Risk Sharing and Adverse Seletion with Asymmetri Information on Risk Preferene Chifeng Dai 1 Department of Eonomis Southern Illinois University Carbondale, IL 62901, USA February 18, 2008 Abstrat We onsider a prinipal-agent relationship where a buyer ontrats with a risk-averse supplier for the prodution of ertain good. At the time of ontrating, both parties share the same inomplete information on ost of prodution, but the supplier is privately informed of his degree of risk aversion. After signing the ontrat and before the prodution, the supplier privately disovers the ost of prodution. We show that a supplier with small degree of risk aversion produes below the first-best supply shedule exept for the lowest and the highest realizations of ost. The prodution distortion inreases for all but the lowest realization of ost as the supplier beomes more risk-averse. The asymmetri information on risk preferene further distorts the supply shedule of a more risk-averse supplier towards a ost-plus ontrat. However, when the buyer is also risk-averse, both types of supplier may produe above the first-best supply shedule. Keywords: Risk Sharing; Adverse Seletion; Risk Preferene JEL lassifiation: D81;D82;D86 1 Tel: 618-453-5347; Fax: 618-453-2717; Email address: dai@siu.edu. I am grateful to David Sappington and Tray Lewis for valuable omments.
1 Introdution It is well understood that prinipal agent relationships often involve simultaneous onsideration of risk sharing, effort motivation (moral hazard), and information revelation (adverse seletion). For example, Zekhauser (1970), Spene and Zekhauser (1971), Holmstrom (1979), Shavell (1979); Grossman and Hart (1983) among others onsider optimal risk sharing under moral hazard; Salanie (1990) studies optimal risk sharing under adverse seletion; Laffont and Rohet (1998), Theilen (2003), and Dai (2007) study optimal risk sharing under both adverse seletion and moral hazard. In all these studies, the equilibrium ontrats losely depend on both the prinipal s and the agent s degree of risk aversion. In reality, prinipals often do not have preise information on agents risk preferene. For example, the owner of a firm typially does not know how risk-averse a manager or a worker is. Similarly, a regulator seldom has perfet information about a firm s degree of risk aversion. In those ases, the agent oneivably an manipulate the prinipal s pereption of his risk preferene. The purpose of this study is to extend the adverse seletion model to settings where the agent is privately informed about his degree of risk aversion. We onsider a prinipal-agent relationship where a buyer ontrats with a risk-averse supplier for the prodution of ertain good. At the time of ontrating, both the buyer and the supplier share the same inomplete information about ost of prodution. However, after signing the ontrat and before the prodution, the supplier an privately disover the ost of prodution. We onsider the optimal risk sharing between the two parties under the ost unertainty when the supplier is privately informed about his degree of risk aversion at the time of ontrating. When both parties share the same information on ost of prodution at the time of ontrating, the effiient supply shedule an be ahieved by a fixed-prie ontrat whih makes the supplier the residual laimant of the prodution, if the supplier is risk-neutral. 1
However, when the supplier is risk-averse, the optimal supply shedule must balane risk sharing and the inentive for the supplier to truthfully reveal his private information on ost of prodution. Consequently, a supplier of small degree of risk aversion is required to produe below the effiient supply shedule exept for the lowest and the highest realizations of the ost. The prodution distortion inreases for all but the lowest realization of the ost as the supplier beomes more risk-averse. When the supplier beomes suffiiently riskaverse, bunhing arises in the supply shedule the supplier is required to produe a onstant level of output for high realizations of ost. When the supplier beomes infinitely riskaverse, the supply shedule onverges to one in a standard adverse seletion problem where the supplier is privately informed about his ost of prodution at the time of ontrating. When the supplier is privately informed about his degree of risk aversion, the buyer must sreen the supplier not only by his marginal ost of prodution but also by his degree of risk aversion. When the buyer is risk-neutral, the optimal ontrat balanes risk sharing and the inentive for the supplier to truthfully reveal both the realization of ost and his degree of risk aversion. Consequently, the supply shedule for the more risk-averse supplier is further distorted towards a ost-plus ontrat in order to limit a less risk-averse supplier s inentive to mimik a more risk-averse one. However, the supply shedule for the less risk-averse supplier is the same as when the supplier s degree of risk aversion is ommon information. When the buyer is also risk-averse, the optimal ontrat must simultaneously balane the buyer s profits with different types of suppliers, risk sharing between the two parties and the supplier s inentives for truthful information revelation. The prodution distortion for both types of supplier dereases as a risk-averse buyer alloates more risk towards the suppliers. Moreover, a risk-averse buyer also redues the prodution distortion for a more risk-averse supplier to smooth her profits with different types of suppliers. As the buyer beomes more risk-averse, the downward prodution distortion for both types of supplier 2
