Risk Aversion in the Nash Bargaining Problem with Uncertainty

Transcription

1 Risk Aversion in the Nash Bargaining Problem with Uncertainty Sanxi Li Hailin Sun Jianye Yan Xunong Yin Abstract We apply the aggregation property of Ientical Shape Harmonic Absolute Risk Aversion ISHARA) utility functions to analyze the comparative statics properties of a bargaining moel with uncertainty. We ientify sufficient an necessary conitions uner which an increase in one s egree of risk aversion benefits/hurts one s opponent. We apply our moel to analyze the problems of bargaining over insurance contracts an bargaining over incentive contracts. Keywors Bargaining, the Nash Solution, ISHARA Preference, Risk Aversion JEL Classification C70, C78 School of Economics, Renmin University of China. Tel: lisanxi@gmail.com Toulouse School of Economics, 21, Allée e Brienne, Toulouse, France. hailinsun@gmail.com Research Center for Applie Finance, School of Banking an Finance, University of International Business & Economics. yanjianye@gmail.com. CAPFPP, Central University of Finance an Economics. yinxunong@gmail.com 1

2 1 Introuction In many real-worl situations, transactions take place through bargaining. Labour markets in most western economies are characterize by collective agreements negotiate between unions an firms; non-unionize workers salaries are commonly set by iniviual negotiation, which is most clearly the case for managerial compensation. Firms negotiate over how to split the profits from a joint venture; buyers an sellers bargain over the price of a prouct; an, in the insurance market, the insurer an his client negotiate over the insurance contract Kihlstrom an Roth 1982). Almost all of the bargaining situations mentione above have something in common they involve uncertainty White 2008). Iniviuals o not know whether an accient will happen when they are bargaining over the insurance contract; the firm an the manager have no iea whether the manager will perform well when they are eciing the manager s compensation package; proucers an retailers are uncertain about the exact eman when they are setting wholesale prices. Compare to the well-unerstoo situation of bargaining with a eterministic outcome, bargaining with a risky outcome is much more ifficult to stuy, especially with regar to the analysis of comparative statistics. For example, a frequently cite proposition in the eterministic bargaining literature asserts that an increase in one s egree of risk aversion improves the welfare of one s opponent. Intuitively, the subjective possibility of strategically reaching isagreement an its costly consequence makes risk aversion isavantageous in bargaining Kannai, 1977; Roth, 1979; Kihlstrom, Roth an Schmeiler, 1981; Sobel, 1981). However, it may fail in the case of a risky outcome an riskless isagreement Roth an Rothblum, 1982), as well as in the case of a risky outcome an risky isagreement Safra, Zhou an Zilcha, 1990). The complexity of the analysis of the comparative statics properties of bargaining moels with risky outcome an risky isagreement impees the application of such moels. This paper provies a simple metho to eal with this situation by focusing on the Ientical Shape Harmonic Absolute Risk Aversion ISHARA) utility functions. The ISHARA assumption uner which risk tolerances are linear in income with ientical slope implies an aggregation property: The sum of the certainty equivalents for the two bargainers is inepenent of the sharing rule that is use as long as the sharing rule is efficient. Therefore, the moel of bargaining over a risky outcome can be reuce to a problem of bargaining over a certainty equivalent a riskless outcome. This transformation allows us to isengage two effects regaring an in- 2

3 crease in one s egree of risk aversion: the bargaining power effect an the net surplus effect. On the one han, a more risk-averse bargainer has weak bargaining power an hence his opponent benefits. On the other han, an increase in one s egree of risk aversion changes the size of the net certainty equivalent the total certainty equivalent of agreement minus the sum of the certainty equivalents of the two bargainers isagreements that the two bargainers are bargaining over. This will benefit resp. hurts) his opponent if the size is increase resp. reuce). Consequently, the welfare of one s opponent will be increase as long as an increase in one s egree of risk aversion increases the net certainty equivalent. The welfare of one s opponent will be reuce if an increase in one s egree of risk aversion significantly reuces the net certainty equivalent. We then apply our moel to analyze two situations: bargaining over an insurance contract an bargaining over an incentive contract. Determining the insurance contract through bargaining between the insurer an the client is justifie if neither of them is small. Kihlstrom an Roth 1982) stuy such a problem with very general utility function an show that an insurer always benefits as the client becomes more risk-averse. However, they analyze only the case of the risk-neutral insurer, although they note that the assumption of the insurer s risk neutrality cannot be justifie in some interesting situations. They argue that subsequent work woul require an extension of their results to the case of the risk-averse insurer. That is precisely the work of this paper. The simple transformation allows us to easily check that their results are still vali in the case of the risk-averse insurer. Another application consiers the problem of bargaining over an incentive contract. Stanar principal-agent moels always assume that the principal offers a take-it-or-leave-it offer. However, as we have alreay argue, it is common that, in real-life situations, both parties hol some bargaining power. In moels of bargaining over incentive contracts, economists show that the istribution of bargaining power between principal an agent has real effects Pitchfor 1998, Balkenborg 2001, Schmitz 2005, Demougin an Helm 2006, Demougin an Helm 2009, Dittrich an Stäter 2011, Yao 2012). However, that literature consiers only the case with a risk-neutral principal an a wealth-constraine risk-neutral agent or a risk-neutral agent with limite liability). This paper complements this literature by consiering bargaining between a risk-neutral principal an a risk-averse agent à la Holmstrom an Milgrom 1987). We show that the bargaining moel preicts the same power of incentives an total surplus as oes the moel in which the principal makes a take-itor-leave-it offer. However, the principal s preference over the agent s egree 3

