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International Review of Law and Economics 32 (2012) 339– 355

Contents lists available at SciVerse ScienceDirect

International Review of Law and Economics

x ante versus ex post expectation damages

hiyong Liua,∗, Ronen Avrahamb

Donald W. Scott College of Business, Indiana State University, Terre Haute, IN 47809, United StatesUniversity of Texas School of Law, 727 E. Dean Keeton St., Austin, TX 78705, United States

r t i c l e i n f o

rticle history:eceived 5 April 2011eceived in revised form 27 July 2012ccepted 27 July 2012

EL classification:0128286

eywords:reach of contractsymmetric informationxpectation interestenegotiation

a b s t r a c t

What information should courts utilize when assessing contract damages? Should they award damagesthat were rationally foreseeable at the ex ante stage (ex ante expected damages)? Or should they awarddamages at the ex post level, incorporating new information revealed after contracting (ex post actualdamages)? In practice courts have varied between the two approaches, awarding damages equal to thelower, or the higher, of the two measures of damages. This article shows that ex ante expectation damagesare more efficient than ex post actual damages through a simple model of costly litigation for contractbreach, where there are either costs of verifying the breach victim’s ex post damages, or general litiga-tion costs such as attorneys’ fees. Courts should award foreseeable flat damages, rather than seeking expost accuracy and awarding actual damages, because actual damages lead to distortions in breach incen-tives once we take parties’ litigation decisions as endogenous. With costly litigation, ex post expectationdamages may cause over-performance or under-performance depending on whether the American orthe English rule applies and on the size of the litigation cost. We find that regardless of the direction ofthe distortion, actual damages induce inefficiency. Ex ante damages are more efficient because of theinsensitivity of parties’ litigation decisions to their ex post private information under fixed damages. Ourresults are robust when accounting for renegotiation.

© 2012 Elsevier Inc. All rights reserved.

. Introduction

Imagine A breaches a contract he signed with B, but it turns out that the contract would have been a losing contract for B. Can B stillecover damages from A? On the one hand, A after all breached the contract and should pay the foreseeable damages to B. On the otherand, B ended up suffering no loss, in fact she may have even benefited from the breach!

In the famous case of Bush v. Canfield,1 the seller failed to deliver barrels of superfine wheat flour which the buyer agreed to purchaset $7 per barrel. It was proven in court that on the day of delivery, May 1, 1812, the price fell to $5.50 per barrel. The Supreme Court ofrrors held it did not matter whether the buyer would have benefited from the breach ex post, what mattered when the seller breachedas an ex ante evaluation of the damages. More modern cases have come to similar results.2

On the other hand, different courts have held that if a contract breach does not harm the other party when viewed ex post, then therean be no damages. In another case frequently assigned in first-year contracts classes, Acme Mills & Elevator Co. v. Johnson, the Kentucky

ourt found that because the price of wheat on the open market at the time of delivery ($.975 per bushel) was less than the contract price$1.03 per bushel), the buyer, Acme Mills, suffered no actual damages from the seller’s breach and should therefore not be compensated.3

ther courts have reached similar conclusions.4

∗ Corresponding author.E-mail addresses: john.liu@indstate.edu (Z. Liu), ravraham@law.utexas.edu (R. Avraham).

1 Supreme Court of Errors 2 Conn. 485 (1818).2 For example, in United States ex rel. Coastal Steel Erectors, Inc. v. Algernon Blair, Inc., the Fourth Circuit found that the subcontractor was entitled to recover from the general

ontractor for the value of the work performed regardless of the fact that his ex post expectation damages if the general contractor had not breached the contract wereegative. In other words the subcontractor benefited from the contract breach but could still recover! [United States v. Algernon Blair, Inc., 479 F.2d 638, 640–41 (4th Cir.973)]. For a list of many cases which disregard the actual loss see Simon, infra note 5, at 85 n.32.3 Acme Mills & Elevator Co. v. Johnson, 141 Ky. 718 (1911).4 See Bastian v. Petren Resources Corp. 892 F.2d 680 (1990) (Judge Posner refused to compensate investors in oil and gas limited partnership whose investments were wiped

ut and who invested only because of fraud by promoters, on the ground that their investments would have been wiped out anyway due to market conditions); Truitt v.vangel Temple, Inc. 486 A.2d 1169 (D.C. 1984) (court awarded no damages to a promisee who entered a contract later with better terms than the contract breached by theromisor), General Supply & Equipment Co., Inc. v. Phillips, 490 S.W.2d 913 (Tex. App. 1972) (Texas appellate court remanded a case because, inter alia, the jury did not take

144-8188/$ – see front matter © 2012 Elsevier Inc. All rights reserved.ttp://dx.doi.org/10.1016/j.irle.2012.07.004

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40 Z. Liu, R. Avraham / International Review of Law and Economics 32 (2012) 339– 355

These examples challenge us to consider what information courts should utilize in assessing contract damages: should they awardamages equal to the ex ante rationally foreseeable level (ex ante expected damages), or should they award damages at the ex post level,aking into account new information revealed after contracting (ex post actual damages)? As revealed in the above cases, courts vary betweenhe ex ante and the ex post measures of expectation damages. But courts have done worse than just randomly choosing between the two

easures. In the past, courts used the ex ante measure as a “floor” which over-compensates non-breaching parties with low ex post losses,ut where parties with large losses are allowed to recover their actual ex post losses. We call this the “higher of the two” approach, andhat seems to be the approach taken by the 1932 Restatement (First) of Contracts.5 In contrast, the 1981 Restatement (Second) of Contractsmployed the “lower of the two” approach.6 If the foreseeable damages are lower than actual damages, courts often limit the award tohe foreseeable damages. The rule of Hadley v. Baxendale is just one famous manifestation of this principle. However, if the foreseeableamages are higher than actual damages, courts often award only the lower actual damages. This principle is embedded, for example, in theenalty doctrine for liquidated damages which adjusts down stipulated damages if the actual harm manifested is lower. More generally,

t is reflected in the compensation principle, which encourages courts to seek accuracy in awarding damages in order for the promisee toecover the full amount, and only the full amount, reasonably necessary to make him or her whole.7

We analyze the efficiency of the two damages measures through a simple model of contract breach, accounting for litigation costs suchs attorneys’ fees or costs associated with verifying the breach victim’s actual damages. In our model, parties sign a contract for trading anndivisible good or service, whose valuation to the buyer and cost to the seller are random at the time of contracting, but their distributionsre common knowledge. Later, the buyer learns the valuation, and the seller learns the cost, but both will be private information. Theeller then decides whether to breach, and if the seller breaches then the buyer decides whether to sue. Litigation is costly, and we willonsider both the American and English rules. We recommend that generally courts should ignore ex post information and commit toimply awarding ex ante expected damages. Thus, courts should abandon either the “lower of the two” or the “higher of the two” approach.n the example above, B should recover damages foreseeable at the time A and B entered the contract even though her actual damages areero (in fact negative!).

Our recommendation might look obvious because of informational costs to enforce the remedies. When the distribution of potentialamages is common knowledge (as we assume in this paper), the ex ante expectation damages (simply equal to the mean of the foreseeableistribution of damages) have a much lower informational requirement, saving courts and parties the costs associated with verifying thelaintiff’s actual harm (Avraham & Liu, 2012). Sometimes, however, especially if a thick market exists, it might be cheaper to verify thelaintiff’s actual harm (when ex post market prices are readily available and it is difficult to determine the ex ante foreseeable harm). Inuch cases the ex post regime might be better.

Our recommendation that courts use the ex ante expectation damages measure is not just based upon the informational costs advantages.ore interestingly, it is based on incentives grounds. Specifically, we argue that the ex post expectation damages regime induces self-

election among victims of breach with respect to the decision of whether to litigate the breach, and this self-selection effect (which doesot exist under ex ante expectation damages) leads to distortions in performance incentives and hence inefficiencies. Victims with zeroor negative) actual damages would choose to not sue because their payoff from litigation would be nil. This option to not sue embeddedn actual damages has two countervailing effects on breach incentives, only one of which was identified in the literature before. First, theres the missing zero-loss plaintiffs effect (discussed in Avraham & Liu, 2012) where the low end of the distribution of potential plaintiffs isruncated because victims whose actual losses are zero or negative choose not to litigate. As a result, the breaching party faces expectedamages with a mean8 higher than the mean of the original damages distribution (the distribution under the assumption that all victimsould be potential plaintiffs). Hence, interestingly, the missing zero-loss plaintiffs effect leads to over-performance.

The other effect, which was not discussed in Avraham and Liu (2012), is the missing medium-loss plaintiffs effect which occurs whenitigation is costly. Under the ex post expectation damages regime the breaching party avoids liability to victims with medium losses ashose victims choose not to sue even though their losses are positive because, given litigation costs, their expected net recovery is negative.he missing medium-loss plaintiffs effect leads to under-performance.

How would the over-performance (due to the missing zero-loss plaintiffs effect) and under-performance (due to the missing medium-oss plaintiffs effect) play out? Whereas in Avraham and Liu (2012) we argued that ex post expectation damages will always lead tover-performance, we show here that with costly litigation ex post expectation damages can also lead to an equilibrium of under-erformance. Nevertheless, no matter which direction of distortion to the performance incentives (over- or under-performance) thathe ex post expectation damages would induce, the distortion renders the ex post damages inferior to the ex ante expectation damages.

Our model endogenizes the non-breaching party’s decision of whether to litigate based on the information revealed only ex post. Thearge volume of previous literature on comparative efficiency of contract remedies typically assumes an informational structure such

hat the breach victim will always sue for a certain type of remedy.9 This assumes away the possibility that a privately informed, non-reaching party may choose not to file a lawsuit if the expected payoff from litigation is negative (either due to ex post market conditionsr due to litigation costs). In a simplified model without litigation cost, Avraham and Liu (2012) show that seeking accuracy in the ex post

nto account lower damages actually suffered when it determined the buyer’s consequential damages). See also Bowlay Logging Ltd. v. Domtar Ltd. (B.C.C.A. 1982) (Canadianourt refused to award high damages to the promisee after the promisor showed that the promisee would have lost more had the promisor performed). For a list of manyases which resort to actual loss see Simon, infra note 5, at 85 n.32.