dereases. When the buyer is suffiiently risk-averse, both types of suppliers produe above the effiient level of output. de Mezza and Webb (2000) and Jullien, Salanie and Salanie (2007) study the optimal insurane ontrats under moral hazard when insurane ustomers risk preferene are their private information. Landsberger and Meilijson (1994) onsider a prinipal-agent setting with one risk neutral monopolisti insurer and one risk-averse agent who is privately informed about his degree of risk aversion. Smart (2000) studies a sreening game in a ompetitive insurane market in whih insurane ustomers differ with respet to both aident probability and degree of risk aversion. In ontrast to these studies, we onsider a prinipal-agent relationship where suppliers differ with respet to both ost of prodution and degree of risk aversion. The rest of the paper is organized as follows. Setion 2 desribes the entral elements of the model. As a benhmark, Setion 3 presents the optimal ontrat when the supplier s degree of risk aversion is ommon information. Setion 4 examines the optimal ontrat when the supplier is privately informed about his degree of risk aversion. Setion 5 summarizes our main findings and onludes the paper with future researh diretions. The proofs of all formal onlusions are in the Appendix. 2 The model A buyer ontrats with a supplier to obtain some quantity, q 0, of a good. The buyer s valuation of q is V (q), andv ( ) is a smooth, inreasing, and onave funtion. The buyer s net surplus is W = V (q) T, where T is the buyer s payment to the supplier. The supplier s total ost of produing q is C = q, where is the supplier s marginal/average ost of prodution. Hene, the supplier s profit isπ = T q. The utility funtion of the supplier, U( ), belongs to some smooth one-dimensional 3
family of utility funtions that is ranked aording to the Arrow-Prat measure of risk aversion: for any wealth level π, U 0 (ρ, π)/u 00 (ρ, π) is inreasing with ρ. Thus,ρ measures the the supplier s degree of risk aversion. The supplier s degree of risk aversion is unknown to the buyer. However, it is ommon knowledge that the supplier s degree of risk aversion, ρ, belongs to the two point support {ρ l,ρ h } with ρ h >ρ l and Pr(ρ = ρ l )=α (therefore Pr(ρ = ρ h )=1 α). The supplier s marginal ost of prodution,, is unertain at the time of ontrating. However, both the buyer and the supplier know that the realization of follows a uniform distribution between and. After ontrating with the buyer and before the prodution takes plae, the supplier privately disovers the realization of. The timing and ontratual relation between the buyer and the supplier are as follows: (1) the supplier privately learns his degree of risk aversion ρ; (2) the buyer offers the supplier a set of ontrat menus M n = {T n (),q n ()} onditional on the supplier s degree of risk aversion n, wheren = l, h, and his eventual marginal ost ; (3) the supplier selets his preferred menu M n given his private information on ρ; (4) the supplier disovers, and selets a desired option (T n (),q n ()) from the seleted menu M n ; (5) exhange takes plae aording to the ontrat terms. 3 Common Information on Risk Preferene As a benhmark, in this setion we disuss the optimal ontrat when the supplier s degree of risk aversion is ommon information. When the buyer is risk-neutral, her optimization problem is hoosing {T (),q()} to maximize Z [V (q()) T ()]f()d, (1) 4
where f() is the probability density funtion of. Denote =, thenf() =1/. A ontrat is feasible (or implementable) provided if it is inentive ompatible and individually rational. Inentive ompatibility requires that the ontrat indues the supplier to truthfully report his realization of marginal ost, i.e., π( i i ) > π( i j ) for i 6= j, (2) where π( i i ) and π( i j ) denote the supplier s respetive profits from hoosing options (T ( i ),q( i )) and (T ( j ),q( j )) when the realization of his marginal ost in fat is i. Individual rationality requires that the supplier s expeted utility from entering the ontrat must be nonnegative, i.e., E[U] = Z U(T () q())f()d > 0. (3) When the supplier is risk-neutral, it is well known that the optimal ontrat {T (),q()} takes the following form: T =argmax q() T () =V (q()) T,where (4) Z [V (q()) q()]f()d. (5) The optimal ontrat in essene makes the supplier the residual laimant of his prodution. Under the ontrat, the supplier produes the effiient amount of goods (i.e., V 0 (q()) =,) based on the realization of his marginal ost. Moreover, the supplier reeives zero rent in R expetation as E[U] =Max [(V (q()) T ) q()]f()d =0under the ontrat. q() Under the above optimal ontrat, the supplier bears the entire risk of ost unertainty as the residual laimant of the prodution. However, when the supplier is risk-averse, the optimal ontrat must balane risk sharing and the inentive for the supplier to truthfully 5