4 of risk aversion is quite ifferent. If the principal hols all the bargaining power, he is worse off when the agent becomes more risk-averse, as proviing incentive becomes more costly an, hence, the total surplus is reuce. When the contract is etermine through bargaining, this result may not hol because an increase in the agent s egree of risk aversion has two effects. On the one han, it reuces the total surplus; on the other han, it also reuces the agent s bargaining power. For a sufficiently riskless prouction process, the first effect is ominate by the secon one, leaing to a higher utility for the principal. The paper is organize as follows. Section 2 lays out the basic moel. Section 3 provies the solution. Section 4 applies the basic moel to the problems of bargaining over insurance an incentive contract. We conclue in Section 5. 2 The Nash Bargaining Game Two bargainers are bargaining over a risky outcome Ỹ. Bargainer i has vnm utility function u i w) : [0, ) R, i = 1, 2. The bargaining game is efine by a pair S, ), where S = {Eu 1 sỹ )), Eu 2Ỹ sỹ ))) 0 sy ) Y } is the set of unanimously agree) feasible expecte utility payoffs to the bargainers, = Eu 1 ỹ 1 ), Eu 2 ỹ 2 )) is the isagreement point, sy ) is the risk sharing rule that maps each realize value of Ỹ to bargainer 1 s iniviual share, an ỹ i is bargainer i s isagreement payoff. We allow Ỹ an ỹ is to be egenerate ranom variables, i.e., riskless variables. If none of Ỹ an ỹ is is egenerate, we are in the case of risky agreement an risky isagreement; if Ỹ is egenerate, we are in the case of riskless agreement an risky isagreement; if ỹ i s are egenerate, we are in the case of risky agreement an riskless isagreement. Here, Ỹ an ỹ is can be inepenent or correlate. The Nash solution will specify risk-sharing rules ŝy ), which solves the following problem: ) P1 max Eu 1 sỹ )) Eu 1ỹ 1 )) Eu 2 Ỹ sỹ )) Eu 2ỹ 2 )), sy ) an yiels the bargaining outcomes F 1 S, ) = EuŝỸ )), F 2S, ) = EuỸ ŝỹ )) for bargainer 1 an 2 respectively. The question that is central to this paper is: will bargainer 1 be better off if bargainer 2 becomes more risk-averse, i.e., his utility function becomes v 2 c), with v 2 c) > u 2 c) v 2 c) u 2 c)? 4

