Contracting with private knowledge of signal quality

Transcription

1 RAND Journal of Economic Vol. 41, No. 2, Summer 2010 pp Contracting with private knowledge of ignal quality Leon Yang Chu and David E. M. Sappington We characterize the optimal procurement contract in a etting where a upplier ha privileged knowledge of the quality of a public ignal about hi production cot. The optimal contract exhibit important difference with tandard contract in advere election etting. For intance, the contract induce output both above and below firt-bet level. Furthermore, the induced output may not vary with the realized public ignal unle the ignal quality i ufficiently pronounced. In addition, output may increae a expected cot increae. 1. Introduction The literature provide many ueful model of procurement in etting where a upplier ha better knowledge of hi production cot than doe the buyer. The varied model in the literature adopt different aumption about the buyer information. Some model aume that the buyer never oberve any informative ignal about the upplier cot. 1 Other model aume that the buyer can ultimately oberve an imperfect ignal of the upplier cot, 2 where the accuracy of the ignal i common knowledge. 3 The latter approach may be more realitic than the former becaue imperfect ignal about production cot often are readily available in practice. For example, higher retail gaoline price typically imply higher operating cot for a upplier of tranportation (e.g., ambulance) ervice, jut a higher price for oil, coal, or other key input can ignal increaed operating cot for electricity producer. Although uch ignal often are readily oberved, their exact implication Univerity of Southern California; leonyzhu@uc.edu. Univerity of Florida; apping@ufl.edu. We are particularly grateful to the editor, David Martimort, and two anonymou referee for extremely inightful comment and uggetion. We alo thank Sanford Berg, Roger Blair, Mark Jamion, and Ted Kury for their helpful aitance. 1 Thee model reflect Baron and Myeron (1982) claic framework. 2 See Baron and Beanko (1987) and Caillaud, Guenerie, and Rey (1992), for example. 3 Some model that reflect Laffont and Tirole (1986) pioneering work aume that the buyer can oberve the upplier cot perfectly. However, the buyer i unable to determine how hard the upplier ha worked to ecure the oberved cot in thee model. 244 Copyright C 2010, RAND.

2 CHU AND SAPPINGTON / 245 for the production cot of a pecific upplier eldom are common knowledge. In practice, a upplier typically ha uperior knowledge about the mix of input he employ, about hi ability to witch among input a their price change, and about the long-term contract he ha negotiated to ecure input. Conequently, the upplier often i better able than a buyer to ae the exact implication of public ignal about likely production cot. The purpoe of thi reearch i to analyze the optimal procurement contract in a etting where a upplier ha better knowledge than a buyer about the implication of a public ignal for the upplier production cot. In particular, the upplier know from the outet of hi interaction with the buyer the amount by which hi production cot change a a public ignal about thee cot change. The buyer, in contrat, doe not know the exact implication of the ignal for the upplier cot. However, the buyer ultimately oberve the ignal. Therefore, when formulating a procurement contract, the buyer can pecify how the term of the contract eventually will be updated to reflect any new information provided by the ignal, if uch updating i deemed to be deirable. Contract often pecify how term of trade will be adjuted to reflect prevailing potential indicator of production cot. For example, contract between the U.S. Veteran Adminitration (VA) and upplier of ambulance ervice to VA hopital tate how payment to upplier will be adjuted a the prevailing average retail price of regular unleaded gaoline change. The adjutment i relatively traightforward. The VA firt pecifie a bae per-mile payment (p b )for ambulance ervice. It then etimate the fraction (η) of the bae payment that i accounted for by the cot of gaoline. Next, the VA compare the average price of gaoline during the current contract period (g) and a pecified bae price of gaoline (g b ). The proportionate change in the price of gaoline ( g g b g b ) i then multiplied by η p b to derive an adjutment (a = η p b [ g g b g b ]) to the bae per-mile payment for ambulance ervice. The total per-mile payment for the current contract period i p b + a. 4 Contract for electricity upply alo commonly include a fuel adjutment claue (FAC) that link payment for electricity to an index of relevant fuel price. The FAC that the South Carolina Public Service Authority commonly employ in regulating retail electricity price i illutrative. Thi FAC applie an adjutment factor (A) to each cutomer monthly electricity conumption that reflect difference between the upplier prevailing fuel cot per unit of electricity production and a correponding bae fuel cot. Formally, A = Fr K r F b K b, where F r denote the upplier total expenditure on fuel in the mot recent three-month period, F b denote the upplier correponding expenditure on fuel in the pecified bae period, K r denote the number of kilowatt hour (kwh) of electricity that the upplier old in the mot recent three-month period, and K b denote the upplier correponding ale of electricity in the pecified bae period. A upplier of electricity that operate under thi FAC adjut each cutomer monthly bill by the product of thi adjutment factor, A, and the cutomer electricity conumption during the month in quetion. 5 Although contract that pecify how term of trade will be updated to reflect prevailing indicator of likely production cot can be relatively imple in practice, the optimal deign of uch contract can be quite ubtle. We find that in etting where a upplier i privately informed about the implication of a realized public ignal for hi production cot, new conideration and new qualitative feature of optimal procurement contract arie. For example, the upplier 4 To illutrate, uppoe the bae payment for ambulance ervice i $2.60 per mile. Further uppoe the price of gaoline increae from $2.50/gallon in the bae period to $3.00/gallon in the current period. In addition, uppoe that gaoline cot are etimated to account for 10% of the bae payment. In thi cae, after adjuting for the oberved increae in the price of gaoline, the VA would pay $2.65 ( $ [.10][$2.60][ ) per mile in the current contract period 2.50 (U.S. Veteran Adminitration, 2006). 5 To illutrate, uppoe the upplier expenditure on fuel increae from $1 million in the bae period to $1.1 million in the mot recent three-month period. Further uppoe the upplier ale of electricity increae from 92 million kwh in the bae period to 92.5 million kwh in the mot recent three-month period. Finally, uppoe a particular conumer of electricity purchae 1000 kwh of electricity in the preent month. Then thi cutomer bill for the month will increae by $1.02 due to riing fuel cot. Thi adjutment i the product of 1000 kwh and A = $.00102/kWh, becaue [ 1.1 ] = (South Carolina Public Service Authority, 2004)