5 David Simon, A Critique of the Treatment of Market Damages in the Restatement (Second) of Contracts, 81 Colum. L. Rev. 80, 81 (1981) (“For over 150 years, under establishedrinciples of common law as well as applicable codes, market damages have served as a standard minimum measure of recovery whenever a market was available to fix thealue of the missing performance. . . On the other hand, if the plaintiff’s actual loss was greater than market, he may recover such additional loss”). Simon argued that the932 Restatement (First) has adopted that approach but that the 1981 Restatement (Second) has deviated from it. Id.6 See id (“[i]n principleRestatement (Second) would not allow market damages as an automatic minimum award, but would limit the award to the lesser of market damages

r the actual economic loss of the particular plaintiff.”).7 24 Samuel Williston & Richard A. Lord, A Treatise on the Law of Contracts § 64:1 (4th ed. 2001); 22 Am. Jur. 2d Damages § 27 (1988); 25 C.J.S. Damages § 21.8 The mean of the truncated damages distribution where only high-damages victims remain in the pool, see Fig. 2 in Section 3.9 See, e.g., Birmingham (1969), Barton (1972), Goetz and Scott (1977), Schwartz (1979), Ulen (1984), Shavell (1980, 1984, 2004), Miceli (2004), and Schwartz and Scott

2008), among many others. Edlin and Schwartz (2003) and Mahoney (2000) provide excellent surveys of this literature.

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Z. Liu, R. Avraham / International Review of Law and Economics 32 (2012) 339– 355 341

amages assessment leads to over-performance, once one takes into account the breach victim’s option to not sue. However, Avrahamnd Liu (2012) do not account for litigation costs either in terms of attorneys’ fees or costs associated with verifying the victim’s actualoss. Litigation cost is characteristically an important factor affecting parties’ decisions of whether to file a suit (see, e.g. Mendell (1983),ng (1987), Polinsky and Rubinfeld (1988), Hughes (1995), Hylton (1990, 2002), and Drahozal and Hylton (2003)).10 Thus, in this articlee introduce positive litigation costs into the model, and ask whether courts should still apply flat ex ante expectation damages instead

f ex post actual damages that are adaptive to post-contracting information. As stated above, costly litigation changes the equilibrium,nd ex post expectation damages may induce under-performance (in contrast to Avraham and Liu (2012) where ex post damages lead tover-performance). Nonetheless, we show here that even when accounting for verification costs and/or attorneys’ fees, ex ante expectationamages are still typically more efficient than ex post actual damages. Seeking ex post accuracy in determining damages, as court so ofteno, actually distorts breach incentives.

Interestingly, and in contrast to Avraham and Liu (2012), we are able to derive conditions where ex post expectation damages are superioro ex ante expectation damages. This occurs under the English rule, when renegotiation is costless and litigation costs are sufficiently highelative to the expected trade surplus. Therefore, our recommendation to award fixed ex ante damages is especially strong in the U.S.here the English rule does not typically apply.

One needs to distinguish our results from the well-known case of Hadley v. Baxendale, where the court argued that ex ante expectationamages efficiently motivate disclosure of pre-contractual private information.11 In our model parties to the contract have no private

nformation at the contracting stage. The advantage of ex ante expectation damages over actual damages in our model emerges because exnte damages have lower informational demand and actual damages distort incentives to breach due to the non-breaching party’s optiono not file a lawsuit.

Our results have some further implications as well. First, when there is a need to incentivize the non-breaching party to minimize herx post loss, for example, by seeking another trading partner in a timely fashion, our recommendation strengthens. The reason is simple:y awarding ex ante expectation damages courts avoid diluting the non-breaching party’s incentives to minimize her ex post losses. Hadhe non-breaching party been guaranteed ex post damages, her incentive to “cover” in the market would have not been optimal.12

Second, our results provide yet another argument against the penalty doctrine for the liquidated damages, only in conditions unexploredefore in the literature. When parties have symmetric incomplete information upon contracting, stipulated damages shall be enforced byourts, no matter whether ex post the damages seem to be over- or under-compensatory.13

Third, our results help explain the rationale behind section 2-713 of the U.C.C. which has been widely criticized. Section 2-713 sets auyer’s damages for non-delivery to equal the difference between the market price at the time of delivery and the contract price. But, asany critics have observed, that measure might over-compensate buyers whose real ex post valuation is low.14 In contrast, we argue that

s long as courts always employ the ex ante approach, then section 2-713 is defensible, even though it might sometimes over-compensateuyers.15

Fourth, our results are relevant to a recent trend among a few legal economists to favor the disgorgement remedy. For example, Richardrooks has recently argued that disgorgement of the promisor’s profits is as efficient a remedy as expectation damages.16 A 2011 paper byteve Thel and Peter Siegelman takes it further and argues that requiring the breaching party to not only pay the other party’s expectationamages but also to disgorge any additional profits is the preferred contract remedy.17 This disgorgement theory involves an ex post

alculation of damages, which, in the conditions we investigate in our model, will not lead to efficient results.

Fifth, our results imply that section 39 of the proposed Restatement (Third) of Restitution and Unjust Enrichment will create improperncentives for breach.18 Section 39 would make disgorgement available in circumstances where “a deliberate breach of contract results

10 Choi and Triantis (2008) made an interesting point that parties may intentionally write ambiguous clauses in contracts that invite costly verification and litigation. Inther words, parties may deliberately increase verification and/or litigation costs in contract design for the benefit of screening the promisees on their incentives to sue, andnhancing expected sanction against a breaching promisor through the credible litigation threat. Unlike their endogenous litigation/verification costs framework, our papertudies performance incentives in situations where litigation/verification costs are exogenously given.11 See Ayres and Gertner (1989), Bebchuk and Shavell (1991), and Adler (1999).12 The actions of the non-breaching party might be impacted by the mitigation principle of contract damages, but when this theory applies is not always clear and sheould have no independent economic incentives to mitigate. See Goetz and Scott (1983) (“The duty to mitigate is a universally accepted principle of contract law. . .”).

13 Penalty doctrine has been a puzzle against the rationality assumption and freedom to contract (Posner, 1979). A great deal of literature (see. e.g., Rea, Jr., 1984) has beenevoted to explaining the efficiency justifications for applying the penalty doctrine in contract law (see an excellent survey by De Geest and Wuyts, 2000). Penalty clausesould be used to (1) insure the contracting parties against idiosyncratic harm that is difficult to quantify (Goetz and Scott, 1977); (2) strategically extract surplus from thirdarties (Diamond and Maskin, 1979; Aghion and Bolton, 1987; Chung, 1992); (3) provide a credible signal of private information (Posner, 1977); (4) provide punitive sanctionshen the apprehension rate is less than one (Polinsky and Shavell, 1998), and other purposes. Stole (1992) proves that with asymmetric information upon contracting the

quilibrium liquidated damages will not be over-compensatory, thus penalty doctrine would make sense in such contracting environments. However, when information isymmetric upon contracting and private information only develops after parties signing the contract, our paper implies that the penalty doctrine should not apply in suchases.14 2 William D. Hawkland, Uniform Commercial Code Series § 2-713:1 (2002) (“Sometimes the buyer may no longer need the goods that the seller has refused to deliver,nd, therefore, he does not want to buy others in their stead. In these cases, cover is not an appropriate remedy, but the buyer is entitled to recover damages under section-713 that are measured by the difference between the market price and the contract price,”); Simon, supra note 5, at 81 (“The objective value fixed by the market representsnly a rough approximation of lost gains, and may often exceed the subjective loss of the particular plaintiff. Nevertheless, the prevailing view is that the market measurehould be adhered to even if the plaintiff’s actual loss was less.”).15 You may see a representative opinion against section 2-713 from Robert Childres, Buyer’s Remedies: The Danger of ection 2-713, 72 Nw. U. L. Rev. 837 (1978) (“Section 2-713f the Uniform Commercial Code should not have been enacted and should be repealed. Given the small chance of repeal, the observer should be made aware that the Code’scheme for buyer’s remedies is sensible only if section 2-713 is ignored. This is so, for section 2-713 fails because it is a hypothetical remedy; it lacks any relevant relationo damages actually suffered”). A defense of section 2-713 that is consistent with the implication of our results is Ellen A. Peters, Remedies for Breach of Contracts Relating tohe Sale of Goods Under the Uniform Commercial Code: A Roadmap for Article Two, 73 Yale L.J. 199, 253, 259 (1963) (“Perhaps it is misleading to think of the market-contractormula as a device for the measurement of damages. . . An alternative way of looking at market-contract is to view this differential as a statutory liquidated damages clause,ather than as an effort to calculate actual losses.”).16 See Ref. Brooks (2006).17 Steve Thel & Peter Siegelman, You Do Have to Keep Your Promises: A Disgorgement Theory of Contract Remedies, 52 Wm. & Mary L. Rev. 1181, 1224–25 (2011) (offering aormative defense of disgorgement).18 Restatement (Third) of Restitution & Unjust Enrichment § 39 (2011).

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42 Z. Liu, R. Avraham / International Review of Law and Economics 32 (2012) 339– 355

n profit to the defaulting promisor and the available damage remedy affords inadequate protection to the promisee’s contractualntitlement.”19 This would entail an ex post evaluation of the profit to be disgorged from the breaching party and given to the non-breachingarty, which our model shows will not lead to efficient results.20

Our work should be distinguished from previous literature. Ayres and Talley (1995) argue that untailored liability rules induce moreredible signaling of private information in price bargaining, and thus facilitate more efficient trade. Scott and Triantis (2006) argue thathe equilibrium contract formation depends on the relative informational advantages of contracting parties (at ex ante) versus of the courtat ex post). Our analysis instead focuses on the parties’ endogenous decision of whether to litigate the breach, given costly litigation. Inarticular, we are assuming that parties have no informational advantage vis-a-vis the court at the contracting stage.

