Liability Insurance: Equilibrium Contracts under Monopoly and Competition.

Transcription

1 Liability Insurance: Equilibrium Contracts uner Monopoly an Competition. Jorge Lemus, Emil Temnyalov, an John L. Turner April 24, 2017 Abstract In thir-party liability lawsuits (e.g., patent infringement or prouct liability), a thir party emans compensation for amages from an agent. Verifying the harm is costly, so liability insurance is valuable for the agent because it improves its bargaining position when negotiating a settlement. We show that equilibrium contracts uner averse selection iffer ramatically from existing results on firstparty insurance: in a competitive market, only an inefficient pooling equilibrium may exist; in a monopolistic setting, the insurer offers at most two contracts which uner-insure low-risk types an may inuce high-risk types to litigate. JEL Coe: D82, G22, K1, K4. Keywors: insurance, averse selection, liability, litigation, ex post moral hazar. We appreciate comments by Dan Bernhart, Yeon-Koo Che, In-Koo Cho, Bruno Jullien, Stefan Krasa, an seminar auiences at IIOC (2016), ESAM (2016), EARIE (2016), LACEA (2016), APIOC (2016), AETW (2017), Monash University, University of Illinois Urbana-Champaign, an University of Texas A&M. University of Illinois Urbana-Champaign, Department of Economics. jalemus@illinois.eu University Technology Syney, Business School. emil.temnyalov@uts.eu.au University of Georgia, Department of Economics. jlturner@uga.eu. 1

2 1 Introuction Thir-party liability insurance is funamentally ifferent from first-party insurance: in the former setting, an agent buys insurance to protect against liability for loss or amage cause to a thir party (e.g., patent infringement, prouct liability, employment-relate liability, or malpractice); whereas in the latter setting, an agent buys insurance to protect itself against losses (e.g., health, life, or property insurance). In liability insurance, claims for compensation require costly assignment of responsibility between the policy holer an a thir party a court must etermine whether the agent is responsible for the loss incurre by the thir party. A large theoretical literature on insurance an averse selection incluing the seminal work of Rothschil an Stiglitz (1976) an Stiglitz (1977) stuies first-party insurance. In reality, however, thir-party insurance is pervasive an nonetheless is not well unerstoo. In this paper we analyze markets for liability insurance contracts that eliver value to risk-neutral agents. Importantly, the value of liability insurance is istinct from that of first-party insurance, where agents instea buy insurance to reuce risks of wealth losses. Most liability lawsuits are settle out of court to avoi the costs involve in the legal process, an liability insurance is valuable in part because it improves the agent s payoff from negotiating a settlement with a thir party. We consier a setting in which an agent buys insurance that covers litigation costs an/or amages. At the time of contracting, the probability that the agent will be liable to a thir party for amages (the agent s type) is imperfectly known. If an when a thir party subsequently sues the agent for amages, the agent an thir party may bargain over a settlement or litigate. Agents that settle introuce no costs to the insurer, whereas agents that litigate introuce strictly positive costs to the insurer. The ex-post ecision to settle or to litigate creates a iscontinuity in the insurer s cost function, which ramatically changes the equilibrium contracts compare to the existing literature on first-party insurance uner averse selection. We stuy two canonical market structures: a perfectly competitive market for liability insurance with free entry, following Rothschil an Stiglitz (1976); an a mechanism esign setting in which a monopolist esigns an prices insurance contracts. We stuy two information environments: symmetric information, where neither the agent nor the insurer know the agent s probability of liability; an asymmetric information, where the 2

3 agent alone is privately informe. For both market structures, we fin that contracts for thir-party insurance iffer significantly from stanar first-party insurance contracts. First, in a competitive market uner asymmetric information, we fin that for any istribution of types there can only be pooling equilibria, an any such equilibrium never inuces litigation an features uner-insurance. Secon, with a single seller an regular type istributions, we show that in any optimal mechanism at most two contracts are offere in equilibrium one that covers legal costs only, an one that covers legal costs an partially covers amages payments. We also show that amages insurance is more generous, an inuces more litigation, uner symmetric information than uner asymmetric information. Our results on the equilibrium of a competitive market for thir-party liability insurance contrast sharply with the seminal work of Rothschil an Stiglitz (1976), where only separating contracts are offere in equilibrium, ue to cream skimming. With first-party insurance, in a caniate pooling equilibrium, an insurer woul be able to profitably eviate by offering a contract that only attracts types who generate positive surplus, which unermines the cross-subsiization neee to sustain the pooling equilibrium. In contrast, with thir-party insurance, cross-subsiization is not necessary as long as insurance oes not inuce litigation by any type that buys it. This enables pooling to survive in equilibrium. In aition, the cream skimming effect is reverse. A separating equilibrium in a competitive market for liability insurance requires contracts sol at ifferent prices, because otherwise types woul pool on the more generous insurance. But for a contract to sell for a positive price an yiel zero profit, it must attract types that settle an types that litigate. Such a contract cannot survive in equilibrium, because it oes require cross-subsiization an is therefore cream-skimme by another contract that only attracts types that settle. This implies that a separating equilibrium oes not exist. Similar to Rothschil an Stiglitz (1976), however, we fin that averse selection estroys the possibility of equilibrium altogether, when there are too few high-risk types of agents. For the most part, our finings thus inicate that the canonical moel of averse selection in markets for insurance applies only to first-party insurance. In particular, thir-party liability insurance requires a richer moel that also consiers the effect of insurance on an agent s ex post actions. Our results on the optimal mechanism with a single seller also iffer from existing results on insurance contracts, such as in Chae an Schlee (2012), where the optimal menu 3

