Optimal Contrating with Unknown Risk Preferene Chifeng Dai Department of Eonomis Southern Illinois University Carbondale, IL 62901, USA Abstrat In environments of unertainty risk sharing is often an important element of eonomi ontrats. We onsider a setting where a buyer and a risk-averse supplier ontrat for the prodution of some good under ost unertainty. At the time of ontrating, both parties have inomplete information on ost of prodution. However, after ontrating and before prodution, the supplier an privately disover the realization of ost. We study the supply ontrat that optimally balanes risk sharing and information revelation when the supplier is privately informed of its risk preferene. We find that all types of supplier ould produe either below or above the effiient supply shedule depending on the buyer s risk preferene. Moreover, "inflexible rules" rather than "disretion" arise for some range of ost realizations as a solution to the onflit between risk sharing and information revelation. Keywords: Unertainty; Risk Sharing; Asymmetri Information; Risk Preferene JEL Classifiation: D81; D82; D86
1 Introdution Eonomi parties often ontrat in environments of unertainty. For example, manufaturers often ontrat with retailers with unertain market demand; and health insurers often ontrat with physiians with unertain ost of treating patients. In these environments of unertainty, risk sharing is often an important element of eonomi ontrats. Optimal risk sharing has also drawn substantial attention in the ageny literature. For example, Zekhauser (1970), Spene and Zekhauser (1971), Holmstrom (1979), Shavell (1979), and 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 (2008) study optimal risk sharing under both adverse seletion and moral hazard. In all these studies, the equilibrium ontrats losely depend on the risk preferene of the ontrating parties. In reality, ontrating parties seldom have perfet information on eah other s risk preferene. For example, the owner of a firm typially has little knowledge about its employees degree of risk aversion; and US ompanies often do not have preise information on the risk preferene of their Chinese suppliers. In those ases, ontrating parties may have inentive to manipulate others pereption of their risk preferene. For example, the suppliers may exaggerate their vulnerability to risk in order to seure more favorable ontat terms. The purpose of our study is to investigate optimal risk sharing in vertial relationships where ontrating parties are privately informed of their 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 have inomplete information about ost of prodution. However, after signing the ontrat and before the prodution, the supplier an privately disover the realization of ost. 1
When the parties risk preferene are ommon information, both parties will have symmetri information at the time of ontrating. Although the supplier an apture information rent after signing the ontrat due to its private information on the realization of ost, the buyer an fully extrat the expeted information rent at the time of ontrating. It is well known that effiient outomes ould be ahieved in this ase through a fixed-prie ontrat if the supplier were risk-neutral. However, when the supplier is risk-averse, the ertainty equivalent of the supplier s ex post information rent ould be different for the two parties, and the buyer may not be able to fully extrat the ex post information rent at the time of ontrating. In this ase, the optimal supply shedule must balane risk sharing and the supplier s inentive to truthfully reveal its ost realization. We show that, when the buyer is risk-neutral, a supplier of small degree of risk aversion supplies less than the effiient level of good exept for the lowest and the highest realizations of the ost. More importantly, when the supplier beomes suffiiently risk-averse, optimal ontrat is haraterized by "rules" rather than "disretion" for some range of ost realizations. In other words, the supplier is required to produe a onstant level of output for some range of ost realizations. "Rules" arise as an optimal solution to the onflit between risk sharing and the supplier s inentive to exaggerate its ost realization. Whenthesupplierisprivatelyinformedofitsdegreeofriskaversion,thebuyermust sreen the supplier not only by its realization of ost but also by its degree of risk aversion. We demonstrate that the properties of the optimal ontrat ritially depends on the buyer s risk preferene. When the buyer is risk-neutral, the optimal ontrat simply balanes risk sharing and the inentive for the supplier to truthfully reveal both the realization of ost and its degree of risk aversion. Sine the ertainty equivalent of a given amount of ex post information rent is larger for a less risk-averse supplier, a less risk-averse supplier an always mimi a more risk-averse one and enjoy positive expeted utility. Consequently, under the optimal ontrat, the supply shedule for the more risk-averse supplier is distorted further 2
downwards to further redue the supplier s ex post information rent. Doing so limits a less risk-averse supplier s inentive to mimi a more risk-averse one. We show that the supplier s private information on its risk preferene aggregates the onflit between risk sharing and information revelation. Consequently, "rules" arise more frequently for high realizations of ost in optimal ontrat ompared to under ommon information on risk preferene. When the buyer is also risk-averse, the optimal ontrat must simultaneously balane the buyer s surplus from different types of suppliers, risk sharing between the two parties, and the supplier s inentives for truthful information revelation. Under the optimal ontrat, the downward distortion in prodution dereases for both types of suppliers 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 its surplus from different types of suppliers. Consequently, when the buyer is suffiiently risk-averse, both types of suppliers an produe above the effiient supply shedule. Inflexible rules are ommonly observed in vertial relationships. For example, in the apparel industry, retailers are often required to make firm, SKU-speifi orderswellinad- vane of the beginning of the selling season despite demonstrable advantages to in-season replenishment; in the eletronis industry, flexibility for reorders is often restrited within some prespeified limits of original foreasts (Barnes-Shuster et al., 2002). Our analysis suggests that the seemingly ineffiient inflexible rules an be an optimal solution to the onflit between risk sharing and information revelation in these environments of unertainty. Lewis and Sappington (1989a, 1989b) among others study the optimality of "inflexible rules" in ageny ontrats when agents fae "ountervailing inentives", i.e., agents have inentive to either understate or overstate their private information depending on its realization. In ontrast, we show that the "inflexible rules" an arise, in the absene of ountervailing inentives, as an optimal solution to the onflit between risk sharing and 3
information revelation. de Mezza and Webb (2000) and Jullien, Salanie and Salanie (2007) study the optimal insurane ontrats under moral hazard when insurane ustomers are privately informed of their risk preferene. Landsberger and Meilijson (1994) onsider the optimal insurane ontrat between one risk-neutral monopolisti insurer and one risk-averse agent who is privately informed of 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 the above studies, we onsider the optimal supply ontrat when suppliers differ with respet to both ost of prodution and degree of risk aversion. Our study also relates to the literature on multi-period mehanism design. Riordan and Sappington (1987), Courty and Li (2000), Dai et al. (2006), and Eső and Szentes (2007) study two-period models where risk-neutral agents learn payoff-relevant private information in both periods. They analyze the optimal revelation mehanism where the ontrat is signed in the first period before the agent disovers his seond period private information. In ontrast to these artiles, we study the optimal supply ontrat between risk-averse parties. We investigate the interation between risk sharing and information revelation in the optimal supply ontrat. 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 buyer is risk-neutral and the supplier s degree of risk aversion is ommon information. Setion 4 examines the optimal ontrat when the supplier is privately informed of its 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. 4
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 surplus is W = V (q) T,whereT 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 is π = T q. The utility of the supplier belongs to some smooth one-dimensional family of utility funtions F = U ρ ( ) that are ranked aording to the Arrow-Prat measure of risk aversion: U 00 ρ (π)/u 0 ρ(π) is inreasing with ρ for any wealth level π. 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 {l, h} with h>l, Pr(ρ = l) =α l and Pr(ρ = h) =α h = 1 α l. The supplier s marginal ost of prodution,, is unertain at the time of ontrating. However, it is ommon knowledge that the distribution of follows an absolutely ontinuous and stritly inreasing umulative distribution funtion F () on [, ]. After ontrating with the buyer and before the prodution takes plae, the supplier privately disovers the realization of. We assume that the distribution of satisfies the following regularity ondition: d[ + F ()/f()]/d > 0. The ondition is ommonly imposed in ageny literature to ensure that the equilibrium supply shedule to be monotonially dereasing in. As we demonstrate later in our analysis, the ondition does not ensure suh property in equilibrium supply shedule in our model. The timing and ontratual relation between the buyer and the supplier are as follows: 5
(1) the supplier privately learns its degree of risk aversion ρ; (2) the buyer offers the supplier a set of ontrat menus M ρ = {T ρ (),q ρ ()} onditional on the supplier s degree of risk aversion ρ and its realization of marginal ost ; (3) the supplier selets its preferred menu M ρ given its private information on ρ; (4) the supplier disovers, and selets a desired option (T ρ (),q ρ ()) from the seleted menu M ρ ; (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 buyer is riskneutral and the supplier s degree of risk aversion is ommon information. When the buyer is risk-neutral, its optimization problem is hoosing {T ρ (),q ρ ()} to maximize its expeted surplus: Z [V (q ρ ()) T ρ ()]df (), (1) for ρ = l, h. A ontrat is feasible (or implementable) provided it is inentive ompatible and individually rational. Inentive ompatibility requires that the ontrat indues eah type of supplier to truthfully report its realization of marginal ost, i.e., π ρ ( i i ) > π ρ ( i j ) for i 6= j, (2) where π ρ ( i i ) and π ρ ( i j ) denote the supplier s respetive profits from hoosing options (T ρ ( i ),q ρ ( i )) and (T ρ ( j ),q ρ ( j )) when the realization of its marginal ost is i. Individual rationality requires that the expeted utility from the ontrat for eah type of 6
supplier must be nonnegative, i.e., E[U ρ (M ρ )] = Z U ρ (T ρ () q ρ ())f()d > 0. (3) Proposition 1 desribes the general properties of the optimal ontrat when the buyer is risk-neutral and the supplier s risk preferene is ommon information. Proposition 1 The optimal ontrat has the following properties: for ρ = l, h, (a) E[U ρ (M ρ )] = U ρ(π ρ ())df () =0; (b) In no bunhing region, q ρ () is given by [V 0 (q ρ ()) ]f() =F () D ρ (), (4) where D ρ () = U 0 ρ(π ρ (x))df (x) U 0 ρ(π ρ (x))df (x) ; (5) () There exists σ suh that omplete or partial bunhing ours for some interval [ 0, ] when ρ>σ. Proof. See appendix. When the supplier s degree of risk aversion is ommon information, both parties have symmetri information at the time of ontrating. Consequently, although the supplier an apture ex post information rent from its 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 realizations of. (Note that it is the differene in T () that provides the inentive for the supplier to truthfully 7
reveal its 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 show below, the supplier s ex post information rent would be ostless to the buyer and the effiient outome would be ahieved if the supplier were risk-neutral. However, when the supplier is risk-averse, the optimal supply shedule must balane risk sharing and the inentive for truthful information revelation. Equation (4) demonstrates the intuition. When the supplier s realization of marginal ost is e, raisingq ρ (e) by δq will in expetation inrease the supplier s prodution effiieny by [V 0 (q ρ (e)) e]f(e)δq. However, the inrease in q ρ (e) will also raise the supplier s ex post information rent by δq when <e. Consequently, in expetation the inrease in q ρ (e) raises the supplier s ex post information rent by F (e)δq. When the supplier is risk-averse, the buyer an only redue T ρ () for all realizations of by δqd ρ (e) in order to indue the supplier s partiipation. Notie that δq R U 0 ρ(π ρ (x))df (x) is the inrease in the supplier s expeted utility as a result of the inrease in ex post information rent, and U 0 ρ(π ρ (x))df (x) istheinreaseinthesupplier s expeted utility as a result of one unit of inrease in T ρ () for all realizations of. Therefore, δqd ρ (e) is the ertainty equivalent of the inrease in ex post information rent for the supplier. At the optimum, the buyer s marginal benefit ofraisingq ρ (e) must equal its marginal ost of doing so, whih yields equation (4). When the supplier is risk-neutral, i.e., u 00 =0, D ρ (e) =F (e), whih means the ertainty equivalent of the inrease in information rent is F (e) 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 for all realizations of by exatly F (e)δq. In that ase, the right-hand side of equation (4) beomes zero, and V 0 (q ρ ()) =. The optimal ontrat 8
q First-best solution Seond-best solution ' Figure 1: Optimal supply shedule with partial bunhing in [ 0, ]. would be a fixed prie ontrat, and the supplier would always supply the effiient level of good. When the supplier is of small degree of risk aversion, the optimal supply shedule is stritly dereasing in in [, ]. Then, equation (4) suggests that D ρ () =F () and V 0 (q ρ ()) = at and. In other words, the supplier delivers the effiient amount of good at and. Notie that Dρ() 0 =Uρ(π 0 ρ ())f()/ U 0 ρ(π ρ (x))df (x) and Uρ(π 0 ρ ())/ = Uρ 00 (π ρ ())q ρ () > 0. Therefore, the sign of f() D 0 () must hange one and only one with f() D 0 () > 0 and f() D 0 () < 0. SineF () D ρ () =0at and, it suggests that F () D ρ () > 0 on (, ). Consequently, equation (4) suggests that the supplier deliverslessthantheeffiient amount of good on (, ). When the supplier beomes suffiiently risk-averse, the monotoniity ondition (q ρ () is 9
non-inreasing) beomes onstraining. In this ase, "inflexible rules" arise as an optimal solution to the onflit between risk sharing and the supplier s inentive to exaggerate its ost realization. In other word, bunhing ours and the supplier is required to produe a onstant level of good in some interval [ 0, ] where < 0 <. Whenf() does not hange rapidly (for example, is uniformly distributed), bunhing ours for the entire interval [ 0, ]. 1 On the other hand, when f() does hange rapidly, bunhing ould our for some ranges of in the interval [ 0, ]. Figure 1 demonstrates the optimal supply shedule with partial bunhing in some interval [ 0, ]. When the supplier onverges to infinitely risk-averse, D ρ () onverges to zero for [, ). Then equation (4) onverges to [V 0 (q ρ ()) ]f() =F () (6) for [, ), whih is the well known solution for a standard adverse seletion problem where the supplier is privately informed of its marginal ost of prodution at the time of ontrating. This is beause the supplier will partiipate in the ontrat only if it is guaranteed nonnegative profit for all realizations of when it is infinitely risk-averse. Consequently our model beomes equivalent to one that the supplier is perfetly informed of its 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. 1 With a onstant absolute risk aversion (CARA) utility funtion and an uniform distribution of, Salanie (1990) and Laffont and Rohet (1998) also show that omplete bunhing arises in some interval [, ] where < <. 10
4 Asymmetri Information on Risk Preferene 4.1 A Risk-Neutral Buyer When the supplier is privately informed of its degree of risk aversion, the buyer must sreen the supplier not only by its realization of ost but also by its degree of risk aversion. When the buyer is risk-neutral, the buyer s optimization problem is hoosing a set of ontrat menus M ρ = {T ρ (),q ρ ()} to maximize Z {α l [V (q l ()) T l ()] + α h [V (q h ()) T h ()]} f()d, (7) subjet to E[U ρ (M ρ )] = Z U ρ (T ρ () q ρ ())f()d > 0; (8) π ρ ( i i ) > π ρ ( i j ) for i 6= j ; and (9) E[U ρ (M ρ )] > E[U ρ (M s )], (10) where ρ = l, h, s = l, h, andρ 6= s. The onditions (8) and (9) ensure the supplier s partiipation and its truthful revelation of marginal ost regardless of its degree of risk aversion; and ondition (10) guarantees the supplier truthfully reveals its degree of risk aversion. Proposition 2 desribes the general properties of the optimal ontrat when the buyer is risk-neutral and the supplier is privately informed of its degree of risk aversion. Proposition 2 When the buyer is risk-neutral, the optimal ontrat has the following properties : 11
(a) E[U l (M l )] >E[U h (M h )] = 0; (b) In no bunhing region, the optimal supply shedule for the less risk-averse supplier is haraterized by [V 0 (q l ()) ]f() =F () D l (); and (11) the optimal supply shedule for the more risk-averse supplier is haraterized by α h [V 0 (q h ()) ]f() =α h [F () D h ()] + α l G(), (12) where D ρ () is defined by (5) for ρ = l, h and G() U 0 l (π h(x))df (x) U 0 l (π l(x))df (x) U 0 h (π h(x))df (x) U 0 l (π h(x))df (x) U 0 h (π h(x))df (x) U 0 l (π l(x))df (x). (13) () There exists σ 0 ρ suh that omplete or partial bunhing ours for some interval [ 0 ρ, ] when ρ>σ 0 ρ for ρ = l, h. 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 preferene. However, the utility funtion of a less risk-averse supplier is an inreasing and onvex transformation of that of a more risk-averse supplier, and the less risk-averse supplier an always 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 its truthful revelation of its 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 its 12
realization of marginal ost, as in the ase of ommon information on risk preferene. 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 tradeoff risk sharing, the supplier s inentives to truthfully reveal its realization of ost, and the less risk-averse supplier s inentive to truthfully reveal its degree of risk aversion. Equation (12) demonstrates the tradeoff. When the more risk-averse supplier s realization of marginal ost is e, raisingq h (e) by δq will in expetation inrease the prodution effiieny by α h [V 0 (q h (e)) e]f(e)δq where α h is probability that the supplier is more risk-averse. However, the inrease in q h (e) will also raise the more risk-averse supplier s ex post information rent by δq when <e. Consequently, in expetation it inreases the more risk-averse supplier s ex post information rent by δq R U 0 h (π h(x))df (x). In addition, the inrease in q h (e) will also raise the less riskaverse supplier s rent by δq R U 0 l (π h(x))df (x) when it mimis the more risk-averse one. The ertainty equivalents of the above ex post information rents for both types of suppliers are δq R U 0 h (π h(x))df (x)/ U 0 h (π h(x))df (x) and δq R U 0 l (π h(x))df (x)/ U 0 l (π l(x))df (x), respetively. Notie that U 0 h (π h(x))df (x) and U 0 l (π l(x))df (x) are the inreases in expeted utilities resulting from one unit inrease in all possible states for both types of suppliers, respetively. In antiipation of the supplier s ex post information rent, at the time of ontrating the buyer an redue the more risk-averse supplier s payments by the amount of δq R U 0 h (π h(x))df (x)/ U 0 h (π h(x))df (x) for all realizations of marginal ost. Doing so fully extrats the more risk-averse supplier s ex post information rent. The redution in payments for the more risk-averse supplier also redues the less risk-averse supplier s rent from exaggerating its degree of risk aversion by δq R U 0 h (π h(x))df (x)/ U 0 h (π h(x))df (x)) U 0 l (π h(x))df (x). Consequently, the buyer must inrease the less risk-averse supplier s payment for all real- 13
izations of by δqg(e) as a result of the inrease in q h (e). Doingsoprovidestheless risk-averse supplier just enough inentive to truthfully reveal its degree of risk aversion. At the optimum, the supplier s marginal benefit ofraisingq h (e) must equal its marginal ost of doing so, whih yields equation (12). Notie that αg() (whih is positive on (, ) as shown in the proof of Corollary 1) is the effet of the supplier s private information on its risk preferene. In order to limit a less risk-averse supplier s inentive to exaggerate its 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 Corollary 1, the more risk-averse supplier produes below the seond-best supply shedule. Corollary 1 Under the optimal ontrat, the more risk-averse supplier s supply shedule is below the seond-best level. Proof. See Appendix. As either type of supplier beomes suffiiently risk-averse, the monotoniity ondition (q ρ () is non-inreasing) beomes onstraining and "inflexible rules" arise in the optimal ontrat, similar to the ase of ommon information on risk preferene. Then, the supplier is required to produe a onstant level of good in some interval [ 0 ρ, ] where < 0 ρ < and ρ = l, h. However, as we show in Corollary 2, the supplier s private information on itsriskpreferene aggregatestheonflit between risk sharing and information revelation. Consequently, "inflexible rules" arise more frequently for high realizations of ost ompared to the ase of ommon information on risk preferene. Corollary 2 "Inflexible rules" arise more frequently for high realizations of ost ompared to ommon information on risk preferene. Proof. See Appendix. 14
Suppose that one type of supplier is risk-neutral and the other type of supplier is infinitely risk-averse. Then equation (12) beomes α h [V 0 (q h ()) ]f() =F (). (14) A diret omparison between equations (6) and (14) demonstrates the effet on the optimal ontrat of the supplier s private information on its risk preferene. An inrease in q h (e) by δq inreases the more risk-averse supplier s prodution effiieny by [V 0 (q(e)) e]f(e)δq regardless whether the supplier is privately informed of its risk preferene. However, with private information on risk preferene, an inrease in q h (e) by δq inreases the ex post information rent for not only the more risk-averse supplier but also the less risk-averse supplier by F (e). The ertainty equivalent of the ex post information rent is zero for the more riskaverse supplier, whih means that the buyer annot extrat any of the ex post rent at the time of ontrating. Consequently, with private information on risk preferene, 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 surplus from different types of suppliers, in addition to the tradeoff between risk sharing and the inentives for the supplier to truthfully reveal both its realization of ost and its degree of risk aversion. Suppose the buyer s utility funtion U( ) also belongs to some smooth one-dimensional family of utility funtions that is ranked aording to the Arrow-Prat measure of risk aversion ρ. The buyer s optimization problem is hoosing a set of ontrat menus M ρ = 15
{T ρ (),q ρ ()} for ρ = l, h to maximize E[U] = Z {α l U(W l ()) + α h U(W h ())} f()d (15) subjet to onditions (8), (9), and (10), where W l () =V (q l ()) T l () and W h () = V (q h ()) T h (). Proposition 2 desribes the properties of the optimal ontrat when both the buyer and the supplier are risk-averse. Proposition 3 When both the buyer and the supplier are risk-averse, the optimal ontrat has the following properties: (a) E[U l (M l )] >E[U h (M h )] = 0; (b) In no bunhing region, the optimal supply shedule for the less risk-averse supplier is haraterized by α l U 0 (W l ())[V 0 (q l ()) ]f() α l η l + α h η h = α l η l α l η l + α h η h ( U ) 0 (W l (x))df (x) D l () ; (16) η l and the optimal optimal supply shedule for the more risk-averse supplier is haraterized by α h U 0 (W h ())[V 0 (q h ()) ]f() α l η l + α h η h = α h η h α l η l + α h η h ( U ) 0 (W h (x))df (x) D h () (17) η h α l η + l G(), (18) α l η l + α h η h where η l U 0 (W l (x))df (x) and η h U 0 (W h (x))df (x). Proof. See Appendix. 16
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 its private information on its degree of risk aversion. However, the optimal supply shedule is profoundly different ompared to the ase when the buyer is risk-neutral. For the less risk-averse supplier, when the realization of marginal ost is e, raising q l (e) by δq will inrease W l (e) by [V 0 (q l (e)) e]δq whih inreases the buyer s ertainty equivalent by δqα l U 0 (W l (e))[v 0 (q l (e)) e]f(e)/(α l η l + α h η h ).Notethatα l η l + α h η h is the inrease in the buyer s expeted utility resulting from one unit inrease in its surplus for all possible events. On the other hand, the inrease in q l (e) will also raise the less risk-averse supplier s ex post information rent by δq when <e. For the buyer, the additional ex post information rent is equivalent to a redution of R U 0 (W l (x))df (x)/η l in W l () for realizations of. For the less risk-averse supplier, the ertainty equivalent of the additional ex post information rent is D l (e) as disussed earlier. Therefore, at the time of ontrating the buyer an optimally redue the less risk-averse supplier s payments by δqd l (e) for all realizations of. Consequently, the right-hand side of (16) is the effet of the additional ex post information rent on the buyer s ertainty equivalent. Notie that, depending on the relative sizes of R U 0 (W l (x))df (x)/η l and D l (e), whih in turn depends on the relative degree of risk aversion between the two parties, the righthand side of (16) an be either positive or negative and the optimal supply shedule an be either below or above the effiient level. For example, when the supplier onverges to risk-neutral, D l () onverges to F (). In the optimal ontrat, W l () must be dereasing and therefore U 0 (W l ()) must be inreasing in. Moreover, R U 0 (W l (x))df (x)/η l = F () at and. Then it an be readily show that U 0 (W l (x))df (x)/η l <F() and the right-hand side of equation (16) is negative on (, ). Consequently, the optimal supply shedule is above the effiient level on (, ). On the other hand, by ontinuity the optimal supply shedule must be below the effiient 17
level when the buyer onverges to risk-neutral, based on our analysis of a risk-neutral buyer in the previous setion. The buyer s risk aversion has a different impat on the more risk-averse supplier s supply shedule. Equation (17) demonstrates how the optimal supply shedule for the more riskaverse supplier balanes risk sharing, inentives for information revelation, and the buyer s surplus from different types of suppliers. The ertainty equivalent of the additional surplus for the buyer from inreasing q h (e) by δq is δqα h U 0 (W h (e))[v 0 (q h (e)) e]f(e)/(α l η l + α h η h ). However, the inrease in q h (e) also inreases the ex post information rent for both types of suppliers. For the buyer, the additional ex post information rent is equivalent to a redution of R U 0 (W h (x))df (x)/η h in W h () for all realizations of. For the more risk-averse supplier, the ertainty equivalent of the additional ex post information rent is D h (e). Therefore, the additional ex post information rent eventually redues the buyer s surplus from a more risk-averse supplier by R U 0 (W h (x))df (x)/η h D h (e) for all realizations of. We have shown earlier that, as a result of the inrease in q h (e), the buyer must also inrease the less risk-averse supplier s payments by G(e)δq for all realizations of to indue its truthful revelation of ost realizations. Consequently, the inrease in q h (e) also redues the buyer s surplus from a less risk-averse supplier by G(e)δq for all realizations of. Therefore, the right-hand side of (17) is the overall effet of the additional ex post information rent on the buyer s ertainty equivalent. Notie that the ratio α ρ η ρ /(α l η l + α h η h ) for ρ = l, h measures how the risk-averse buyer weights the surplus from different types of suppliers. Similar to the ase of risk-neutral buyer, in order to restrit a less risk-averse supplier s inentive to exaggerate its degree of risk aversion, the buyer distorts the more risk-averse supplier s ontrat towards a ost plus ontrat ompared to the ontrat for the less riskaverse supplier. However, the distortion is smaller ompared to the ase of a risk-neutral buyer as U 0 (W l (x))df (x) < U 0 (W h (x))df (x), i.e., the distortion for a more risk- 18
q q l q h First-best solution Figure 2: Optimal ontrat with a highly risk-averse buyer. 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 also an be either below or above the effiient level depending on the relative degree of risk aversion between the buyer and the supplier. Figure 2 demonstrates an optimal ontrat with both types of suppliers produe above the effiient supply shedule. We summarize the above property of the optimal ontrat with a risk-averse buyer in Corollary 3. Corollary 3 Depending on the relative degree of risk aversion between the buyer and the supplier, the optimal supply shedule for both types of suppliers an be either below or above the effiient level. 19
5 Conlusion Eonomi parties often ontrat in environments of unertainty, and risk sharing is an important element of many eonomi ontrats. We study the supply ontrat between a buyer and a risk-averse supplier under ost unertainty when the supplier is privately informed of its risk preferene. We show that the optimal ontrat simultaneously balanes risk sharing, inentives for information revelation, and the buyer s expeted surplus with different types of suppliers. When the buyer is risk-neutral, all types of supplier generally produe below the effiient level of output. Moreover, the supply shedule for a more risk-averse supplier is distorted further downwards to indue a less risk-averse supplier s revelation of its risk preferene. However, when the buyer is also risk-averse, both types of suppliers may produe above the effiient supply shedule. The supplier s private information on its risk preferene also aggregates the onflit between risk sharing and information revelation. Consequently, the optimal ontrat is often haraterized by "inflexible rules" rather than "disretion" for some range of ost realizations. Therefore, our analysis shows, in the absene of transation osts, bounded rationality, and ountervailing inentives, seemingly ineffiient "inflexible rules" an arise in optimal ontrats in environments of unertainty. Our researh ould be extended in several diretions. For example, although the supplier s information on ost of prodution is inomplete at the time of ontrating, the supplier ould be better informed of its future osts than the buyer is. Moreover, suppliers with different levels of expertise might have different foreasts of future osts at the time of ontrating. The optimal ontrats in these situations merit further investigation. 20
6 Appendix 6.1 Proof of Proposition 1 A well known haraterization of feasible ontrats is the following: (a) T 0 () =q 0 (); (b) q() is non inreasing; () EU 0. Therefore, the buyer s optimization problem an be written as an optimal ontrol problem with state variables T () and q() and ontrol variable q 0 () =z: Max Z [V (q()) T ()]df (), (A1) subjet to q 0 () =z T 0 () =z q 0 () 0; and (A2) (A3) (A4) Z U(T () q())df () > 0. (A5) The Hamiltonian is H =[V (q) T ]f()+μz + λz + θu(π)f(). (A6) The neessary onditions are given by H z = μ + λ > 0, z6 0, and (μ + λ) z =0; (A7) 21
λ 0 = H q = [V 0 (q) θu 0 (π)]f(); μ 0 = H T = [ 1+θU0 (π)]f(); and λ() =λ() =μ() =μ() =0. (A8) (A9) (A10) From (A9) and (A10), μ() μ() = Z [1 θu 0 (π)]d =0. (A11) Therefore, θ = 1 U 0 (π())df (). (A12) Define h() =μ+λ. From ondition (A7), on any interval where q is stritly dereasing, h() and λ() must be zero. So h 0 () =μ + μ 0 + λ 0 =0,whihleadsto μ = μ 0 λ 0. (A13) Sine λ 0 () =0, substituting (A9) into the above equation for μ 0 and λ 0,wehave μ = Z [1 θu 0 (π)]f(x)dx =[V 0 (q) ]f(x). (A14) Substituting (A12) into the above equation for θ, we have [V 0 (q) ]f() =F () U 0 (π(x))df (x) U 0 (π(x))df (x). (A15) 22
Sine 1 U 0 (π())/ U 0 (π(x))df (x) is stritly dereasing in, its sign hanges one and only one with 1 U 0 (π())/ U 0 (π(x))df (x) > 0 (< 0)at = ( = ). Moreover, F () U 0 (π(x))df (x)/ U 0 (π(x))df (x) equals zero at and. Therefore, F () U 0 (π(x))df (x)/ U 0 (π(x))df (x) must be positive on (, ). From (A15), we have dv 0 (q) d =2 R U 0 (π()) U 0 (π(x))df (x) f 0 () [F () U 0 (π(x))df (x) f 2 R () U 0 (π(x))df (x) ]. (A16) Sine U 0 (π(x))df (x)/ U 0 (π(x))df (x) equals zero at and U 0 (π())/ U 0 (π(x))df (x) < 1 in the neighborhood of, (A16) suggests dv 0 (q)d > 0 and the monotoniity ondition is not onstraining in the neighborhood of. When the supplier onverges to risk-neutral, U 0 (π())/ U 0 (π(x))df (x) onverges to 1, U 0 (π(x))df (x)/ U 0 (π(x))df (x) onverges to F (), anddv 0 (q)d onverges to 1. In that ase, (A16) suggests that dv 0 (q)d > 0 and q is stritly dereasing on [, ]. For any monotonially dereasing shedule of q() satisfying (A15), a lower bound for U 0 (π())/ U 0 (π(x))df (x) and F () U 0 (π(x))df (x)/ U 0 (π(x))df (x), respetively, an be obtained by setting q() =q() for all [, ]. It an be readily shown that the lower bound for U 0 (π())/ U 0 (π(x))df (x) is great than 2 and therefore U 0 (π())/ U 0 (π(x))df (x) > 2 when the supplier is suffiiently risk averse. In that ase, dv 0 (q)d < 0 and bunhing oursforsomeentireintervalof[ 0, ] if f 0 () = 0. However, if f 0 ()/f 2 () is suffiiently large for some ranges of, dv 0 (q)d < 0 an our at the above lower bound of F () U 0 (π(x))df (x)/ U 0 (π(x))df (x) for some middle range of., i.e., bunhing an our for some middle range of. For any bunhing range [ 1, 2 ] in [ 0, ], λ( 1 )=λ( 2 )=0for ontinuity. Therefore, 23
from (A8) we have Z 2 1 [V 0 U 0 (π()) (q) U ]d =0. 0 (π(x))df (x) Moreover, q( 1 ) and q( 2 ) are determined by (A15). 6.2 Proof of Proposition 2 The Hamiltonian is H = α[v (q l ()) T l ()]f()+(1 α)[v (q h ()) T h ()]f() (A17) +μ l z l + μ h z h + λ l z l + λ h z h + θu h (π h )f()+β[u l (π l ) U h (π h )]f(), (A18) where μ l, μ h, λ l, λ h, θ, 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; (A19) H z = μ h + λ h > 0, z6 0, and (μ h + λ h ) z l =0; (A20) λ 0 l = H q l = [αv 0 (q l ) βu 0 l(π l )]f(); (A21) λ 0 h = H q h = [(1 α)v 0 (q h ) θu 0 h(π h ) + βu 0 l(π h )]f(); μ 0 l = H T l = [ α + βu 0 l (π l )]f(); μ 0 h = H T h = [ (1 α)+θu 0 h(π h ) βu 0 l(π h )]f(); and λ ρ () =λ ρ () =μ ρ () =μ ρ () =0,whereρ = l, h. (A22) (A23) (A24) (A25) 24
From the transversality ondition (A25) and equation (A23), μ l () μ l () = Z [α βu 0 l(π l )]df () =0, (A26) whih provides β = α U 0 l (π l)df (x) > 0. (A27) From equation (A24) and the transversality ondition (A25), μ h () μ h () = Z [(1 α) θu 0 h(π h )+βu 0 l(π h )]d =0, (A28) whih provides θ = 1 α U 0 h (π h(x))df (x) α U 0 l (π h(x))df (x) U 0 h (π h(x))df (x) U 0 l (π l(x))df (x). (A29) When q ρ is stritly dereasing in, wehave h 0 ρ() =μ ρ + μ 0 ρ + λ 0 ρ =0or μ ρ = μ 0 ρ λ 0 ρ. (A30) Then substituting μ 0 ρ and λ 0 ρ into (A30), we have μ l = Z αul 0 [α (π l) U 0 l (π l)df (x) ]dz = α[v 0 (q l ()) ]f() (A31) 25
and μ h = Z [(1 α) θe ρ h π h ρ h + βe ρ l π h ρ l ]dx (A32) " = (1 α) F () U h 0 (π # h(x))df (x) U 0 h (π h(x))df (x) = (1 α)[v 0 (q h ()) ], + αg() where G() U 0 l (π h(x))df (x) U 0 l (π l(x))df (x) U 0 h (π h(x))df (x) U 0 l (π h(x))df (x) U 0 h (π h(x))df (x) U 0 l (π l(x))df (x). (A33) 6.3 Proof of Corollary 1 Sine where G() U 0 l (π h(x))df (x) U 0 l (π l(x))df (x) " R() = G 0 () = U 0 h (π h(x))df (x) U 0 l (π h(x))df (x) U 0 h (π h(x))df (x) U 0 l (π l(x))df (x), U 0 l (π h(x))df (x) U 0 l (π l(x))df (x) R()f(), # Ul 0(π 0 h()) U 0 l (π h(x))df (x) Uh (π h()) U 0 h (π h(x))df (x) (A34) (A35) (A36) 26
Sine U h ( ) must be a stritly onave transformation of U l ( ), there exists a stritly onave funtion Y ( ) suh that U h ( ) Y (U l ( )). Therefore, " # 1 R() = U 0 l (π h(x))df (x) Y 0 (U l (π h ())) Y 0 (U l (π h (x)))ul 0(π Ul 0 (π h ()). h(x))df (x) (A37) Notie that π h () is stritly dereasing in, onsequently Y 0 (U l (π h ())) is stritly inreasing in. Therefore, there exists some 0 suh that R() > 0 in [, 0 ) and R() < 0 in ( 0, ]. Therefore, G 0 () > 0 in [, 0 ) and G 0 () < 0 in ( 0, ]. Moreover, G() =G() =0. Consequently, G() > 0 on (, ). 6.4 Proof of Corollary 2 From (A32), we have dv 0 (q h ()) d ( = (1 α) 2 +αg 0 (). R Uh 0 (π h()) U 0 h (π h(x))df (x) f 0 () [F () U h 0 (π ) h(x))df (x) f 2 R () U 0 h (π h(x))df (x) ] (A38) As we show in the proof of Corollary 1, there exists some 0 suh that G 0 () > 0 in [, 0 ) and G 0 () < 0 in ( 0, ]. Therefore, the monotoniity ondition is not onstraining in the neighborhood of. However, for any given F (), bunhing is more likely to our in the interval ( 0, ] ompared to the ase of ommon information on risk preferene. 27
6.5 Proof of Proposition 3 The Hamiltonian is H = {α l U l (W l ()) + α h U h (W h ())}f()+μ l z l + μ h z h (A39) +λ l z l + λ h z h + θu h (π h )f()+β[u l (π l ) U h (π h )]f(), where μ l, μ h, λ l, λ h, θ, andβ are the Lagrange multipliers. The neessary onditions are given by H z l = μ l + λ l > 0, z l 6 0, and (μ l + λ l ) z l =0; (A40) H z h = μ h + λ h > 0, z6 0, and (μ h + λ h ) z l =0; (A41) λ 0 l = H q l = [α l U 0 (W l )V 0 (q l ) βu 0 l(π l )]f(); (A42) λ 0 h = H q h = [α h U 0 (W h )V 0 (q h ) θu 0 h(π h ) + βu 0 l (π h )]f(); μ 0 l = H T l = [ α l U 0 (W l )+βu 0 l (π l )]f(); μ 0 h = H T h = [ α h U 0 (W h )+θu 0 h(π h ) βu 0 l(π h )]f(); and λ ρ () =λ ρ () =μ ρ () =μ ρ () =0,whereρ = l, h. (A43) (A44) (A45) (A46) From the transversality ondition (A46) and equation (A44), μ l () μ l () = Z [α l U 0 (W l ) βu 0 l(π l )]dx =0, (A47) 28
whih provides β = α l U 0 (W l )df (x) U 0 l (π. (A48) l)df (x) From the transversality ondition (A46) and equation (A45), μ h () μ h () = Z [α h U 0 (W h ) θu 0 h(π h )+βu 0 l(π h )]df (x) =0, (A49) whih provides θ = α h U 0 (W h )df (x) U 0 h (π h(x))df (x) α l U 0 (W l )df (x) U 0 l (π h(x))df (x) U 0 h (π h(x))df (x) U 0 l (π l(x))df (x). (A50) When q ρ is stritly dereasing in, wehave h 0 ρ() =μ ρ + μ 0 ρ + λ 0 ρ =0or μ ρ = μ 0 ρ λ 0 ρ. (A51) Then substituting μ 0 ρ and λ 0 ρ into (A51), we have μ l = = Z Z [α l U 0 (W l ) βu 0 l(π l )]dx (A52) [α l α lul 0(π l) U 0 (W l )df (x) U 0 l (π ]dx l)df (x) = α l U 0 (W l )[V 0 (q l ()) ], 29
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