PROCUREMENT CONTRACTS: THEORY VS. PRACTICE. Leon Yang Chu* and David E. M. Sappington** Abstract

Transcription

1 PROCUREMENT CONTRACTS: THEORY VS. PRACTICE by Leon Yang Cu* and David E. M. Sappington** Abstract La ont and Tirole s (1986) classic model of procurement under asymmetric information predicts tat optimal contracts will always entail some cost saring and tat payments will be a convex function of realized cost. In contrast, pure cost-reimbursement contracts are common in practice, as are contracts in wic payments are a concave function of realized cost. We consider a straigtforward extension of La ont and Tirole s model tat admits optimal contracts of te forms tat prevail in practice. Te extension simply allows te supplier to be able to reduce production costs more easily wen costs are initially ig tan wen tey are initially low. December 2007 * University of Soutern California. ** University of Florida. We are grateful to Jean Tirole, two anonymous referees, and te co-editor, Jay Pil Coi, for very elpful comments and observations.

2 1 Introduction. Real-world procurement contracts take on a variety of forms. Rogerson (1992) describes four distinct types of contracts commonly employed by te U.S. Department of Defense: (1) pure xed price (PFP) contracts, in wic te supplier receives a single, xed payment for te procured item, regardless of te supplier s realized cost; (2) pure cost reimbursement (PCR) contracts, in wic te payment made to te supplier is precisely te supplier s realized cost of producing te item; (3) incentive xed price (IFP) contracts, in wic payment to te supplier increases wit realized cost up to a tresold cost level, and is capped at tis tresold level; and (4) incentive cost reimbursement (ICR) contracts, in wic payment to te supplier again increases wit realized cost up to a tresold, and ten re ects realized cost exactly above te tresold. 1 Tese four types of contracts are illustrated in Figures 1 4. Notice from Figures 3 and 4 tat payment is a concave function of realized cost under an IFP contract, wile payment is a convex function of realized cost under an ICR contract. Under te procurement contract illustrated in Figure 5, payment is a concave function of realized cost in some regions and a convex function of cost in oter regions. Tis structure parallels te reward structure regulated utilities commonly face wen tey operate under incentive regulation. Extremely ig pro t (corresponding to unit cost below c 1 ) and extremely low pro t (corresponding to unit cost above c 2 ) typically are not politically acceptable in regulated industries. Consequently, allowed pro t is bounded above and below, corresponding to payments (revenues) tat vary dollar for dollar wit costs as costs decline below c 1 and as costs increase above c 2 in Figure 5. In contrast, to motivate te rm to reduce its operating costs, incentive regulation implements pro t saring for intermediate pro t realizations (corresponding to cost realizations c 2 (c 1 ; c 2 ) in Figure 5). By allowing 1 Te Department of Defense (DOD) assigns di erent names to tese four types of contracts. As Rogerson (1992, p. 11) reports, te DOD refers to: (1) PFP contracts as rm xed price (FFP) contracts; (2) PCR contracts at cost plus xed fee (CPFF) contracts; (3) IFP contracts as cost plus incentive fee (CPIF) contracts tat revert to FFP contracts above a tresold cost level; and (4) ICR contracts as CPIF contracts tat revert to CPFF contracts above a tresold cost level. 1

3 te rm to retain some, but not all, of te cost reduction it implements in te form of iger pro t, incentive regulation can secure bene ts for te regulated rm and consumers alike. 2 Despite te ric variety of procurement and regulatory contracts tat are observed in practice, te classic (and still te standard) economic model of procurement under asymmetric information admits only convex contracts like te ICR contracts illustrated in Figure 4. Except in trivial cases of limited interest, La ont and Tirole (LT) s (1986) classic model of procurement does not admit PFP or PCR contracts. Te primary model on wic LT focus teir analysis also does not admit concave contracts like te IFP contract illustrated in Figure 3. Tis model also does not produce contracts wit bot concave and convex regions, like te contract depicted in Figure 5. Te purpose of tis researc is to consider a simple generalization of LT s classic procurement model tat admits a broader class of contract forms, including tose commonly observed in practice. 3 Te generalization as an intuitive interpretation: te supplier can reduce production costs more easily wen initial cost (te component of cost beyond te supplier s control) is ig tan wen it is low. In tis sense, we allow initial cost and te supplier s cost-reducing e ort to be substitutes in reducing nal production costs. Intuitively, one migt envision a feasible range of nal cost realizations. If costs are initially close to te lower bound of tis range, furter cost reductions are relatively di cult to acieve. In contrast, if costs are initially quite ig, some cost reduction is not di cult to acieve. Calkley and Malcomson (CM) (2002) analyze a model of tis type in a ealt care setting. Te e ort tat a ealt care provider devotes to reducing treatment costs in CM s model is more e ective at reducing tese costs te more severely ill te patient is (and tus te more costly te patient would be to care for in te absence of cost-reducing e ort). 2 Di erent incentive regulation plans incorporate di erent forms of pro t saring for intermediate pro t realizations. See Sappington (2002), for example, for a discussion of te di erent types of incentive regulation plans tat ave been implemented in te telecommunications industry. 3 More fundamental modi cations of LT s model also can admit optimal contracts tat include tose commonly observed in practice. Tese modi cations include te introduction of transactions costs and renegotiation costs (e.g., Bajari and Tadelis, 2001) and preferences for simple contracts (e.g., Rogerson, 2003; Cu and Sappington, 2007). 2