4 iscriminates among ifferent agent types. In sharp contrast we fin that the insurer will offer at most one contract that covers amages. In fact, the insurer s problem of esigning a menu of liability insurance contracts is one of mechanism esign with a nonifferentiable value function, 1 where the non-ifferentiability arises because the agent has a non-contractible ex-post action to settle or litigate. This choice introuces a novel type of ex-post moral hazar that oes not appear in first-party insurance, because the insurance changes the agent s incentives to settle, which enters the seller s mechanism esign problem as an aitional ex post incentive constraint. We fin that in general the insurer wants to fully cover the legal costs of all agent types, an to partially cover the amages payments of a subset of relatively high ( riskier ) types. The solution generally features istortions at the top, in aition to the more familiar type of istortion at the bottom, an in fact the optimal mechanism is not necessarily efficient for the highest type. In some cases, the optimal contract may inuce inefficient litigation in equilibrium, where in the absence of insurance there woul have been no litigation. This points to novel potentially negative welfare effects of liability insurance. Finally, the monopoly insurer s problem of choosing the level of amages bears some similarities to a single-price monopolist s choice of prouct quality, e.g., (Spence, 1975, 1976). Higher coverage for amages raises the willingness-to-pay of all agents that buy insurance, an raises the insurer s costs by inucing more litigation. We fin that the level of amages covere uner asymmetric information is (weakly) higher when information is symmetric. Intuitively, a monopolist insurer selling to uninforme agents cares about how the level of amages insurance affects the willingness-to-pay conitional on each possible level of the probability of liability. This is similar to the social planner s concern about the average marginal effect of prouct quality on willingness to pay. Uner asymmetric information, in contrast, a monopolist insurer cares about how the level of amages affects the willingness-to-pay of the marginal type of agent that buys insurance. This is similar to the monopolist s concern about the marginal marginal effect of prouct quality on willingness to pay. We fin that the marginal effect of increasing amages coverage is higher for agents with higher willingness-to-pay i.e., the average marginal is higher so the monopolist optimally chooses higher amages coverage uner symmetric information. 1 Non-ifferentiability in mechanism esign is stuie by Carbajal an Ely (2013). 4

5 Motivating Example of Liability Insurance: Patent Infringement Patent litigation in the Unite States increase after the establishment of the Court of Appeals for the Feeral Circuit, in 1982, an further surge after 2004 (Bessen et al., 2015; Tucker, 2016). This surge which increase the number of cases from about 2,500 to 5,000 per year was largely been riven by litigation initiate by patent assertion entities ( PAEs ), also calle patent trolls. 2 Patent litigation is costly for firms an entrepreneurs (Bessen et al., 2011). Although markets for patent litigation insurance have existe in the Unite States since the 1980s, the recent increase in patent litigation has spurre more growth an activity in the market. Firms such as RPX Corporation, IPISC, Triology, an InsureCast now offer insurance to entrepreneurs an firms to cover some fraction of the legal costs or amages that they may have to pay as efenants in an infringement lawsuit. 3 These companies offer both offensive an efensive insurance contracts. The former is use by patent owners to pay for the cost of enforcing their patents, whereas the latter is use by proucing firms accuse of patent infringement to cover the legal costs an penalties impose by a court following a lawsuit. 4 A cornerstone feature of these contracts is the freeom of the policy holer to ecie whether to settle or to litigate: 5 The Policy Holer controls the lawsuit. The Company may suggest reliable an preferre counsel to the Insure but the Insure ultimately chooses [...] The Insure ictates the settlement terms, if any, not the Company. The market for patent insurance has also been active in Europe. 6 A 2006 stuy for the European Commission propose to make patent insurance manatory for small-tomeium-size enterprises. 7 Fuentes et al. (2009) stuy the trae-offs of this proposal. 2 See, for example, Chien (2009) an Tucker (2016). 3 For other companies offering Patent Infringement Insurance, please visit: Infringement-Lawsuits.pf 4 To see specific etails on some of the contracts, visit the following links: Trilogy Insurance:

6 2 Literature Review To the best of our knowlege, our paper is the first to stuy thir-party liability insurance markets uner averse selection. Our work relates to work on insurance in both law (Schwarcz an Siegelman, 2015) an economics (Dionne, 2013). First-party insurance markets with perfect competition have been extensively stuie, beginning with Rothschil an Stiglitz (1976). Subsequent work e.g., Wilson (1977); Miyazaki (1977); Riley (1979); Crocker an Snow (1985); Azeveo an Gottlieb (2017) shows that alternative equilibrium concepts change both the set of contracts that survive in equilibrium an the welfare implications of averse selection. But generally equilibrium contracting is the same across the ifferent concepts for the cases where equilibrium exists in the Rothschil-Stiglitz setting an pooling equilibria o not exist in those cases. In our setting, pooling equilibria obtain uner precisely this notion of equilibrium. The framework of Stiglitz (1977) to stuy the problem of a monopoly insurer uner averse selection is generalize by Chae an Schlee (2012). We use mechanism esign tools to erive the optimal monopoly menu of contracts. Because the agent s ex-post choice of settlement or litigation creates a non-ifferentiability in the value of insurance an the cost of the insurer, our analysis relates to the work of Carbajal an Ely (2013) on optimal mechanisms with non-ifferentiable value functions. Our work also relates to the literature on optimal contracting uner averse selection an moral hazar (Picar, 1987; Guesnerie et al., 1989). The key riving force in our moel is the improve bargaining position of an insure agent. Kirstein (2000), Van Velthoven an van Wijck (2001), Kirstein an Rickman (2004), an Llobet et al. (2012) have shown that riskneutral buyers may value insurance, because it makes litigation creible or it improves the policy holer s bargaining position. However, none of these papers stuy equilibrium uner averse selection or the optimal monopoly contract, an their results are qualitatively very ifferent from ours. Townsen (1979) also shows that contracts ajust when it is costly to verify an agent s private information. In an insurance context, this work helps explain why contracts inclue euctibles that reuce the frequency that an agent files a claim. In our setting, the agent is not privately informe about the true state of liability. Verification requires litigation. As a result, insurance in our context provies value even in cases where the agent chooses to settle out of court rather than verify the state. 6

