database. Subjects may have participated in other economic experiments but not the speciic treatments conducted or this paper. Nine sessions involving 170 subjects were conducted between March- October 2013 under an approved IRB protocol. All payos are given in points and these points accumulate across periods. Points were converted into U.S. dollars at the rate o 30 points=$1. This method o payment is airly common in repeated game experiments (e.g. Bó (2005)). Average pay was slightly over 25 USD per-session, with a range rom $15 to $38, which includes a 5 dollar show-up ee. We spent considerable time (usually minutes) at the beginning o each session covering the instructions, control questions, and practice rounds. 14 Moreover, we also wanted to have subjects play enough periods during the live periods to ensure that they had adequate time to learn. 15 Consequently, the average session lasted about three hours in total, including instructions, questionnaire, trial periods, post experimental payouts and post experimental demographic questionnaire. Our average hourly payouts were consistent with average hourly rates o other experiments conducted in the same 13 We recruited more subjects or the 0.5 treatments because the expected length o supergames are shorter. Thus, it was likely that we would have exhausted matches more requently in the 0.5 sessions i we did not recruit more subjects. The dierences in group size should not create an imbalance in group reputation eects since we implemented stranger matching. 14 Instructions were typically read aloud with subjects ollowing along using paper copies. 15 We were airly conident that subjects had become quite competent ater some experience because, with the exception o one or two outliers per session, most were making trades airly quickly ater the irst ive or so initial periods. 25