reveal his marginal ost of prodution. Lemma 1 desribes the general properties of the optimal ontrat when the supplier s degree of risk aversion is ommon information. Lemma 1 The optimal ontrat must be of the following form, for some in [,] and q > 0: (a) E[U] = 1 (b) q() is given by R U(π())d =0; R 1 [V 0 (q()) ] = U 0 (π(x))df (x) R U 0 (π(x))df (x) (6) on [, ) and q() =q on [, ]. Proof. See appendix. At the time of ontrating both the supplier and the buyer fae the same unertainty about the marginal ost of prodution. Consequently, although the supplier an apture information rent from his private information on the realization of after signing the ontrat, the buyer an fully extrat the expeted information rent at the time of ontrating by reduing the level of transfer payments T () for all realization of. (Note that it is the differene in T () that provides the inentive for the supplier to truthfully reveal his marginal ost.) Consequently, the supplier reeives zero expeted utility under the optimal ontrat. Given that the buyer an fully extrat the supplier s ex post information rent at the time of ontrating, the buyer does not fae the traditional trade-off between rent extration and prodution effiieny as in Baron and Myerson (1982). As we have shown earlier, the supplier s ex post information rent would be ostless to the buyer and the effiient outome 6
would be ahieved if the supplier were risk-neutral. However, when the supplier is riskaverse, the optimal supply shedule must balane risk sharing and the inentive for truthful information revelation. Equation (6) demonstrates the intuition. When the supplier s realization of marginal ost is b, raising q(b) by δq will in expetation inrease the supplier s prodution effiieny by [V 0 (q(b)) b]δq/ where 1/ is the probability that = b. However, the inrease in q(b) will also raise the supplier s ex post information rent by δq when <b. Consequently, in expetation the inrease in q(b) raises the supplier s ex post information rent by ( )δq/, where( )/ is the probability that <b. When the supplier is risk-averse, the buyer an only redue T () for all realization of by δq R U 0 (π(x))dx/ R U 0 (π(x))dx in order to keep the supplier s expeted utility unhanged. Notie that δq R U 0 (π(x))dx is the inrease in the supplier s expeted utility as a result of the inreased ex post information rent, and R U 0 (π(x))dx is the supplier s additional expeted utility as a result of one unit of inrease in T () for all realization of. Consequently, δq R U 0 (π(x))dx/ R U 0 (π(x))dx is the ertainty equivalent of the supplier s inreased additional information rent. At the optimum, the supplier s marginal benefit of raising q(b) must equal her marginal ost of doing so, whih yields equation (6). When the supplier is risk-neutral, i.e., u 00 =0, R U 0 (π(x))dx/ R U 0 (π(x))dx =( )/, whih means the the ertainty equivalent of the supplier s inreased additional information rent is the same for both the buyer and the seller. Consequently, the buyer an fully extrat the supplier s expeted ex post information rent by reduing the transfer payments under all realization of by exatly ( )/. Inthatase,theright-handside of equation (6) beomes zero, and V 0 (q()) =. The optimal ontrat would be a fixed prie ontrat, and the supplier would always supply the effiient level of goods. Denote the term on the left-hand side of equation (6) as D(). When the optimal supply shedule is stritly dereasing in in [, ], i.e., = (no bunhing), equation (6) suggests that D() =0and V 0 (q()) = at and. Therefore, the supplier delivers the effiient 7
amount of goods at and. Furthermore,D 00 () =U 00 (π())q()/ R U 0 (π(x))dx < 0, whih implies that D() is onave on (, ). SineD() =D() =0, the onavity suggests that D() > 0 on (, ). Consequently, the supplier delivers less than the effiient amount of goods on (, ). To fully demonstrate the effet of risk aversion on the optimal ontrat, we assume that the supplier has a onstant absolute risk aversion (CARA) utility funtion, u(x) =1 e ρx with ρ>0 and the buyer have a quadrati value funtion, V (q) =aq bq 2,witha>>0 and b>0. 1 Lemma 2 demonstrates the effet of risk aversion on the optimal supply shedule in the no-bunhing region. Lemma 2 When there is no bunhing, the supply shedule, q(), dereaseswiththesupplier sdegreeofriskaversion,ρ, on(, ). Proof. See Appendix. Lemma 2 suggests that, when the supplier beomes more risk-averse, the buyer optimally redues the supplier s exposure to ost unertainty by distorting the supply shedule on (, ) downwards. As ρ beomes inreasingly large, the monotoniity ondition that requires q 0 () 6 0 a neessary ondition for the supplier to truthfully reveal his realization of marginal ost may beome onstraining. Consequently, the optimal supply shedule may involve bunhing as the buyer beomes suffiiently risk-averse. Lemma 3 fully haraterizes the effet of the supplier s degree of risk aversion on the optimal ontrat. Lemma 3 There exists ρ with ρ > 0, suhthat 1 Tehnially, our analysis in this ase is similar to Salanie (1990) where a risk-neutral produer ontrats with a risk-averse retailer. 8