7 Shavell (1982) stuies the effect of liability insurance on ex-ante moral hazar (eman for care) in a moel without ex-post bargaining. In contrast, we focus on ex-post bargaining given the equilibrium contracts uner ifferent market structures. Meurer (1992) investigates why it may be optimal for the insurer to offer a contract where it controls the litigation an settlement process on behalf of the insure, espite a potential conflict of interest. Motivate by patent litigation insurance, we focus instea on the case where the insure controls litigation an settlement The literature on offensive patent insurance is also relate. Llobet et al. (2012) an Buzzacchi an Scellato (2008) stuy insurance that covers a fraction of the patentee s litigation costs. This type of insurance makes patent infringement threats creible, which may eter entry by imitators. Duchene (2015) shows that with private information, patent holers may opt not to buy it because of an inability to sharply signal an avoi pooling equilibria. In our setting, by contrast, there is no gain to the insuring party from making litigation threats creible an the insurer is expose to significant losses when litigation occurs. Both factors affect equilibrium contracting. Historically, markets for thir-party insurance have been more volatile than first-party insurance markets. In 1986 in the Unite States, for example, premiums rose sharply an some insurers ecline to sell certain types of coverage. In the wake of this crisis, Priest (1987), Winter (1991) an Harrington an Danzon (1994) analyze how liability insurance iffers from other kin of insurance in particular, the ifficulty insurers have in forecasting liability losses. Unlike our setting, these papers o not focus on the role of insurance in shaping subsequent bargaining. Our work also relates to the literature on lawyer s contingent fees. Uner such contracts, lawyers charge lower upfront fees but they keep part of any payments aware. The agent that hire the lawyer may not receive the full litigation outcome. Dana an Spier (1993) show that contingency fees help solve an agency problem. Intuitively, an attorney who is pai using a contingency fee has stronger incentives to provie accurate information to her client about the strength of the case. Rubinfel an Scotchmer (1993) stuy a Rothschil-Stiglitz-style competition moel, an make the point that clients with high-quality cases can signal their cases strength by selecting hourly fees, while attorneys can signal their ability by requesting contingency fees. Gravelle an Waterson (1993) make similar points. Finally, Hay an Spier (1998) an Spier (2007) review the large literature on litigation an settlement. 7

8 3 Moel Consier a risk-neutral agent (A), or a firm, who sells a prouct or provies a service. The agent may cause amage or harm to a thir party (TP), thereby creating a legal liability. Only a court can verify whether or not the harm has occurre. To cover the legal costs an amages if the court etermines that the harm in fact has occurre, the agent may purchase thir-party liability insurance from a risk neutral insurer (I). Going to trial is costly: A pays a cost c A > 0 an TP pays c > 0. If the court etermines that the agent is liable, the agent must make a payment to the thir party. The agent s type is p [0, 1], which is the probability that the agent is foun liable. Figure 1 escribes the timing of the moel. t = 1 t = 2 t = 3 t = 4 A purchases a liability insurance policy α from I. TP sues A A an TP bargain to negotiate a settlement an to avoi litigation If there is no agreement, A an TP go to trial Figure 1: Timing of the events in the moel. At t = 1, the agent consiers buying liability insurance. Insurance contracts are efine by α = ( α L, α D ), where the insurer will pay α L to cover the litigation costs an α D to cover amages. 8 The set of contracts that the insurer can offer is A = {( α L, α D ) : α L [0, c A ], α D [0, ]}. After a lawsuit is file, at t = 2, the agent an the thir party can assess the probability of liability p, so they bargain at t = 3 uner complete information. 9 The agent s bargaining payoff at t = 3 epens on the probability of liability p, the insurance contract it has bought, an the ecision to settle or to go to trial. The thir party oes not have a creible litigation threat if p c. For a given agent s type p an contract 8 In Online Appenix B.2 we allow for insurance policies to also cover settlements. We show that in the optimal contract that insurer oes not cover settlements, so the main results of our paper are unaffecte. 9 In Online Appenix B.3 we iscuss the case of bargaining uner incomplete information. 8

9 α A, the agent s expecte payoff from litigation at t = 4 is (c A α L ) p( α D ) if p c, V (p, α, L) = 0 if p < c. (1) Notice the importance of the litigation costs in Equation 1: if c A = c = 0, this is precisely the Rothschil an Stiglitz (1976) framework uner risk neutrality. 10 At t = 3, the agent an the thir party Nash-bargain over a fee to settle the lawsuit. The agent s bargaining power is θ [0, 1]. Since bargaining occurs uner complete information, settlement occurs with probability one if an only if the bargaining surplus is non-negative. Notably, when the agent oes not have insurance, the bargaining surplus is c A + c > 0, so settlement always occurs. With insurance α, the bargaining surplus is instea S B = c + c A α L p α D. If α D = 0, S B is positive an inepenent of the liability probability, so there is always settlement. However, because the settlement fee is proportional to the joint surplus, the agent pays a lower settlement fee when is covere by insurance having insurance improves the agent s bargaining position. Within the class of contracts with α D = 0, the contract that maximizes the value of insurance for the agent is α L = c A. If α D > 0, the bargaining surplus epens on the probability of liability an coul be negative. In that case, the parties go to trial. In particular, the bargaining surplus is negative for agents with a probability of liability p larger than p c + c A α L α D. (2) If c p p, settlement increases the joint surplus, an the agent s payoff is V (p, α, S) = (c A ˆα L ) p( ˆα D ) + θs B = V (p, 0, S) + (1 θ)( α L + p α D ) If p > p, settlement ecreases the joint surplus, so litigation becomes unavoiable. In 10 In Rothschil an Stiglitz (1976), an iniviual has an initial wealth of W an will suffer a loss of with probability p. Consier an insurance policy that pays α 2 if the loss occurs. Uner risk neutrality, we can normalize W = 0. The agent s utility net of the cost of the policy is V (p, α 2 ) = p( α 2 ). 9