4 CM focus on te welfare gains tat an optimal contract can secure relative to te simple payment structures tat are commonly employed in te ealt care industry. In contrast, we focus on te di erent contract structures tat can be optimal in settings were ig initial costs are associated wit increased potential for cost reduction. We nd tat tis simple and arguably reasonable and intuitive generalization of LT s model admits optimal contracts tat are concave (like IFP contracts) and tat ave bot concave and convex regions (as in Figure 5). Te generalization also allows PCR contracts (wic induce no cost-reducing e ort from te supplier) to be optimal, even in settings were cost-reducing e ort is e cient, and so would always be secured absent asymmetric information about innate cost. 4 Our presentation of tese ndings proceeds as follows. Section 2 describes te key features of te formal model we analyze. Section 3 reviews te key tecnical details of our analysis. Section 4 illustrates te variety of contracts tat can arise in our model, and identi es te factors tat in uence te sape of optimal procurement contracts. Section 5 provides a concluding discussion. Te proofs of our formal conclusions are outlined in te Appendix. A more detailed tecnical appendix is available upon request. 2 Te Model. A buyer seeks to minimize te expected cost of procuring a single unit of a commodity from a monopoly supplier. Te supplier will only deliver te commodity if e derives nonnegative utility from doing so. Te supplier is privately informed about is innate cost of production () from te outset of is interaction wit te buyer. Te buyer s beliefs about are captured by te density function, wic as support on te interval [; ]. For expositional ease, we assume 1 F () > 0 for all 2 [; ), were F () is te corresponding cumulative distribution function. 5 In contrast to innate cost, wic is observed only by te supplier, te supplier s nal production cost is observed publicly. Final cost (c) is te 4 In bot our model and LT s model, a PFP contract is optimal only if te buyer sares te supplier s knowledge of initial production cost. 5 Te key qualitative conclusions drawn below also old if F () = 1 for some 2 (; ), but te proofs are more tedious in tis case. 3

5 di erence between innate cost and te supplier s cost-reducing e ort, e. Formally, c = e. Te supplier s utility, u(), is te di erence between te payment (p ()) e receives from te buyer and te sum of is nal production cost (c) and te personal cost e incurs in delivering cost-reducing e ort. 6 C(e; ) will denote te supplier s personal (unobserved) cost of delivering e ort e wen innate cost is. To re ect diminising returns to cost-reducing e ort, we follow LT in assuming C(e; ) is a strictly increasing, strictly convex function of e for e > 0, wit C(0; ) = 0. 7 In contrast to LT, we allow for te possibility tat te supplier may nd it less onerous to reduce production costs wen innate costs are ig tan wen tey are low. In particular, bot te supplier s total and marginal cost of e ort decline as innate cost increases. Formally, C 2 (e; ) < 0 and C 12 (e; ) < 0 for all e > 0, were subscripts denote partial derivatives, ere and trougout. 8 A lower innate cost is not an unequivocal blessing for te supplier in tis setting. A lower implies tat te supplier can avoid te very igest production costs e ortlessly, peraps by directly importing cost-saving innovations discovered in earlier researc projects, for example. However, te lower secured from relatively extensive earlier researc also implies tat incremental cost reductions will be more onerous for te supplier. Tis may be te case, for example, if te failure of te relatively extensive earlier researc to discover even more pronounced cost savings implies tat suc savings will be di cult to acieve. Aside from associating more favorable innate cost realizations wit greater di culty in acieving incremental cost reductions, our model parallels LT s model exactly. A formal statement of bot problems is facilitated by te following notation. p(^) will denote te 6 Tus, we assume te supplier pays for is nal production costs. LT assume te buyer pays tese costs. Te particular convention tat is adopted is inconsequential. 7 Negative e ort (e < 0) is feasible. As LT do implicitly, we assume C(e;) = 0 for e < 0, so te supplier is able to in ate is innate cost at will, but enjoys no direct increase in utility from doing so. La ont and Tirole (1992) allow te supplier to bene t directly from cost in ation (so negative e ort entails perquisites, for example). Cu and Sappington (2008) analyze a setting in wic te supplier s ability to in ate is innate cost is limited. 8 C i () denotes te partial derivative of C() wit respect to its i t argument. C ij () denotes te corresponding second partial derivative wit respect to te i t and j t arguments of C(). 4

6 payment te buyer delivers to te supplier wen te supplier reports is innate cost to be ^, provided nal cost is ^ e(^). If te supplier wit innate cost reports is innate cost to be ^, e must deliver e ort e(^ j ) = e(^) + ^ to ensure nal cost ^ e(^). Te buyer s problem in our model, labelled [BP], can be stated formally as follows: Minimize p(); e() R p() df () (1) subject to, for all ; ^ 2 ; : u() p() [ e()] C(e(); ) = 0 ; and (2) u() = u(^ j ) p(^) ^ e(^) i C(e(^ j ); ) ; (3) were e(^ j ) = e(^) + ^: (4) Expression (1) re ects te buyer s goal of minimizing expected procurement cost. Te participation constraints in (2) ensure tat te supplier secures non-negative utility wenever e trutfully reports is innate cost. 9 Te incentive compatibility constraints (3) ensure tat te supplier always reports is innate cost trutfully. 10 Expression (4) de nes te e ort te supplier wit innate cost must supply to secure nal cost ^ e(^). 3 Tecnical Analysis. Te purpose of tis section is two-fold: (1) to outline te key steps in solving [BP]; and (2) to explain ow te distinguising assumption of our model tat it is less onerous for te supplier to implement cost reductions wen costs are initially ig complicates te tecnical analysis. Te reader wo is primarily interested in our qualitative ndings may prefer to skip muc of te tecnical analysis in tis section. To solve [BP], we follow LT by replacing te global incentive compatibility (GIC) constraints (3) wit te corresponding local incentive compatibility (LIC) constraints. Te LIC 9 Tus, te supplier s reservation utility is normalized to zero. 10 From te revelation principle (e.g., Myerson, 1979), tis formulation is witout loss of generality. 5