5 3 Solution with ISHARA utility functions Problem P1 concerning risk-sharing rules is not easy to solve. We hence focus on the case of Ientical Shape Harmonic Absolute Risk Aversion ISHARA) utility functions, because this kin of utility functions avois the problem that the total surplus changes with the risk-sharing rule. Following Schulhofer- Wohl 2006), we give the following efinition of ISHARA: Definition 1 The two bargainers have ISHARA preferences if their utility functions are given by u i c) = c θ i) 1 σ, i = 1, 2, where σ 0 is common to 1 σ both bargainers an θ i is bargainer i s iniviual parameter. Notice that the constant absolute risk aversion is a special case in the limit as σ goes to infinity with θ/σ fixe. Aggregation Property. It is well known that with ISHARA utility functions, the Pareto frontier in the monetary-equivalent space is a straight line an the monetary value of the joint pie is istribution-free, i.e., the sum of the two bargainers certainty equivalents is constant for any efficient risk sharing rule an oes not epen on the weights given to the bargainers see Schulhofer-Wohl 2006 for a proof). We call this property the aggregation property. Thus the Nash solution to bargaining with risky outcomes an risky isagreement points can be viewe as the ivision of a fixe amount of certainty equivalent between ))) two risk-averse bargainers. Formally, enote C 1 = u 1 1 Eu 1 s Ỹ as bargainer 1 s certainty equivalent, ))) C 2 = u 1 2 Eu 2 Ỹ s Ỹ as bargainer 2 s certainty equivalent. Then, for any efficient risk sharing rule s Y ), we must have C 1 + C 2 C, where C is constant, representing the total certainty equivalent bargaine over by the two bargainers. Denote C1 u 1 1 Eu 1 ỹ 1 )), C2 u 1 2 Eu 2 ỹ 2 )) as the two bargainers isagreement payoffs in monetary terms. The net surplus in terms of certainty equivalent is NC = C C1 + C2) > 0. Henceforth, whenever we say the size of the pie, we refer to the net surplus NC. Since the Nash solution is Pareto optimal an satisfies the axiom of inepenence of irrelevant alternatives, we can restrict our attention to the Pareto frontier which, uner this transformation, is given by S p = {u 1 C 1 ), u 2 C 2 )) C 1 0, C 2 0, C 1 + C 2 = C} 1. Because each bargainer shoul obtain at least his 1 Inepenence of irrelevant alternatives means that the solution to the bargaining problem oes not change if the utility possibilities set is unfavorably altere such that the isagreement point is unchange an the original solution remains feasible. That is, if S, ) an S, ) are bargaining problems an S S, an the solution of S, ) also belongs to S, then the two bargaining problems have the same solution. 5

6 isagreement utility, we can further restrict our attention to S p = {u 1 C 1 ), u 2 C 2 )) C 1 C1, C 2 C2, C 1 + C 2 = C}, which, using the expression of NC, can be rewritten as S p = { u 1 C1 + x), u 2 C2 + NC x) ) 0 x NC}. It can be easily prove that there exists a unique Nash solution on S P, an the solution in the certainty-equivalent space) can be obtaine from the following maximization problem: P2 max 0 x NC u1 C 1 + x ) u 1 C 1 )) u2 C 2 + NC x ) u 2 C 2 ))). Thus, we have transforme the bargaining moel with risky agreement an risky isagreement into a bargaining moel with riskless agreement an riskless isagreement. Denote w 1 c) = u 1 C 1 + c ) an w 2 c) = u 2 C 2 + c ). The above bargaining problem can be viewe as two bargainers, whose utility functions are w 1 c) an w 2 c) with isagreement payoffs zero, are bargaining over a riskless pie NC. P2 max w 1 x) w 1 0)) w 2 NC x) w 2 0))). 0 x NC Denote the solution as x #. The F.O.C. with respect to x gives: w 1 x # ) [w 2 NC x # ) w 2 0)] w 2 NC x # ) [w 1 x # ) w 1 0)] = 0, 1) which, after rearranging, yiels: w 1 x # ) w 1 0) w 1 x # ) = w ) 2 NC x # w 2 0), 2) w 2 NC x # ) i.e., the ratio of each bargainer s net share of the pie in terms of expecte utility to marginal utility shoul be equal. Now consier the effect of replacing bargainer 2 s preference with a more risk-averse utility function v 2. The increase in risk aversion has two effects. First, it reuces the sum of the certainty equivalent. Denote the reuce amount as C = C C, where we use the superscript to enote the corresponing variables in the new bargaining game between bargainer u 1 an bargainer v 2. Secon, it also reuces the isagreement certainty equivalent of bargainer 2. Denote the reuce amount as C2 = C2 C2. The reuce amount of the size of the pie the net surplus) is hence NC = NC NC = C C2. When NC > 0, the size of the pie ecreases after the replacement; when NC < 0, the size of the pie increases after the replacement. The solution x solves the following problem: P3 max 0 x NC w 1 x) w 1 0)) w 2 NC x) w 2 0))), 6