Kaplow (1994) and Kaplow and Shavell (1996) argue that if the decision maker is informed about the harm when making the decision,ore accurate ex post damages assessment will improve efficiency. However, if the decision maker does not know the harm when making

he decision, more accuracy in damages assessment will not improve efficiency but waste resources in establishing the actual damages,ecause greater precision ex post in damages assessment “cannot improve the earlier decision” (Kaplow, 1994, p. 314). Our model fits intohe uninformed decision-maker scenario in the Kaplow and Shavell model as we assume that the seller does not know the buyer’s privatealuation when deciding whether to breach the contract. However, our argument against ex post damages is based on a different rationalen such an informational environment: under ex post damages a privately informed victim with low damages may choose not to sue, andhis actually pushes up the expected punishment in litigation (recall the missing zero-loss plaintiffs effect and see Fig. 2 in Section 3), thuseading to over-performance. Therefore, in an incomplete contracting context when the decision maker is uninformed, greater precisionx post not only does not improve the ex ante decision as pointed out by Kaplow and Shavell, but actually exacerbates the ex ante decisionue to the potential victims’ self-selection in litigation decisions.

Ben-Shahar and Bernstein (2000) find that in repeated transactions an aggrieved party may not file a suit if doing so requires disclosuref private information, thus hurting her future competitive position.21 In our one-shot model, an aggrieved party may choose not to sueue to her negative net payoff from litigation regardless of any secrecy interest.

Bowels (2008) recently surveyed the literature on ex ante versus ex post measures of commercial damages, and argued that in anfficient market ex post damages (which he called hindsight) result in overcompensation of the plaintiff. He found hindsight may workor or against the plaintiff, depending on the litigation cost, expected economic profit, and the probability and size of negative outcomes.owels correctly pointed out the disallowance of negative damages in contract law22 and its effect on compensation, but he did not formallyompare the joint payoff under ex ante versus under ex post damages, from which we derive our main results. Also Bowels did not considerenegotiation.

In our model with renegotiation the seller may strategically breach (breach when performance is profitable) to extract surplus fromhe buyer through renegotiation. Luelfesmann (2009) illustrates a similar point in a model with investments. A seller may strategicallynderinvest so as to render ex post trade being inefficient, then he could extract surplus via renegotiation. For simplicity we assume aostless discovery process in litigation. Schwartz and Watson (2004) show that the (endogenous) cost of contracting and renegotiationffects parties’ choice of initial contractual forms (simple versus complex contracts) and relation-specific investments, and therefore theontracting parties exhibit preferences over legal rules and other factors that affect the costs of contracting and renegotiation.

The rest of the paper is organized as follows: In Section 2, we present a simple model of buyer–seller contracting with costly litigation.n Section 3, we compare the efficiency of ex ante versus ex post expectation damages for the case with positive verification costs. Thenn Section 4, we account for parties’ general litigation costs. For each remedy we analyze both cases with and without renegotiation, and

e also compare the American rule with the English rule of litigation cost shifting for each case. In Section 5, we summarize our resultsnd conclude. Appendix A provides a proof that the buyer’s participation constraint in the seller’s optimization problem is binding. Inppendices B–D, we provide detailed specifications of equilibrium prices and expected joint payoffs under various regimes.

. The model

At Time 1 two risk neutral parties (Buyer B and Seller S) enter a contract with agreed price, p, for the sale of a single indivisible goodr service. The seller receives payment upon performance at Time 2. There is uncertainty at Time 1 regarding the value of the contract foroth parties. Specifically, the seller’s cost, c ∈ [0, c], is drawn from a distribution F with density f. The buyer’s valuation, v ∈ [0, v], is drawnrom a distribution G with density g. The commonly known distributions F(·) and G(·) are independent, continuous and twice differentiable.

enote �c:=∫ c

0cdF(c), and �v:=

∫ v0

vdG(v). Both c and v are finite, with c > �v, and v > �c . Between Time 1 and Time 2 both parties learnheir private valuations and the seller decides whether to breach the contract. Each party’s respective valuation is unobservable to thether party. We first assume that parties commit not to renegotiate the contract ex post; later we relax this assumption and allow forenegotiation at the litigation stage. If there is a breach at Time 2, then at Time 3 parties may litigate the case under the given remedy. In

ontrast to Avraham and Liu (2012), we assume that litigation is costly. There are either costs of verifying ex post damages, denoted by

∈ R+, when the enforced remedy requires information about the victim’s ex post damages; or there are general litigation costs, such as

ttorneys’ fees. We denote these general litigation costs by l ∈ R+. Fig. 1 presents the time line.23

19 Id.20 Comment (d), Illustration 1 is an example of how this proposed section would wrongly require ex post damages calculations. See id. at Illustration 1.21 Illustrating with a tort model, Polinsky and Shavell (1998) have noticed earlier that litigation cost may discourage victims to sue, and that with costly litigationompensatory damages under strict liability do not generally result in socially optimal outcomes.22 In a recent paper Adler (2008) provides an analysis of the potential benefit from allowing negative damages (paid by the non-breaching party to the breaching party) inontract law. Allowing for negative damages will strengthen the incentives to not sue. Our analysis takes the prohibition of negative damages as given, and then formallyompares the efficiency of ex ante versus ex post expectation damages in a costly litigation environment.23 You may wonder why parties contract at ex ante stage when neither has private information, and why they do not wait to write a contract until they learn more information.

rationale for the value of ex ante contracting is to avoid the efficiency loss stemming from asymmetric information which parties will face in interim contracting (contractingfter they learn private information). For an example of comparing optimal ex ante contract and optimal interim contract, see Avraham and Liu (2004).

Z. Liu, R. Avraham / International Review of Law and Economics 32 (2012) 339– 355 343

Time 1 1.5 2 3

Parties sign a contract

Parties learn private information

S decides whether to breach. If S breaches, B decides whether to sue. If B files a lawsuit, evidence is

Court enforces the default remedy in contract if there is

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Fig. 1. Time-line for the model.

Without loss of generality, and for simplicity, we assume that the seller has full bargaining power, an innocuous assumption in ouretup with no private information at the time of contracting.24 We compare parties’ joint expected payoff under ex ante versus ex postxpectation damages. Under ex ante damages, courts commit to awarding fixed damages which were foreseeable at the ex ante stage,ithout incorporating new information revealed later. Under ex post expectation damages courts commit to accurately accessing actualamages by fully incorporating new information. We assume that with some cost the buyer’s valuation is verifiable to the court and thusctual damages can be completely assessed. In contrast, the seller’s private information is unverifiable.25 The following notations are used:D ex ante expectation damages;D ex post expectation damages, or actual damages;r(·) seller’s breach threshold under actual damages given his private information;Ri(R = ED or AD; i = B or S) party i’s expected payoff under remedy R;

�R joint expected payoff under remedy R;ubscript r regime with renegotiation;(∼) at top of notations under the American (English) rule;

∈R+ cost of verifying ex post damages in litigation;i ∈ R

+ (i = b or s) parties’ general litigation cost if they bring a lawsuit.

. Analysis of the case with positive verification costs

.1. The case of no renegotiation

First, we focus on the case that parties commit to not renegotiating the contract after they learn new information.26 In this section, wessume that all litigation costs fall on verifying the buyer’s ex post damages, if enforcing a remedy requires that information. We assumehat the verification cost, ˇ, is constant and does not depend on which party bears the cost. A natural implication of costly verification ishat enforcing ex ante expectation damages is less costly than enforcing actual damages, because the former does not require verifying thex post damages.27

.1.1. Ex ante expectation damagesUnder ex ante expectation damages (ED) the court commits to awarding fixed damages at the ex ante rationally foreseeable level (which

s �v − p in our model), and does not adjust the damages award according to new information revealed in litigation. This remedy guaranteeshe victim of breach an expected payoff from contract performance (calculated using ex ante commonly observed information). Notice thathe equilibrium price under this remedy must be no greater than �v, otherwise the buyer’s expected payoff from the seller’s performanceor from litigation over breach), �v − p, would be negative, hence she will never sign such a contract. Thus, p ≤ �v, which implies that theuyer always sues upon breach. As a result, the seller breaches only if c > �v. The seller’s optimization problem is:

maxp

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v

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0

(�v − c)dF(c). (1)

24 For the general case of both parties sharing some bargaining power, the parties still maximize the joint expected payoff when writing the contract since no one has anyrivate information at that stage. Thus, our basic results remain, and the change is the distribution of expected surplus between parties. Moreover, in our model we have nox ante investment. If there is investment, the bargaining power assumption will affect the investment incentives (the “holdup” problem, see, e.g., Luelfesmann, 2009).25 Verifiable damages enable the ex post expectation damages remedy to be operational; otherwise the court needs to overcome the problem of asymmetric informationnd credibility of signaling. See Usman (2002), Sanchirico and Triantis (2008), and Bernardo, Talley and Welch (2000). On the other hand, if both parties’ information wereerifiable, it would have been trivial for the court to determine a first-best allocation, no matter what remedies the parties had contracted for.26 Renegotiation has costs, especially when there is asymmetric information. Additionally, some contingencies may arise where one party needs to make the decisionuickly, leaving no time for renegotiation. We analyze the case with renegotiation in Section 3.2.27 In our model the distributions of valuation and costs are common knowledge, thus the ex ante expected damages are straightforwardly identified as the mean of theamages distribution. If it is not common knowledge but rather only the parties to the contract knew the distributions, then parties may need to verify the ex ante expectationamages to the court. However, even in such a case, it is still reasonable to assume that verifying the distribution (or range) of damages (for the purpose of enforcing ex antexpected damages) would be less costly than verifying a specific value of damages (for the purpose of enforcing actual damages) to the court. As long as the verification costnder ex ante damages is relatively much cheaper than under actual damages (which is true in our model), the results in this section would not change. Nonetheless, therere situations in which estimating ex ante expected damages is more difficult than assessing ex post actual damages, for example, when there are ambiguities or asymmetricnformation at the contracting stage regarding the distribution of potential payoff and damages. Such situations are interesting but not the focus of our current article.