7 constraints ensure tat te supplier will not exaggerate or understate is innate cost sligtly (i.e., locally), but do not preclude pronounced misrepresentation of. We will denote by [BP] 0 te buyer s problem were te LIC constraints replace te GIC constraints. To solve [BP], we rst solve [BP] 0. Ten we determine if te GIC constraints are satis ed at te identi ed solution to [BP] 0. If te GIC constraints are satis ed (as tey will be wen te su cient condition presented in Lemma 4 is satis ed), ten te identi ed solution to [BP] 0 is te solution to [BP]. If te GIC constraints are not satis ed, ten te identi ed solution to [BP] 0 will not constitute te solution to [BP], and so te solution must be identi ed by oter means (suc as tose employed in te setting of Finding 2 b Te LIC constraints b j b= = 0 for all ; b 2 [; ]). Furtermore, (4) implies = 1. Terefore, (3) implies tat wen te LIC constraints are satis ed, te supplier s utility must increase wit at te j) b j b= = u 0 () = [C 1 (e(); ) + C 2 (e(); )] for all 2 ; : Wen u 0 () 0 for all 2 ;, te supplier s participation constraints in (2) will be satis ed for all 2 [; ] as long as u() = 0. Consequently, te buyer s utility given innate R cost can be expressed as u() = u 0 ()d in tis case. Terefore, from expressions (2) and (5), te payment from te buyer to te supplier wit innate cost can be written as: p() = e() + C(e(); ) + R [C 1 (e(); ) + C 2 (e(); )]d. (6) Substituting te expression in (6) into (1) and integrating by parts reveals tat [BP] 0 can be written as: Minimize e() P R n o e() + C(e(); ) + [C 1 (e(); ) + C 2 (e(); )] F () df (): (7) Let R e()+c(e(); )+[C 1 (e(); )+C 2 (e(); )] F (). Di erentiation of = 1 + C 1(e(); ) + [C 11 (e(); ) + C 12 (e(); )] F (), and (8) 6

8 @ 2 2 = C 11 () + [C 111 () + C 112 ()] F (). (9) Recall tat te supplier s ability to reduce is operating costs does not vary wit is innate cost in LT s model, and so C 2 (e; ) = 0 for all. Terefore, it is apparent from (5) tat u 0 () 0 for all in te basic model on wic LT focus teir analysis. 11 Furtermore, C 12 () = 0, and so (8) implies tat 1 + C 1 (e(); ) 0 for all at te optimum in LT s basic model. Consequently, te supplier s equilibrium e ort supply never exceeds te rstbest e ort (e ()), wic is te e ort tat minimizes te sum of realized production costs and te supplier s e ort cost. (Formally, e () arg min f e + C(e; )g.) In addition, e C 112 () = 0 in LT s basic model. Terefore, (9) reveals tat wen C 111 () is eiter positive or su ciently small in absolute value, expected payment (P ) is a convex function of e ort in LT s basic model. Tese convenient features of LT s basic model do not necessarily arise ere. If jc 2 (e; )j is su ciently large, (5) implies tat te supplier s equilibrium utility could increase as innate cost () increases (because iger realizations of result in reduced e ort costs for te supplier). Furtermore, if jc 12 ()j is su ciently large, (8) implies tat te supplier could optimally be induced to deliver more tan te rst-best level of e ort. In addition, if C 112 () is negative and su ciently large in absolute value, (9) reveals tat expected payment is not necessarily a convex function of e ort at te solution to [BP] 0. To render te analysis tractable wile maintaining a close parallel to LT s basic model, we will impose su cient structure to ensure tat te predominant e ect of an increase in innate cost () is to increase te supplier s operating costs and reduce is equilibrium utility. Tis approac allows us to demonstrate tat optimal procurement contracts can take on a variety of forms even in a model tat maintains tese distinguising features of LT s model. Two qualitative restrictions ensure tat te supplier s equilibrium costs increase and tat is 11 Tis basic model is te one in wic te solution to [BP] 0 is te solution to [BP]. LT do not examine in detail te convexity of te optimal payment scedule in teir model wen te solution to [BP] 0 is not te solution to [BP]. 7

9 equilibrium utility declines as increases. First, cost-reducing e ort is su ciently onerous for te supplier tat relatively little e ort is induced in equilibrium. Second, wile te supplier s marginal cost of e ort declines substantially as innate cost increases, te decline is not too pronounced. Tis structure is imposed most simply by introducing te following tree assumptions. Assumption 1. C(e; ) = K i e 2 were K > 0. Assumption 2. K 1 : Assumption 3. 2 [ 2; 1] : Assumption 1 simpli es te ensuing analysis by presuming C() to be a quadratic function of e ort (e). Assumption 2 ensures tat e ort costs are relatively large. Assumption 3 requires to be negative, and so lower innate costs are associated wit iger total and marginal costs of e ort. Te bounds tat Assumption 3 imposes on ensure tat te supplier s e ort costs decline substantially as is innate cost increases, but tat te decline is not so pronounced as to cause te supplier s equilibrium utility to increase. Terefore, te supplier as a systematic incentive to exaggerate is innate cost wen Assumptions 1-3 old, as e does in LT s model. 12 Unless oterwise noted, Assumptions 1-3 will be maintained trougout te ensuing analysis. Lemma 1 con rms tat te supplier s equilibrium utility declines as is innate cost increases at te solution to [BP] 0 wen Assumptions 1-3 old, just as in LT s basic model. Lemma 1 also identi es an upper bound on te supplier s equilibrium level of cost-reducing e ort. Lemma 1. u 0 () 0 and e() for all 2 ; at te solution to [BP ] If a reduction in were to increase C() more substantially, countervailing incentives (e.g., Lewis and Sappington, 1989; Maggi and Rodriguez-Clare, 1995; Jullien, 2000) could arise in tat te supplier migt wis to understate is innate cost in order to exaggerate is e ort costs. We impose assumptions tat are su cient to preclude countervailing incentives in order to simplify te tecnical analysis and to ensure tat te basic structure in our model parallels te structure in LT s basic model. 8