7 where NC = C C1 C2 is the net surplus in the bargaining game between bargainer u 1 an bargainer v 2, an w2 c) = v 2 C 2 + c ). The bargaining game can be viewe as two bargainers, whose utility functions are w 1 c) an w2 c), an who are bargaining over a riskless pie NC, with isagreement payoffs zero. Similarly, as in solving P2, the F.O.C. yiels w 1 x ) w 1 0) w 1 x ) = w 2 NC x ) w2 0). 3) w2 NC x ) Bargainer 1 prefers to bargain with bargainer v 2 rather than with bargainer u 2 if x x #, which is the case iff w 1 x ) [w 2 NC x ) w 2 0)] w 2 NC x ) [w 1 x ) w 1 0)] 0, which, after rearranging, yiels w 2 NC x ) w 2 0) w 2 NC x ) w 1 x ) w 1 0). w 1 x ) Substitute equation 3) into the above inequality, an we get the necessary an sufficient conition of x x # : w 2 NC x ) w 2 0) w 2 NC x ) which can be rewritten as: w 2 NC x ) w2 0), 4) w2 NC x ) [ w 2 NC x ) w 2 0) w 2 NC x ) +[ w 2 NC x ) w2 0) w2 NC x ) w 2 NC x ) w2 0) ] w2 NC x ) w 2 NC x ) w2 0) ] 0. 5) w2 NC x ) An increase in one s egree of risk aversion has two effects on one s opponent s welfare. First, when one becomes more risk-averse, one s bargaining power will change. The term in the first square bracket reflects this bargaining power effect, because it keeps the net certainty equivalent unchange. Secon, the net surplus also changes as one becomes more risk-averse. This net surplus effect is reflecte by the terms in the secon square bracket. The following lemma tells us that the sign of the first bracket is always negative. Lemma 1 w 2NC x ) w 2 0) w w 2 2 NC x ) w2 0) NC x ) w 2 NC x ) 0. 7

8 Proof. Denote δ = NC x. The inequality is equivalent to δ 0 w 2 c) δ w 2 δ) c 0 w w 2 c) 2 δ) c w 2 c) w 2 δ) w 2 c) w2 δ), c < δ w 2 c) w2 c) w 2 δ), c < δ, w2 δ) which hols if w 2 c) is increasing in c. w c) 2 w 2 c) c w 2 c) = w 2 c) w2 c) 2 > 0, 2 c) w 2 c) w 2 c) w w 2 c) w w 2 c) < 2 c) w2 c). Because bargainer v 2 is more risk-averse than bargainer u 2, we have w 2 c) u 2C2 +c) u 2C2 2 C 2 +c) w 2 c) = v +c) < v 2C 2 +c). Moreover, our assumption that σ 0 implies v 2 exhibits Decreasing Absolute Risk Aversion property, an hence v 2 C 2 +c) v 2 C2 +c) v 2C 2 2 c) w +c) = ue to w C c) 2 < C2. 2 v 2C 2 +c) < Notice that equation 4) is equivalent to Lemma 1, given that NC = NC. The above lemma states that an increase in bargainer 2 s egree of risk aversion, if it oesn t affect the net bargaining surplus, i.e., C = C2, will make bargainer 1 better off. This result is consistent with the prevailing preictions on the Nash solution with risk-averse bargainers: risk aversion benefits one s opponent Kihlstrom, Roth, an Schmeiler, 1981; Roth, 1979, among others). Disagreement has costly consequences, an the esire to avoi the risk of isagreement is reflecte in the final bargaining outcome. A more risk-averse bargainer has a stronger esire to avoi such risk, an hence is willing to give up more share uring the bargaining in orer to facilitate reaching an agreement. Lemma 2 w 2 NC x ) w2 0) is increasing in NC. w2 NC x ) Proof. The result is straightforwar by noticing that w2 NC x ) w2 0) is increasing in NC an that w2 NC x ) is ecreasing in NC. 8