10 this case, the agent s payoff becomes V (p, α, L) = c A p + α L + p α D. Insurance allows an agent that settles to capture more of the bargaining surplus: it increases the payoff of a low-risk type by improving its bargaining position. The agent only captures a fraction (1 θ) of the savings inuce by a better bargaining position. 11 High-risk agents go to trial an part of their expenses are covere by insurance. The cost of insurance jumps iscontinuously at p = p, because the insurer pays no claims uner settlement but pays strictly positive claims when litigation occurs. Figure 12 summarizes the effects of insurance on the ecision to reach a settlement No threat Lower settlement fee Litigation c p 1 p Figure 2: The effect of insurance on litigation for ifferent types of agents. Lemma 1. Consier an insurance policy α = ( α L, α D ) A an p as efine in (2). The agent s willingness to pay for insurance, W (p, α), an the expecte cost for the insurer of proviing policy α to an agent of type p, K(p, α), are given by W (p, α) = (1 θ)(c + c A ) + (p p ) α D (1 θ) if p p (1 θ)(c + c A ) + (p p ) α, D if p > p (3) K(p, α) = 0 if p p c + c A + (p p ) α. D if p > p (4) All the proofs omitte in the text are in the appenix. Equation (3) shows that the willingness to pay is a continuous an convex function of p with a kink at p. Also, it epens on α L implicitly through the efinition of p. From equations (3) an (4) it is easy to see that the willingness to pay for insurance is always less than the cost of proviing it for high risk types that choose to litigate, 11 Notice the insurance oes not provie any value for an agent that has all the bargaining power (θ = 1) an settles, because the agent alreay captures all the bargaining surplus. 12 The agent faces no threat for p < c. For the remainer of the paper we restrict attention to p c. 10

11 i.e., for types p > p. In fact, the ifference between the willingness to pay an cost is exactly θ(c + c A ) for p > p. Figure 3 epicts the willingness to pay an the cost of proviing an insurance contract α to an agent of type p. c + c A K(p, α) W (p, α) (1 θ)(c + c A ) p c p Figure 3: W (p, α) is type p s willingness to pay for insurance policy α. The cost to the insurer of proviing the coverage prescribe by policy α for an agent of type p is given by K(p, α). Type p is inifferent between settlement an litigation. Corollary 1. We have: 1. The willingness to pay for contract ( α L, α D ) = (c A, 0) is (1 θ)c A. 2. For any p > p an for any policy α we have K(p, α) W (p, α) = θ(c + c A ). The intuition for Corollary 1 is the following. A contract that fully covers litigation costs but oes not cover amages always inuces the agent to settle. From the thir party s perspective, when the agent oes not pay for its own litigation costs, the agent has litigation costs equal to zero at the time of negotiating a settlement. This improves the agent s bargaining position an the thir party is unable to capture (1 θ)c A in bargaining rents. The reuction in the bargaining surplus lowers the settlement fee the agent pays the thir party, which is precisely the amount the agent is willing to pay for an insurance policy that fully covers litigation costs but oes not cover amages. The secon part of Corollary 1 shows that when the agent litigates instea of settling, the joint surplus of the insurer an the agent ecreases by θ(c+c A ), which is the bargaining surplus that woul have been capture by the agent in the settlement negotiation. Although the insurance contracts we consier are generally characterize by two parameters, some contracts are weakly ominate from the insurer s perspective. 11

12 Lemma 2. An insurance contract α = ( α L, α D ) with α L < c A is weakly ominate by an alternative contract with ˆα L = c A. By Lemma 2, the space of contracts can be characterize by the single parameter p, representing the contract (ˆα L, ˆα D ) = ( c A, c p ). We allow for p = +, representing the contract (c A, 0) that fully covers litigation costs, but oes not cover amages. Hence, from now on, we can ientify the space of contracts with p [ c, ]. We now re-write equations (3) an (4), the value of a contract p to an agent of type p, an the insurer s cost of proviing a contract p for an agent of type p, using the single parameter p to characterize ifferent contracts, W (p, p ) = K(p, p ) = [c A + c pp ] (1 θ) if p p [c A + c pp ], (5) θ(c + c A ) if p > p 0 if p p c A + c p if p > p. (6) p With this change in notation, it is easy to see that willingness to pay for insurance p increases faster with p when insurance is more generous (i.e., p is lower). Figure 4 shows two policy contracts p 1 an p 2 with p 2 > p 1. For any type p, W (p, p 1) > W (p, p 2) an that W (p, p 1) W (p, p 2) is increasing in p. Corollary 2. Let W (p, p ) = W (p, 1 p ). Then, W (p, p ) is supermoular. 12

13 W (p, p 1) W (p, p 2) (1 θ)(c + c A ) c p 1 p 2 1 p Figure 4: Willingness to pay for two insurance policy contracts inexe by p 1 an p Complete Information As a benchmark, we first consier the case of complete information. With complete information, an insurer sells the contract that most improves the bargaining position of the agent without inucing litigation. This is optimal in the case of competition or monopoly, although the prices of the policies iffer in the two cases. Lemma 3. For a monopoly or uner perfect competition, if the insurer(s) can observe p, the optimal insurance policy is α (p) = ( c A, p) c, a contract that fully covers the litigation expenses, partially covers amages, an oes not inuce litigation. A competitive market offers this policy for free an a monopolist charges (1 θ)(c + c A ). Proof. The optimal contract must inuce each agent to reach a settlement agreement, because the insurer incur loses by selling a policy that inuces litigation. When all agents settle, the insurer oes not incur costs, hence, either a monopolist or a competitive market offer the contract α (p) = ( c A, p) c that maximizes the agent s willingness to pay uner settlement. The monopolist extracts all the surplus an sells it at price (1 θ)(c + c A ). A competitive market offers this policy for free. 13