10 Lemma 2 caracterizes te supplier s e ort (e 0 ()) at te solution to [BP] 0. Lemma 2. e 0 () = maxf0; 1 2K 2K 1+ F () g F () i at te solution to [BP ] 0 : Lemma 3 reports tat te e ort identi ed in Lemma 2 never exceeds te rst-best e ort, just as induced e ort never exceeds te rst-best e ort in LT s basic model. Lemma 3. e 0 () e () = 1 2K i jj for all 2 ;. Lemmas 1-3 caracterize te solution to [BP] 0, wic imposes only te local incentive compatibility (LIC) constraints. However, our ultimate concern is wit te solution to [BP], wic imposes te relevant global incentive compatibility (GIC) constraints. Te GIC constraints will be satis ed at te solution to [BP] 0 b b 0 for all b < b b 0 for all b >. (10) Te rst and second inequalities in (10), respectively, ensure tat te agent wit innate cost will not gain by understating or overstating. Lemma 4 provides a su cient condition for te inequalities in (10) to old at te solution to [BP] 0. Lemma 4. If e 0 0() 1 for all 2 ;, ten e 0 () is te e ort supply at te solution to [BP ]. Lemma 4 reveals tat if realized production costs ( e 0 ()) increase wit at te solution to [BP] 0, ten tis solution will constitute te solution to [BP]. 13 Te analysis of Finding 2 in section 4 reveals ow te solution to [BP] is identi ed wen production costs do not increase wit systematically (because e 0 0() > 1 for some ) at te solution to [BP] 0. 4 Findings. 13 Te fact tat d d n o F () 0 for all 2 [; ] is instrumental in ensuring tat e 0 0() 1, and so realized production costs increase wit, in LT s model. Tis monotone inverse azard rate condition does not ensure tat e 0 0() 1 in our model, as te analysis of Finding 2 reveals. 9

11 We now illustrate some of te ric variety of optimal procurement contracts tat can arise even under te restrictions imposed by Assumptions 1-3. Finding 1 provides one set of conditions under wic a pure cost reimbursement (PCR) contract is optimal. Under te speci ed conditions, te buyer optimally induces no cost-reducing e ort from te supplier even toug te e ort would systematically reduce total production costs. Finding 1. Suppose = 1, = 2; and K = : Ten p(c) = c for all cost realizations, c, and so e() = 0 for all 2 ; at te solution to [BP]. Notice from Lemma 3 tat te rst-best e ort, e 2[ ]2 () =, is strictly positive for 5[ ] all 2 (; ] in te setting of Finding 1. Yet te buyer optimally implements a PCR contract, wic induces no cost-reducing e ort from te supplier. Tis decision re ects te well-known trade-o between rent and e ciency, modi ed to account for te cost structure under consideration ere. Te buyer can always reduce er procurement costs wen innate costs () are ig by inducing te supplier to deliver some cost-reducing e ort (e): E ort is motivated by saring wit te supplier some of te cost savings tat is e ort secures. Tis saring is implemented via a payment in excess of realized cost over a range of s. Of course, payment in excess of cost over a range of ig s provides rent to te supplier wen te smaller innate costs are realized. Te rent arises because te supplier can always substitute is good fortune (low innate cost) for te costly e ort tat te supplier must deliver wen is ig to acieve any speci ed nal cost level. To avoid awarding te supplier excessive rent wen is low, te buyer may coose to induce less tan te rst-best e ort from te supplier wen is ig. Indeed, te buyer may optimally induce no cost-reducing e ort from te supplier for te igest innate cost realizations. Te buyer can do so by implementing a cost reimbursement contract tat pays te supplier exactly is realized production costs wen relatively ig production costs are observed. Because it provides te supplier wit no incentive to deliver cost-reducing e ort for te igest realizations, suc a contract promotes relatively ig 10

12 procurement costs for tese innate cost realizations. However, te contract can reduce overall expected procurement costs by limiting te rent tat te supplier commands wen te smaller realizations of arise. In LT s model, it is never optimal to implement cost reimbursement over te entire range of realizations. Tis is te case because te rst-best e ort supply does not vary wit in LT s model, since C() is independent of. Consequently, a policy tat never induced any cost-reducing e ort from te supplier would incur substantial e ciency losses (i.e., e C(e )) for all realizations wile providing very limited gains for te smaller realizations. Tese gains would be limited because as te supplier s innate cost declines toward, te likeliood of even lower innate cost realizations becomes small, and so te reduction in te supplier s expected rent tat is secured by implementing cost reimbursement becomes small relative to te e ciency losses associated wit no cost-reducing e ort. In contrast to LT s model, te rst-best e ort supply declines as declines in our model. As te supplier s innate cost declines, it becomes more di cult for im to implement furter cost reductions (since C 2 (e; ) < 0 and C 12 () < 0), and so te e cient level of cost reduction declines. Terefore, te increase in procurement cost tat arises wen te supplier delivers no cost-reducing e ort becomes less pronounced as declines. Consequently, even toug te reduction in te supplier s expected rent tat is secured by inducing no cost-reducing e ort from te supplier declines as declines, te cost of inducing tis e ort distortion (i.e., te increase in te sum of production and e ort costs above te e cient level) also declines as declines. In te setting of Finding 1 were all smaller realizations of are equally likely and were jj is relatively large, te reduction in expected rent tat is secured by inducing no cost-reducing e ort continues to exceed te (diminising) losses from te ine ciently small e ort supply as declines toward its lower bound. Consequently, te supplier optimally implements a pure cost reimbursement contract (as illustrated in Figure 2) in tis setting even toug suc a contract is never optimal in te basic setting considered by LT As te proof of Finding 1 reveals, tis is te case even wen te solution to [BP] 0 is te solution to [BP]. Finding 2 considers a setting in wic te solution to [BP] 0 is not te solution to [BP] (as in te analysis 11