9 Thus, the term in the secon square bracket of 5), reflecting the net surplus effect, is negative if NC > NC. It states an intuitive result: bargainer 1 will be better off as the size of the pie increases. Combining lemma 1 an lemma 2 yiels the following proposition: Proposition 1 An increase in one s egree of risk aversion benefits one s opponent if the net certainty equivalent increases. It hurts one s opponent only if the net certainty equivalent ecreases significantly, i.e., when it outweighs the opponent s benefit from the increase of relative bargaining power. As bargainer 2 becomes more risk-averse, the total certainty equivalent will ecrease significantly when the agreement income Ỹ is highly risky. Bargainer 2 s total certainty equivalent will ecrease significantly when his/her isagreement income ỹ 2 is highly risky. The net certainty equivalent is more likely to increase when Ỹ is not risky an ỹ 2 is highly risky; while it will ecrease when Ỹ is highly risky an ỹ 2 is not risky. In the case of riskless agreement an risky isagreement, the total certainty equivalent oes not change, while bargainer 2 s certainty equivalent of isagreement ecreases as he/she becomes more risk-averse. Thus the net surplus increases an hence benefits bargainer 1. In the case of risky agreement an riskless isagreement, the net certainty equivalent ecreases an bargainer 1 may become worse off if Ỹ is very risky. In the case of risky agreement an risky isagreement, whether the net certainty equivalent increases or ecreases epens on the relative riskiness of Ỹ an ỹ 2. Finally, the change in the size of the pie also epens on the relative egree of risk aversion between the two bargainers. If bargainer 1 is much less risk-averse than bargainer 2, then bargainer 1 bears most of the risk. Thus, an increase in bargainer 2 s egree of risk aversion woul not change the total certainty equivalent very much. In the extreme case, where bargainer 1 is risk-neutral, the total certainty equivalent remains unchange. Therefore, the size of the pie increases as bargainer 2 s certainty equivalent of isagreement ecreases. Similar arguments tell us that the size of the pie will be reuce if bargainer 2 s egree of risk aversion is much less than that of bargainer 1. We summarize these results as a corollary of Proposition 1. Corollary 1 1) With riskless agreement, an increase in a bargainer s egree of risk aversion always increases his/her opponent s welfare. 2) With risky agreement, when the egree of a bargainer s risk aversion increases, its impact on his/her opponent s welfare is ambiguous. Specifically, an increase in a bargainer s egree of risk aversion is more likely to ecrease resp. increase) his/her opponent s welfare in the following three situations, ceteris paribus: 9

10 2-a) the agreement is highly resp. less) risky; 2-b) the bargainer s isagreement is less resp. highly) risky; 2-c) the bargainer s egree of risk aversion is much less resp. higher) than that of his/her opponent. We illustrate the above proposition with the following example. Example 1 Consier the case where two bargainers have CARA utility function u i c) = 1 exp r ic) r i, i = 1, 2. Assume, Ỹ N µ, σ2 ), ỹ i N µ i, σi 2 ). The specific assumption allow us to write C = µ R 2 σ2, C1 = µ 1 r 1 2 σ1 2 an C2 = µ 2 r 2 2 σ2. 2 The net certainty equivalent is given by NC = C C1 C2. An increases in r 2 will benefit resp. hurts) bargainer 1 if f r 2, NC) = u 2C2 +NC x) u 2C2 ) u 2C2 +NC x) x. r 2 f r 2, NC) = is increasing resp. ecreasing) in r 2. Denote δ = NC f r 2, NC) + r 2 NC f r 2, NC) NC r 2 = r2 δe r2δ e ) [ r 2δ + e r 2δ 1 σ2 2 R r 2 2 = r2 δe r2δ e ) [ r 2δ + e r 2δ 1 σ2 2 r 2 2 )] σ 2 r 2 r 2 1 r 1 + r 2 ) 2 σ2 )]. It is easy to prove that 1+r 2 δe r2δ e r2δ > 0. Therefore, we have r 2 f r 2, NC) > 0 when σ2 2 r2 1 σ 2 > 0, which is more likely to be the case if σ r 1 +r 2 ) 2 2 is large, σ 2 is small an that r 2 is much larger than r 1. That r 2 f r 2, NC) < 0 occurs only if σ2 2 r2 1 σ 2 < 0. Consier r 1 +r 2 ) 2 the case with riskless isagreement where σ2 2 = 0. r 2 f r 2, NC) < 0 will be the case if σ 2 > ˆσ 2, with ˆσ 2 = 21+r 2/r 1 ) 2 1+r 2 δe r 2 δ e r 2 δ ) r 2. Notice that ˆσ 2 e r 2 δ is increasing in r 2 /r 1, which means that r 2 f r 2, NC) < 0 is easier to be satisfie if r 1 is much larger than r 2. Discussion The symmetric Nash bargaining moel that we have iscusse in the paper can be easily extene to the case of asymmetric Nash bargaining. The asymmetric Nash solution will specify risk-sharing rules ŝy ), which solves the following problem: 1 α max Eu 1 sỹ )) Eu 1ỹ 1 )) α Eu 2 Ỹ sỹ )) Eu 2ỹ 2 ))), sy ) 10