3 246 / THE RAND JOURNAL OF ECONOMICS incentive to mirepreent hi private information about the relevance (or quality ) of the public ignal i not traightforward. Furthermore, the optimal procurement contract often induce output above the efficient level when the ignal ugget that cot are higher than initially anticipated, and below the efficient level when the ignal indicate that cot are lower than originally thought. The optimal contract may alo call for increaed output a the upplier unit production cot increae. In addition, the optimal contract may fail to adjut the upplier output to the new cot information revealed by the public ignal for an extenive range of reported quality realization. In thi ene, optimal contract may exhibit ome contractual incompletene. The upplier incentive to mirepreent the quality of the public ignal i not traightforward becaue the upplier mut report the quality of the ignal before oberving the precie realization of the ignal. Conequently, although an undertatement of ignal quality implie an (advantageou) exaggeration of unit cot if the ignal ultimately indicate that unit cot i relatively low, the ame undertatement of ignal quality amount to a (diadvantageou) undertatement of unit cot if the ignal ultimately ugget that unit cot i relatively high. On balance, the upplier predominant incentive i to undertate the quality of the public ignal when output i adjuted to reflect the realized ignal. Undertatement of the prevailing ignal quality erve to exaggerate expected total cot more when ignal of relatively low unit cot arie (and relatively large output are induced) than expected cot are undertated when ignal of relatively high unit cot are ultimately oberved (and relatively mall output are induced). To limit the upplier incentive to undertate the quality of the public ignal, the buyer reduce the upplier output below the efficient level when ignal of relatively low unit cot are oberved. Thi output reduction reduce the extent to which an undertatement of the quality of a low unit cot ignal exaggerate total production cot. The buyer alo increae the upplier output above the efficient level when ignal of relatively high unit cot are oberved. Thi increae in output increae the extent to which an undertatement of the quality of a high unit cot ignal undertate total production cot. Both form of output ditortion limit the upplier anticipated gain from undertating the true quality of the public ignal. Under plauible condition, the optimal output ditortion can be o pronounced that the public ignal i ignored altogether for a broad range of realized ignal qualitie. In particular, for the lowet ignal qualitie, the upplier i required to produce the ame level of output for all realization of the public ignal. Total urplu i reduced when the information conveyed by the ignal i ignored. However, the reduced urplu i more than offet by the correponding reduction in the rent that the upplier command from hi uperior knowledge of the quality of the public ignal. In eence, ignoring the public ignal here play a role imilar to the role played by hutdown (i.e., no production) in more traditional advere election model. We develop thee finding a follow. Section 2 decribe the key element of our model. Section 3 conider a benchmark etting in which the buyer hare the upplier knowledge of the quality of the public ignal. Section 4 preent and explain our main finding. Section 5 conider the poibility that the term of the original contract might be renegotiated after the realization of the public ignal i oberved. Section 6 review the implication of our finding and ugget direction for future reearch. The Appendix provide proof of the formal concluion in the text. Before proceeding, we note briefly how our analyi relate to other in the literature. Our analyi differ from tandard model of monitoring or auditing (e.g., Dye, 1986; Baron and Beanko, 1987) in part becaue the public ignal in our model i cotle and i alway available for contracting. More importantly, the upplier in our model ha uperior information from the outet about the quality of the public ignal. Dai, Lewi, and Lopomo (2006) model, like our, conider a etting in which a buyer i initially uncertain about the extent of the prevailing information aymmetry. In Dai et al. model, the upplier i initially privately informed about the ditribution of hi operating cot, and he ubequently learn hi operating cot before producing. No public ignal about operating cot

4 CHU AND SAPPINGTON / 247 arie in Dai et al. model, and o the author do not explore the optimal ue of a public ignal, which i the focu of our analyi. 6 Szalay (2009) alo conider varying degree of information aymmetry between a buyer and a upplier. Szalay add a moral hazard problem to the advere election problem by allowing the upplier unoberved effort to determine the extent of hi uperior knowledge of hi production cot. 7 The information aymmetry i exogenou in our model, and the aymmetry pertain to the quality of a public ignal (which i not preent in Szalay model). 