344 Z. Liu, R. Avraham / International Review of Law and Economics 32 (2012) 339– 355

|0

a

3(ablcptad

h

staTwvtpesp

c

te

Fig. 2. Truncated distribution of damages under actual damages and the American rule.

As it turns out, ED is also the welfare-maximizing money damages remedy (see Avraham & Liu, 2012).28 We now turn to analyzingctual damages under both the American and the English rules.

.1.2.1. The American rule. Assume here that the default remedy in contract is ex post expectation damages, which we call Actual DamagesAD). This remedy compensates the victim of breach such that her ex post payoff would be as if the contract were performed. We firstssume the American rule applies, which implies the plaintiff (B in our case) bears her own cost of verifying damages, ˇ. Therefore, uponreach the buyer will sue for damages only if v > p + ˇ. With probability G(p + ˇ), shown as the shaded area in Fig. 2 above, buyers with

ow valuation of the good will not file a suit upon breach. Some of these buyers (with v < p) may in fact feel lucky the seller breached theontract as it saved them the obligation to pay for a good they do not value much. The other group of these low-valuation buyers (with

< v ≤ p + ˇ) does suffer some loss from breach, but the damages are not large enough to cover the cost of litigation. As shown in Fig. 2,he buyers’ self-selection in litigation decisions under actual damages leads to a truncated distribution of potential damages in litigation. If

suit is filed, the expected damages that a breaching seller faces in litigation are E(v|v ≥ p + ˇ) − p, the mean of the truncated damagesistribution, which is obviously greater than the ex ante expectation damages, �v − p.29 The seller’s payoff from performance is p−c; while

is expected payoff from breach is∫ v

p+ˇ(p − v)dG(v). Therefore, the seller will breach if and only if:

c > p +∫ v

p+ˇ

(v − p)dG(v):=Br(p, ˇ). (2)

Br(p, ˇ) = p +∫ v

p+ˇ(v − p)dG(v) > p, i.e. the seller’s breach threshold is greater than the contract price, which implies that the seller in

ome cases (when p < c < Br(p, ˇ)) voluntarily performs even when performance is unprofitable. This seems counter-intuitive since underhe American rule the seller does not bear the cost of verifying damages in case of a breach. Furthermore, as was just explained, there is

positive probability that the buyer would decide not to sue at all upon breach; thus, the seller might avoid paying damages altogether.he reason that the seller sometimes voluntarily performs at a loss is twofold: First, because under the American rule the buyer is lessilling to challenge breach, the seller indeed can escape paying damages in some cases. However, in many of these cases (for buyers with

< p) the seller would have had to “pay” negative damages (had they been allowed). Thus, the seller is, in a way, worse off from the facthat those buyers do not sue. Second, in those cases in which the buyer does file a lawsuit the expected damages that the seller needs toay are higher. These two features together characterize the truncated distribution of damages, which deters the seller from breaching,ven under the American rule. Interestingly, as will be shown below, when we allow for renegotiation, this result changes. Specifically, wehow that when there is renegotiation under actual damages and the American rule, the seller may strategically breach (i.e. breach whenerformance is profitable) to extract surplus from the buyer.

The seller’s optimization problem is:

Maxp�ADS =

∫ Br(p,ˇ)

0

(p − c)dF(c) +∫ c

Br(p,ˇ)

∫ v

p+ˇ

(p − v)dG(v)dF(c)

AD

∫ Br(p,ˇ) ∫ c ∫ v

s.t. �B =0

(�v − p)dF(c) +Br(p,ˇ) p+ˇ

(v − p − ˇ)dG(v)dF(c) ≥ 0.

28 Under money damages remedies, the breach of contract is irreversible. The only question is to determine the monetary compensation to the non-breaching party. Inontrast, under non-money damages such as specific performance, the breach is reversible and the court could order performance of the contract.29 Of course, this does not directly imply that the seller’s expected payoff from breach is smaller under AD than under ED, since on the other hand we have to considerhe potential avoidance of damages under AD to those breach victims that do not sue. The breach thresholds under AD vs. under ED will be compared after we solve for thequilibrium price.

b

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3T

T

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ttfsamtlt

Z. Liu, R. Avraham / International Review of Law and Economics 32 (2012) 339– 355 345

The Kuhn–Tucker conditions for the seller’s optimization problem entail that in equilibrium the buyer’s participation constraint muste binding, i.e. �AD

B = 0. (The proof is in Appendix A.) The equilibrium price and joint payoff are defined by:

pAD = �v +[1 − F(Br(pAD, ˇ))]

∫ vpAD+ˇ

(v − �v − ˇ)dG(v)

1 − G(pAD + ˇ) + F(Br(pAD, ˇ))G(pAD + ˇ); (3)

j�AD =

∫ Br(pAD,ˇ)

0

(�v − c)dF(c) − ˇ[1 − F(Br(pAD, ˇ))][1 − G(pAD + ˇ)]. (4)

We compare the breach threshold induced by ED with the one induced by AD under the American rule:

�Br(ˇ):=Br(pAD, ˇ) − �v =∫ pAD

0

(pAD − v)dG(v) −∫ pAD+ˇ

pAD

(v − pAD)dG(v). (5)

Denote ˇ∗ such that �Br(ˇ∗):=0. In other words, given the equilibrium price under AD, when = ˇ∗, the breach threshold inducedy AD is the same as the one induced by ED. It is easy to see that when → 0, �Br(ˇ) > 0. That is, when the cost of assessing actualamages is negligible, then as we have demonstrated in Avraham and Liu (2012), actual damages lead to under-breach— from the ex anteerspective, the seller breaches less often than optimal due to the buyer’s self-selection with respect to the decision to litigate and theesulting truncated distribution of damages. When < ˇ∗, �Br(ˇ) > 0, the seller breaches less often under AD than under ED. When

> ˇ∗, �Br(ˇ) < 0, the seller breaches more often under AD than under ED. This is quite intuitive: when the litigation cost for the buyers high, the seller breaches more frequently because he rationally expects a higher chance of no suit being filed. In contrast to Avraham andiu (2012), the introduction of verification cost under actual damages plays an interesting role in balancing two countervailing effects of breachictims’ self-selection on the decision to litigate. The first effect is the aforementioned truncated damages distribution effect, which increasesxpected damages payment, and thus weakens the seller’s incentive to breach. The other effect is the potential avoidance of liability toome buyers (whose valuation is between p and p + ˇ) due to the barrier of litigation cost, which encourages the seller to breach. Whenamages verification is cheap or free, as shown in Avraham and Liu (2012), the first effect dominates and the seller breaches less often thanptimal under actual damages. But when the American rule applies and verification is costly, which of the two effects dominates dependsn the magnitude of the verification cost. There exists a critical level of verification cost that distinguishes the cases where the truncatedamages effect or the avoidance of liability effect dominates.

.1.2.2. The English rule. We now assume the English rule applies, under which the breaching party (S) bears the verification cost (ˇ).hen at Time 3 upon breach, the buyer will sue for damages if v > p. Hence, the seller’s expected payoff from breach is

∫ vp

(p − v − ˇ)dG(v).

herefore he will breach if:

c > p +v∫p

(v − p + ˇ)dG(v):=Br(p, ˇ). (6)

rom (6) we have Br(p, ˇ) > p, which implies that the seller in some cases voluntarily performs at a loss. The reason is twofold. First, asefore, the truncation of damages distribution deters breach under actual damages. Second, under the English rule the breaching partyears the verification cost in litigation, which further discourages his incentives to breach.

We will now compare the breach threshold induced by ED with the one induced by AD under the English rule: �Br(ˇ):=Br(p, ˇ) − �v =p

0

(p − v)dG (v) + ˇ[1 − G(p)]. It is easy to see that �Br(ˇ) > 0. Under the English rule, the seller breaches less often under AD than under

D. We have the following Lemma:

emma 1. When verification of actual damages is costly and parties commit to not renegotiate the contract, under the American rule, theelative expected frequency of breach under AD versus under ED depends on the size of verification cost. There exists a critical level of verificationost, ˇ∗, such that if = ˇ∗, the expected frequency of breach is the same under both remedies; if > ˇ∗, the seller breaches more often underD than under ED; if < ˇ∗, the seller breaches less often under AD than under ED. Under the English rule, the expected frequency of breach is

ower under AD than under ED, and this does not depend on the size of verification cost.

As can be seen from Lemma 1, when there is litigation cost (in the current case, taking the form of the cost of verifying damages),he truncation of damages distribution in litigation (see Fig. 2) leads to under-breach when the English rule applies. However, in contrasto Avraham and Liu (2012), it does not necessarily lead to under-breach when the American rule applies; instead, the relative expectedrequency of breach under AD vs. under ED depends on the size of the litigation cost. There exists a threshold level of verification cost (ˇ∗)uch that when the verification cost equals that level, AD and ED induce the same expected frequency of breach. When the American rulepplies and litigation is more costly than the threshold level AD leads to over-breach in the equilibrium, as the high litigation costs induce

ore low-type buyers (including those with positive payoff from contract performance but negative payoff from the costly litigation)

o choose to not sue, and the avoidance-of-liability effect under AD outweighs the truncation-of-damages-distribution effect. When theitigation is less costly than the threshold level, the truncated damages distribution effect dominates and AD reduces the seller’s incentiveso breach (relative to the welfare-maximizing money damages, ED).