13 Finding 2 reveals anoter contract form an incentive xed price contract like te one depicted in Figure 3 tat can be an optimal contract ere, but not in LT s basic model. Finding 2 considers a setting were te iger innate cost realizations are relatively likely and te lower innate cost realizations are relatively unlikely. In suc a setting, te supplier will optimally induce te supplier to deliver cost-reducing e ort wen suc e ort is least costly for te supplier (i.e., for te iger realizations). Altoug te e ort provides rent to te supplier wen te lower s arise, tese realizations are not very likely, and so te corresponding expected rent is limited. Finding 2. Suppose = 3[ ]2 [ ] ; = 2, and K = 5 4 [ ] 1 : Ten p(c) = c + b for all c c 0 and p(c) = c 0 + b for all c > c 0 at te solution to [BP], were c 0 = 1[3 p 2 7] + 1 [p 7 1] and b is a strictly positive constant. Consequently, 2 8 >< 0 for 2 [; c 0 ) e() = at te solution to [BP ]. >: c 0 for 2 [c 0 ; ] Notice tat te buyer optimally induces te supplier to realize te same production cost, c 0, for all innate costs in excess of tis level (i.e., for all 2 [c 0 ; ]) in te setting of Finding 2. Tis policy re ects a compromise between te buyer s preferred outcome and feasible outcomes. Ideally, te buyer would like to induce substantial cost-reducing e ort for te igest innate cost realizations and relatively little e ort for te intermediate realizations in te setting of Finding 2. Tis is te case because te igest realizations are relatively likely compared to te intermediate realizations and because cost-reducing e ort becomes substantially less onerous for te supplier as increases due to te relatively large value of jj. To induce er preferred pattern of cost-reducing e ort, te buyer would ave to o er payments tat are relatively insensitive to realized cost for te igest realizations but tat track realized costs more closely for te intermediate realizations. Tis payment structure, coupled wit e ort supply costs tat decline substantially as increases, would induce e ort of Guesnerie and La ont (1984), for example). 12

14 tat increases so rapidly wit tat nal cost ( e()) would decline as increases over a range of te iger innate cost realizations at te solution to [BP] 0. Suc a reward structure would create an incentive for a supplier wit an intermediate realization (wo can secure a low nal cost wit little or no e ort supply) to exaggerate is true innate cost realization substantially. 15 To prevent suc innate cost exaggeration, te buyer would ave to induce less costreducing e ort for te igest realizations (by making payments track costs more closely) and induce more cost-reducing e ort for te intermediate realizations (by making payments track costs less closely). To eliminate all incentives to exaggerate, tis process would ave to continue until te reward structure induced te same nal cost realization for te intermediate and te igest realizations, as it does under te optimal policy described in Finding 2. Te lowest innate cost realizations are relatively unlikely in te setting of Finding 2. Terefore, te losses tat arise wen te supplier is induced to deliver no cost-reducing e ort for te lowest realizations are relatively unlikely to be incurred. For tis reason, and because te rst-best e ort level declines rapidly as declines since jj is relatively large, te expected gains from inducing no cost-reducing e ort from te supplier for all of te lowest realizations outweig te corresponding expected costs. Terefore, te buyer optimally implements cost reimbursement (wic induces no cost-reducing e ort) for all of te smaller innate cost realizations in te setting of Finding 2. If te lowest innate cost realizations were less likely tan tey are in te setting of Finding 2, te buyer would be less concerned about te rent tat te supplier secures wen tese smallest realizations occur. Consequently, te buyer would optimally induce some costreducing e ort from te supplier wen te intermediate innate cost realizations arise. As 15 Tecnically, te local second-order conditions are not satis ed for all 2 ; at te solution to [BP] 0, u( b b 2 > 0 for some. Consequently, altoug te LIC constraints b j b= = 0 for all ; b 2 [ ; ], trutful reporting of provides a local minimum rater tan a local maximum of te supplier s utility. 13

15 a result, te disparity in induced e ort supply (and corresponding compensation) between te intermediate and te igest realizations would be less pronounced tan it is under te buyer s ideal reward structure in te setting of Finding 2. Consequently, te supplier s incentive to exaggerate intermediate realizations would be eliminated, and so te optimal contract would specify payments tat: (i) increase wit costs dollar for dollar for te lower cost realizations; and (ii) increase wit costs at a decreasing rate for te iger cost realizations. It is readily sown tat te optimal procurement contract assumes tis concave structure if, for example, = 2, K = 5[ ] 1 5[ ]4, and =. 4 [ ] 5 Te pattern of cost-reducing e ort described in Finding 2 stands in contrast to te e ort supply induced in LT s basic model. In tat model, cost-reducing e ort declines as increases, and no e ort may be induced over a range of te iger innate cost realizations. In te setting of Finding 2, e ort increases wit for te largest innate cost realizations (i.e., for 2 [c 0 ; ]) and no e ort is induced for te smallest innate cost realizations. Tese di erences, of course, re ect te fact tat e ort costs increase as declines ere, wile e ort costs are independent of in LT s model. Finding 3 considers a setting were intermediate innate cost realizations are relatively likely wile te lowest and igest realizations are relatively unlikely. Tis setting gives rise to an optimal procurement contract like te one depicted in Figure 5, were payments increase wit costs dollar for dollar for te extreme cost realizations, wile cost saring is introduced for intermediate cost realizations. Finding 3. Suppose = 2, K = 1 30[ ] 4 [ ], and =. [ 8 ] 6 0 for 2 [; >< 1 ) Ten at te solution to [BP ]: e() = e 0 () for 2 [ 1 ; 2 ) >: 0 for 2 ( 2 ; ]; were 1 p p ; and p It is readily veri ed tat e 0 () = [ 15x 2 +20x 6]x 2[9 10x] i [ ] for 2 [ 1 ; 2 ) in te setting of Finding 3, 14