2. Element of the model We analyze the interaction between two rik-neutral partie, a buyer and a upplier of a particular product or ervice (e.g., electricity or tranportation ervice). The upplier will contract with the buyer a long a he anticipate nonnegative profit from doing o. At the time they ign a binding contract, the buyer and the upplier both ae the upplier expected unit cot of production to be c 0. The two partie know that updated public information, that i, a ignal, about the upplier unit cot will arrive after the contract i igned but before a final deciion i made regarding the amount of output that the upplier will deliver to the buyer. Both partie alo know that the upplier ha better knowledge than the buyer about the implication of the public ignal for the upplier production cot. Thee implication are ummarized in what we call the quality of the public ignal. The upplier final expected unit cot of production, c(, q i ), varie with both the realization of the ignal [, ] and the quality of the ignal q i {q 1,...,q n } Q. 8 To draw the harpet ditinction between our model and mot other procurement model in the literature, we aume that the upplier uperior information pertain olely to the implication of the public ignal for hi unit production cot, and not to the ignal realization that i likely to be oberved. To illutrate, an electricity upplier might not be able to predict the price of coal more accurately than a large indutrial buyer of electricity. However, the upplier i better informed about the extent to which hi production cot increae a the price of coal increae. Formally, we aume that the ditribution of the public ignal i independent of the ignal quality. 9 The ditribution of q and are common knowledge. We normalize the realization of the public ignal, [, ], o that they reflect the extent to which ex ante belief about the upplier unit cot are revied. In particular, we et < 0 and > 0 and aume that a poitive ignal, (0, ], indicate that the upplier unit cot exceed the initial aement of unit cot, c 0. In contrat, a negative ignal, [, 0), indicate that the upplier unit cot i le than c 0. The ignal realization = 0 indicate no change in ex ante belief about the upplier unit cot of production, o c(0, q i ) = c 0 for all q i Q. We will denote by f () the denity function for the public ignal,. The upplier ex ante expected unit cot given ignal quality q i i c(, q i) f ()d = c 0 for all q i Q. The extent to which ex ante belief about the upplier unit cot are revied following the obervation of an realization other than 0 varie with the quality of the ignal. A higher ignal 6 Dai et al. (2006) how that the upplier privileged knowledge of the ditribution of hi operating cot alway benefit the upplier, but may harm the buyer. Johnon and Myatt (2006) undertake a related invetigation of a firm preference regarding the ditribution of (privately known) cutomer preference. 7 See Lewi and Sappington (1997) and Crémer, Khalil, and Rochet (1998), for example, for related invetigation in le general environment. 8 A dicrete ditribution for the ignal quality facilitate analye of pecial cae of interet. The key qualitative concluion drawn below perit when the ditribution of q i continuou. The concluion drawn below alo hold if the ditribution of i dicrete. 9 If the public ignal were informative about q, the buyer could eliminate the rent of the rik-neutral upplier while inducing him to reveal q truthfully. See Crémer and McLean (1985), Riordan and Sappington (1988), and Gary-Bobo and Spiegel (2006), for example.

5 248 / THE RAND JOURNAL OF ECONOMICS quality i aociated with a higher expected unit cot c(, q) when the ignal i poitive and a lower expected unit cot when the ignal i negative. 10 Formally, for all q i > q j, c(, q i ) > c(, q j ) when > 0 and c(, q i ) < c(, q j ) when < 0. (1) Holding contant the quality of the ignal, the upplier expected unit cot increae a the ignal realization increae, that i, c (, q i ) 0 for all q i Q. 11 Thi inequality i aumed to hold trictly unle otherwie noted. To illutrate, uppoe the quality of the public ignal (q i ) i the probability that the ignal i informative, wherea the ignal i completely uninformative with probability 1 q i. Further uppoe that when it i informative, the ignal realization i the actual change in the upplier unit cot. In thi etting, the upplier expected unit cot given ignal of quality q i i c(, q i ) = q i [c 0 + ] + [1 q i ]c 0 = c 0 + q i. Although the upplier doe not know initially which ignal realization ultimately will be oberved, he doe know the quality of the public ignal from the outet of hi interaction with the buyer. In contrat, the buyer never oberve the quality of the ignal. The buyer ex ante aement of the probability that the ignal quality i q i i φ i (0, 1). The buyer valuation of output x i given by the trictly increaing, trictly concave function V(x). The buyer eek to maximize the difference between the value of the output he receive from the upplier and her payment to the upplier. To avoid unintereting corner olution, we aume that the buyer valuation of the initial unit of output i ufficiently large relative to it expected cot, for example, lim x 0 V (x) =. The timing in the model i a follow. Firt, the upplier learn the realization of the ignal quality. Then the buyer and upplier ign a binding contract. After the realization of the public ignal i oberved, the upplier deliver the output pecified in the prevailing contract. Finally, the buyer pay the upplier, a per the term of the contract. The buyer problem, [BP], in thi etting i the following: Maximize {x(,q i ), T (,q i )} ubject to, for all q i, q j Q: (q i q i ) (q i q i ) n φ i [ V (x(, q i )) T (, q i ) ] f () d (2) i=1 [ T (, q i ) c(, q i ) x(, q i ) ] f () d 0 ; and (3) [ T (, q j ) c(, q i ) x(, q j ) ] f () d (q j q i ). (4) Expreion (2) reflect the buyer deire to maximize the expected difference between the value of the output he ecure from the upplier and the payment he make to the upplier. The participation contraint in (3) guarantee the upplier nonnegative expected profit for all quality realization. The incentive compatibility contraint in (4) enure that the upplier truthfully reveal the prevailing ignal quality The firt-bet etting To undertand the olution to [BP], it i helpful to conider the buyer optimal trategy in the benchmark etting where he hare the upplier knowledge of the ignal quality from the outet of their interaction. In thi firt-bet etting, the buyer problem, given ignal of quality 10 Becaue the buyer and upplier are both rik neutral, the concluion in our model are the ame whether c(, q) reflect the upplier actual unit cot given and q or hi expected unit cot given and q. 11 Here and throughout the enuing analyi, c ( ) denote the partial derivative of c( ) with repect to. 12 The revelation principle (e.g., Myeron, 1979) enure that thi formulation i without lo of generality.