14 Uner complete information, there is always settlement an the effect of insurance is to reuces the bargaining surplus. By taking the agent s incentive to litigate to the absolute brink with amages insurance α D (p) = c, the optimal insurance contract extracts all bargaining surplus from the thir party. Effectively, insurance uner p complete information transfers rents from the thir party to the insurer (in the case of monopoly) or to the agent (in the case of perfect competition). Thir-party insurance contracts an first-party insurance contracts have significant ifferences. First-party insurance contracts have no value for risk neutral iniviuals since all value comes from risk reuction. Thir-party insurance contracts, in contrast, are valuable for risk neutral iniviuals because there is costly verification of the harm. This verification gives rise to settlement negotiations an insurance as value within that framework, as long as the thir party has some bargaining power. 3.2 Symmetric Information (No Averse Selection) Consier the problem of selling insurance when the insurer an the agent are both uninforme about p but they know its istribution F : [0, 1] [0, 1]. 13 In this instance, every agent is ex-ante ientical, an because there are no externalities among agents, there is no reason to offer more than a single insurance policy. The expecte willingness to pay for liability insurance contract ˆp is E p [W (p, ˆp)]. A monopolist prices this policy at P M = E p [W (p, ˆp)] an extracts all the ex-ante value of uninforme agents. Hence, the profit maximizing contract for the monopolist solves: p arg max Ψ SI (ˆp) E p [W (p, ˆp) K(p, ˆp)]. (7) c ˆp In a perfectly competitive market firms must break even, so if insurance contract ˆp is offere in equilibrium its price must be P C (ˆp) = E p [K(p, ˆp)]. Agents buy this contract as long as E p [W (p, ˆp)] P C (ˆp). Thus, the only contract that is offere in equilibrium must also be the solution to (7). A perfectly competitive market an the monopolist sell the same contract at ifferent prices. The next proposition characterizes the contract offere to an agent that is uninforme about its type when buying insurance. 13 In the context of efensive patent insurance, a firm an an insurer know that the firm potentially infringes on some patents, but they o not know the scope of the threat (patent thickets). 14

15 Proposition 1. Let both the agent an the insurer know F ( ) but be uninforme about p. Then, the liability insurance policy offere by a monopolist or a perfectly competitive market is p characterize by the solution to: 14 ˆp p arg max ˆp [ c, ] Ψ SI (ˆp) = (1 θ) c/ [ c A + cpˆp ] F (p) θ(c + c A )[1 F (ˆp)]. (8) The price of the contract uner perfect competition is P C (p ) = E p [K(p, p )] an uner monopoly is P M (p ) = E p [W (p, p )]. Equation (8) in Proposition 1 shows that the optimal contract balances two forces. Only an agent of type ˆp receives efficient insurance uner contract ˆp. Type p ˆp is uner-insure by this contract The insurer s marginal cost for these types is zero. Types p > ˆp litigate an their willingness to pay rises more with p than types below ˆp there is a kink in the eman at ˆp. However, the marginal cost of insurance is positive for these types, an excees willingness to pay by θ(c + c A ). This amount is exactly what the agent woul have capture in a settlement, an therefore cannot be price by the insurer. For a given istribution of types, these effects have ifferent weights represente by areas A an B in Figure 5. Area A is the gain in joint surplus from types that settle an correspons to the term (1 θ) ˆp [ ] ca + cpˆp F (p) in equation (8). Area B is the loss in joint surplus from types that litigate an correspons to the term θ(c + c A )[1 F (ˆp)] in equation (8). 14 Ψ SI (ˆp) is upper semi-continuous: it is obvious when F ( ) is continuous; when F ( ) is not continuous (e.g., iscrete types), u.s.c. follows from our assumption that the agent settles when inifferent. ˆp = + correspons to the contract that oes not cover amages. However, Ψ SI ( ) ecreasing for ˆp > 1, so a solution must lie in the compact interval [ c, 1]. This guarantees existence of a solution. c/ 15

16 c + c A B (1 θ)(c + c A ) A c ˆp p Figure 5: The soli area (in blue) represents the gainsan the ashe area (in re) represents the losses of contract ˆp < 1. To help characterize the solution to this problem, we consier smooth istributions for which the ensity may equal zero only at the bounaries of the support. Assumption 1. Let F ( ) be twice-continuously ifferentiable, with probability ensity f(p) > 0 for all p (0, 1). Consier the erivative of the objective function in equation (8): Ψ SI(p ) = (1 θ)(c + c A )f(p c ) (1 θ) pf (p) + θ(c + c }{{} (p ) 2 A )f(p ). (9) }{{} marginal type c/ marginal type }{{} infra-marginal types p Increasing coverage has an effect on the marginal type an infra-marginal types. First, the marginal type p gets the efficient level of insurance an extracts the full bargaining surplus (c+c A ) from the thir party. The gain of the marginal type is shown in equation (9) in two ifferent places: a gain from the improve bargaining position of the marginal type (1 θ)(c + c A ); an a gain from avoiing a loss of θ(c + c A ) in bargaining surplus ha the marginal type gone to court. Secon, the infra-marginal types p < p receive a level of insurance further away from the efficient level, inucing a loss in the joint surplus of the insurer an agent. The optimal contract either preclues litigation entirely (p = 1) or balances the gain 16