(a) For ρ<ρ, there is no bunhing and q() is given by R 1 [V 0 (q()) ] = e ρπ dx R e ρπ dx (7) for all on [, ]; [, ]. (b) For ρ>ρ, q() is given by equation (7) on some interval [, ) and is onstant on Proof. See Appendix. Lemma 3 suggests that bunhing arises in the optimal ontrat when the supplier beomes suffiiently risk-averse. In the optimal ontrat, the supply shedule is stritly dereasing in the realization of marginal ost for small value of marginal ost but is onstant for all realizations of marginal ost above whose value depends on ρ. Notie that when the supplier is infinitely risk-averse, i.e., ρ onverges to infinity, equation (7) beomes 1 [V 0 (q()) ] =. (8) It is the well known solution for a standard adverse seletion problem where the supplier is privately informed about his marginal ost of prodution at the time of ontrating. This is beause the supplier will partiipate in the ontrat only if he is guaranteed nonnegative profit for all realization of when he is infinitely risk-averse. Consequently our model beomes equivalent to one that the supplier is perfetly informed about his marginal ost at the time of ontrating. For later use, we all the optimal supply shedule when the supplier s degree of risk aversion is ommon information the seond-best supply shedule. 9
a 2b q First-best solution a 2b Inreasing ρ Infinite risk aversion a + 2 2b Figure 1: The omparative statis of the seond-best supply shedule as ρ inreases. 10
4 Asymmetri Information on Risk Preferene 4.1 A Risk-Neutral Buyer When the supplier is privately informed about his degree of risk aversion, it is possible for him to manipulate the buyer s pereption of his risk aversion. In this ase, the buyer must sreen the supplier not only by his marginal ost of prodution but also by his degree of risk aversion. When the buyer is risk-neutral, the buyer s optimization problem is hoosing a set of ontrat menus M n = {T n (),q n ()} for n = l, h to maximize Z {α[v (q l ()) T l ()] + (1 α)[v (q h ()) T h ()]} f()d, (9) subjet to E[U(ρ n,m n )] = Z U(T n () q n ())f()d > 0; (10) π n ( i i ) > π n ( i j ) for i 6= j ; and (11) E[U(ρ n,m n )] > E[U(ρ s,m s )], (12) where n = l, h, s = l, h, and n 6= s. While onditions (10) and (11) ensure the supplier s partiipation and truthful report of his marginal ost regardless of his degree of risk aversion, ondition (12) guarantees that the supplier truthfully reveals his degree of risk aversion. Proposition 1 desribes the general properties of the optimal ontrat when the supplier is privately informed about his degree of risk aversion. 11
Proposition 1 The optimal ontrat has the following properties : (a) E[U(ρ l,m l )] >E[U(ρ h,m h )] = 0; (b) In no bunhing region, the optimal supply shedule for the less risk-averse supplier is haraterized by 1 [V 0 (q l ()) ] = Z ( 1 e ρlπl R e ρ l π l dx ) dz; and (13) the optimal supply shedule for the more risk-averse supplier is haraterized by 1 α [V 0 (q h ()) ] = Z { 1 α e ρ h π h R + R αe ρlπh e ρ h π }dz. (14) h dx e ρ l π l dx Proof. See Appendix. Under the optimal ontrat, the buyer an fully extrat the more risk-averse supplier s ex post information rent by adjusting the level of payments for all realizations of marginal ost as in the ase of ommon information on risk aversion. However, with CARA utility funtions, the utility funtion of a less risk-averse supplier is an inreasing and onvex transformation of that of a more risk-averse supplier, and in equilibrium the less riskaverse supplier an enjoy positive expeted utility by mimiking a more risk-averse one. Consequently, the optimal ontrat provides a less risk-averse supplier positive expeted utility to indue his truthful revelation of his degree of risk aversion. Under the optimal ontrat, the supply shedule for the less risk-averse supplier optimally balanes risk sharing and the inentive for the supplier to truthfully reveal his realization of marginal ost, as in the ase of ommon information on risk aversion. Consequently, the less risk-averse supplier produes aording to the seond-best supply shedule. However, the supply shedule for the more risk-averse supplier now must simultaneously 12