6 CHU AND SAPPINGTON / 249 FIGURE 1 UNDERUSE OF THE SIGNAL x( ) x*(,q) x*(0, ) 0 x(,q) q i, i to chooe output x to maximize V (x) c(, q i ) x while limiting the upplier rent to zero. Therefore, when ignal of quality q i i realized, the buyer will induce the firt-bet output, x (, q i ), defined by V (x (, q i )) = c(, q i ). (5) Equation (5) indicate that in the firt-bet etting, output i expanded to the point where it marginal value to the buyer i equal to it expected unit cot, given the realized ignal which i known to be of quality q i. In particular, output i alway trictly above (repectively, below) it ex ante efficient level, x (0, ), when the ignal indicate that the upplier unit cot i le than (repectively, greater than) initially thought. The dahed line in Figure 1 illutrate a continuum of firt-bet output. For future reference, Figure 1 and 2 preent other output continua that the buyer might induce. The olid line in Figure 1 depict output that arie when the public ignal i underued, wherea the olid line in Figure 2 illutrate output that arie when the ignal i miued. The public ignal i aid to be underued when induced output i alway intermediate between it ex ante efficient level and it firt-bet level for all realization. The ignal i aid to be miued when induced output i reduced below it ex ante efficient level following an indication of reduced cot and/or increaed above it ex ante efficient level following an indication of increaed cot. More formally, we have the following definition. Definition. The public ignal i underued when q = q i if: (i) x(, q i ) [x (0, ), x (, q i )] for all [, 0]; (ii) x(, q i ) [x (, q i ), x (0, )] for all [0, ]; and (iii) it i not the cae that x(, q i ) = x (, q i ) for all [, ]. Definition. The public ignal i miued when q = q i if: (i) x(, q i ) < x (0, )forome [, 0); and/or (ii) x(, q i ) > x (0, ) forome (0, ]. A the analyi in Section 4 demontrate, the buyer often will underue or miue the public ignal in order to limit the rent the upplier can ecure by undertating the prevailing ignal quality.

7 250 / THE RAND JOURNAL OF ECONOMICS FIGURE 2 MISUSE OF THE SIGNAL x( ) x*(,q) x*(0, ) 0 x(,q) 4. Finding The firt-bet outcome i not feaible when the buyer i privately informed about the quality of the public ignal from the outet of hi relationhip with the buyer. Thi concluion follow from Lemma 1, which report that if the public ignal i not miued, then the upplier expected production cot decline a the quality of the public ignal increae. Lemma 1. x(, q j ) c(, q i ) f () d, the upplier expected cot of producing output continuum x(, q j ), i nonincreaing in q i when the ignal i not miued, that i, when x(, q j ) x (0, ) for all [, 0] and x(, q j ) x (0, ) for all [0, ]. Lemma 1 reflect the fact that the upplier expected total cot of production can be viewed a a weighted average of hi unit production cot, where the weight are induced output level. When the ignal i not miued: (i) unit cot that decline a q increae (thoe aociated with < 0) are weighted relatively heavily; and (ii) unit cot that increae a q increae (thoe aociated with > 0) receive relatively little weight. Therefore, the upplier expected total cot of production decline a the ignal quality increae when the ignal i not miued. 13 Conequently, the firtbet outcome i not feaible becaue the upplier can ecure trictly poitive rent when the ignal quality exceed q 1 by agreeing to produce the firt-bet output continuum, x (, q 1 ), in return for the aociated firt-bet compenation. When the public ignal i underued, the upplier continue to produce le output a the realized ignal,, increae, jut a in the firt-bet etting. Conequently, the upplier expected total production cot decline (and o hi rent increae) a the ignal quality increae. However, becaue underue of the ignal render induced output le enitive to the realized ignal than it 13 If the public ignal i not ued when q = q i,thati,ifx(, q i ) doe not vary with, then the upplier expected total cot of producing x(, q i )whenq = q j i x(, q i)c(, q j ) f () d = c 0 x(, q i ), which doe not vary with q j. Conequently, the upplier rent under the {T (, q i ), x(, q i )} contract doe not vary with the prevailing ignal quality.