11 where the parameter α measures the bargaining power of each bargainer. A higher α means that bargainer 1 has higher bargaining power. A natural question is how bargainer 1 s bargaining power α affects the property of comparative statistics. As α increases, bargainer 1 obtains most of the pie. Hence, the net surplus effect cause by an increase in bargainer 2 s risk aversion becomes more relevant, while the bargaining power effect cause by an increase in bargainer 2 s risk aversion becomes less relevant. In the extreme case where α 1, only the net surplus effect exists; hence, whether bargainer 1 is betters offer epens only on whether or not an increase in bargainer 2 s risk aversion increases the net surplus. 4 Applications 4.1 Bargaining Over Insurance Contracts In this section, we apply our moel to stuy insurance contracts reache through bargaining. Although Kihlstrom an Roth 1982) have alreay use this moel, they consier only the case with a risk-neutral insurer. The assumption of a risk-neutral insurer is appropriate if the insurer insures many risks inepenent of the one being analyze an, hence, iversifies these risks. In other interesting situations, however, the assumption of insurer risk neutrality can not be justifie Kihlstrom an Roth 1982). In this section we apply our basic moel an provie a simple metho to reconsier their situation but with a risk-averse insurer Consier a situation with two iniviuals: a client an an insurer. Both the insurer an the client are risk-averse an have ISHARA utility functions: u i c) = c θ i) 1 σ, i = I, C, where I, C represent insurer an client. 1 σ The client faces a possible financial loss. His wealth is a binary ranom variable: { w C > 0 with probability v w C = w C L > 0 with probability 1 v. The insurer s wealth is w I an he is not face with the possibility of any exogenous losses. Assume that the insurer has sufficient wealth to provie complete coverage in any case. The insurer agrees to insure the client an bear some of the buren of the client s loss in the event that such a loss occurs. His wealth is x Il = w I A if the loss occurs an x In = w I + p 11

12 if the loss oes not occur. With this insurance contract in force, the client s wealth is if the loss occurs an x Cl = w C L + A x Cn = w C p. if the loss oes not occur. Let s first consier the case of a risk-neutral insurer an a risk-averse client. In a competitive insurance market, the client is completely insure, an the insurer s expecte wealth is equal to w I. The competitive equilibrium contract A, p) is unchange by an increase in the client s risk aversion, an is etermine by the following two equations L A = p, vp 1 v) A = 0. Now we assume that the insurance contracts are reache through Nash bargaining. Pareto optimality of the Nash solution requires that the riskneutral bear all the risks. That is, the client is completely insure. Thus, the total surplus C is given by C = w C + w I 1 v) L, regarless of the egree of the client s risk aversion. The client s isagreement payoff is C2 u 1 C Eu C w C )) in monetary terms. As the client becomes more risk-averse, C2 ecreases. Thus, an increase in the client s risk aversion increases the net certainty equivalent that the insurer an the client are bargaining over because it oes not change the total certainty equivalent, but reuces the client s isagreement certainty e- quivalent. It follows from proposition 1 that the insurer is better off, which is the result of theorem 4.1 in Kihlstrom an Roth 1982). Moreover, that the insurer is better off means that 1 v) p va increases. Because the client is completely insure, we have L A = p. It follows immeiately that p increases an A ecreases: A more risk-averse client pays a higher premium an receives less coverage of his potential loss. Now we turn to the case in which both the insurer an the client are riskaverse. The total certainty equivalent is C = w 1 Ew w C + w I )), where w is the representative s utility function, with w = c θ C θ I ) 1 σ. The isagreement 1 σ payoff of the client is equal to C2 u 1 C Eu C w C )), where u C = c θ C) 1 σ. 1 σ 12

13 Lemma 3 The net certainty equivalent increases as the client becomes more risk-averse. Proof. We nee to prove NC θ C = C θ C C 2 θ C 0. Using the specific formula of u C an w, we know that C 2 = [ E w C θ C ) 1 σ] 1 1 σ + θ C an that C = [ E w C + w I θ I θ C ) 1 σ] 1 1 σ + θ I + θ C. Notice that the insurer has sufficient wealth an, hence, w I θ I > 0. The above two equations imply that we will be one if we can prove 2 C θ C θ I 0. From the above equation, we have C θ C = [ E w C + w I θ I θ C ) 1 σ] σ 1 σ E w C + w I θ I θ C ) σ + 1, an, therefore, 2 C θ C θ I = σ{ [ E w C + w I θ I θ C ) 1 σ] 2σ 1 1 σ E w C + w I θ I θ C ) σ) 2 [ E w C + w I θ I θ C ) 1 σ] σ 1 σ E w C + w I θ I θ C ) σ 1 } = σ [ 2σ 1 1 σ E w C + w I θ I θ C ) 1 σ] { E wc + w I θ I θ C ) σ) 2 E w C + w I θ I θ C ) 1 σ E w C + w I θ I θ C ) σ 1 } 0, where the last inequality hols as a irect application of Cauchy-Schwarz inequality. Notice that an increase in the client s egree of risk aversion reuces both the total surplus an the client s certainty equivalent of isagreement. In the case of no isagreement, the certainty equivalent C 2 is reuce significantly because the client alone bear all the risk. In the case of agreement, however, the total certainty equivalent C is reuce only slightly because the insurer share some of the risk. As a result, the reuce amount of C is much less than the reuce amount of C 2, an, therefore, the net certainty equivalent increases. The above lemma, together with Proposition 1, immeiately gives the following proposition: 13