trade-off risk sharing, the supplier s inentives to truthfully reveal his marginal ost of prodution, and the less risk-averse supplier s inentive to truthfully reveal his degree of risk aversion. To demonstrate the trade-off, we rewrite equation (14) as R 1 α [V 0 ( ) (q h ()) ] =(1 α)[ e ρ h π h dz R ]+αg(), (15) e ρ h π h dx where G() R e ρ l π h dz/ R e ρ l π l dx R e ρ h π h dz/ R e ρ h π h dx. When the more risk-averse supplier s realization of marginal ost is b, raisingq h (b) by δq will in expetation inrease the prodution effiieny by (1 α)[v 0 (q(b)) b]δq/ where 1 α is probability that the supplier is more risk-averse. However, the inrease in q h (b) will also raise the more risk-averse supplier s ex post information rent by δq when <b. Consequently, in expetation it inreases the more risk-averse supplier s ex post information rent by δq R e ρ h π h dz. In addition, the inrease in qh (b) will also inrease the less risk-averse supplier s rent from mimiking the more risk-averse one by δq R e ρ l π h dz in expetation. The ertainty equivalents of the above ex post information rents for both types of suppliers are δq R R e ρ h π h R dz/ e ρ h π h R dx and δq e ρ l π h dz/ e ρ l π l dx, respetively. Notie that R R e ρ h π h dx and e ρ l π l dx are the marginal utilities of one unit of inrease in ertainty equivalent for both types of suppliers, respetively. In antiipation of the supplier s information rent, at the time of ontrating the buyer an redue both types of suppliers payments by δq R R e ρ h π h dz/ e ρ h π h dx for all realizations of marginal ost. Doing so fully extrats the more risk-averse supplier s ex post information rent and provides the less risk-averse supplier just enough inentive to truthfully reveal his degree of risk aversion. At the optimum, the supplier s marginal benefit of raising q h (b) must equal her marginal ost of doing so, whih yields equation (15). 13
Notie that αg() (whih is positive on (, ) asshownintheproofofproposition5) is the effet of asymmetri information on the supplier s risk aversion. In order to limit a less risk-averse supplier s rent of exaggerating his degree of risk aversion, the buyer further distorts the more risk-averse supplier s ontrat towards a ost-plus ontrat. Consequently, as we show in Proposition 2, the more risk-averse supplier produes below the seond-best supply shedule. Proposition 2 Under the optimal ontrat, the more risk-averse supplier s supply shedule is below the seond-best level. Proof. See Appendix. Similar to the ase of ommon information on risk aversion, bunhing arises in the optimal ontrat as either type of supplier beomes inreasingly risk-averse. In the optimal ontrat with bunhing, the supply shedule is stritly dereasing in the realization of marginal ost for small value of marginal ost but is onstant for all realizations of marginal ost above ertain ritial value of marginal ost. The ritial value of marginal ost depends on the degree of risk aversion of both types of suppliers. Proposition 3 desribes the properties of the optimal ontrat in that ase. Proposition 3 When both types of suppliers beome suffiiently risk-averse, there exist some h and l in [, ] that the optimal supply shedule is onstant over [ h, ] for the more risk-averse supplier and [ l, ] for the less risk-averse supplier; The supply shedules for thenon-bunhingregionsaredeterminedbyequation(14)forthemorerisk-aversesupplier and by equation (13) for the less risk-averse supplier. Proof. The proof is similar to that of Lemma 3 and is therefore omitted. 14
Suppose that one type of supplier is risk-neutral and the other type of supplier is infinitely risk-averse. Then equation (15) beomes 1 α [V 0 (q h ()) ] =. (16) A diret omparison between equations (8) and (16) demonstrates the effet on the optimal ontrat of asymmetri information on the supplier s risk aversion. An inrease in q h (b) by δq inreases the more risk-averse suppliers prodution effiieny by [V 0 (q(b)) b]δq/ regardless whether he is privately informed about his risk aversion. However, with asymmetri information on risk aversion, an inrease in q h (b) by δq inreases the ex post information rent for not only the more risk-averse supplier but also the less risk-averse supplier by ( )/. The ertainty equivalent of the ex post information rent is zero for the more risk-averse supplier, whih means that the buyer annot extrat any of the ex post rent at the time of ontrating. Consequently, with asymmetri information on risk aversion, the more risk-averse supplier s supply shedule is further distorted towards a ost plus ontrat. 4.2 A Risk-Averse Buyer When the buyer is also risk-averse, the optimal ontrat must balane the buyer s profits with different types of suppliers, in addition to the tradeoff among risk sharing and the inentives for the supplier to truthfully reveal both his marginal ost of prodution and his degree of risk aversion. Suppose that the buyer has a onstant absolute risk aversion (CARA) utility funtion, u(x) =1 e ρ b x with ρ b > 0. The buyer s optimization problem is hoosing a set of ontrat menus M n = {T n (),q n ()} for n = l, h to maximize E[U b ]= Z α[1 e ρ b W l ]+(1 α)[1 e ρ b W h ] ª f()d (17) 15