8 CHU AND SAPPINGTON / 251 i in the firt-bet etting, the upplier rent increae le rapidly a q increae than it would if the firt-bet output were induced. 14 Miue of the ignal can further reduce the rate at which the upplier rent increae with q. When the ignal i miued, induced output can increae a increae for ome realization. Conequently, in calculating expected total cot, the (output-baed) weight placed on the unit cot that increae a q increae (thoe aociated with > 0) can be relatively large, wherea the weight placed on the unit cot that decline with q (thoe aociated with < 0) can be relatively mall. Therefore, expected total cot can increae a the ignal quality increae when the ignal i miued. When expected total cot increae with q, an undertatement of q can contitute an undertatement of expected total cot, which i diadvantageou for the upplier. Hence, miue of the ignal can limit the upplier incentive to undertate the prevailing ignal quality. 15 The binary etting (n = 2). The buyer optimal ue of the public ignal i characterized mot readily in the imple binary etting where Q ={q 1, q 2 }. Propoition 1 report that the buyer will alway underue or miue the ignal in thi etting when the lowet ignal quality (q 1 ) i realized in order to limit the rent the upplier can ecure by undertating the prevailing ignal quality. In contrat, the principal implement the firt-bet output for all realization of the public ignal when q = q 2 (becaue the upplier ha no incentive to exaggerate the prevailing ignal quality). The upplier earn no rent when q = q 1, and earn exactly the rent he could earn by undertating the true ignal quality when q = q 2. Thu, when the ignal quality i binary, the olution to [BP] exhibit the key feature of baic advere election problem (in which countervailing incentive are not preent). 16 For expoitional eae, we will refer to thee feature a tandard feature of advere election problem. 17 Propoition 1. Suppoe n = 2. Then at the olution to [BP]: (i) x(, q 1 ) < x (, q 1 ) < x (, q 2 ) = x(, q 2 ) for all [, 0); x(0, q 1 ) = x (0, ) = x(0, q 2 ); and x(, q 1 ) > x (, q 1 ) > x (, q 2 ) = x(, q 2 ) for all (0, ]; (ii) (q 1 q 1 ) = 0; and (iii) (q 2 q 2 ) = (q 1 q 2 ) 0. To characterize the buyer optimal ue of the ignal in thi binary etting, let γ 21 denote the Lagrange multiplier aociated with the (q 2 q 2 ) (q 1 q 2 ) incentive compatibility contraint. Then it i readily verified that the outcome continuum x(, q 1 ) at the olution to [BP] in the preent binary etting i determined by V (x(, q 1 )) = c(, q 1 ) [c(, q 1 ) c 0 ] γ 21 Z 21 (), (6) where Z 21 () 1 [ ] c(, q2 ) c 0 [c(, q 1 ) c 0 ]. (7) φ 1 c(, q 1 ) c 0 The term γ 21 Z 21 () in (6) reflect the potential benefit and cot of inducing the upplier to produce an output other than the firt-bet output when ignal of quality q 1 i realized. 18 γ (q i q j ) (q i q i ) = 0 [ c(, q i ) c(, q j )]x(, q i ) f () d + [ c(, q 0 i ) c(, q j )]x(, q i ) f () d. Alo, c(, q i ) c(, q j )a 0whenq j > q i. Therefore, the rent the upplier can command from undertating ignal quality q j a q i decline when x(, q i ) i reduced below x (, q i )for < 0 and increaed above x (, q i )for > 0, a it i when the ignal i underued. 15 In contrat, miue of the ignal can increae the upplier incentive to exaggerate the prevailing ignal quality, a the dicuion later in thi ection emphaize. 16 The dicuion in Section 6 explain how the olution to [BP] differ from the olution to advere election problem in which countervailing incentive arie (e.g., Lewi and Sappington, 1989; Maggi and Rodriquez-Clare, 1995; Jullien, 2000). 17 A noted in the Introduction, our focu i on advere election problem with exogenou information tructure. The olution to advere election problem in which the prevailing information aymmetry i endogenou can exhibit ditinct feature. See Szalay (2009), for example. 18 Recall from (5) that the firt-bet output given ignal of quality q 1 i characterized by V (x(, q 1 )) = c(, q 1 ).