17 of the marginal type versus the average loss of the infra-marginal types. To further unerstan when it is optimal to offer a contract that inuces litigation, we efine the elasticity of ensity. Definition 1. For istributions satisfying Assumption 1, the elasticity of ensity is η(p) = pf (p) f(p). It is easy to see that the following ientity hols Ψ SI(p)p 2 + 2Ψ SI(p)p = pf(p) [ η(p) c ] A + θc. c A + c c A + c Thus, if p is an interior solution of problem (8), the first an secon orer conitions, Ψ SI(p ) = 0 an Ψ SI(p ) < 0, respectively, imply η(p ) < ( 1 + c A + θc c A + c The elasticity of ensity provies us with a sufficient conition for a unique solution of problem (8). Lemma 4. Uner Assumption 1, the solution to problem (8) is unique an equal to p = 1 if for all p [ c, 1] we have η(p) ( 1 + c A + θc c A + c ) ).. For any convex istribution F ( ), η(p) 0 for all p. By Lemma 4, the unique optimal contract preclues litigation by setting p = 1. When the ensity function is increasing, the marginal gain ominates the infra-marginal loss, i.e., it is suboptimal to sell insurance generous enough to inuce litigation by risky types. Intuitively, it is also optimal to preclue litigation when F (p) is milly concave. There are many istributions where the solution to (8) inuces litigation for some types. In such cases, η(p) allows us to provie a sufficient conition for uniqueness. Lemma 5. Uner Assumption 1, let p < 1 be such that Ψ SI(p ) = 0 an Ψ SI(p ) < 0. 17

18 Then, p is the unique interior solution if η(p) ( 1 + c A + θc c + c A ), for all p [p, 1] When p < 1, the insurer targets a particular type p with perfect insurance an enures litigation by types p > p an inefficient insurance for types p < p. In targeting, the insurer seeks a sufficiently low level of relative litigation risk associate with type p. 15 When the elasticity of ensity falls with p an the ensity of a high-risk types is low, intuitively, the insurer prefers to inuce some litigation. We have the following result. Corollary 3. If η(p) is non-increasing an f(1) < unique p ( c, 1) that solves (8). (1 θ)c c A + c 1 c/ pf (p), there exists a Proof. When Ψ SI(1) < 0, there exists p < 1 that solves (8). Since η(p ) < ( ) 1 + c A+θc c+c A an η(p) is non-increasing, the sufficient conition for uniqueness in Lemma 5 hols. Figure 6 shows the gains an losses of a contract p < 1 relative to p = 1. The gain of p < 1 comes from offering insurance that is closer to the efficient level, so every type below p is willing to pay more for this contract. The losses come from two sources. First, the cost of proviing insurance is larger than the willingness to pay for types above p, thus the insurer incurs a net loss for types above p. Secon, there is an opportunity cost of offering p < 1 instea of p = 1. With p = 1 all types settle an the insurer oes not incur costs. The balance, of course, epens on the istribution of types. It is immeiate from the figure that if the ensity of types in a neighborhoo of p = 1 is small, the gain is larger than the loss an hence p < 1 ominates p = η( ) is analogous to the Arrow-Pratt coefficient of relative risk aversion when the Bernouilli utility function is u(x) F (x). A large coefficient of relative risk aversion implies that the ecision-maker has very little to gain by gambling. In our environment, a large negative η(p) means that the insurer wants a lower p, because it has very little to lose from gambling on relatively unlikely litigation. 18

19 c + c A Loss (1 θ)(c + c A ) Gain c p 1 p Figure 6: The soli area (in blue) represents the gains of contract p < 1 relative to the contract p = 1 an the ashe area (in re) represents the losses. The following two families of istributions help illustrate our results. Example 1. The unique optimal contract for an uninforme agent is 1. p = 1 if F (p) = p α, α > p < 1 if F (p) = 1 (1 p) α, α > 1. Figure 7 illustrates these families of istributions. Figure 7(a) shows the ensity of the cf F (p) = p α, which allocates significant probability mass to the highest-risk types for all α. For these istributions, η(p) 1 for all p an α, so by Lemma 4, it is optimal to set p = 1. Figure 7(b) shows ensity of the cf F (p) = 1 (1 p) α for α > 1, showing meager mass aroun p = 1. For these istributions, it is easy to show that Ψ SI(1) < 0 because f(1) = 0. Therefore, the solution must be p < 1. Even more, because η(p) is ecreasing for this istribution, we know the solution must be unique. Another way to think about the problem is that the insurer wishes to target the ense part of the istribution with optimal insurance. Consier a iscrete istribution with only two types. Definition 2 (Two-types case). Let p {p L, p H }, such that c < p L < p H 1, an suppose the type istribution is Pr(p = p H ) = λ an Pr(p = p L ) = 1 λ. 19

20 f(p) = αp α 1 f(p) = α(1 p) α 1 α = 0.65 α = 3 α = 4 α = 2 α = 1 α = 1 (a) Family F (p) = p α 1 p α = 2 (b) Family F (p) = 1 (1 p) α 1 p Figure 7: (a) Density for the family F (x) = x α for ifferent values of α. (b) Density for the family F (x) = 1 (1 x) α for ifferent values of α 1. From Proposition 1 it is easy to see that with two types, the optimal contract is either p = p L or p = p H. Which of these contracts is optimal epens on the fraction of types. When the proportion of high-risk types is relatively large, λ > λ Lit SI (1 θ)c(p H p L ) p H (c + c A ) + (1 θ)c(p H p L ), then the optimal contract is p = p H an targets types p H. However, when λ is small, the optimal contract is p = p L. 16 Consier now comparative statics. We have the following results. 17 Lemma 6. p is non-ecreasing in c A an θ, an is non-increasing in. Lemma 6 follows from Topkis monotonicity theorem. An increase in the agent s litigation cost c A increases the opportunity cost of litigation. The gain from increasing the number of types that settle is unambiguously higher, so p is non-ecreasing in c A. An increase in the agent s bargaining power ecreases the insurer s ability to profit from insurance: the willingness to pay for insurance falls but the cost of insurance is the same. Thus p is non-ecreasing in θ because an increase in the agent s bargaining power oes not change the surplus gain of the marginal type, but it reuces the surplus loss of the infra-marginal types. An increase in amages increases the number of 16 The etails of this case is in Online Appenix B As the two-type case suggests, problem (8) may have multiple solutions, e.g., a continuous istribution with non-monotonic η(p). We interpret monotonicity of p as reflecting the strong set orer. 20