16 Te buyer cooses not to induce any cost-reducing e ort from te supplier for te iger innate cost realizations ( 2 2 ; ) in te setting of Finding 3. Altoug suc e ort would reduce procurement costs for te iger innate cost realizations, it would increase procurement costs for te intermediate realizations by allowing te supplier to secure rent. Because te intermediate s are relatively likely in te setting of Finding 3, te buyer optimally reduces rent for tese realizations by implementing cost reimbursement (and tus inducing no cost-reducing e ort) for te igest realizations. Te buyer also induces no cost-reducing e ort for te lowest innate cost realizations ( 2 ; 1 ). E ort is relatively costly for te supplier wen tese lower innate costs are realized, and so te costs of e ort (including te rent generated by expanded e ort) outweig its bene ts in tis region. In contrast, te buyer induces cost-reducing e ort by implementing cost saring for te intermediate innate cost realizations ( 2 [ 1 ; 2 )) in te setting of Finding 3. Altoug te cost saring for intermediate s admits rent for te lower s, tese innate cost realizations are relatively unlikely in te setting of Finding 3, and so expected rent is relatively limited. Consequently, te expected reduction in procurement cost introduced by cost saring for intermediate s outweigs te corresponding expected increase in procurement cost for te lower s, and so suc cost saring is optimal. Of course, wen < 0 is su ciently close to zero (or wen > 0), convex contracts of te type identi ed by LT and illustrated in Figure 4 also can be optimal. In tis case, e ort costs do not increase muc as declines toward. Consequently, for te reasons identi ed by LT, it will often be optimal for te buyer to induce te supplier to deliver considerable cost-reducing e ort wen te lowest s arise. Tis pattern of e ort supply is optimally induced by a convex payment structure in wic payments vary little wit observed costs for te lowest cost realizations and track costs more closely for te igest cost realizations. In summary, optimal procurement contracts may be convex, as LT s basic model predicts. were x. 15

17 However, as Findings 1-3 illustrate, optimal procurement contracts can take on a variety of oter forms (suc as tose depicted in Figures 2, 3, and 5) wen ig initial costs are associated wit su ciently ric opportunities for cost reduction Conclusion. We ave demonstrated tat a simple extension of LT s classic procurement model admits a variety of contract forms tat are employed in practice. Te extension considers settings in wic te supplier nds it less onerous to reduce production costs wen is innate cost () is ig tan wen it is low. In tese settings, te buyer would induce more cost-reducing e ort from te supplier as increases if were observed publicly. However, in standard fasion, asymmetric knowledge of makes it optimal to induce less e ort as increases, ceteris paribus, in order to reduce te supplier s rent. Tese con icting forces give rise to a wide variety of optimal procurement contracts. To illustrate, wen te igest realizations of are particularly likely and wen it becomes substantially less onerous for te supplier to reduce is production costs as increases, te buyer will optimally induce te supplier to deliver substantial cost-reducing e ort for te igest realizations and little or no e ort for te smallest realizations. Tis e ort supply is optimally induced wit a concave payment structure, in contrast to te convex contracts tat are always optimal in LT s basic model (were te supplier s e ort costs do not vary wit ). Under conditions suc as tose identi ed in Finding 2, te optimal contract can be a simple piecewise linear contract of te form illustrated in Figure 3. More generally, optimal procurement contracts may be concave but not piecewise linear, tey may be convex, or tey may contain bot convex and concave regions. Te optimal procurement contract may also be linear, as in a pure cost reimbursement contract. Te form of te optimal contract re ects te intuitive and classic trade-o between rent and e ciency, adjusted to account for te possibility tat cost reduction becomes more onerous 17 As noted above, a pure xed price contract like te one illustrated in Figure 1 will only be optimal (bot ere and in LT s model) if te buyer sares te supplier s knowledge of from te outset of teir relationsip. 16

18 as initial costs decline. Altoug te extension of LT s model tat we ave considered is a simple one, te extension complicates considerably te caracterization of optimal procurement contracts. We ave analyzed optimal contract design in structured settings tat ensure strong parallels wit LT s basic model. Future researc migt analyze alternative settings, including tose in wic te supplier s e ort costs decline more rapidly as innate cost () increases, so tat te supplier s equilibrium utility increases wit over some or all relevant ranges. Future researc also migt proceed beyond te igly structured settings tat we ave analyzed to caracterize optimal procurement contracts more generally. Te examples we analyzed suggest tat a pure cost reimbursement contract (as illustrated in Figure 2) will tend to be optimal wen te potential for furter cost reduction declines fairly rapidly as innate cost declines wile te likeliood of te lowest innate cost realizations is relatively pronounced. In contrast, an incentive xed price contract like te one depicted in Figure 3 will tend to be optimal wen te iger innate cost realizations are substantially more likely tan te lower innate cost realizations. In addition, a contract wit linear segments for te igest and lowest cost realizations and a concave structure for intermediate cost realizations (as illustrated in Figure 5) may be optimal wen te intermediate realizations are particularly likely. General conditions under wic tese conclusions old remain to be speci ed. A more general speci cation of ow te properties of optimal procurement contracts are a ected by te interaction between te buyer s initial beliefs and te intensity of te inverse relationsip between innate costs and te potential for cost reduction also awaits future researc. 17

19 Payment Final Cost Figure 1. A Pure Fixed Price Contract. Payment o 45 Final Cost Figure 2. A Pure Cost-Reimbursement Contract.