subjet to onditions (10), (11), and (12), where W l = V (q l ()) T l () and W h = V (q h ()) T h (). Proposition 4 desribes the properties of the optimal ontrat when both the buyer and the supplier are risk-averse. Proposition 4 When both the buyer and the supplier are risk-averse, the optimal ontrat has the following properties: (a) E[U(ρ l,m l )] >E[U(ρ h,m h )] = 0; (b) In no bunhing region, the optimal supply shedule for the less risk-averse supplier is haraterized by e ρ b W l CE [V 0 (q l ()) ] = Z ( e ρ b W l CE ) e ρ b W l dx CE R dz; (18) e ρ l π l dx R e ρlπl and the optimal optimal supply shedule for the more risk-averse supplier is haraterized by (1 α)e ρ b W h [V 0 (q h ()) ] = CE Z { (1 α)e ρ b W h CE where CE α R e ρ b W l dx +(1 α) R e ρ b W h dx. R e ρ b W l dx CE R R e ρhπh e ρ l π }dz, l dx e ρ h π h dx + αe ρ l π h (19) Proof. See Appendix. Under the optimal ontrat, the more risk-averse supplier still reeives zero expeted utility, and the less risk-averse supplier still reeives positive expeted utility due to his private information on his degree of risk aversion. However, the optimal supply shedule is profoundly different ompared with the ase when the buyer is risk neutral. 16
For the less risk-averse supplier, when the realization of marginal ost is b, raisingq l (b) by δq will in expetation inrease W (b) by [V 0 (q l (b)) b]δq whih inreases the buyer s ertainty equivalent by δqα[v 0 (q l (b)) b]e ρ b W /CE. Note that CE istheinreaseinthe buyer sexpetedsurplusasaresultofoneunitinreaseinherprofits for all possible events. On the other hand, the inrease in q l (b) will also raise the less risk-averse supplier s ex post information rent by δq when <b. The ertainty equivalent of the additional ex post information rent is δq R R e ρ h π h dz/ e ρ h π h dx for the supplier. Therefore, therefore at the time of ontrating the buyer an optimally redue the supplier s payments under all realization of by δq R e ρ h π h dz/ R e ρ h π h dx. However, the buyer s ertainty equivalent of the supplier s additional ex post information rent is δqα R e ρ b W l dz/ce and her ertainty equivalent of the amount that an be extrated from the supplier at the time of ontrating is δqα R R e ρ l π l R dz e ρ b W l dx/ce e ρ l π l dx. Hene depending on the relative sizes of these two ertainty equivalents for the buyer, whih in turn depends on the relative degree of risk aversion between the two parties, the optimal supply shedule an be either above or below the effiient level. For example, when ρ l onverges to zero, i.e., the supplier onverges to risk-neutral, R R e ρ l π l dx/ e ρ l π l dx onverges to ( )/ and the latter ertainty equivalent onverges to δqα( ) R e ρ b W l dx/(ce). In the optimal ontrat, Wl must be non-inreasing on [, ]. Then, we have ( ) R R e ρ b W l dx/ > e ρ b W l dz and the right-hand side of equation (18) is non-positive. Consequently, the optimal supply shedule is above or at the effiient level on [, ]. On the other hand, based on our analysis of a risk-neutral buyer in the previous setion, by ontinuity the optimal supply shedule must be below the effiient level when the buyer onverges to risk-neutral. The buyer s risk aversion has a different impat on the more risk-averse supplier s supply shedule. Equation (19) demonstrates how the optimal supply shedule for the more risk- 17
averse supplier balanes risk sharing, inentives for truthful revelation, and the buyer s profits with different types of suppliers. The ertainty equivalent of the additional profits of inreasing q h () by δq is (1 α)[v 0 (q h ()) ]e ρ b W h /CE. However, the inrease in q h () also inreases the ex post information rent for both types of suppliers. The ertainty equivalent for the buyer of the more risk-averse supplier s additional ex post information rent is (1 α)δq R e ρ b W h dz/ce. We have shown earlier that the less risk-averse supplier s ertainty equivalent of the additional ex post information rent is δq R R e ρ l π h dz/ e ρ l π l dx, the ertainty equivalent of whih for the buyer is αδq R R e ρ l π h R dz e ρ b W l R dx/ce e ρ l π l dx. Note that α e ρ b W l dx/ce is the ertainty equivalent for the buyer of one unit inrease in profits under a less risk-averse supplier for all realizations of marginal osts. Therefore, it measures how the risk-averse buyer values additional profits under the less-risk averse supplier. Reall that, in antiipation of the additional ex post information rents for both types of suppliers, the buyer an redue both types of suppliers payments for all realizations of marginal ost by the amount equal to the more risk-averse supplier s ertainty equivalent of the additional rent. Therefore, as indiated by the right-hand of Equation (19), the marginal ost of inreasing q h () is the sum of the ertainty equivalents for the buyer of both types of suppliers additional rents (whih are (1 α)δq R e ρ b W h dz/ce and α R R e ρ l π h R dz e ρ b W l dx/ce e ρ l π l dx, respetively) minus the more risk-averse supplier s ertainty equivalent of the additional rent R R e ρ h π h dz/ e ρ h π h dx. Nonetheless, the less risk-averse supplier reeives positive information rent from his private information on his degree of risk aversion. Similar to the ase of risk-neutral buyer, in order to restrit a less risk-averse supplier s rent of exaggerating his degree of risk aversion, the buyer distorts the more risk-averse supplier s ontrat towards a ost plus ontrat ompared with the ontrat for the less risk-averse supplier. However, the distortion is smaller ompared with 18