3

T

Pf

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3

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sqq

r

dp

46 Z. Liu, R. Avraham / International Review of Law and Economics 32 (2012) 339– 355

It can be shown that the buyer’s participation constraint is binding, i.e. �ADB =

∫ Br(p,ˇ)0

(�v − p)dF(c) +∫ c

Br(p,ˇ)

∫ vp

(v − p)dG(v)dF(c) = 0.

he equilibrium price and joint surplus are defined by:

pAD = �v +[1 − F(Br(pAD, ˇ))]

∫ vpAD (v − �v)dG(v)

1 − G(pAD) + F(Br(pAD, ˇ))G(pAD); (7)

j�AD =

∫ Br(p,ˇ)

0

(�v − c)dF(c) − ˇ[1 − F(Br(pAD, ˇ))][1 − G(pAD)]. (8)

The following proposition holds:

roposition 1. Assume that parties commit not to renegotiate the contract, and that verifying buyer’s actual damages is costly. Then theollowing holds:

(i) Ex ante expectation damages are more efficient than actual damages, under both the English rule and the American rule.(ii) Under actual damages (with either the English or the American rule) the seller sometimes voluntarily performs at a loss.iii) The American rule is superior to the English rule under the actual damages remedy if:∫ Br(pAD,ˇ)

Br(pAD,ˇ)

(�v − c)dF(c) ≥ ˇ

{[1 − F(Br(pAD, ˇ))][1 − G(pAD + ˇ)]

−[1 − F(Br(pAD, ˇ))][1 − G(pAD)]

}. (9)

roof. (i) Under the American rule, the difference between the joint expected payoffs under actual damages and ex ante expectationamages is:

j�AD − j�

ED =∫ Br(pAD,ˇ)

�v

(�v − c)dF(c) − ˇ[1 − F(Br(pAD, ˇ))][1 − G(pAD + ˇ)] < 0,

ince the first item is negative, and the second item is positive. We can prove for the case under the English rule in a similar way.(ii) Follows from the fact that the seller’s breach thresholds under both rules are larger than the contract price.(iii) Follows from comparison of the expected joint payoffs.

ED is superior to AD. This is not only due to the deadweight loss embedded in the verification costs ( > 0) under actual damages, butlso because the incentives to breach are distorted relative to the ED remedy, which provides the best ex ante incentives to breach amongll money damages remedies. Even though as reflected in Lemma 1, AD may induce over-breach in the equilibrium when the Americanule applies and litigation is quite costly, a result different from the one in Avraham and Liu (2012), which showed an under-breachquilibrium induced by AD in a framework absent of litigation cost, this does not change the efficiency comparison between AD and ED.ither under-breach or over-breach induced by AD harms efficiency, as compared to the optimal ex ante money damages (ED).

If parties were to specify a liquidated damages (LD) clause in the contract, they will set the liquidated damages equal to the optimal exnte money damages, given the parties had symmetric incomplete information at the contracting stage (see, e.g. Avraham & Liu, 2006).herefore, with symmetric incomplete information at contracting stage and no renegotiation, LD would be equivalent to ED. A well-knownrinciple in enforcing liquidated damages is the penalty doctrine, under which courts often invalidate over-compensatory LD clauses inontracts. Typically to determine whether an LD clause is over-compensatory, a court would compare it with the actual harm caused byhe breach. Our result here shows no support for the penalty doctrine when the parties had symmetric incomplete information uponontracting30 and there is no renegotiation. In such contracting environments, our result rather suggests that courts should honor the LDamages clause, no matter whether it is over-compensatory or not.

.2. Accounting for renegotiation

So far we have assumed that parties commit to not renegotiate the contract. Now we relax this assumption and allow for renegotiation.arties sign a contract with a default price, p, and anticipate that pre-trial renegotiation may take place after the discovery process in whichhe buyer’s damages are revealed.31 S (with full bargaining power by assumption) makes a take-it-or-leave-it offer in the renegotiation. If

accepts the offer, the game is over. If renegotiation breaks down, the court will enforce the default remedy. Fig. 3 presents the time-linef the game with renegotiation.

Given that the buyer’s actual damages are revealed through the discovery process, the seller’s optimal renegotiation strategy is quitetraightforward regardless of the contract remedy: If v ≥ c, the seller will seek to trade at a price which guarantees the buyer her status

uo payoff from trial. If v < c, the seller will seek to breach the contract, paying money damages to ensure the buyer obtains her statusuo payoff from trial. Notice that, assuming the litigation occurs, this simple renegotiation scheme would ensure efficient trade.

The seller observes the buyer’s actual damages through the discovery process, but the verification of damages is only required whenenegotiation fails and the court-enforced remedy requires knowing the buyer’s ex post damages. In other words, discovery reveals actual

30 This assumes the asymmetric information arises only after contracting.31 Since the discovery process will unveil one party’s private information, from the ex ante perspective, parties would prefer to renegotiate after rather than before theiscovery, because according to the bargaining theory (see, e.g., Stahl, 1972), bargaining with one-sided asymmetric information can lead to first best allocation if therivately-informed party makes the offer.

Z. Liu, R. Avraham / International Review of Law and Economics 32 (2012) 339– 355 347

Time 1 1.5 2 3

Parties sign a contract

Par ties learn pri vate information and S deci des whether to breach

If S breaches B decides whether to sue; if B files a laws uit, evidence is collec ted in a

Parties rene gotiate; if no renegotiation ag reement is achieved, court enf orces

dtn

3

asp

Ica

33ola

f

ˇ

Smbrtv

st

3gv

ct

v

dt

a

c

C

b

i

discovery pr ocess the def ault remedy

Fig. 3. Time-line for the model with renegotiation.

amages, but does not necessarily verify them.32 Therefore, if renegotiation fails, there is an extra cost to enforce actual damages due tohe positive verification costs. However, positive verification costs have no impact on enforcing ex ante expectation damages that requireo ex post information.

.2.1. Ex ante expectation damages with renegotiationSince there is no need to verify actual damages under the ex ante expectation damages, the introduction of verification costs does not

ffect the renegotiation outcome under the ex ante expectation damages, as specified in Avraham and Liu (2012). With renegotiation theeller may strategically breach (breach when performance is profitable) to extract surplus from the buyer.33 By Avraham and Liu (2012),EDr = �v, and the joint expected payoff is:

j�EDr =∫ c

0

∫ v

min(c,v)

(v − c)dG(v)dF(c). (10)

The equilibrium price is the same as under ED with no renegotiation, but the joint payoff is increased with the opportunity to renegotiate.n fact, as can be seen from (10), with post discovery renegotiation, ex ante expectation damages induce first best allocation. Under thease of no renegotiation, ED is the best money damages remedy; while under the case with renegotiation, EDr is the best remedy amongll remedies, including money damages and specific performance.34

.2.2. Actual damages with renegotiation

.2.2.1. The American rule. Under the American rule, the victim of breach bears her own cost of verifying ex post damages. At Time 3, Sbserves B’s damages through the discovery process. In renegotiation the buyer will accept an offer only if she is guaranteed a payoff of ateast v − p − ˇ, which is her payoff if she rejects the offer and proceeds to trial. As a result the seller’s optimal strategy is to offer to trade at

higher price p + if c ≤ v + ˇ; and to not make any renegotiation offer otherwise. Anticipating the strategies at Time 3, the buyer will sue

or damages at Time 2 only if v > p + ˇ. Hence, S’s expected payoff from breach is:∫ min(v,max(p+ˇ,c−ˇ))

p+ˇ(p − v)dG(v) +

∫ vmin(v,max(p+ˇ,c−ˇ))

(p + − c)dG(v). Therefore, the seller will breach when35:

c > p − 1 − G(p + ˇ)G(p + ˇ)

ˇ:=Brr(p, ˇ). (11)

ince Brr(p, ˇ) < p, we conclude that the ADr regime (under the American rule) sometimes induces strategic breach (breach when perfor-ance is profitable). With a strategic breach there is a positive probability the buyer will file a lawsuit, thus leading to disclosure of the

uyer’s private information through the discovery process in litigation. As a result the seller can extract some surplus from the buyer inenegotiation. Furthermore, under the American rule the verification cost is borne by the buyer. Hence, breach may be more attractive tohe seller than performance even when performance is profitable. This is in contrast with the case of no renegotiation, where AD inducesoluntary performance at a loss in some cases.

Also, as can be seen from the description of the renegotiation strategies, under AD the renegotiation opportunity and the inducedtrategic breach do not lead to first-best allocation, in contrast to the case under ED. Under AD, the verification cost becomes a wedgehwarting efficient ex post allocation.

.2.2.2. The English rule. Under the English rule the seller bears the verification cost. At Time 3 the buyer will accept an offer only if ituarantees her a payoff of at least v–p. Accordingly, the seller’s optimal strategy in renegotiation is to offer to trade at the original price p if

≥ c–ˇ; and to breach and not make any renegotiation offer otherwise. Anticipating the litigation and renegotiation outcome in Time 3, the

32 For simplicity, we assumed away the cost of discovery. Introducing a positive cost of discovery would have a consequence similar to the effect of introducing positiveost of verifying actual damages, which is analyzed in the text. However, if the analysis is not restricted to simple fixed price contracts, Schwartz and Watson (2004) showhat parties may endogenously influence the costs of contracting and renegotiation, which affect their choice of initial contractual forms (simple versus complex contracts).33 Luelfesmann (2009) illustrates in a model with investments that a party may strategically underinvest so as to render ex post trade inefficient, and then to extract surplusia renegotiation with the other party.34 First best will not be attained if there is a cost of discovery. However, the main result and implication are quite general even when discovery is costly. When the cost ofiscovery is small relative to the joint expected surplus from renegotiation, the seller is still motivated to strategically breach in order to extract information and surplushrough renegotiation.35 There are three cases: (1) c < p + 2ˇ. In this case c − < p + ˇ, if the buyer sues upon breach (i.e. v > p + ˇ), then c − < p + < v, the seller would offer to tradet price p + ˇ, and his expected payoff from breach is [1 − G(p + ˇ)](p + − c). Comparing this with his payoff from performance, p − c, the seller will breach only when

> p − [(1 − G(p + ˇ))/G(p + ˇ)]ˇ. (2) p + 2 ≤ c < v + ˇ. In this case p + ≤ c − < v. The seller’s expected payoff from breach is∫ c−ˇ

p+ˇ(p − v)dG(v) +

∫ v

c−ˇ(p + − c)dG(v).

omparing this with his payoff from performance, p–c, the seller will always breach in this region. (3) c ≥ v + ˇ. In this case c − ≥ v. The seller’s expected payoff from

reach is∫ v

p+ˇ(p − v)dG(v). Comparing this with his payoff from performance, p–c, the seller will always breach in this region. Therefore, in sum, the seller will breach only

f c > p − [(1 − G(p + ˇ))/G(p + ˇ)]ˇ.