21 agents expose to creible liability claims. Thus the number of infra-marginal types increases an therefore p weakly ecreases. The effect of the thir-party s litigation cost c is ambiguous, because it increases both the surplus gain of the marginal type an the loss in surplus of the infra-marginal types. 3.3 Asymmetric Information (Averse Selection) Perfect Competition Suppose agents are privately informe about the probability of liability, an the market for insurance is perfectly competitive. There is a perfectly elastic supply of potential insurers capable of freely entering an selling insurance. We follow Rothschil an Stiglitz (1976) in specifying that equilibrium requires insurer profit be zero in equilibrium an that there is no possibility of a profitable eviation by an alternative insurer. That is, there is no contract that an entrant coul offer that woul earn a strictly positive profit. The equilibrium price epens on how much litigation is inuce by the insurance contracts. If an insurance policy inuces all types that buy it to settle, its price must be zero in equilibrium, because the insurer proviing the policy bears no cost. In contrast, if the insurance inuces litigation for some types, then Corollary 1 shows that the insurer earns a negative profit on the group of agents who litigate. Hence, to break even, the insurer must earn a strictly positive profit on the other group of agents. Hence, any pooling contract that inuces litigation requires cross-subsiization, an cannot survive in equilibrium. Proposition 2. For any istribution F ( ), a single pooling contract that inuces litigation cannot be offere in equilibrium in a perfectly competitive market. Intuitively, an alternative, slightly less generous contract coul be offere to attract only types that settle (which oes not impose any cost on the insurer) an coul be sol at a slightly lower, but positive price. This intuition is similar to the cream skimming argument in Rothschil an Stiglitz (1976). Cream-skimming also preclues the possibility of any separating equilibrium. Theorem 1. For any istribution F ( ), a separating equilibrium oes not exist in a perfectly competitive market. 21

22 The intuition of these result is easiest to see with two types. Suppose agents can be lowrisk (type p 1 ) or high-risk (type p 2 ), with p 1 < p 2. To separate types in equilibrium, an insurer must sell contracts with ifferent amages coverage p at ifferent prices. With common prices, all types woul buy the more generous coverage. This rules out two contracts that preclue litigation an are sol for a price of zero. Inee, to earn zero profit with two contracts that each generate trae, some types must litigate, some types must settle, an the types that settle must pay strictly positive prices (while generating no costs). The reason is that the willingness to pay of types that litigate is below the insurer s cost, so the insurer inevitably loses money on these types. The insurer must therefore earn money from types that settle. But given these requirements, an alternative insurer can then attract some types that settle, by offering a slightly less generous contract at a slightly lower price. This generates positive profits because all switching types settle. This cream-skimming intuition therefore unermines any such separating equilibrium. The result in Theorem 1 contrasts with Rothschil an Stiglitz (1976), where a separating equilibrium oes exist provie there are a sufficiently high number of high-risk types. Also in contrast to Rothschil an Stiglitz (1976), we now show that a simple pooling equilibrium may exist in this market. From Proposition 2 an Theorem 1, the only possible equilibrium is a pooling equilibrium that oes not inuce litigation. Theorem 2. Let p such that F (p ) = 1. A pooling equilibrium exists if an only if max p [ c,p ) (1 θ)c (p p) pp The pooling equilibrium contract is p sol at price zero. [ max p [ c, p] p[1 F ( p)] p c A + cp p ] F (p) p + 0. Theorems 2 an 1 in combination say that in a perfectly competitive market for liability insurance, only a pooling equilibrium can exist, an its existence will epen on the istribution of types. Intuitively, the conition in Theorem 2 says that a pooling equilibrium exists as long as the istribution of high-risk types is such that any eviation woul inuce such losses that it is not profitable to offer a contract that inuces litigation. This conition is relate to the conition for inucing litigation uner symmetric information. Proposition 3. If the optimal liability insurance contract uner symmetric informa- 22

23 tion, enote by p, satisfies F (p ) = 1, then there exists a pooling equilibrium with F (p ) = 1 in a competitive market with averse selection. The intuition for this result can be seen in Figure 6. The joint gains from p < 1 relative to p = 1 are higher for a monopoly uner symmetric information than for a eviating insurer in a competitive market. This is because the monopolist offers only one contract, so the agent s outsie option is to not buy liability insurance. In contrast, when contract p = 1 is offere in a competitive market, any eviation must take into account that only types that prefer the eviating contract p over p = 1 will buy it. Therefore, the gain from eviating from p = 1 in a competitive market is weakly lower than in the case of monopoly. However, the losses are the same an equal to θ(c A + c)[1 F ( p)]. Hence, whenever p = 1 is optimal for a monopolist uner symmetric information, no insurer fins that eviating from p = 1 is profitable. Proposition 3 paire with Lemma 4 from the previous section, implies that a pooling equilibrium with p = 1 exists whenever η(p) ( ) 1 + c A+θc c A +c. The conitions neee for a pooling equilibrium are weaker than the sufficient conitions for p = 1 uner symmetric information, however. Consier again the two-type case: there is a mass λ of high-risk types p H an a mass (1 λ) of low-risk types p L. The caniate for pooling equilibrium is to sell contract p = p H to all types at price zero. This contract oes not inuce litigation. Applying the conition in Theorem 2, it is easy to see that the only eviation to consier is p = p = p L. Therefore, in this case the conition is equivalent to λ λ P ool AI (1 θ)c(ph p L )p L. p H (c A p L + cp H ) When the population consists primarily of p H types, then a free contract that targets these types is an equilibrium. The p L types will also buy this contract. There is no way to cream skim, because any better contract offere to p L types also attracts too many litigious p H types. Consistent with Proposition 3, it is easy to show that λ Lit λ P ool AI SI >. Hence, if λ is high enough so that p = p H uner symmetric information, then a pooling equilibrium obtains for contract p = p H uner competition with asymmetric information. 23