20 Payment Final Cost Figure 3. A Concave Procurement Contract. Payment o 45 Figure 4. A Convex Procurement Contract. Final Cost

21 Payment c c 1 2 Final Cost Figure 5. A Procurement Contract wit Concave and Convex Regions.

22 Sketc of Proof of Lemma 1. Appendix From (5) in te text, u 0 () 0 for all 2 [; ] if C 1 (e; ) + C 2 (e; ) 0 for all 2 [; ]. It is readily sown tat wen Assumption 1 olds: C 1 (e; ) + C 2 (e; ) 0, e() 2[ ] jj. (A1) Te Lemma is ten proved by sowing tat for any feasible contract, an alternative contract wit e() ( 2[ ] jj by Assumption 3) for all 2 [; ] can be constructed tat as lower expected payment tan te original contract. Te alternative contract is of te form f(; ) j 2 [; ]g [ f(c i ; p i ) j p i ; c i ; i 2 Ig, were f(c i ; p i ) j i 2 Ig is te set of cost-payment pairs tat constitute te original contract. Altoug te alternative contract may induce a iger realized production cost by reducing te supplier s equilibrium e ort supply, te corresponding reduction in payment reduces expected procurement costs. Recall tat te buyer s task in problem [BP] 0 is to minimize P = R R(e())dF (), were: R(e()) e() + C(e(); ) + [C 1 (e; ) + C 2 (e; )] F () : (A2) Wen Assumption 1 olds: " F () = e2k " # 1 + F (), and 2 2 = 2K " # 1 + F (). (A4) Tese expressions are useful in proving Observation 1 wic, in turn, is useful in proving Lemma 2. Observation 1. e() = 0 at te solution to [BP] 0 if and only if 1 + 2K i F () 0: Sketc of Proof. Recall from Lemma 1 tat u 0 () 0 and e() for all 2 [; ] at te solution to [BP] 0 wen Assumptions 1 3 old. For a xed, te term in (A4) is 18

23 independent of e. Because < 0, tis term can be negative, in wic case expected payment is a concave function of e under te optimal contract. Consequently, te optimal e will be at a boundary: eiter 0 or. (Altoug e < 0 is feasible, te supplier always delivers non-negative e ort. Negative e ort increases nal production cost at least as rapidly as it increases te payment from te buyer, and so is not advantageous for te supplier.) It is readily e = 0 > = 0 at e = 1 [ ] wen 2K 2 point on [ 2 i 1 + i F () 2K 1 2K 1+ F () F () i > < 0. Terefore, since R() attains its critical ; ] rater tan on [0; ], te symmetry of te quadratic function R() 2 implies tat R()j e = 0 R()j e = under te speci ed condition. Since e = 0 under tis i condition, (A3) = 1 + 2K F () If te term in (A4) is positive, expected payment is a convex function of e under te optimal contract. In tis case, if te expression in (A3) is non-negative at e = 0, ten e is optimally 0. = 1 + 2K te expression in (A3) is negative at e = 0 (so i F () 1 + 2K optimally positive. Terefore, e ort is optimally 0 if and only if 0, from (A3). In contrast, if i F () < 0 ), ten e ort is i 1 + 2K F () 0. Sketc of Proof of Lemma 2. From Observation 1, e = 0 if 1 1 2K i F () 2K i F () 0. Observation 1 also implies tat if > 0, ten te optimal e is te value of e at wic te expression in (A3) is zero. Tis value is as speci ed in te Lemma. Sketc of Proof of Lemma 3. From Observation 1, e 0 () = 0 if 1 i 1 2K F () 0. 2K i F () 0. Terefore, e 0() e () wen It is readily sown tat wen 1 2K i F () > 0: 19

24 e 0 () e (), 2K " # jj. (A5) Te inequality in (A5) olds because: (1) K [ i +1; and (2) [ ] +1 [ ] Assumption 3. ] 1 by Assumption 2, and so K i +1 1 jj since and jj 2 by 2 Sketc of Proof of Lemma 4. Te proof proceeds by sowing tat under te 0 for ^ 0 for ^ > (A6) at te solution to [BP] 0. Wen (A6) olds, te global incentive compatibility (GIC) constraints will be satis ed at te solution to [BP] 0, and so tis solution will constitute te solution to [BP]. It is readily sown tat wen ^ = [1 e 0 (^)]C 1 (e(^j 0 ); 0 )j ^. (A7) Because e 0 0() 1 (by assumption), 1 e 0 (^) 0. Terefore, to (A7) implies tat it will su ce to sow C 1 (e(^j 0 )); 0 )j ^ 0. From Assumption 1: 0, " # " # ^ C 1 (e(^j 0 )); 0 )j ^ 0, 2K e(^j) 2K e(^j^) 0 (A8), " # e(^) " # " # ^ e(^) + [ ^] 0: (A9) (A9) follows from (A8) because e(^j) = e(^) + ^ (and so e(^j^) = e(^)) from (4) in te text. Because ^ < and < 0, te expression in (A9) is decreasing in e(^). Since e(^) e (^), (A9) will old if: 20