3

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c

fibdpa

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48 Z. Liu, R. Avraham / International Review of Law and Economics 32 (2012) 339– 355

uyer in Time 2 will sue for damages only if v>p. So the seller’s expected payoff from breach is:∫ max(c−ˇ,p)

p(p − v − ˇ)dG(v) +

∫ vmax(c−ˇ,p)(p −

)dG(v). Therefore, he will breach when c > p:=Brr(p, �).36

As can be seen from the above breach threshold, under ADr with the English rule there is neither strategic breach nor strategic per-ormance. This is quite different from the case under the American rule, and it is the result of two countervailing effects of breach: one isnformation disclosure via the discovery process, which allows the seller to take advantage of renegotiation to extract surplus from theuyer; and the second effect is that the seller will now have to bear the verification cost if the buyer sues and the renegotiation breaksown. These two countervailing effects cancel out each other under ADr with the English rule, so that the seller breaches if and only iferformance is unprofitable. Additionally, as can be seen from the renegotiation strategies, under the English rule, the verification costgain handicaps a fully efficient ex post allocation under AD.

We have shown that with renegotiation EDr attains first best. Therefore, ADr ≺ EDr. (The equilibrium prices and joint payoffs under ADr

ith either the American or the English rule are specified in Appendix B) The result is summarized in Proposition 2:

roposition 2. Assuming that parties may renegotiate the contract ex post and that verifying ex post damages is costly, the following holds:

(i) EDr is the first best remedy, unconditionally superior to ADr.(ii) The opportunity for renegotiation enhances efficiency when the default remedy for breach is ex ante expectation damages; in contrast,

the opportunity to renegotiate may make the actual damages more or less efficient depending on the verification cost and valuation/costdistributions.

iii) Under ADr with the American rule, the seller sometimes strategically breaches in order to extract surplus from the buyer, while he does notdo so under the English rule.

In sum, we have demonstrated that when verifying ex post damages is costly, both with and without renegotiation, courts would doetter to stick to the fixed ex ante foreseeable expectation damages rather than seek ex post accuracy in determining damages.

. Analysis of the case with general litigation costs

.1. The case of no renegotiation

We now assume that the parties’ general litigation costs (such as attorneys’ fees) are lb and ls for the buyer and seller, respectively. Forimplicity in this section we assume that there is no verification cost of ex post damages. The following notations are used (again, we useat (tilde) at top to denote the American (English) rule):

EDl—ex ante expectation damages regime with positive litigation costs; similarly, we denote actual damages regime as ADl;Brl(p)—the seller’s breach threshold under actual damages with positive litigation costs, given price p.

.1.1. Ex ante expectation damages

.1.1.1. The American rule. We first assume that the default remedy is ex ante expectation damages, and that parties do not renegotiatehe contract ex post. When the American rule applies, each party bears his (her) own litigation costs. To solve for the equilibrium pricender this regime, we consider two cases distinguished by the buyer’s litigation decision upon breach. Case (1) p > �v − lb. In this case, sincehe buyer’s payoff from litigation, �v − p − lb, is negative, she never sues upon breach. As a result, the seller breaches if c > p. The seller’sptimization problem in this price region (we denote the price in this region as p(1)) is:

Maxp(1) �EDl

S =∫ p(1)

0

(p(1) − c)dF(c),

s.t. �EDl

B =∫ p(1)

0

(�v − p(1))dF(c) ≥ 0 and p(1) > �v − lb.

Obviously, p(1)∗ = �v, and the seller’s expected payoff is �(1)S =

∫ �v

0(�v − c)dF(c).

Case (2) p ≤ �v − lb. In this case, since the buyer’s payoff from litigation, �v − p − lb, is non-negative, she always sues upon breach. As aesult, the seller breaches only if c > �v + ls. His optimization problem in this price region (denoted as p(2)) is:

Maxp(2) �EDl

S =∫ �v+ls

0

(p(2) − c)dF(c) +∫ c

�v+ls

(p(2) − �v − ls)dF(c),

∫ ∫

s.t. �EDl

B =�v+ls

0

(�v − p(2))dF(c) +c

�v+ls

(�v − p(2) − lb)dF(c) ≥ 0

36 There are three cases: (1) c ≤ p. In this case c − < p, if the buyer sues upon breach (i.e., v > p), then c − < p < v, the seller would offer to trade, and his expectedayoff from breach is [1 − G(p)](p − c). Comparing this with his payoff from performance, p − c, the seller will always perform in this region. (2) p < c < p + ˇ. In this case

− < p, the seller’s expected payoff from breach is [1 − G(p)](p − c). Comparing this with his payoff from performance, p − c, the seller will always breach in this region

since now p − c). (3) c ≥ p + ˇ. In this case c − ≥ p. The seller’s expected payoff from breach is∫ c−ˇ

p(p − v − ˇ)dG(v) +

∫ v

c−ˇ(p − c)dG(v). Comparing this with his payoff

rom performance, p − c, the seller will always breach in this region. Therefore, in sum, the seller will breach only if c > p.

Z. Liu, R. Avraham / International Review of Law and Economics 32 (2012) 339– 355 349

and p(2) ≤ �v − lb.

Obviously, in this region p(2)∗ = �v − lb, and the seller’s expected payoff is �(2)S =

∫ �v+ls0

(�v − lb − c)dF(c) −∫ c

�v+ls(lb + ls)dF(c).

It is simple to show that �(1)S > �(2)

S . Therefore, the optimal price pEDl = �v, and the joint expected payoff is:

j�EDl

=∫ �v

0

(�v − c)dF(c). (12)

This is the same as the joint expected payoff under the optimal money damages in the case of no litigation costs, as was illustrated inSection 3.

4.1.1.2. The English rule. Now assume the English rule applies, so that S bears both parties’ litigation costs. Similarly to the above, it can beshown that the joint expected payoff under the English rule is:

j�EDl

=∫ �v

0

(�v − c)dF(c) = j�EDl

. (13)

Therefore, under ex ante expectation damages with general litigation costs, the equilibrium joint payoff is the same under both the Englishand the American rule.

We now compare ED with the welfare-maximizing money damages when there are general litigation costs. To determine the optimalcourt-imposed (money) damages, we assume that given some court-imposed damages, the buyer’s optimal litigation threshold is y (i.e.she will sue upon breach only when v ≥ y); while the seller’s optimal breach threshold is x (i.e. he will breach when c ≥ x). Then the jointexpected payoff is j� =

∫ x

0(�v − c)dF(c) − [1 − F(x)][1 − G(y)](lb + ls). The court will choose money damages such that the induced breach

and litigation thresholds maximize the joint expected payoff. Simple analysis yields a solution: x∗ = �v; y∗ = v. As we showed above, exante expectation damages induce these decision thresholds precisely. Hence ED is also the optimal money damages remedy, under boththe American and the English rules. In both cases the buyer never files a lawsuit, so parties do not incur litigation costs.

4.1.2. Actual damagesWe delegate to Appendix C a detailed specification of equilibrium under actual damages with positive litigation costs (when either the

American or the English rule applies). Since ED are the welfare-maximizing money damages, as shown in Section 4.1.1, they are superiorto ex post actual damages. Proposition 3 summarizes:

Proposition 3. Assuming that parties commit to not renegotiate the contract after learning new information, and that there are positive litigationcosts, the following holds:

(i) ED are superior to AD under both the American rule and the English rule.(ii) ED are the welfare-maximizing money damages.

Fig. 4. Numerical comparison of EDr and ADr ’s efficiency under the American and the English rule. (a) Difference in expected joint payoffs between EDr and ADr under theAmerican rule. (b) Difference in expected joint payoffs between EDr and ADr under the English rule.

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50 Z. Liu, R. Avraham / International Review of Law and Economics 32 (2012) 339– 355

.2. The case with renegotiation

We now assume that the parties may renegotiate the contract, in the same way as in Section 3.2. Our analysis yields that there is nolear-cut analytical solution to the comparison between the efficiency of the two remedies when there are positive litigation costs andenegotiation.37 We therefore conduct a numerical analysis instead. We assume that the buyer’s valuation is uniformly distributed on [0,1],nd the seller’s cost is uniformly distributed on [0, a], where a runs from 0.5 to 1. First we fix lb = 0.125, and let ls be uniformly distributedn [0, 0.25]. Fig. 4 shows the difference between the expected joint payoffs under EDr and under ADr when the American rule (Fig. 4(a))r the English rule (Fig. 4(b)) applies. The y-axis represents a, the upper bound of S’s cost distribution. The x-axis represents ls, the seller’sitigation cost. The z-axis represents the difference in expected joint payoffs under the two remedies (EDr and ADr). (We do not present thether figure where we fix ls = 0.125, and let lb ∼ U[0, 0.25], which yields similar results.)