14 Proposition 2 The insurer, whether he is risk-neutral or risk-averse, benefits as the client becomes more risk-averse. Calculating the Bargaine Insurance Contract. Now we illustrate how we can calculate the bargaine contract A, p) from the transforme problem. First, from equation 1), we can calculate the exact net certainty equivalent that the insurer gets x # ) an the client gets NC x # ). Then, the contract A, p) can be calculate from the efinition of certainty equivalent, which is given by the following two equations: vu C w C p) + 1 v) u C w C L + A) = u C C 2 + NC x #), vu I w I + p) + 1 v) u I w I A) = u I wi + x #). 4.2 Bargaining Over the Incentive Contract Stanar principal-agent moels often assume that the principal offers takeit-or-leave-it contracts to the agent. Consequently, the principal obtains all the surplus of the transaction. A irect result is that the principal suffers from an increase in the egree of the agent s risk aversion, because the cost of proviing higher incentive increases as the agent becomes more risk-averse. However, in many real-worl situations, both parties hol some bargaining power an, thus, the contracting involves bargaining. For example, many labour market situations are characterize by bargaining between workers an firms Demougin an Helm 2006). We will prove in this section that bargaining will significantly change the property of comparative statistics. In particular, we will show that the principal may benefit if the agent becomes more risk-averse. Consier the case in which a risk-neutral principal is bargaining with a risk-averse agent over an incentive contract. The principal hires the agent to prouce output. The agent has CARA utility function with absolute riskaverse coefficient r: u x) = 1 exp rx). The agent can exert costly effort to r increase output. The output is y = e + ε, where ε N 0, σ 2 ), with σ 2 representing the riskiness involve in the prouction process an, e representing the effort exerte by the agent. The effort cost is c e) = e2. 2 Contract. The effort is not observable. The only observable an contractible variable is the output y. Assume that the contract that the two parties are bargaining over is linear: w = w 0 + αy, 14

15 where w 0 is the fixe salary an α is the power of incentive. The timing is as follows. First, the two parties engage in a Nash bargaining process an bargain over the contract w 0, α). If they reach no agreement, then the game is over an both of them get nothing. If they sign a contract, then the agent chooses his effort. Finally, output is realize an the contract is execute. Given the contract, the agent chooses e to maximize his certainty equivalent C A = w 0 + αe rα2 σ 2 e F.O.C with respect to e gives the following incentive-compatible conition IC : e = α. For contract w 0, α), the total certainty equivalent of the principal an the agent is C = e e2 2 rα2 σ 2. 2 Substituting the IC conition into the expression of C an C A, we obtain C α) = α 1+rσ2 )α 2 as a function of α an C 2 A α, w 0 ) = w rσ2 )α 2 as 2 a function of α an w 0. Notice that both the principal an the agent get nothing if no agreement is reache. Hence, the Nash Bargaining solution is given by the following problem max C α) C A α, w 0 )) u C A α, w 0 )). w 0,α The solution of Nash bargaining implies that two parties will choose α to maximize the total certainty equivalent C α). Otherwise, suppose that the solution is w 0, α ), while there exists α such that C α ) > C α ). Then one can choose a proper w 0 such that C A α, w 0) = C A α, w 0). Obviously, w 0, α ) gives a higher value of C α) C A α, w 0 )) u C A α, w 0 )), contraicting that w 0, α ) is the Nash solution. The first-orer conition of C α) = 0 immeiately gives α = rσ 2. The net certainty equivalent is equal to the total surplus an is given by 1 NC = C = 1 + rσ 1 ) 2 ) 2 1 rσ rσ rσ 2 = rσ. 2 15