9 252 / THE RAND JOURNAL OF ECONOMICS FIGURE 3 UNIT COSTS IN THE SETTING OF COROLLARY 1 c(,q) c 0 + ½ c(,q 2 ) c(,q 1 ) c 0 -½ 0 ½ c 0 ½ reflect the rate at which the buyer expected utility increae a the (q 2 q 2 ) (q 1 q 2 )incentive compatibility contraint i relaxed. c(,q 2) c 0 [c(,q 1 ) c 0 ] c(,q 1 ) c 0 reflect the proportionate difference in the rate at which a change in x(, q 1 ) change the upplier total cot when the ignal quality i q 2 rather than q 1. Thu, thi term capture the potential for reducing the upplier rent from mirepreenting q 2 a q 1 by altering x(, q 1 ) and adjuting payment o a to leave (q 1 q 1 ) unchanged. φ 1 i the probability that the reduction in total urplu caued by ditorting output away from their firt-bet level when q = q 1 will be incurred. A thi probability increae, the expected cot of uch output ditortion increae, which tend to reduce their magnitude, ceteri paribu. Equation (6) and (7) imply that the extent to which the buyer optimally underue or miue the ignal varie with the magnitude of Z 21 (). 19 Thi magnitude, in turn, varie according to the precie manner in which expectation about the upplier unit cot are revied a the public ignal varie. To illutrate, uppoe [ 1, 1 ] and the upplier unit cot i { c0 + for q = q 2 c(, q) = (8) c 0 + [gn()] 2 for q = q 1. When q = q 2 in thi etting, the upplier expected unit cot i updated in linear fahion after oberving the realization of the public ignal. The updating i ytematically le pronounced when q = q 1. For both ignal qualitie, c(, ) exceed c 0 when > 0 and c(, ) i le than c 0 when < 0. However, c(, q 1 )icloertoc 0 than i c(, q 2 ) for all 0, a illutrated in Figure 3. In thi ene, the ignal i ytematically more informative about the upplier unit cot when q = q 2 than when q = q 1. The quadratic formulation for c(, q 1 ) i employed for analytic implicity. 19 When < 0, c(, q 2 ) < c(, q 1 ) < c 0. Therefore, from (6) and (7), Z 21 () > 0andV (x(, q 1 )) c(, q 1 ), which implie x(, q 1 ) x (, q 1 ). When > 0, c(, q 2 ) > c(, q 1 ) > c 0. Therefore, (6) and (7) imply that Z 21 () > 0and V (x(, q 1 )) c(, q 1 ), and o x(, q 1 ) x (, q 1 ). 20 gn() denote the ign of.

10 CHU AND SAPPINGTON / 253 When the cot tructure i a pecified in (8), Z 21 () = 2, which decline a increae 2 above 0 and a decline below 0. Therefore, Z 21 () i mot pronounced for the ignal realization cloet to 0. Conequently, to bet limit the upplier rent when q = q 2, the buyer miue the ignal for a range of realization jut below 0 and jut above 0, a Corollary 1 report. Corollary 1. Suppoe n = 2, i ditributed uniformly on [ 1, 1 ], V ( ) i quadratic, φ = 3,φ 4 2 = 1, and c(, q) i a pecified in (8). Then at the olution to [BP]: (i) x(, q 4 1) < x (0, )for (.25, 0), x(0, q 1 ) = x (0, ), and x(, q 1 ) > x (0, )for (0,.25); and (ii) (q 2 q 2 ) > 0. The miue of the ignal that the upplier implement when (.25, 0) (0,.25) and q = q 1 in the etting of Corollary 1 limit, but doe not eliminate, the upplier rent when q = q 2. More extenive miue of the ignal could eliminate the upplier rent, but would further reduce total expected urplu when q = q 1 (by implementing relatively large level of output when unit production cot are high and relatively mall level of output when unit production cot are low). Becaue the lower ignal quality (q 1 ) i relatively likely in the etting of Corollary 1, the buyer optimally afford ome rent to the upplier when q = q 2 rather than reduce total expected urplu unduly when q = q 1. When the ignal i le likely to be of the lowet quality (q 1 ), the buyer may implement more extenive miue of the ignal. By doing o, the buyer can limit the upplier rent more effectively when q = q 2, a Corollary 2 report. Corollary 2. Suppoe the etting i a decribed in Corollary 1 except that φ 1 = φ 2 = 1. Then at 2 the olution to [BP]: (i) x(, q 1 ) < x (0, ) for (.36, 0), x(0, q 1 ) = x (0, ), and x(, q 1 ) > x (0, ) for (0,.36); and (ii) (q 2 q 2 ) = 0. The more extenive miue of the ignal in the etting of Corollary 2 reduce total urplu when q = q 1. However, the aociated reduction in the upplier rent when q = q 2 i particularly valuable to the buyer becaue q 2 i relatively likely in thi etting. Corollary 2 reveal that the upplier rent may be eliminated for all realization of hi private information at the olution to [BP]. Thi i the cae even when the upplier alway deliver a trictly poitive level of output (that varie with the realization of the public ignal). Thu, even in imple binary etting, the olution to [BP] can entail feature that do not arie in baic advere election problem. Additional nontandard concluion can emerge in more general etting. More general etting (n 2). To illutrate the additional nontandard feature that can arie at the olution to [BP], conider the trinary etting decribed in Propoition 2. Propoition 2. Suppoe n = 3, i ditributed uniformly on [ 1, 1 ], V ( ) i quadratic, φ = 0.1,φ 2 = 0.2,φ 3 = 0.7, and c 0 + for q = q 3 c(, q) = c 0 + [gn()] 2 for q = q 1 (9) ω() c(, q 1 ) + [1 ω()] c(, q 3 ) forq = q 2, where ω() = Then at the olution to [BP]: (i) (q 1 q 1 ) > (q 2 q 2 ) = (q 3 q 3 ) = 0; (ii) the (q 1 q 1 ) (q 2 q 1 ) incentive compatibility contraint bind; (iii) the ignal i miued (i.e., x(, ) < x (0, ) for < 0 and x(, ) > x (0, ) for > 0) for <.36 when q = q 1 and for <.37 when q = q 2 ; and (iv) x(, q 3 ) = x (, q 3 ) for all [ 1 2, 1 2 ]. Propoition 2 reveal the nontandard pattern of rent and binding incentive compatibility contraint that can arie more generally at the olution to [BP]. The cot tructure in (9) conit of the two cot tructure in Corollarie 1 and 2 and an additional intermediate cot tructure, c(, q 2 ), that i a weighted average of the other two tructure. The weight placed on the more informative cot tructure, c(, q 3 ), decline a increae. Such a weighting might reflect, for