As can be seen from Fig. 4, under the American rule with renegotiation the ex ante expectation damages remedy is more efficient thanctual damages, even when litigation is costly. This is true even if courts can accurately assess actual damages without any verification cost.nder the English rule, EDr is more efficient than ADr, unless the litigation cost is sufficiently high relative to the expected joint surplus

rom trade.

. Conclusion

In an incomplete contracts framework with post-contracting private information, we have shown that when litigation is costly (eitherecause of costs of verifying ex post damages, or because of general litigation costs such as attorneys’ fees), ex post expectation damagesay lead to over-performance (as in Avraham & Liu, 2012) or under-performance. We find that ex ante expectation damages are almost

lways more efficient than ex post actual damages. As demonstrated, ex post accuracy in damages assessment does not help, but ratherinders the effort of motivating efficient incentives to breach. Therefore courts should commit to awarding fixed ex ante foreseeablexpected damages, rather than seek accuracy by adapting damages to ex post information. The prevalent approach of picking either theower or the higher amount between actual damages and ex ante foreseeable expectation damages is not supported even when litigationosts, verification costs, and renegotiation are considered. Our recommendation to award foreseeable damages strengthens when there is

need to incentivize the promisee to minimize her harm by seeking another party to contract with. We find one exception, and that ishen the English rule applies, renegotiation is costless, and litigation costs are sufficiently high relative to the expected joint surplus from

rade; then ex post damages may be superior.One may wonder whether courts are able to handle the determination of the ex ante expectation damages. We start by observing that in

ategories of cases where the determination of ex ante damages proves difficult yet ex post market price is readily available, it is plausiblehat ex post expectation damages might be superior because the savings in damages determination costs can outweigh the benefit frometter performance incentives that the ex ante regime provides. This could be true, for example, in situations where thick markets exist.

n many other categories of cases, however, courts may well find it easier to estimate the ex ante expectation damages. Arguably, provingxed ex ante damages in court should not be much different from proving lost future profits. In both cases, courts are presented withvidence about the distribution of future damages. In the lost profits case, it is future damages vis-a-vis the time of breach; whereas inhe fixed ex ante damages case, it is future damages vis-a-vis the point of contracting. Considering that many courts allow recovery of lostuture profits due to a breach in both established business cases (e.g. Denny Const., Inc. v. City and County of Denver, 2009) and in newusiness cases (e.g. Chung v. Kaonohi Center, 1980), estimating ex ante expectation damages should not prove too difficult a task.

Furthermore, other contract doctrines already employ the ex ante perspective, including in determining expectation damages. First,ourts use the ex ante perspective when they enforce doctrines such as mistake, impossibility, and frustration. Second, when awardingctual ex post expectation damages, courts typically limit damages to the foreseeable results from a breach at the time the contract wasade. Indeed, the ex ante damage calculation exercise is required in order to operationalize Hadley v. Baxendale (1854). Third, when

eviewing liquidated damages clauses, courts are required, under the penalty doctrine, to limit these damages to the reasonable ex antestimation of the non-breaching party’s expectation interest.38

Against the backdrop of these fundamental, commonplace doctrines, the argument that courts are incapable of calculating ex antexpectation damages is unconvincing. In light of the gains in efficiency promoted by incentives under ex ante foreseeable damages, it is aask courts should be willing to undertake.

cknowledgements

We are grateful to Ian Ayres, Oren Bar-Gill, Omri Ben-Shahar, Helmut Bester, Richard Brooks, Albert Choi, Daniel Göller, Marty Grace,eff Harper, Avery Katz, Steve Lamb, Yue Qiao, Alan Schwartz, Alexander Stremitzer, Ajay Subramanian, John Tatom, Terrie Troxel, Ericlm, Bill Warfel, Jian Wei, Abraham Wickelgren, Adam Winship and seminar participants at University of Texas, Georgia State University,

ndiana State University, University of Virginia, Shandong University, the annual meetings of American Law and Economics Association,nd Asian Law and Economics Association for their helpful comments on various drafts of this paper. We also appreciate two anonymous

eferees for their very helpful comments and suggestions. The usual disclaimer applies.

37 The detailed specifications of equilibrium price and joint payoff in this case are delegated to Appendix D.38 Indeed, since in our model there is no pre-contractual private information, the court’s task of determining ex ante damages is no different from the contracting parties’ask of writing a mutually agreed liquidated damages clause in the contract based on information commonly observed by the parties at ex ante stage.

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Z. Liu, R. Avraham / International Review of Law and Economics 32 (2012) 339– 355 351

ppendix A. Proof of binding constraint in equilibrium under AD

roof. Let � be the multiplier for the constraint. Then the Lagrangian for the seller’s optimization program is L = �ADS + ��AD

B . Therst-order conditions are:

Lp = (1 − �)F(Br(p, ˇ)) + [1 − F(Br(p, ˇ))]{(1 − �)[1 − G(p + ˇ)] + ˇg(p + ˇ)} + �f (Br(p, ˇ))∂Br(p, ˇ)

∂p

[ +

∫ p+ˇ

0

(v − p − ˇ)dG(v)

]= 0; (A1)

L� = �ADB =

∫ Br(p,ˇ)

0

(�v − p)dF(c) +∫ c

Br(p,ˇ)

∫ v

p+ˇ

(v − p − ˇ)dG(v)dF(c) ≥ 0; (A2)

� ≥ 0; �L� = ��ADB = 0. (A3)

We claim that the buyer’s participation constraint is binding, i.e. her expected payoff is zero in equilibrium. Otherwise, �ADB > 0, which

mplies � = 0 by (A3). Then (A1) simplifies to F(Br(p, ˇ)) + [1 − F(Br(p, ˇ))][1 − G(p + ˇ) + ˇg(p + ˇ)] = 0. This is a contradiction since theeft hand side is always positive. Therefore, we have

�ADB =

∫ Br(p,ˇ)

0

(�v − p)dF(c) +∫ c

Br(p,ˇ)

∫ v

p+ˇ

(v − p − ˇ)dG(v)dF(c) = 0.

ppendix B. Equilibrium price and expected joint payoff under actual damages with renegotiation when verification is costly

.1. Under the American rule

The seller’s optimization problem is:

Maxp�ADrS =

∫ Brr (p,ˇ)

0

(p − c)dF(C) +∫ c

Brr (p,ˇ)

⎡⎢⎢⎣∫ min(v,max(p+ˇ,c−ˇ))

p+ˇ

(p − v)dG(v)

+∫ v

min(v,max(p+ˇ,c−ˇ))

(p + − c)dG(v)

⎤⎥⎥⎦dF(c)

s.t. �ADrB =

∫ Brr (p,ˇ)

0

(�v − p)dF(c) +∫ c

Brr (p,ˇ)

∫ v

p+ˇ

(v − p − ˇ)dG(v)dF(c) ≥ 0.

It can be shown that in equilibrium the constraint is binding and the price and joint surplus are given as follows:

pADr = �v +[1 − F(Brr(pADr , ˇ))]

∫ vpADr +ˇ

(v − �v − ˇ)dG(v)

1 − G(pADr + ˇ) + F(Brr(pADr , ˇ))G(pADr + ˇ); (A4)

j�ADr =

∫ Brr (pADr ,ˇ)

0

(�v − c)dF(c) +∫ c

Brr (pADr ,ˇ)

⎡⎢⎢⎢⎣∫ v

min(v,max(pADr +ˇ,c−ˇ))

(v − c)dG(v)

−∫ min(v,max(pADr +ˇ,c−ˇ))

pADr +ˇ

ˇdG(v)

⎤⎥⎥⎥⎦dF(c). (A5)

.2. Under the English rule

It can be shown that in equilibrium the buyer’s participation constraint is binding, i.e. �ADrB =

∫ p

0(�v − c)dF(c) +

c

p

∫ v

p

(v − p)dG(v)dF(c) = 0, and the price and expected joint surplus are:

pADr = �v +[1 − F(pADr )]

∫ vpADr

(v − �v)dG(v)

1 − G(pADr ) + F(pADr )G(pADr ); (A6)⎡∫ v ⎤

j�ADr =

∫ pADr

0

(�vv − c)dF(c) +∫ c

pADr

⎢⎢⎣ min(v,max(c−ˇ,p))

(v − c)dG(v)

−∫ min(v,max(c−ˇ,p))

pADr

ˇdG(v)

⎥⎥⎦dF(c). (A7)

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52 Z. Liu, R. Avraham / International Review of Law and Economics 32 (2012) 339– 355

ppendix C. Equilibrium price and expected joint payoff under actual damages when litigation is costly and there is noenegotiation

.1. Under the American rule

Here we assume the default remedy is actual damages, and that the American rule applies. Then at Time 3 if the seller breaches the

ontract, the buyer will sue only if v > p + lb, so the seller’s expected payoff from breach is:∫ v

p+lb(p − v − ls)dG(v). Therefore, he will breach

hen

c > p +∫ v

p+lb

(v − p + ls)dG(v) := Brl(p).

It is obvious that Brl(p) > p, which implies that the seller sometimes voluntarily delivers at a loss. The seller’s optimization problem is:

Maxp�ADl

S =∫ Br

l(p)

0

(p − c)dF(c) +∫ c

Brl(p)

∫ v

p+lb

(p − v − ls)dG(v)dF(c)

s.t. �ADl

B =∫ Brl(p)

0

(�v − p)dF(c) +∫ c

Brl(p)

∫ v

p+lb

(v − p − lb)dG(v)dF(c) ≥ 0.

In equilibrium the constraint is binding. The price and joint surplus are given in the following expressions:

pADl = �v +[1 − F(Br

l(pADl

))]∫ v

pADl +lb(v − �v − lb)dG(v)

1 − G(pADl + lb) + F(Brl(pADl ))G(pADl + lb)

; (A8)

j�ADl

=∫ Br

l(pADl

)

0

(�v − c)dF(c) − [1 − F(Brl(pADl

))][1 − G(pADl + lb)](lb + ls). (A9)

.2. Under the English rule

Now assume that the English rule applies, hence S bears both parties’ litigation costs. At Time 3 the buyer will sue for actual damages

nly if v > p, so the seller’s expected payoff if he breaches is:∫ v

p(p − v − lb − ls)dG(v). Therefore, he will breach when

c > p +∫ v

p

(v − p + lb + ls)dG(v):=Brl(p).