24 3.3.2 Monopoly Now consier a monopolist insurer. When agents have private information about their type, a monopolist may offer a menu of contracts, or a mechanism, to maximize profits. By the revelation principle we can restrict attention to irect mechanisms that are incentive compatible. Our mechanism esign problem, however, presents a subtle complication. For a given contract p, the willingness to pay an the cost for the monopolist are not ifferentiable at the point p = p. Carbajal an Ely (2013) stuy quasi-linear settings with nonifferentiable valuations. In this case, the envelope theorem characterization may lea to a range of possible payoffs as a function of the allocation rule. The problem pointe out in Carbajal an Ely (2013) is that, although the valuation may be non-ifferentiable at one point (which has zero-measure), the mechanism may allocate a non-zero measure set of types to the non-ifferentiable point. The marginal valuation is not pointientifie at the non-ifferentiable point, because it belongs to an interval (the subifferential instea of the erivative). In our context, however, the optimal mechanism allocates at most one type to the non-ifferentiable point; hence, we can apply the envelope theorem to erive the optimal mechanism. Before we present the main result of this section, we erive a series of results that are useful to characterize the optimal menu of contracts. Instea of inexing contracts by p [ c, ], we efine x(p ) = 1 [ ] 0, p c to be the allocation, which correspons to x c = ˆα D. The insurer offers a irect revelation mechanism such that for each reporte type p, the agent receives allocation x(p) at price T (p). The payoff for an agent of type p that reports p is given by: U(p, p) = Ŵ (p, x( p)) T ( p), where (1 θ)(cpx + c A ) px 1, Ŵ (p, x) = cpx + c A θ(c + c A ) px > 1. Notice that when θ = 0, this is the classic quasilinear environment. When θ > 0, the agent s payoff has a non-ifferentiable point (a kink) whenever xp = 1. 24

25 The insurer s cost of serving type p with allocation x is 0 px 1, K(p, x) = cpx + c A px > 1. The insurer s cost has a kink whenever xp = 1, regarless of the value of θ. The problem of the insurer is to choose the functions x( ) an T ( ) to solve: max P ( ), x( ) 1 c/ T (p)f (p) {p:px(p)>1} [c A + cpx(p)]f (p) subject to p arg max Ŵ (p, x(p )) T (p ) (IC) p U(p, p) 0 (IR) As is stanar in the mechanism esign literature, when the valuation satisfies supermoularity, the allocation features a monotonicity property. Lemma 7. In an incentive compatible mechanism x( ) must be non-ecreasing. Lemma 7 shows that the supermoularity of the willingness to pay implies that incentive compatibility requires that higher types receive weakly more generous insurance. The next lemma shows that given the non-ecreasing property of the optimal allocation, there exists at most one type that receives the efficient amount of amages coverage. 18 Lemma 8. In the optimal menu of contract px(p) = 1 for at most one p [ c, 1]. Proof. Suppose there exist p 1 > p 2 > 0 such that p 1 x(p 1 ) = p 2 x(p 2 ) = 1. Then, x(p 1 ) = 1 p 1 < 1 p 2 = x(p 2 ). This contraicts Lemma 7. Lemma 8 shows further that at most one type will get efficient amages coverage. We can now use the envelope theorem an erive a unique payoff function for the optimal allocation, because the set of types for which the erivative of the payoff is not efine has measure zero for all incentive compatible contracts. The next lemma shows that 18 Interestingly, this will not be in general the type at the top, but the type at the kink. 25

26 the non-ecreasing property of the optimal allocation implies that there must be a threshol type, ˆp, that is inifferent between settlement an litigation. Lemma 9. Suppose that in the optimal allocation px(p) > 1. Then, for p > p we must have p x(p ) > 1. Proof. Suppose that p > p, px(p) > 1, an that (by contraiction) p x(p ) 1. Then, p x(p ) < px(p). This contraicts that x(p ) x(p) in the optimal contract. Lemmas 8 an 9 allows us to characterize the optimal contract as a threshol strategy: There exists ˆp [ c, 1] such that for all types p ˆp there is settlement an for types p > ˆp there is litigation. Assumption 2. Let G(p) = p an from below. 1 F (p) f(p) an assume that G( ) crosses zero only once The class of istributions that satisfy Assumption 2 is larger than the class of regular istributions (i.e., when G( ) is increasing everywhere). The following Theorem characterizes the optimal menu of contracts offere by a monopolist facing an agent with private information regaring the risk of liability. Theorem 3. For any istribution satisfying Assumption 2, let p be the solution to 1 F ( p) p =. Define p as f( p) p arg max ˆp [ p,1] Ψ AI (ˆp) (1 θ)c A F ( p) + (1 θ) [ 1 ˆp [ ˆp p c A + cˆp θ(c + c A ) + cˆp ( )] 1 F (p) f(p)p. f(p) ( p 1 F (p) f(p) )] f(p)p The optimal menu of contracts offere by a monopolist insurer consist of (at most) two contracts: 1) (c A, 0) sol at price T (p) = (1 θ)c A for types p p; 2) Contract ( c A, c p ) sol at price T (p) = (1 θ) ( ca + c p p ) for types p > p. First, we fin a type p that partitions types into those with positive an negative virtual surplus. Unlike the stanar setting, where the mechanism exclues types with 26