11 254 / THE RAND JOURNAL OF ECONOMICS example, a etting in which the upplier ha hedged againt the rik of large variation in the cot of a key input, but not againt the rik of minor variation. The buyer hold the upplier to zero rent for the two highet-quality realization in thi trinary etting (where q 2 and q 3 are relatively likely), jut a he eliminate the upplier rent for both quality realization in the binary etting of Corollary 2. She doe o in part by miuing the ignal for both of the maller quality realization (q 1 and q 2 ). The upplier earn rent in the etting of Propoition 2 when q = q 1. Thi i the cae becaue the miue of the ignal that i implemented when q = q 2 in order to reduce the upplier rent when q = q 3 erve to increae the upplier expected profit when q = q 1. Recall from the dicuion following Lemma 1 that miue of the ignal can increae the upplier incentive to exaggerate the prevailing ignal quality. Thi i the cae becaue when induced output increae a increae, an exaggeration of ignal quality contitute an exaggeration of expected total production cot, which can generate rent for the upplier. In the etting of Propoition 2, miue of the ignal under the q 2 contract caue the (q 1 q 1 ) (q 2 q 1 ) incentive compatibility contraint to bind upward at the olution to [BP]. It doe o becaue the upplier unit cot are ufficiently different when q = q 1 from when q = q 2 (i.e., c(, q 2 ) c(, q 1 ) i ufficiently large) for the maller realization for which the ignal i miued under the q 2 contract. 21 Thi relatively pronounced difference in unit cot arie in the etting of Propoition 2 becaue ω( ) i an increaing function of. Conequently, although c(, q 2 ) may approximate c(, q 1 ) fairly cloely for the larger realization of, it doe not do o for the maller realization. 22 The buyer could reduce the miue of the ignal when q = q 2 in order to reduce the upplier rent when q = q 1. However, doing o would increae the upplier rent when q = q 3. (Recall Lemma 1.) Conequently, a long a q 3 i ufficiently likely compared to q 1, the buyer optimally hold the upplier to zero rent when q = q 3 and allow the upplier to ecure rent in the (relatively unlikely) event that q = q Thee nontandard pattern of binding participation and incentive compatibility contraint reflect the fact that the critical information aymmetry in [BP] can differ in important way from the correponding aymmetry in tandard advere election problem. In tandard problem, the buyer i privately informed from the outet about hi unit production cot c i {c 1,...,c n }. Thi cot i a calar, and each poible unit cot can be viewed a a convex combination of the lowet (c n ) and the highet (c 1 ) poible unit cot, that i, c i = α i c n + [1 α i ]c 1 where 0 = α 1 < <α n = 1. In contrat, the buyer private information in [BP] pertain to a continuum of unit cot realization c(, q i ) that i not necearily a imple convex combination of the two extreme uch continua. Even in the pecial etting of Propoition 2 where the intermediate cot tructure c(, q 2 ) i a weighted average of cot tructure c(, q 1 ) and c(, q 3 ), the weighting varie with the ultimate realization of the public ignal. Conequently, a change in q i need not correpond to a change in the calar weight that are applied ytematically to the extreme cot continua for all realization of the public ignal. In thi ene, [BP] doe not necearily impoe on the critical information aymmetry between buyer and upplier the ame tructure that i impoed in tandard advere election problem. When correponding tructure i impoed in [BP], the olution to [BP] exhibit the key feature of the olution to tandard advere election problem, a Propoition 3 report. The 21 The (q 2 q 2 ) (q 1 q 2 )and (q 3 q 3 ) (q 1 q 3 ) incentive compatibility contraint alo bind (downward) at the olution to [BP] in the etting of Propoition It can be hown that if ω() did not vary with, the upplier would earn zero rent for all realization of the ignal quality under the tructure and probabilitie pecified in Propoition The buyer will implement different qualitative ditortion in the q 2 contract if ω() = 1 + β and β>.925 in 2 2 the etting of Propoition 2. In particular, the buyer will miue the ignal for intermediate realization rather than for the mallet realization. Thee ditortion erve to limit the upplier rent when q = q 1 by curtailing miue of the ignal for thoe realization of the public ignal for which c(, q 2 ) c(, q 1 ) i particularly pronounced.