the ase of a risk-neutral buyer as the distortion for a more risk-averse supplier beomes more ostly and the information rent for a less risk-averse supplier beomes less important to a risk-averse buyer. It an be readily shown that the optimal supply shedule for a more risk-averse supplier an also be either below or above the effiient level depending on the relative degree of risk aversion between the buyer and the supplier. We summarize this property in Proposition 5. Proposition 5 Dependingontherelativedegreeofriskaversionbetweenthebuyerand the supplier, the optimal supply shedule for both types of suppliers an be either below or above the effiient level. 5 Conlusion We extend the standard adverse seletion model to settings where the supplier is privately informed of his degree of risk aversion. The optimal ontrat simultaneously balanes risk sharing, inentives for information revelation, and the buyer s expeted profits with different types of suppliers. A supplier with small degree of risk aversion produes below the first-best supply shedule exept for the lowest and the highest realizations of ost. The prodution distortion inreases for all but the lowest realization of ost as the supplier beomes more risk-averse. The asymmetri information on risk preferene further distorts the supply shedule of a more risk-averse supplier towards a ost-plus ontrat. However, when the buyer is also risk-averse, both types of supplier may produe above the effiient supply shedule. Our researh ould be extended in several diretions. For example, although the supplier s information on ost of prodution is imperfet at the time of ontrating, the supplier 19
ould be better informed about potential osts than the buyer. Then the optimal ontrat must sreen the firm not only by its degree of risk aversion but also by its information regarding ost of prodution at the time of ontrating. The optimal ontrat in this situation merits further investigation. 6 Appendix 6.1 Proof of Lemma 1 A well known haraterization of feasible ontrats is the following: (a) T 0 () =q 0 (); (b) q() is non inreasing; () EU 0. Therefore, we an rewrite the buyer s optimization problem as an optimal ontrol problem with state variables T () and q() and ontrol variable q 0 () =z: Max 1 Z [V (q()) T ()]d, (A1) subjet to 1 Z q 0 () =z T 0 () =z q 0 () 0; and U(T () q())d > 0. (A2) (A3) (A4) (A5) The Hamiltonian is H =[V (q) T ]+μz + λz + θu(π). (A6) 20
The neessary onditions are given by H z = μ + λ > 0, z6 0, and (μ + λ) z =0; (A7) λ 0 = H q = [V 0 (q) θu 0 (π)]; μ 0 = H T = [ 1+θU0 (π)]; and λ() =λ() =μ() =μ() =0. (A8) (A9) (A10) From the transversality ondition (A8) and equation (A9), μ() μ() = Z [1 θu 0 (π)]d =0. (A11) Therefore, θ = R U 0 (π())d. (A12) Define h() =μ+λ. From ondition (A7), on any interval where q is stritly dereasing, h() must be zero. So h 0 () =μ + μ 0 + λ 0 =0,whihleadsto μ = μ 0 λ 0. (A13) Substituting equations (A8) and (A9) into the above equation for μ 0 and λ 0,wehave μ = Z [1 θu 0 (π)]dx = V 0 (q). (A14) 21
Substituting equation (A12) into the above equation for θ, we have R 1 [V 0 (q) ] = U 0 (π(x))dx R U 0 (π(x))dx. (A15) Next we show that q 0 () =0an only our on some interval [, ], and the solution is stritly dereasing on [, ). Suppose that there exist 1, 2, and 3 suh that q is onstant on ( 1, 2 ) and stritly dereasing on ( 2, 3 ). Sine q is onstant on ( 1, 2 ) on ( 2, 3 ), h( + 2 )=h 0 ( + 2 )=h 00 ( + 2 )=0.Furthermore, h 0 () = μ + μ 0 + λ 0 (A16) = Z [1 θu 0 (π)]dx [V 0 (q) ]. Hene, 0 = h 00 ( + 2 )=[1 θu 0 (π( + 2 ))] [V 00 (q( + 2 ))q 0 ( + 2 ) 1] (A17) < 2 θu 0 (π( 2 )) = h 00 ( 2 ), as V 00 (q( + 2 )) < 0, q 0 ( + 2 ) < 0, andq 0 ( 2 )=0. Moreover, h 000 () =θu 00 (π())q() < 0 (A18) on ( 1, 2 ), whih together with (A17) implies that h 00 () is positive (i.e., h() is onvex,) on ( 1, 2 ).Sineh( + 2 )=0, h() is onvex on ( 1, 2 ) means h( 1 ) > 0. Ash() is ontinuous, it h() > 0 must be true for some < 1. Sine μ() =0by the transversality ondition, there is a ontradition. 22
6.2 Proof of Lemma 2 With CARA utility funtion, when there is no bunhing, q() is given by equation (7) for all on [, ]. Differentiating both sides of equation (7) with respet to, wehave 1 e ρπ() = 2bq 0 () 1 and q 0 () =(e ρπ() 2)/2b. Sineπ 0 () = q(), π 00 () = q 0 () =(2 e ρπ() )/2b. (A19) Assume for some (ρ 0, 0 ), A q/ ρ = 2 π()/ ρ > 0. SineA(ρ 0,)=A(ρ 0, ) = 0 and A(ρ 0, ) is smooth, A must admit an interior positive maximum on (, ), i.e., A(ρ 0, m ) > 0; (A20) A(ρ 0, m ) =0; and (A21) 2 A(ρ 0, m ) 2 6 0. (A22) From equation (A17), A(ρ 0, m ) = 3 π( m ) 2 ρ = e ρπ 2b ρπ ρ ; and (A23) 2 A(ρ 0, m ) = 4 π( m ) 2 3 ρ = 1 2b [e ρπ ρq ρπ ρ + 2 (ρπ) e ρπ ρ ]. (A24) Equations (7) and (A21) together imply that ρπ/ ρ =0and 2 A(ρ 0, m ) 2 = 1 2b e ρπ 2 (ρπ) ρ. (A25) 23