25 " # e (^) " # " # ^ e (^) + [ ^] 0, (A10), ^ (A11) is derived from (A10) by dividing all terms by Because ^ < and K [ " # jj " # jj ^ 1 + 2K[ ^] 0: (A11) ] 1, (A11) will old if: i e (^) (= ^ i ^ 2K i 1 ). " # jj " # jj ^ J(^) 1 + 2[ ^][ ] 1 0: (A12) Notice tat J() = 0: Furtermore: J 0 (^) = jj[^ ] jj 1 [ ] jj 2[ ] jj 1 [ ] jj 0. (A13) Te inequality in (A13) olds because jj 2, ^, and jj 1. Because J 0 (^) 0 and J() = 0, we know tat (A12) olds for all ^ <. Terefore, C 1 (e(^j 0 )); 0 )j ^ 0. Te proof for te case were ^ > is analogous. Sketc of Proof of Finding 1. In te setting of Finding 1: 2K " # 1 + F () = 2K " # 2 [1 2] = 5[ ] 2[ ] 2 < 0: (A14) Terefore, from (A4), expected payment is a concave function of e. Consequently, as sown in te proof of Observation 1, e() is optimally 0 for all 2 [; ]. Furtermore, it is readily veri ed tat te GIC constraints are satis ed at tis solution, so it is indeed te solution to [BP]. Sketc of Proof of Finding 2. In te setting of Finding 2, 1 + i i F () 2 = 1 3 > 0. Terefore, expected payment is a 21

26 convex function of e, from (A4). Furtermore, 1 2K i F () > 0, i > 0 in tis setting. Terefore, Observation 1 and Lemma 2 imply tat at te solution to [BP] 0 : (i) no e ort is optimally induced on [; 1 + 5]; and (ii) te optimal e ort on [ ; ] is: e 0 () = [ ][ ] i = 6 5 [ ] 2 [ ] [ ]. (A15) However, te relevant local second-order conditions u( b b 2 < 0 for all ) are not satis ed at tis solution to [BP] 0. Tis is te case 2 u(^j) Furtermore, Assumption 1 2 j^= < 0, [C 11 (e; ) + C 12 (e; )][e 0 () 1] < 0. (A16) C 11 (e 0 (); ) + C 12 (e 0 (); ) = 2K " # 1 Also, (A15) implies tat at te identi ed solution, for 2 ( ; ]: 6 6 e 0 0() = 12 [ ] 5 [ ] 1 > jj e 0 () > 0: (A17) 1 = 1: (A18) (A17) and (A18) imply tat (A16) does not old at te identi ed solution in tis setting. Te optimal contract is identi ed by considering te following alternative formulation of te buyer s problem, called [AF]: Minimize e() Z e() + C(e(); ) + [C 1 (e; ) + C 2 (e; )] F () df () subject to: 1 e( 1 ) 2 e( 2 ) for all 1 2. Te proof proceeds by demonstrating tat te following two conclusions old in te present setting: Conclusion 1. For any feasible solution to [BP], tere is a solution to [AF] tat ensures lower expected payment for te buyer. Conclusion 2. Te solution to [AF] satis es te GIC constraints and is of te form identi ed 22

27 in te Finding. Consequently, te solution identi ed in te Finding is te solution to [BP] in tis setting. Conclusion 1 is proved as follows. For any e ort function ^e() tat satis es te GIC constraints, de ne ~e() suc tat ~e() = maxf e (); sup f 0 j 0 ;^e( 0 )e ( 0 )g [ 0 ^e( 0 )]g. It is readily veri ed tat ~e() is weakly increasing in. Furtermore, it can be sown tat j~e() e 0 ()j j^e() e 0 ()j for all because ^e() satis es te GIC constraints. Consequently, since expected payment (R()) is a convex, quadratic function of e tat attains its minimum value at e 0, R(~e()) R(^e()) for all. Conclusion 2 is proved as follows. For any feasible solution to [AF], ^e(), de ne 0 2 ( ; ] as te realization of for wic: 6 6 ^e() 0 e 0 ( 0 ) > e 0 () for 2 0 ;, and ^e() 0 e 0 ( 0 ) < e 0 ( 0 ) for 2 ( ; 0). Suc a 0 exists as long as ^e() e 0 () because ^c() (recall ^e() is a feasible solution to [AF]) and c 0 () ^e() is (weakly) increasing in e 0 () is a decreasing function of for ; (since e 0 0() 1 in tis region). If ^e() > e 0 () (so c 0 () = e 0 () > ^c() = ^e()), de ne 0 as. Now consider te following feasible solution to [AF]: 8 < 0 for < ~e() = 0 e( 0 ) : [ 0 e 0 ( 0 )] for 0 e( 0 ). It can be sown tat j^e() e 0 ()j j~e() e 0 ()j for all 2 ;. Because R(e()) is a convex, quadratic function of e() tat attains its minimum value at e 0 (), ~e() secures a lower value of R() tan ^e(). Finally, te identi ed solution to [AF] can be sown to satisfy te GIC constraints using an approac similar to te approac employed in te proof of Lemma 4. Sketc of Proof of Finding 3. 23

28 It is readily sown tat 1 2k i F () > 0 in tis setting if and only if 2 ( 1 ; 2 ). Terefore, from Observation 1, no e ort is optimally induced for te lowest and te igest innate cost realizations at te solution to [BP] 0. Lemma 2 also implies tat e 0 () = i [ 15x 2 +20x 6]x [ ] for 2 ( 2[9 10x] 1 ; 2 ), were x. It is tedious but straigtforward to verify tat e 0 0() < 1. Terefore, by Lemmas 3 and 4, te GIC constraints are satis ed, and e 0 () is te solution to [BP]. 24

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