It is obvious that Brl(p) > p, implying that the seller sometimes voluntarily delivers at a loss. It can be shown that the equilibrium price

nd joint surplus are the following:

pADl = �v +[1 − F(Br

l(pADl

))]∫ v

pADl (v − �v)dG(v)

1 − G(pADl ) + F(Brl(pADl ))G(pADl )

; (A10)

j�ADl

=∫ Br

l(pADl

)

0

(�v − c)dF(c) − [1 − F(Brl(pADl

))][1 − G(pADl)](lb + ls). (A11)

ppendix D. Equilibrium prices and joint payoffs under ex ante expectation damages and actual damages when litigation isostly and parties may renegotiate the contract ex post

For simplicity we will focus on the cases where c ≤ v.

.1. Ex ante expectation damages with renegotiation

.1.1. The American ruleNow the default remedy is ex ante expectation damages. Parties may renegotiate the contract ex post. First assume that the American

ule applies. As we discussed before, there are two cases under this regime distinguished by the buyer’s litigation decision upon breach:Case (1): p > �v − lb. In this price region, as we have proven, p(1)∗ = �v, and the seller’s expected payoff is �(1)∗

S =∫ �v

0(�v − c)dF(c).

Case (2): p ≤ �v − lb. In this case, since the buyer’s payoff from litigation, �v − p − lb, is non-negative, she always sues upon breach.iven that the litigation costs are sunk at the renegotiation stage, the buyer will only accept an offer with guaranteed payoff of at leastv − p. Therefore, the seller’s optimal renegotiation strategy is to offer to trade at price p − �v + v when v ≥ c; and to not make any offer

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Z. Liu, R. Avraham / International Review of Law and Economics 32 (2012) 339– 355 353

hen v < c. Anticipating the strategies in Time 3, the seller at Time 2 chooses to breach when∫ c

0(c − v)dG(v) > ls.39 The left hand side of

he inequality is the expected renegotiation surplus he can extract from the buyer, and the right hand side is the litigation cost. Denote:

ϕ(c):=∫ c

0

(c − v)dG(v) − ls (A12)

nd let ϕ(c∗) = 0. Obviously, ϕ′(c) > 0. Therefore, the seller breaches when c > c∗. The seller’s optimization problem in this price regiondenoted as p(2)) is:

Maxp�EDl

rS =

∫ c∗

0

(p − c)dF(c) +∫ c

c∗

⎡⎢⎢⎣∫ c

0

(p − �v)dG(v)

+∫ v

c

(p − �v + v − c)dG(v) − ls

⎤⎥⎥⎦dF(c)

s.t. �EDl

rB =

∫ c∗

0

(�v − p)dF(c) +∫ c

c∗(�v − p − lb)dF(c) ≥ 0 and p ≤ �v − lb.

Obviously, p(2)∗ = �v − lb. The seller’s expected payoff in this price region is �(2)∗S =

∫ c∗

0(�v − lb − c)dF (c) +

c

c∗

[∫ vc

(v − c)dG(v) − lb − ls

]dF(c). The joint expected payoff is:

j�EDl

r = max(�(1)∗S , 1{�(2)∗

S>�(1)∗

X}[�

(2)∗S + lbF(c∗)]), (A13)

here the indicator function 1{ϕ} = 1 if the statement ϕ is true, and 0 otherwise.

.1.2. The English ruleUnder ex ante expectation damages and the English rule, the equilibrium price must satisfy p ≤ �v. Otherwise, if p > �v, the buyer’s

xpected payoff would always be negative (either when the seller performs or when the seller breaches), and she would never have signeduch a contract. Hence, given p ≤ �v, the buyer always sues upon breach and she will only accept a renegotiation offer with guaranteedayoff of at least �v − p. The seller’s optimal strategy is to offer to trade at price p − �v + v when v ≥ c; and to not make any offer at theenegotiation stage when v < c. Anticipating the strategies at Time 3, the seller at Time 2 chooses to breach when

∫ c

0(c − v)dG(v) > lb + ls.40

he left hand side of this inequality is the expected renegotiation surplus he can extract from the buyer, and the right hand side is theeller’s litigation cost. Denote:

ı(c):=∫ c

0

(c − v)dG(v) − lb − ls, (A14)

nd let ı(c∗) = 0.41 Obviously, ı′(c) > 0. Therefore, the seller breaches when c > c∗. The seller’s optimization problem in this price regiondenoted as p(2)) is:

Maxp�EDl

rS =

∫ c∗

0

(p − c)dF(c) +∫ c

c∗

⎡⎢⎢⎣∫ c

0

(p − �v)dG(v)

+∫ v

c

(p − �v + v − c)dG(v) − lb − ls

⎤⎥⎥⎦dF(c)

s.t. �EDl

rB = �v − p ≥ 0 and p ≤ �v.

Obviously, p(2)∗ = �v. The joint expected payoff (which is the same as the seller’s expected payoff) is:

j�EDl

r =∫ c∗

0

(�v − c)dF(c) +∫ c

c∗

[∫ v

c

(v − c)dG(v) − lb − ls

]dF(c). (A15)

.2. Actual damages with renegotiation

.2.1. The American rule

Now the default remedy is actual damages. Parties may renegotiate the contract ex post. First assume that the American rule applies.

t Time 3 the buyer’s damages are observed through the discovery process. She will accept an offer only if it guarantees her a payoff noess than v − p. Given this, the seller’s optimal strategy is to renegotiate and offer delivery at price p when v ≥ c; and to not make any offer

39 Because his payoff from performance is p − c; while his expected payoff from breach is∫ c

0(p − �v)dG(v) +

∫ v

c(p − �v + v − c)dG(v) − ls = p − �v +

∫ v

c(v − c)dG(v) − ls. He

ill breach when the payoff from breach exceeds the one from performance.40 Because his payoff from performance is p − c; while his expected payoff from breach is

∫ c

0(p − �v)dG(v) +

∫ v

c(p − �v + v − c)dG(v) − lb − ls = p − �v +

∫ v

c(v − c)dG(v) −

b − ls. He will breach when the payoff from breach exceeds the one from performance.41 From the definitions it is straightforward to see that c∗ > c∗ .

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54 Z. Liu, R. Avraham / International Review of Law and Economics 32 (2012) 339– 355

hen v < c. Anticipating the strategies in Time 3, the buyer will sue for damages upon breach at Time 2 only if v > p + lb. The seller’s payoff

rom performance is p–c; and his expected payoff if he breaches is:∫ max(c,p+lb)

p+lb(p − v − ls)dG(v) +

∫ vmax(c,p+lb)

(p − c − ls)dG(v). Therefore, the

eller will breach when

c >1

G(max(c, p + lb))

[∫ p+lb

0

pdG(v) +∫ v

p+lb

lsdG(v) +∫ max(c,p+lb)

p+lb

vdG(v)

]:=Brl

r(p). (A16)

The seller’s optimization problem is:

Maxp�ADl

rS =

∫ Brl

r (p)

0

(p − c)dF(c) +∫ c

Brl

r (p)

⎡⎢⎢⎢⎣∫ max(c,p+lb)

p+lb

(p − v − ls)dG(v)

+∫ v

max(c,p+lb)

(p − c − ls)dG(v)

⎤⎥⎥⎥⎦dF(c);

s.t. �ADl

rB =

∫ Brl

r (p)

0

(�v − p)dF(c) +∫ c

Brl

r (p)

∫ v

p+lb

(v − p − lb)dG(v)dF(c) ≥ 0.

It can be shown that the constraint is binding, and the equilibrium price and expected joint payoff are:

pADlr = �v +

[1 − F(Brl

r(pADlr ))]

∫ v

pADlr +lb

(v − �v − lb)dG(v)

1 − G(pADlr + lb) + F(Br

l

r(pADlr ))G(pADl

r + lb)(A17)

j�ADl

r =∫ Br

l

r (pADlr )

0

(�v − c)dF(c) +∫ c

Brl

r (pADlr )

∫ v

max(c,pADlr +lb)

(v − c)dG(v)dF(c) −∫ c

Brl

r (pADlr )

∫ v

pADlr +lb

(lb + ls)dG(v)dF(c). (A18)

.2.2. The English ruleWe now analyze the case where the losing party (i.e. the seller) bears all the litigation costs. At the litigation stage the buyer will accept

n offer only if her guaranteed payoff is at least v − p. Given this, the seller’s optimal strategy is to offer to deliver at price p if v ≥ c; and toot make any offer when v < c. Therefore, at Time 2 the buyer will sue upon breach only if v > p As a result, the seller’s expected payoff from

reach is∫ max(c,p)

p(p − v − lb − ls)dG(v) +

∫ vmax(c,p)

(p − c − lb − ls)dG(v). Therefore, he will breach when

c >1

G(max(c, p))

[∫ p

0

pdG(v) +∫ v

p

(lb + ls)dG(v) +∫ max(c,p)

p

vdG(v)

]:=Brl

r(p). (A19)

It can be shown that the equilibrium price and expected joint payoff are:

pADlr = �v +

[1 − F(Brl

r(pADlr ))

∫ pADlr

0(�v − v)dG(v)]

1 − G(pADlr ) + F(Br

l

r(pADlr ))G(pADl

r ); (A20)

j�ADl

r =∫ Br

l

r (pADlr )

0

(�v − c)dF(c) +∫ c

Brl

r (pADlr )

∫ v

c

(v − c)dG(v)dF(c) −∫ c

Brl

r (pADlr )

∫ v

pADlr

(lb + ls)dG(v)dF(c). (A21)

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