Then equations (A22) and (A25) together require that 2 (ρπ) ρ = ρq ρ = (q + ρ q ρ ) > 0. (A26) Equation (A26) implies q/ ρ < 0, whih ontradits with ondition (A20). 6.3 Proof of Lemma 3 With CARA utility funtion, when there is no bunhing, from equation (6) we have R 1 [V 0 (q()) ] = e ρπ dx R e ρπ dx (A27) on [, ]. When ρ =0, q() is stritly dereasing on [, ]. By ontinuity, the optimal supply shedule is stritly dereasing on [, ] for small ρ. With the quadrati value funtion, equation (A27) provides q 0 () =(e ρπ() 2)/2b > 1/b; and (A28) q 00 () =e ρπ() ρq/2b =(1+bq 0 ())ρq/b. (A29) Therefore, q 00 () > 0, i.e., the supply shedule is stritly onvex on [, ] when there is no bunhing. Hene, the supply shedule will be non-inreasing everywhere if and only if q()/ 6 0. Suppose that q()/ 6 0 for any ρ in the optimal ontrat. Sine q 0 () > 1/b, thegraphofq must stay inside the triangle pitured in Figure 1. Therefore, we have Z q()d > a 2b + b 2 [(a ) (a ) 2b 24 ] 2 = 2b [a ]. (A30) 4
Integrating both sides of equation (A29) with respet to provides Z q 0 () q 0 () = ρ (1 + bq 0 ())q()d b > ρ ½ b 2b > ρ 2b 2 () q 2 () (a 4 )+q2 2 ½ (a 4 )+2a 4 ¾ ¾. (A31) The term on the left-hand side of inequality onverges to infinity as ρ goes to infinity. However, sine 0 >q 0 () > 1/b, q 0 () q 0 () < 1/b, whih is ontradit with ondition (A31). Proposition 2 implies that 2 q(, ρ)/ ρ > 0. Therefore, there exists ρ 1 with ρ 1 > 0, suh that bunhing ours when ρ>ρ 1. 6.4 Proof of Proposition 1 The Hamiltonian is H = α[v (q l ()) T l ()] + (1 α)[v (q h ()) T h ()] (A32) +μ l z l + μ h z h + λ l z l + λ h z h θe ρ h π h + β[e ρ l π h e ρ l π l ], (A33) where μ l, μ h, λ l, λ h,andβ are the Lagrange multipliers. The neessary onditions are given by H z = μ l + λ l > 0, z l 6 0, and (μ l + λ l ) z l =0; (A34) H z = μ h + λ h > 0, z6 0, and (μ h + λ h ) z l =0; (A35) 25
λ 0 l = H q = [αv 0 (q l ) βe ρ l π l ρ l ]; λ 0 h = H q = [(1 α)v 0 (q h ) θe ρ h π h ρ h + βe ρ l π h ρ l ]; μ 0 l = H T = [ α + βe ρ l π l ρ l ]; μ 0 h = H T = [ (1 α)+θe ρ h π h ρ h βe ρ l π h ρ l ]; and λ n () =λ n () =μ n () =μ n () =0,wheren = l, h. (A36) (A37) (A38) (A39) (A40) From the transversality ondition (A40) and equation (A38), μ l () μ l () = Z [α βe ρ l π l ρ l ]d =0, (A41) whih provides β = α ρ l R e ρ l π l d. (A42) From the transversality ondition (A40) and equation (A39), μ l () μ l () = Z [(1 α) θe ρ h π h ρ h + βe ρ l π h ρ l ]d =0, (A43) whih provides θ = 1 ρ l [(1 α)+α R e ρ l π h d R ]= 1 e ρ l π l d ρ l (A44) as R e ρ l π h d = R e ρ l π l d, i.e., onstraint (15) is binding at equilibrium. Sine h 0 n() =μ n + μ 0 n + λ 0 n =0or μ l = μ 0 l λ0 l when q n is stritly dereasing in, 26
we have and μ l = Z [α α e ρ l π l R e ρ l π l dx ]dz = α[v 0 (q l ()) ], (A45) μ h = Z [(1 α) θe ρ h π h ρ h + βe ρ l π h ρ l ]dx (A46) = Z [(1 α) e ρ h π h + α e ρ l π h R e ρ l π l dz ]dx (A47) = (1 α)[v 0 (q h ()) ]. (A48) 6.5 Proof of Proposition 2 Sine G 0 () = e ρ l π h R R e ρhπh e ρ l π, h d e ρ h π h d (A49) G 0 () > 0 if π h > ln R e ρ l π h d ln R e ρ h π h d ρ h ρ l. (A50) Notie that ln R e ρ l π h d ln R e ρ h π h d < 0 as R e ρ l π h d < R e ρ h π h d. Sine π h is stritly dereasing in and R [1 e ρ h π h ]d =0, πh () must be positive. Consequently, G 0 () > 0. Sine π h is monotone in, thesignofg 0 () an hange at most one. Moreover, sine G() =G() =0, the sign of G 0 () must hange at least one. Consequently, G() must be inreasing for some region starting from and beome dereasing for its omplimentary region. As a result, G() > 0 on (, ). 27
6.6 Proof of Proposition 4 The Hamiltonian is H = α[1 e ρ b W l ]+(1 α)[1 e ρ b W h ]+μ l z l +μ h z h +λ l z l +λ h z h θe ρ h π h +β[e ρ l π h e ρ l π l ], (A51) where μ l, μ h, λ l, λ h,andβ are the Lagrange multipliers. The neessary onditions are given by H z = μ l + λ l > 0, z l 6 0, and (μ l + λ l ) z l =0; (A52) H z = μ h + λ h > 0, z6 0, and (μ h + λ h ) z l =0; (A53) λ 0 l = H q l = [αρ b e ρ b W l V 0 (q l ) βe ρ l π l ρ l ]; (A54) λ 0 h = H q h = [(1 α)ρ b e ρ b W h V 0 (q h ) θe ρ h π h ρ h + βe ρ l π h ρ l ]; μ 0 l = H T l = [ αρ b e ρ b W l + βe ρ l π l ρ l ]; μ 0 h = H T h = [ (1 + α)ρ b e ρ b W h + θe ρ h π h ρ h βe ρ l π h ρ l ]; and λ n () =λ n () =μ n () =μ n () =0,wheren = l, h. (A55) (A56) (A57) (A58) From the transversality ondition (A58) and equation (A56), μ l () μ l () = Z [αρ b e ρ b W l βρ l e ρ l π l ]d =0, (A59) 28
whih provides β = αρ R b e ρ b W l d R. ρ l e ρ l π l d (A60) From the transversality ondition (A58) and equation (A57), μ h () μ h () = Z [(1 α)ρ b e ρ b W h θe ρ h π h ρ h + βe ρ l π h ρ l ]d =0, (A61) whih provides θ = αρ R R b e ρ b W l d +(1 α)ρb e ρ b W h d R. (A62) ρ h e ρ h π h d Sine μ l = μ 0 l λ0 l when q n is stritly dereasing in, wehave μ l = Z [αρ b e ρ b W l βe ρ l π l ρ l ]dz (A63) = Z [αρ b e ρ b W l αρ b R R e ρ b W l d e ρ l π l d e ρlπl ]dz (A64) = αρ b e ρ b W l [V 0 (q l ()) ], (A65) and μ h = = Z Z [(1 α)ρ b e ρ b W h θe ρ h π h ρ h + βe ρ l π h ρ l ]dx (A66) [(1 α)ρ b e ρ b W h R CE αρ b R e ρhπh + e ρ b W l d R e ρ l π e ρlπh ]dx (A67) l d e ρ l π l dz = (1 α)ρ b e ρ b W h [V 0 (q h ()) ]. (A68) 29
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