12 CHU AND SAPPINGTON / 255 FIGURE 4 OPTIMAL OUTPUT (x(, q i )FORi < n) WHEN ASSUMPTION 2 HOLDS x ( ) x* (,q i) x (,q i) <0 x* (0, ) x (,q i) q i x* (,q i) >0 propoition refer to Aumption 1, under which each continuum of unit cot realization, c(, q i ) for i = 1,...,n, i a weighted average of the extreme continua, c(, q 1 ) and c(, q n ), where the relevant weight are parameter that do not vary with. Aumption 1. There exit calar 0 = α 1 <α 2 < <α n = 1 uch that for each i = 1,...,n, c(, q i ) = α i c(, q n ) + [1 α i ] c(, q 1 ) for all [, ]. Propoition 3. Suppoe Aumption 1 hold. Then at the olution to [BP]: (i) the (q i q i ) (q i 1 q i ) incentive compatibility contraint bind for i = 2,...,n; (ii) the upward incentive compatibility contraint ( (q i q i ) (q j q i ) for j > i) are not contraining; (iii) (q 1 q 1 ) = 0; (iv) (q i q i ) i a convex, monotone increaing function of q i ; (v) for i = 1, 2,...,n 1:x(, q i ) x(, q i+1 ) x (, q i+1 ) for all [, 0), x(0, q i ) = x (0, ), and x (, q i+1 ) x(, q i+1 ) x(, q i ) for all (0, ]; and (vi) x(, q n ) = x (, q n ) for all [, ]. When Aumption 1 hold, a change in the upplier private information (q i ) contitute a change in the weight that are applied ytematically (for each realization) to the extreme continua of cot realization. Conequently, even though the upplier ha privileged knowledge of the entire continuum of poible unit cot c(, ), thi potential multidimenional information aymmetry effectively collape to a ingle dimenion. Propoition 3 report that tandard tructure i optimally employed to manage the prevailing information aymmetry when the aymmetry i effectively unidimenional, a it i in tandard advere election problem. 24 In particular, the adjacent downward incentive compatibility contraint bind at the olution to [BP] and the correponding upward contraint are not contraining. The upplier earn no rent for the lowet ignal quality, and hi rent (weakly) increae a the ignal quality increae. 25 The buyer limit the upplier rent by reducing output below it firt-bet level when [, 0) and by increaing output above it firt-bet level when (0, ] for all ignal qualitie other than the highet, a illutrated in Figure 4. In tandard fahion, the firt-bet output are induced for all realization of the public ignal when the ignal i of the highet quality. Although Aumption 1 give rie to tandard tructure, it doe not preclude miue of the public ignal. To determine when the buyer will refrain from uch miue, notice that, much a 24 Notice that Aumption 1 i atified trivially when n = 2, which help to explain Propoition It i alo readily hown that the buyer expected utility increae a the ignal quality q i increae when Aumption 1 hold. The buyer expected utility at the olution to [BP] need not increae monotonically with q i more generally.

13 256 / THE RAND JOURNAL OF ECONOMICS in the binary etting earlier in thi of ection, the buyer optimal ue of the ignal when q = q i i determined by [ ] V (x(, q i )) = c(, q i ) [c(, q i ) c 0 ] γ ji Z ji (), (10) where Z ji () 1 φ i [ c(,q j ) c 0 [c(,q i ) c 0 ] c(,q i ) c 0 ] and γ ji i the Lagrange multiplier aociated with the (q j q j ) (q i q j ) incentive compatibility contraint. 26 Equation (10) implie that the optimal ue of the ignal under ignal quality q i varie with Z ji (), which reflect in part the proportionate difference in the rate at which a change in x(, q i ) change the upplier total cot when the ignal quality i q j rather than q i. 27 If Z ji () doe not vary with for any pair of ignal qualitie, q i, q j Q, then the buyer will not find it optimal to implement relatively evere output ditortion for realization near 0 and relatively limited ditortion for other realization, a he doe in the etting of Propoition 2. In particular, although the buyer will underue the ignal to limit the upplier rent, he will not miue the ignal. Thi concluion follow from (10). When Z ji () doe not vary with, 1 γ j i ji Z ji () i a contant, k i, that doe not vary with. In thi cae, (10) can be written a V (x(, q i )) = k i c(, q i ) + [1 k i ]c 0. (11) It can be hown that k i [0, 1] at the olution to [BP] when Z ji () i independent of for all q i, q j Q. Conequently, (11) implie that induced output i alway intermediate between it ex pot efficient level, x (, q i ), and it ex ante efficient level, x (0, ). In other word, the ignal i never miued. Z ji () will be independent of for all q i, q j Q when the upplier expected unit cot i multiplicatively eparable in and q, for example, when Aumption 2 hold. Aumption 2. c(, q i ) = c 0 + q i for all [, ] and for all q i Q. 28 A upplier expected unit cot of production will atify Aumption 2 in many etting of interet, including the etting decribed at the end of Section 2 where q i i the probability that the public ignal i informative. A one further example, uppoe q i i the additional number of unit of a key input that the upplier mut purchae in a pot market (relative to initial expectation) in order to produce each unit of the product in quetion. Further uppoe that the ignal realization () i the difference between the actual unit pot price of the key input and the ex ante expected unit pot price of the input. In thi etting, the upplier expected unit cot of production given ignal of quality q i i c(, q i ) = c 0 + q i. When Aumption 2 hold, the upplier unit cot c(, q i ) can be written a α i [c 0 + q n ] + [1 α i ][c 0 + q 1 ], where α i [0, 1] i a calar and q i = α i q n + [1 α i ]q 1 for i = 1,...,n. Conequently, Aumption 1 hold. Aumption 2 impoe additional tructure by requiring the proportionate change in the upplier unit cot a the ignal quality change ( c(,q j ) c(,q i ) c(,q i ) c 0 j i = q j q i q i ) to be independent of. Thi independence enure that the buyer never miue the ignal, a Propoition 4 report and Figure 4 illutrate. