Optimal Procurement Contracts with Private Knowledge of Cost Uncertainty

Transcription

1 Optimal Procurement Contracts with Private Knowledge of Cost Uncertainty Chifeng Dai Department of Economics Southern Illinois University Carbondale, IL 62901, USA August 2014 Abstract We study optimal procurement contracts in an environment where a risk-averse supplier discovers cost information privately and gradually over time. At the time of contracting, the supplier is privately informed about the cost uncertainty which is characterized by the underlying distribution of cost condition. After contracting and before production, the supplier privately discovers the realization of cost condition. We show that both parties prefer more risk when the supplier is relatively risk tolerant but less risk when the supplier is sufficiently risk-averse. However, the buyer always prefers to contract before the cost uncertainty resolves regardless of the supplier s degree of risk aversion. The nature of the optimal procurement contact depends on the supplier s degree of risk-aversion. A separating contract is optimal when the supplier is either not very risk-averse or infinitely risk-averse; however, a pooling contract, which offers the same contract terms regardless of the cost uncertainty, can be optimal when the supplier is sufficiently risk-averse. Moreover, the optimal production schedule is often characterized by "inflexible rules". Keywords: Procurement Contract; Uncertainty; Risk Aversion; Adverse Selection JEL Classification: D81; D82; D86

2 1 Introduction In many contractual environments, economic parties face uncertainty at the time of contracting but can discover more information after signing a contract. For instance, manufacturers often face uncertainty about the future cost of their inputs when contracting with their downstream buyers 1 ; healthcare providers can only estimate the cost of treating a population of patients when negotiating with health insurers; similarly, retailers often contract with their suppliers based upon predictions of market conditions. After signing their contracts, manufacturers and healthcare providers will discover more information about their costs, and retailers will gain better knowledge of their market conditions. In these scenarios, some natural questions arise: what are the contracting parties preference for the uncertainty at the time of contracting? how do the uncertainty at the time of contracting and the new information after contracting affect the terms of contract given that contracting parties are often asymmetrically informed about the uncertainty and the new information? When is the optimal time of contracting as uncertainty resolves over time? Most studies in contract theory assume that firms are risk-neutral. However, there are many reasons that firms may act as if they were risk-averse in the presence of uncertainty. For example, information imperfections in the financial markets, non-diversified owners, significant bankruptcy costs, nonlinear tax systems, and risk-averse managers with career concerns and/or performance-based compensations are all reasons that may cause firms to behave in a risk-averse manner. Empirically, firms aversion to risk is evidenced by the extent of costly risk management activities conducted by corporations. 2 Nonetheless, there is little research that considers the effects of risk aversion on contractual arrangements. We consider a setting where a risk-neutral buyer and a risk-averse supplier contract 1 In many industries (e.g., paper, agriculture, electronics, textiles), the spot prices of many commodity inputs (e.g., commodity fibers, petrochemicals) fluctuate substantially but there are no futures markets for them. See Li and Kouvelis (1999) for a detailed discussion. 2 See Géczy et al., (1997), Jacque (1981), Mayers and Smith (1982), and Tufano (1996) for instance. 1

3 for the production of some good under cost uncertainty. At the time of contracting, the supplier is privately informed about the cost uncertainty, which is characterized by the underlying distribution of cost condition. Then, after contracting and before production, the supplier privately discovers the realization of cost condition. We show that both parties preferences for uncertainty and the nature of optimal procurement contract all depend on the supplier s degree of risk aversion. The supplier prefers more risk (a riskier distribution of cost condition) when it is relatively risk tolerant, but prefers less risk (a less risky distribution of cost condition) when it is sufficiently risk-averse. The result follows from the fact that the supplier s final profit must be decreasing and convex in the realization of cost condition for the supplier to truthfully reveal the cost condition. Consequently, on one hand, the expected value of the supplier s profit increases as costs become more variable; on the other hand, for any given expected value of profit, the supplier s certainty equivalent of the profit decreases as the profit becomes more variable, and more so as the supplier becomes more risk-averse. Therefore, as the cost condition becomes more variable, the supplier s expected utility increases when it is sufficiently risk-tolerant but decreases when it is sufficiently risk-averse. The optimal procurement contract balances risk sharing and the incentives for the supplier to truthfully reveal both its cost uncertainty and its later discovery of cost condition. A separating contract, which offers different contract terms conditional on the supplier s cost uncertainty, is optimal when the supplier is either not very risk-averse or infinitely risk-averse; however, a pooling contract, which offers the same contract terms regardless of the supplier s cost uncertainty, can be optimal when the supplier is sufficiently risk-averse. In a separating contract, a supplier with favorable risk (a preferred distribution of cost condition) is required to produce below the efficient level of output except for the lowest and the highest realizations of cost condition. When the supplier is sufficiently risk-averse, bunching arises in the production schedule the supplier is required to produce a constant 2

4 level of output for certain ranges of cost condition. The production schedule for a supplier with unfavorable risk (a less preferred distribution of cost condition) is further distorted to limit the information rent of a supplier with favorable risk. It can either over-produce or under-produce compared to the case of common information on cost uncertainty, and the direction of distortion depends on the certainty equivalent of the marginal information rent for each type of supplier as the production level increases for a supplier with unfavorable risk. When the distributions of cost condition are sufficiently different for different types of suppliers, bunching occurs in the production schedule even if the supplier is risk-neutral. We find that the buyer prefers a supplier with a riskier cost distribution when the supplier is sufficiently risk tolerant, because a supplier with a riskier cost distribution generates greater total surplus for the buyer and the supplier to share. However, when the supplier becomes more risk-averse, it demands a larger risk premium for a riskier cost distribution. Consequently, when the supplier is sufficiently risk-averse, the difference in risk premiums can outweigh the difference in total surpluses, and the buyer may prefer contracting with a supplier with a less risky cost distribution. However, the buyer always prefers to contract before the uncertainty resolves, that is, before the supplier discovers the realization of cost condition, regardless of the supplier s degree of risk aversion. Our analysis predicts that, in many situations with uncertainty, it is optimal for economic parties to contract before uncertainty resolves and the optimal contracts are often characterized by inflexible rules. It helps explain some contracting practices commonly observed in many industries. For example, in the apparel industry, retailers are often required to make firm, SKU-specific orders well in advance of the beginning of the selling season despite demonstrable advantages to in-season replenishment; in the electronics industry, flexibility for reorders is often restricted within some prespecified limits of original forecasts (Barnes-Schuster et al., 2002). Our study contributes to the growing literature on optimal dynamic mechanism design, 3

5 which studies optimal contract design in environments where information arrives gradually over time and decisions are made over multiple periods. Baron and Besanko (1984), Riordan and Sappington (1987a), Courty and Li (2000), Dai et al (2006), and Krähmer and Strausz (2008, 2011) study two-period models where a risk-neutral agent learn payoffrelevant private information in both periods. Riordan and Sappington (1987b) and Eső and Szentes (2007) examine similar issues in a two-period setting with multiple agents. Besanko (1985), Battaglini (2005), and Boleslavsky and Said (2013) characterize the optimal mechanism for a single risk-neutral agent who receives private information over time in a infinite time horizon. Board (2007) extends the analysis of Eső and Szentes (2007) to an environment with infinite time horizon where a seller auctions a dynamic option among multiple agents. Pavan, Segal and Toikka (2014) study the design of incentive compatible mechanism in a very general dynamic environment in which multiples agents receive private information over time and decisions are made in multiple periods over an arbitrary time horizon. Similar to these studies, we analyze the optimal mechanism when the supplier receives private information over time in multiple periods. However, we study the optimal procurement contract between a risk-neutral buyer and a risk-averse supplier. We investigate contracting parties preference for uncertainty and the interaction between risk sharing and information revelation in the optimal procurement contract. We show that the nature of the optimal contact depends profoundly on the supplier s degree of risk-aversion. Our research also relates to several studies on adverse selection with risk-averse agents. Salanié (1990) studies vertical contracting between a supplier and a risk-averse retailer who possesses private information on demand conditions. Laffont and Rochet (1998) and Dai (2008) analyze the optimal regulatory policy when a risk-averse firm is privately informed about its cost parameter. In these studies, both parties share the same incomplete information on either the demand condition or the cost parameter at the time of contracting, then agents privately discover the actual demand condition or cost parameter after signing 4

6 a contract. In contrast, the current research considers a more realistic information setting where the supplier has incomplete but better information on its cost condition than the buyer does at the time of contracting, more specifically, the supplier is privately informed about the distribution of cost condition at the time of contracting. de Mezza and Webb (2000) and Jullien, Salanie and Salanie (2007) study the optimal insurance contracts under moral hazard when insurance customers are privately informed of their risk preference. Landsberger and Meilijson (1994) consider the optimal insurance contract between one risk-neutral monopolistic insurer and one risk-averse agent who is privately informed of his degree of risk aversion. Smart (2000) studies a screening game in a competitive insurance market in which insurance customers differ with respect to both accident probability and degree of risk aversion. In contrast to these studies, we consider the optimal procurement contract when the risk-averse supplier is privately informed of its cost condition and the supplier s private information arrives over time. The rest of the paper is organized as follows. Section 2 describes the central elements of the model. As a benchmark, Section 3 presents the optimal contract when the cost uncertainty is common information. Section 4 analyzes the supplier s preference for cost uncertainty, and then examines the optimal procurement contract when the supplier is privately informed about the cost uncertainty. Section 5 investigates the buyer s preference for cost uncertainty. Finally, section 6 summarizes our main findings and concludes the paper with future research directions. 2 The Model A buyer contracts with a supplier to obtain some quantity 0 of a good. The buyer s valuation of is (), and ( ) is a smooth, increasing, and concave function. The buyer s surplus is = (),where is the buyer s payment to the supplier. The supplier s 5

7 total cost of producing is =, where is the supplier s marginal/average cost of production. Hence, the supplier s profit is = The utility function of the supplier, ( ), belongs to some smooth one-dimensional family of utility functions that are ranked according to the Arrow-Pratt measure of risk aversion: for any wealth level, 00 () 0 () is increasing with. Thus, measures the supplier s degree of risk aversion. The supplier s marginal cost of production,, is uncertain at the time of contracting. The cost uncertainty is characterized by the supplier s underlying distribution of. It is common knowledge that follows distribution 0 () with probability 0 and distribution 1 () with probability Both 0 () and 1 () are absolutely continuous and strictly increasing cumulative distribution functions on [ ]. Moreover, R 0() = R 1() and R ( 0() 1 ()) 0 for all [ ] with the strict inequality holding for at least one. Therefore, as shown in Rothschild and Stiglitz (1970) (RS thereafter), distribution 0 () is riskier than distribution 1 () in the sense that 0 () can be obtained from 1 () through a sequence of mean-preserving spreads (MPSs). We also assume that both 0 () and 1 () satisfy the following regularity condition: [+ () ()] > 0 for =0, 1, where () is the probability density function of (). The condition is commonly imposed in agency literature to ensure that the equilibrium production schedule is monotonically decreasing in. However, as we demonstrate later in our analysis, the condition does not ensure such property in equilibrium production schedule in our setting. The supplier is privately informed about its distribution of cost condition, 0 () or 1 (), at the time of contracting. After contracting with the buyer and before the production takes place, the supplier privately discovers the realization of. The timing and contractual relation between the buyer and the supplier are as follows: (1) the supplier privately learns its distribution of cost condition; (2) the buyer offers 6

8 the supplier a set of contract menus = { () ()} conditional on the supplier s underlying distribution of cost condition, where =0, 1 and the realization of cost condition ; (3) the supplier selects its preferred menu given its private information on its cost distribution; (4) the supplier discovers and selects a desired option ( () ()) from the selected menu ; (5) exchange takes place according to the contract terms. 3 Benchmark As a benchmark, we first describe the optimal procurement contract when the cost uncertainty, i.e., the underlying distribution of cost condition is common information. Then our model essentially becomes a classic adverse selection problem (e.g., Baron and Myerson (1982)) where suppliers are only privately informed of the realization of the cost condition, except that the supplier is risk-averse in our setting. Based on the revelation principle (Myerson 1986), we can focus our attention on direct and incentive compatible mechanisms. In a direct mechanism, the supplier is required to submit a report, b, of the realization of cost condition. Then the buyer offers the supplier a contract which specifies the quantity and the payment baseduponthesupplier s report. Therefore, a direct contract is a combination { (b)(b)}. A direct contract is incentive compatible if it is in the supplier s best interest to report the realization of cost condition truthfully under the contract. Let ( ) denote the supplier s profit when the realization of cost condition is and the supplier reports. Then incentive compatibility requires that ( ) > ( ) for 6=, (1) To guarantee the supplier s participation in the incentive compatible contract, it needs to 7

9 be individual rational. The individual rationality condition requires that the supplier s expected utility from the contract must be nonnegative, i.e., [ ]= Z ( () ()) () > 0 =0 1 (2) The buyer s optimization problem is choosing { () ()} to maximize Z [ ( ()) ()] () =0 1 (3) subject to the incentive compatibility condition and the individually rational condition. Since our analysis applies to both distributions of cost condition when the supplier s cost distribution is common information, we suppress the subscript in this section to simplify the exposition. Proposition 1 describes the properties of the optimal contract: Proposition 1 When the cost uncertainty is common information, the optimal contract has the following properties: (a) [] = R (()) () =0; (b) In no bunching region(s), () is given by [ 0 (()) ]() = () R 0 (()) () R 0 (()) () (4) (c) There exists 0 such that complete or partial bunching occurs for some interval [ 0 ] with 0 when 0. When the distribution of cost condition is common information, at the time of contract- 8

10 ing both the buyer and the supplier face the same uncertainty about cost of production. Consequently, although the supplier can capture ex post information rent from its private information on the realization of after signing the contract, the buyer can fully extract the expected information rent at the time of contracting by reducing the level of transfer payments () for all realizations of. (Note that it is the difference in () that induces the supplier to truthfully reveal the realization of cost condition.) Consequently, the supplier receives zero expected utility under the optimal contract. Equation (4) demonstrates how the optimal production schedule balances risk sharing and the incentive for the supplier to truthfully reveal the realization of cost condition. When the supplier s realization of marginal cost is e, raising(e) by will in expectation increase the supplier s production efficiency by [ 0 ((e)) e](e). However, the increase in (e) willalsoraisethesupplier sexpostinformationrentby when e. Consequently, in expectation the increase in (e) raises the supplier s ex post information rent by (e). When the supplier is risk-averse, the buyer can only reduce () for all realizations of by R 0 (()) () R 0 (()) () in order to induce the supplier s participation. Notice that R 0 (()) () is the increase in the supplier s expected utility as a result of the increase in ex post information rent, and R 0 (()) () istheincreaseinthe supplier s expected utility as a result of one unit increase in () for all realizations of. Therefore, R 0 (()) () R 0 (()) () is the certainty equivalent of the increase in the supplier s ex post information rent, and the RHS of (4) measures the marginal cost of raising (e) due to the supplier s risk aversion. When the supplier is risk-neutral, i.e., 00 =0, R 0 (()) () R 0 (()) () = (e), which means the certainty equivalent of the increase in ex post information rent is the same for both the buyer and the seller. Consequently, the buyer can fully extract the supplier s ex post information rent by reducing the transfer payments for all realizations of by exactly (e). In that case, equation (4) provides 0 ((e)) = e, which indicates the supplier always produces the efficient level of goods under the optimal contract. The 9

11 optimal contract is equivalent to a sales contract the buyer charges the supplier an upfront fee equal to the maximum expected total surplus of the production and then pays () for any produced by the supplier, i.e., the supplier is made the residual claimant of its production. When the supplier is relatively risk tolerant, the optimal production schedule is strictly decreasing in in [ ]. Equation (4) suggests that the supplier delivers the efficient amount of goods at and but less than the efficient amount of goods on ( ). When the supplier becomes sufficiently risk-averse, the monotonicity condition (() is non-increasing) becomes constraining and bunching occurs in some interval [ 0 ] where 0, despite the regularity condition [ + () ()] > 0. When () does not change rapidly (for example, is uniformly distributed), bunching occurs for the entire interval [ 0 ]. On the other hand, when () does change rapidly, bunching could occur for some ranges of in the interval [ 0 ]. Figure 1 demonstrates the optimal production schedule with partial bunching in some interval [ 0 ]. For later use, we call the optimal production schedule where the cost uncertainty is common information the second-best production schedule. 10

12 q First-best solution Second-best solution c c' c c Figure 1. Optimal supply schedule with partial bunching in [ 0 ]. When the supplier becomes infinitely risk-averse, R 0 (()) () R 0 (()) () converges to zero for [ ). Then equation (4) converges to [ 0 (()) ]() = () (5) for [ ). Note that equation (5) is the well known solution for a classic adverse selection problem (Baron and Myerson (1982), for example) where the supplier is privately and perfectly informed about its cost of production at the time of contracting. This is because the supplier will participate in the contract only if it is guaranteed nonnegative profitforall realizations of and the buyer cannot extract any of the supplier s ex post information rent when the supplier is infinitely risk-averse. Consequently our model becomes equivalent to one that the supplier is perfectly informed about the cost condition at the time of 11

13 contracting. Figure 2 demonstrates the optimal production schedule when the supplier is infinitely risk-averse. q First-best solution Second-best solution c c c Figure 2. Optimal supply schedule for an infinitely risk-averse supplier. 4 AsymmetricInformationonRisk 4.1 The Supplier s Preference for Risk In this section, we analyze the setting where the supplier is privately informed about the cost uncertainty. We start with a discussion of the supplier s preference for risk in this setting. For single-person problems, RS show that all risk-averse individuals would prefer one risky prospect to the other if the latter can be obtained from the former through a 12

14 sequence of MPSs. However, as we demonstrate below, the supplier s preference for risk in our setting depends on its degree of risk aversion. For any contract to be feasible (or implementable) in our setting, it must satisfy the incentive compatibility condition (1) and the individual rational condition (2). It is well known in the agency literature that the incentive compatibility condition (1), which ensures the supplier s truthful revelation of the cost realization, requires: (i) 0 () = () 6 0; and (ii) () is non-increasing. Notice that (i) and () together suggest that 00 () = 0 () > 0, i.e., the supplier s profit mustbeconvexin for a contract to be incentive compatible. The rationale behind this finding is that as increases there is a less than proportionate increase in the total cost of production, because the supplier responds to the cost increase by decreasing the quantity supplied. Consequently, the supplier s profit isdecreasingin at a decreasing rate, hence the supplier s profit is decreasing and convex in. Given the above properties of feasible contracts, we can demonstrate the supplier s preference for risk for any given feasible contract. We consider the supplier prefers distribution 0 to distribution 1 for any given feasible contract, if 0 [()] > 1 [()], i.e., the expected utility under 0 is no less than that under 1 Lemma 1 For any given feasible contract, thereexists 00 such that the supplier prefers 0 to 1 if 00 ( ) 0 ( ) 00 but prefers 1 to 0 if 00 ( ) 0 ( ) 00. Proof. For a given feasible contract, 0 [()] > 1 [()] if Z (())[ 0 () 1 ()] > 0. (6) 13

15 Moreover, we have Z (())[ 0 () 1 ()] (7) = = = = Z Z Z Z {[ 1 () 0 ()] 0 (()) 0 ()} { 0 (()) 0 ()} Z [ 1 () 0 ()] ½ 00 (())( 0 ()) (()) 00 () Z ½ 0 (())[ 00 (()) 0 (()) (0 ()) ()] Z ¾ [ 0 () 1 ()] ¾ [ 0 () 1 ()] where the second line of (7) follows from integration by part, the third line follows after some algebraic manipulations, the fourth line follows from integration by part and the fact that R [ 0() 1 ()] =0,andthefifth line is the result of some algebraic manipulations. Recall that 00 () = 0 () > 0 for to be feasible. When 00 ( ) 0 ( ) is sufficiently small, we have 00 (())( 0 ()) 2 0 (()) + 00 () > 0 in [ ]. Then (7) suggests that R (())[ 0() 1 ()] > 0 if R [ 0() 1 ()] > 0. When 00 ( ) 0 ( ) is sufficiently large, we have 00 (())( 0 ()) 2 0 (()) + 00 () 6 0 in [ ]. Then (7) suggests that R (())[ 0() 1 ()] 6 0 if R [ 0() 1 ()] > 0. Moreover, for any given feasible contract, 00 (())( 0 ()) 2 0 (()) + 00 () is monotonically increasing in 00 ( ) 0 ( ) at any [ ]. Therefore, there exists 00 such that the supplier prefers 0 to 1 if 00 ( ) 0 ( ) 00 but prefers 1 to 0 if 00 ( ) 0 ( ) 00. Lemma 1 shows that, for any given feasible contract, a risk-averse supplier prefers a riskier cost distribution when it is sufficiently risk-tolerant but prefers a less risky cost distribution when it is sufficiently risk-averse. The intuition behind the finding is the following. For any feasible contract, the supplier s final profit is decreasing and convex in the realization of cost condition. Consequently, on one hand, the expected value of the 14

16 supplier s profit increases as the cost becomes more variable; on the other hand, for any givenexpectedvalueofprofit, the supplier s certainty equivalent of the profit decreases as the profit becomes more variable, and more so as the supplier becomes more risk-averse. Therefore, as the cost condition becomes more variable, the supplier s expected utility increases when it is sufficiently risk-tolerant but decreases when it is sufficiently risk-averse. 4.2 Optimal Procurement Contract In this section, we analyze the optimal procurement contract when the supplier is privately informed about the cost uncertainty. In this case, the buyer must screen the supplier by both its cost distribution and its realization of cost condition. As we have shown earlier, the supplier s preference for risk depends on its degree of risk aversion. Suppose the supplier prefers distribution () to distribution () given its degree of risk aversion, where =0, 1, =0, 1 and 6= To simplify the exposition, we denote suppliers with different levels of risk by the subscript of their cost distributions, i.e., and. Onceagain,wefocusour attention on direct and incentive compatible mechanisms. The buyer s optimization problem is choosing a set of contract menus = { () ()} for =, to maximize Z Z [ ( ()) ()] ()+ [ ( ()) ()] () (8) subject to [( )] = Z ( () ()) () > 0 (9) ( ) > ( ) for 6= and (10) [( )] > [( )] (11) 15

17 where =,, =, and 6= Conditions (9) and (10) ensure the supplier s participation and its truthful revelation of the realization of cost condition regardless of its cost distribution; and condition (11) guarantees that the supplier truthfully reveals its underlying distribution of cost condition. Proposition 2 describes how the nature of the optimal contact depends on the supplier s degree of risk-aversion. Proposition 2 A separating contract, which offers different contract terms conditional on the supplier s cost uncertainty, is optimal when the supplier is either not very risk-averse or infinitely risk-averse; however, a pooling contract, which offers the same contract terms regardless of the supplier s cost uncertainty, can be optimal when the supplier is sufficiently risk-averse. Below we discuss the properties of the optimal separating contract and the optimal pooling contract, respectively The Separating Contract The optimal separating contract is a menu that contains two nonlinear price-quantity schedules such that the supplier first selects a schedule from the menu based on its private information on its cost distribution and then a particular price-quantity combination from the schedule after observing the realization of cost condition. Proposition 3 describes the general properties of the optimal separating contract. Proposition 3 The optimal separating contract has the following properties: (a) [( )] [( )] = 0; 16

18 (b) In no bunching region(s), the optimal production schedule for supplier is characterized by [ 0 ( ()) ] () = () (); (12) and the optimal production schedule for supplier is characterized by [ 0 ( ()) ] () =[ () ()] + R 0 ( ()) () () R 0 ( ()) () R 0 ( ()) () (13) where () R 0 ( ()) () R 0 ( ()) () for = ; (c) Bunching can arise for any range(s) of when the supplier is sufficiently risk-averse and both and () () are sufficiently large. Proof. See Appendix. Under the optimal contract, the buyer can fully extract supplier s ex post information rent by adjusting the level of payments for all realizations of cost condition, as in the case of common information on cost uncertainty. However, supplier can always obtain positive expected utility by pretending to be supplier. Therefore, the optimal contract provides supplier positive expected utility to induce its truthful revelation of its cost distribution. Under the optimal contract, the production schedule for supplier optimally balances risk sharing and the incentive for the supplier to truthfully reveal the realization of the cost condition, as in the case of common information on cost uncertainty. Consequently, supplier produces according to the second-best production schedule. However, the production schedule for supplier must simultaneously balance risk sharing, s incentives to truthfully reveal its realization of cost condition, and the incentive for supplier to truthfully reveal its cost distribution. 17

19 When the realization of cost condition is e for supplier, raising (e) by will in expectation increase the production efficiency by [ 0 ((e)) e] (e). However, the increase in (e) will also raise s ex post information rent by when e. Consequently, in expectation the increase in (e) raises supplier s ex post information rent by (e). As we discussed earlier, in anticipation of the ex post information rent, the buyer can reduce () for all realizations of by R 0 ( ()) () R 0 ( ()) (). Therefore the first term in the RHS of (13) demonstrates the marginal cost of raising (e) due to supplier s risk aversion. The second term in the RHS of (13) demonstrates the effect of asymmetric information on cost uncertainty. On one hand, the increase in (e) raises supplier s ex post information rent by R 0 ( ()) () if supplier pretends to be supplier. Ontheother hand, as the buyer can reduce supplier s payment for all realizations of by (e), it reduces supplier s utility from pretending to be supplier by (e) R 0 ( ()) (). Therefore, the increase in (e) as a whole raises supplier s rent from mimicking supplier by [ R 0 ( ()) () (e) R 0 ( ()) ()] Consequently, in order to prevent supplier from mimicking supplier, the buyer will have to raise supplier s payment for all realizations of by [ R 0 ( ()) () (e) R 0 ( ()) ()] R 0 ( ()) (), where R 0 ( ()) () is the increase in expected utilities resulting from one unit increase in supplier s payment for all realizations of. Notice that the second term in the RHS of (13) can be rewritten as R 0 ( ()) () R 0 ( ()) () "R 0 ( ()) () R 0 ( ()) () R 0 ( ()) () R 0 ( ()) () # (14) The two terms between the middle brackets in (14) are the certainty equivalents of the marginal ex post information rent for a cheating supplier and for supplier, respectively, as () increases. 18

20 Therefore, under the optimal contract, supplier s production schedule is distorted to limit supplier s information rent from mimicking supplier. The direction of the distortion depends on the certainty equivalent of the marginal ex post information rent for each type of supplier as () increases. Supplier produces below (above) the second-best production schedule for the range of where supplier s certainty equivalent of the marginal ex post information rent is larger (smaller) than that of supplier as () increases. When the supplier converges to risk-neutral, it prefers the riskier cost distribution ( 0 ()) as discussed in previous section. As 1 () increases, the certainty equivalents of the marginal ex post information rent for a cheating supplier 0 and for supplier 1 converge to 0 () and 1 (), respectively. Consequently, (14) converges to 0 1 [ 0 () 1 ()]. (15) As an example, we can considering a special class of cost distributions where 0 () is a single mean-preserving spread of 1 (). As shown in RS, there exists ( ) such that 0 () 1 () > (6)0 when 6 (>), (16) i.e., 0 () single-crosses 1 (). Then, as the supplier converges to risk-neutral, supplier 1 produces no more (less) than the second-best level when 6 (>). Moreover, by continuity, there must exist ( ) (which can be different from ) such that supplier 1 produces no more (less) than the second-best level when 6 (>) when the supplier is sufficient risk-tolerant. Figure 1 demonstrates the optimal production schedule for supplier 1 in this case in comparison with the second-best solution. We summarize this finding in Corollary 1. Corollary 1 When the supplier is sufficiently risk-tolerant and 0 () is a single mean- 19

21 preserving spread of 1 (), there exists ( ) such that supplier 1 produces no more (less) than the second-best level when 6 (>) q 1 Third-best solution Second-best solution c c'' c c Figure 3. Optimal supply schedule for supplier 1. In contrast to the case of common information on cost uncertainty, the regularity condition ([+ () ()] > 0 for =0, 1.) cannot ensure supplier 1 s production schedule to be monotonically decreasing in when the supplier is risk-neutral. In fact, if both 0 1 and 0 () 1 () are sufficiently large, bunching occurs in supplier 1 s production schedule even when the supplier is risk-neutral. When the supplier is infinitely risk-averse, the certainty equivalent of the supplier s profit converges to its profit at =. Consequently, () converges to 0 for, where 20

22 =,. Then (12) and (13) converge to [ 0 ( ()) ] () = () and (17) [ 0 ( ()) ] () = (), (18) respectively. Notice that (17) and (18) are the optimal production schedules for both types of supplier, respectively, in a classic adverse selection problem where a risk-neutral supplier is perfectly informed about the cost condition at the time of contracting. The above result is due to three factors that come into play when the supplier becomes infinitely risk-averse. First, the supplier will participate in the contract only if it is guaranteed nonnegative profit for all realizations of. Second, the supplier receives no expected utility from its ex post information rent and therefore the buyer can no longer extract the supplier s ex post information rent at the time of contracting. Third, the supplier becomes indifferent between different distributions of cost condition, since its expected profit from the contract depends solely on its profit at = which equals zero in the optimal contract regardless of the supplier s potential distribution of cost. Consequently, the optimal production schedule for each type of supplier converges to the one where a risk-neutral supplier is perfectly informed about the cost condition at the time of contracting The Pooling Contract. When the supplier becomes sufficiently risk-averse, the tension between risk-sharing and information revelation becomes more constraining, and the above separating contract is no longer always optimal. The idea can be best demonstrated with a special class of cost distributions. Suppose 0 () is uniformly distributed on [ ] and is the expected value of. Moreover, 1 () = 0 ()+(1 ) where 0 1. Therefore, distribution 0 () is a mean 21

23 preserving spread of distribution 1 (). Suppose the supplier is sufficiently risk-averse so that it prefers 1 () to 0 (). When the distribution of cost condition is common information, the optimal production schedule for the supplier is characterized by equation (4) in Proposition 1. A comparison of the optimal production schedules for different cost distributions suggests that supplier 1 produces more output for [ ) but less output for ( ] than suppliers 0 does when R [ 0 ( 1 ()) ] 0 ( 1 ( )). 3 Since 0 () = () in any implementable contract, the comparison suggests that 1 () decreases at a higher rate for [ ) but at a lower rate for ( ] than 0 () does. In other words, supplier 1 s profit is less concave (or more convex) in the realization of than supplier 0 s profit is under optimal contract. However, when the distribution of cost condition becomes private information, the two incentive-compatibility conditions, 0 [( 0 )] > 0 [( 1 )] and 1 [( 1 )] > 1 [( 0 )] jointly require Z [( 0 ()) ] > Z [( 1 ()) ], and ( 1 ( )) Z [( 1 ()) ] > ( 0 ( )) Z [( 0 ()) ] where. In words, the incentive compatibility conditions require that supplier 1 s profit is more concave (or less convex) in the realization of than supplier 0 s profit is. When 1 0 +, i.e., the likelihood that the supplier has cost distribution 1 () is minimal, the rent extraction by a separating contract is outweighed by the efficiency loss resulted from the production distortion necessary to achieve incentive compatibility. Consequently, a pooling contract is optimal when In general, the optimal pooling contract offers a contract menu = { ()()} which 3 A detailed analysis is provided in the proof of Proposition 4 in Appendix. 22

24 maximizes subject to Z Z [ (()) ()] ()+ [ (()) ()] () (19) [] = Z ( () ()) () > 0 (20) ( ) > ( ) for 6= and =. (21) Proposition 4 describes the general properties of the optimal pooling contract. Proposition 4 The optimal pooling contract has the following properties: (a) [( )] [( )] = 0;; (b) In no bunching region(s), () is given by [ 0 (()) ] () = () R 0 (()) () R 0 (()) (), (22) where () = ()+ () and () = ()+ (). (c) There exists 000 such that complete or partial bunching occurs for some interval [ 0 ] with 0 when 000. Therefore, in an optimal pooling contract, a supplier receives the same nonlinear pricequantity schedule regardless of its cost uncertainty. A supplier with a favorable cost distribution receives positive expected utility while a supplier with an unfavorable cost distribution receives zero expected utility. In the apparel industry, retailers are often required to make firm, SKU-specific orders well in advance of the beginning of the selling season despite demonstrable advantages to in-season replenishment; in the electronics industry, flexibility for reorders is often re- 23

25 stricted within some prespecified limits of original forecasts (Barnes-Schuster et al., 2002). Similar practices are also observed in vertical relationships in other industries. Lewis and Sappington (1989a, 1989b) among others show that "inflexible rules" rather than "discretion" can be optimal in agency problems with "countervailing incentives", that is, agents have incentive to either understate or overstate their private information depending on the state of nature. In our setting, "inflexible rules" arise in the optimal contract under very general conditions in the absence of countervailing incentives, that is, the supplier is often required to produce a constant level of output for some ranges of cost conditions. Especially, when the supplier is privately informed about the cost uncertainty, bunching can occur in the equilibrium production schedule for any range of cost condition and for any degree of risk aversion. Moreover, a suppliers, who is sufficiently risk-averse, can be offered the same contract terms regardless of its cost uncertainty. The "rules" arise as an optimal solution to the conflicts among risk-sharing and the supplier s incentives to reveal both the cost uncertainty and the realization of cost condition The Optimal Timing of Contracting Another interesting element of the above contracting practices observed in many industries is the timing of contracting. In our setting, the supplier no longer faces cost uncertainty if the two parties contract after the supplier discovers the realization of cost condition. In that case, on one hand, the buyer could induce the risk-averse supplier s participation without paying a risk premium; on the other hand, it could be more difficult to control the supplier s information rent after the supplier privately discovers the realization of cost condition. Therefore, the optimal time of contracting is not clear. Suppose the two parties contract after the supplier discovers the realization of cost condition. Then, the buyer can do no better than to offer a single contract menu { ()()}. 24

26 Otherwise the supplier will just select the menu containing its best option given its cost condition when multiple contract menus are offered. 4 Therefore, the buyer s optimization problem is choosing { ()()} to maximize Z Z [ (()) ()] ()+ [ (()) ()] () (23) subject to ( () ()) > 0 and (24) ( ) > ( ) for 6=. (25) Notice that, since the supplier observes the realization of cost condition at the time of contracting in this case, it will always participate only if its utility is non-negative for all realizations of cost condition. The optimal procurement contract in this case is characterized by (a) ( () ()) = 0; (b) () is given by [ 0 (()) ][ ()+ ()] = ()+ () (26) A comparison of the buyer s optimization problems, contracting before and after the supplier discovers the realization of cost condition, demonstrates the buyer s preference for the timing of contracting. Notice that the latter optimization problem is the former problem with () = () and a stricter individual rationality condition. Therefore, the buyer always prefers to contract before the supplier discovers the realization of cost condition. In 4 That is, when a set of contract menus { () ()} for =0 1 are offered, a supplier will select the contract menu such that () () = arg max () () So without loss of generality the buyer may consolidate these choices by simply offering the contract menu () () = arg max () () for all 25

27 fact, a comparison of the optimal procurement contracts with different timings of contracting ((17), (18) and (26)) shows that the buyer strictly prefers to contract before the supplier discovers the realization of cost condition even if the supplier is infinitely risk-averse. Proposition 5 The buyer strictly prefers to contract before the supplier discovers the realization of cost condition regardless of the supplier s degree of risk aversion. The intuition behind Proposition 5 is the following. First, when contracting after the supplier discovers the realization of cost condition, the supplier will always participate only if it receives non-negative utility for all realizations of cost condition, and the supplier captures information rent from its private knowledge of the cost condition. In contrast, when contracting before the supplier discovers the realization of cost condition, the supplier will participate if it receives non-negative expected utility given its distribution of cost condition. Although a risk-averse supplier must be afforded a risk premium due to the uncertainty at the time of contracting, the buyer can extract at least part of the supplier s ex post information rent at the time of contracting. Second, the buyer can only offer a single contract menu when contracting after the supplier discovers the realization of cost condition. In contrast, when contracting before the supplier discovers the realization of cost condition, the buyer can tailor the procurement contract to the supplier s potential distribution of cost condition. 5 The Buyer s Preference for Risk Based on the properties of optimal procurement contract, in this section we analyze the buyer s preference for risk in our setting. As we demonstrate below, the buyer s preference for risk also depends on the supplier s degree of risk aversion. 26

28 Define () to be the buyer s maximum expected surplus from contracting with a supplier who draws cost distribution 0 () with probability and cost distribution 1 () with probability 1. Then () = 0 ( 0 ( ) 0 ( )) + (1 ) 1 ( 1 ( ) 1 ( )) (27) where ( ( ) ( )) is the buyer s expected surplus when the supplier draws the distribution of cost () with =0 1. Lemma 2 establishes a key property of () Lemma 2 () is convex in. Proof. Suppose = 0 +(1 ) 00 for, 0 00 [0 1] and [0 1] Then, () = 0 ( 0 ( ) 0 ( )) + (1 ) 1 ( 1 ( ) 1 ( )) (28) = [ 0 +(1 ) 00 ] 0 ( 0 ( ) 0 ( )) + [1 0 (1 ) 00 ] 1 ( 1 ( ) 1 ( )) = { 0 0 ( 0 ( ) 0 ( )) + (1 0 ) 1 ( 1 ( ) 1 ( ))} +(1 ) { 00 0 ( 0 ( ) 0 ( )) + (1 00 ) 1 ( 1 ( ) 1 ( ))} 6 ( 0 )+(1 )( 00 ) Thefourthlinefollowsthefactthatthecontract{ ( ) ( )}, which is optimal for, is implementable but not optimal for 0 or 00. Lemma 3 0 () =0 > 0 when the supplier is risk-neutral. Proof. Define f () = 0 ( 1() 1 ()) + (1 ) 1 ( 1() 1 ()) (29) 27

29 where { 1() 1 ()} is the second-best contract for a supplier with distribution 1 (). Notice that f () is the buyer s expected surplus from the pooling contract { ( ) ( )} = { 1() 1 ()} where =0 1. As we discussed earlier, the second-best contract is a sales contract when the supplier is risk-neutral. Under the sales contract, the buyer collects a upfront payment 1 from the supplier and makes the supplier the residual claimant of its production. Consequently, 1() = arg max{ () 1 } and 1 = (1()) 1 where 1 = R [ ( 1()) 1()] 1 (). It can be readily shown that both 0 (1() 1 ()) and 1 (1() 1 ()) are convex in. Then, 0 ( 1() 1 ()) > 1 ( 1() 1 ()) following the proof of Lemma 1. Therefore, f 0 (0) = 0 ( 1() 1 ()) 1 ( 1() 1 ()) > 0. Since () =f () at =0and () > f () for 0 by definition, we have 0 (0) > f 0 (0) > 0. Lemma2andLemma3togetherimplythat () is increasing in when the supplier is risk-neutral. Therefore, the buyer prefers a supplier with a a riskier distribution of cost condition when the supplier is risk-neutral. The intuition is that a supplier with a riskier cost distribution generates greater total surplus for the buyer and the supplier to share. This is partially negated by the fact that the supplier earns more information rent when it is privately informed about the cost distribution. However, by distorting the contracts to reduce information rents, the buyer can capture at least a portion of the extra surplus generated by a riskier cost distribution. However, as the supplier becomes more risk-averse, a supplier with a riskier cost distribution demands a larger risk premium. When the supplier is sufficient risk-averse, the difference in risk premiums outweighs the difference in total surpluses, and the buyer prefers contracting with a supplier with a less risky cost distribution. Lemma 4 provides a sufficient condition for the buyer to prefer a supplier with a less risky cost distribution. Lemma 4 When the supplier is infinitely risk-averse, 0 () 0 if [ 0 () 0 ()] > 0, 28

30 2 [ 0 () 0 ()] 2 > 0 and 000 () Proof. As shown in the previous section, when the supplier is infinitely risk-averse, () = 0 ( 0() 0 ()) + (1 ) 1 ( 1() 1 ()) (30) under the optimal contract, where { () ()} is the second-best contract for a supplier with distribution () and =0 1. Then 0 () = 0 ( 0() 0 ()) 1 ( 1() 1 ()) (31) = 0 ( 0() 0 ()) 1 ( 0() 0 ()) Z [ ( 0()) 0 ()][ 0 () 1 ()] = = = Z Z Z [ 0 () 1 ()][ 0 ( 0()) 0 0 () 0 0 ()] [ 0 () 1 ()] 0() 0 () 0 0 () ½ [ 0() 0 () 00 0 ()+ ( 0() 0 () ) 0 0 ()] Z ¾ [ 0 () 1 ()] The second line is because 1 ( 0() 0 ()) 1 ( 1() 1 ()) by definition. The fourth line follows from integration by part. The fifth line follows from equations (17) and (18) and 0 0 () = 0 0 () (the standard first order condition for the contract to be incentive compatible). The last line follows from integration by part again. Since R [ 0() 1 ()] > 0 and 0 0 () 6 0, 0 () 0 if 0 00 () 6 0 and [ 0 () 0 ()] > 0. 5 The conditions [ 0 () 0 ()] > 0 and 2 [ 0 () 0 ()] 2 > 0 are satisfied by many distributions, e.g., uniform distribution, normal distribution, and logistic distribution. 29

31 From equations (17)and (18), we have [ 0 ( 0()) ] 0 () = 0 () (32) Total differentiating (32) provides 00 ( 0()) = 1+ ( 0() 0 () ). (33) Therefore, 0 0 () ) 0 () = 1+ ( 0() 00 (0()),and (34) 00 0 () = 2 ( 0() ) ( () 0()) [1 + ( 0() )] ( () 0())0 0 () (35) 00 (0()) 2 Equation (35) suggests that 00 0 () 6 0 if 2 [ 0 () 0 ()] 2 > 0 and 000 () 6 0. Based on the above findings and by continuity, we can summarize the buyer s preference for risk as in Proposition 5. Proposition 6 The buyer prefers a supplier with a riskier cost distribution when the supplier is sufficiently risk tolerant, but may prefer a supplier with a less risky cost distribution when the supplier is sufficiently risk-averse. 6 Conclusion We consider a setting where a risk-neutral buyer and a risk-averse supplier contract for the production of some good under cost uncertainty. At the time of contracting, the supplier is privately informed about the distribution of cost condition. Then, after contracting and before production, the supplier privately discovers the realization of cost condition. 30

32 We analyze contracting parties preference for cost uncertainty and derive the optimal procurement contract in this setting. We show that both the buyer and the supplier prefer more risk (a riskier distribution of cost condition) when the supplier is relatively risk tolerant but less risk (a less risky distribution of cost condition) when the supplier is sufficiently risk-averse. The nature of the optimal contact also depends on the supplier s degree of risk-aversion. A separating contract, which offers different contract terms conditional on the supplier s cost uncertainty, is optimal when the supplier is either not very risk-averse or infinitely risk-averse; however, a pooling contract, which offers the same contract terms regardless of the supplier s cost uncertainty, can be optimal when the supplier is sufficiently risk-averse. Moreover, the buyer always prefers to contract before the supplier discovers the realization of cost condition regardless of the supplier s degree of risk aversion, and the optimal procurement contract is often characterized by "inflexible rules" rather than "discretion" for some ranges of cost condition. "Rules" arise as an optimal solution to the conflicts among risk-sharing and the supplier s incentives to reveal both the cost uncertainty and the realization of cost condition. Our findings help explain some contracting practices commonly observed in many industries regarding the timing and the terms of contracts. Our findings show that inflexible rules can generally arise in the absence of transaction costs, bounded rationality, and countervailing incentives. Our analysis also implies that a supplier has incentive to understate the cost uncertainty when it is relatively risk tolerant but overstate the cost uncertainty when it is relatively risk-averse. Therefore, "countervailing incentives" exist when suppliers are privately informed about their risk preference, which is often the case in reality. From the findings by Lewis and Sappington (1989a, 1989b), it is conceivable that inflexible rules would be more prevalent with such countervailing incentives. The optimal procurement contract that take into account the suppler s private information on risk preference warrants further research. 31