16 Proposition 3 Compare to the case in which the principal has all the bargaining power, the bargaining moel preicts the same power of incentive α) an, hence, the same total surplus. The existing literature on bargaining contracts between principals an agents often assumes a risk-neutral agent with limite liability Pitchfor 1998, Balkenborg 2001, Demougin an Helm 2006). The main result is that the bargaining moel an the take-it-or-leave-it moel preict ifferent incentives. The above proposition is in contrast with this result an provies an example of when bargaining oes not have a real effect. However, as we will show immeiately, the principal s preference over the agent s egree of risk aversion is quite ifferent from the take-it-or-leave-it moel. The above analysis shows that the bargaining moel can be viewe as if the principal an the agent were bargaining over a total surplus C = 1 1, with the outsie option normalize to zero. Hence, we can rewrite 2 1+rσ 2 the problem as: max xu C x) x The first-orer conition gives: from which we get 1 exp r C x)) r x exp r C x)) = 0, x = 1 exp r C x)) 1), r where C = 1 1. Define L = 1 exp r C x)) 1); then, we know that 2 1+rσ 2 r x L 0 iff 0. r r L r = L r + L C C r = 1 [1 + r C x) exp r C x)) exp r C x))] r2 [ ] 1 σ 2 exp r C x)) rσ 2 ) 2 Obviously, for σ 2 close to zero, L is strictly positive. Hence, the principal r benefits from an increase in the agent s risk aversion. Proposition 4 The principal may benefit from or be hurt by an increase in the agent s egree of risk aversion. Specially, he benefits from an increase in the agent s egree of risk aversion if the prouction process is sufficiently riskless. 16

17 An increase in the agent s egree of risk aversion has two effects. On the one han, it reuces the total surplus, which hurts the principal. On the other han, the agent s bargaining power becomes weaker as he becomes more risk-averse, which benefits the principal. For a sufficiently riskless prouction process, the first effect is ominate by the secon one, leaing to a higher utility for the principal. Figure 1 Figure 1 illustrates how the principal s payoff x varies with the agent s egree of risk aversion, given ifferent σ. We can see that for a small value of σ, the principal s payoff is increasing in the agent s egree of risk aversion; for a large value of σ, the principal s payoff is ecreasing in the agent s egree of risk aversion; for a mile value of σ, the principal s payoff is first increasing an then ecreasing in the agent s egree of risk aversion. 5 Concluing Remarks This paper buils a simple Nash bargaining moel with uncertainty. In particular, we ientify the two effects of a change in one bargainer s egree of risk aversion: the bargaining power effect an the net surplus effect. An increase in one bargainer s egree of risk aversion reuces his bargaining 17

18 power, while, at the same time, it changes the net surplus. Whether this benefits his opponent epens on which effect ominates. The simplicity of our moel allows us to apply it in many situations. In an application to bargaining over insurance contracts, we show that the result in Kihlstrom an Roth 1982), which states that a risk-neutral insurer is better off if the insure becomes more risk-averse, is also robust for the case of the risk-averse insurer. Applying our moel to the situation of bargaining over incentive contracts, à la Holmstrom an Milgrom 1987), we show that the principal may benefit as the agent becomes more risk-averse, in contrast to the preiction of the take-it-or-leave-it moel. We believe that our moel has many other applications, because many real-worl bargaining games involve uncertainty. For example, our moel can be applie in situations in which agents form small groups marriage, partnership, etc.) for the purpose of risk sharing. In Inia, families fin suitable men for their aughters from istant villages to reuce correlation in climatic an prouction shocks. A primary concern is about the compositions of such risk-sharing partnerships i.e., whether agents in the group have similar or issimilar risk preferences. An important issue in this literature is the conflict between theory an empirical/experimental eviences. Theoretical matching moels preict negative assortative matching Chiappori an Reny 2006; Legros an Newman 2007; Schulhofer-Wohl 2006). This result is, however, not consistent with the empirical an experimental literature Lam 1988; Charles an Hurst 2003; Di Cagno et al 2012). One possibility to resolve this conflict is to relax the assumption that the risk-sharing rule is etermine by the competitive market in the theoretical moels. Instea, one can assume that agents share their joint risky income through Nash bargaining. If we can show that an agent suffers if his partner becomes more risk-averse uner some conitions, then the resulting matching will be positively assortative. The most risk averse agent will make a proposal to the most risk averse partner, who is happy to accept the offer. As a result, agents in the group have similar preferences, which is consistent with empirical an experimental evience. 18

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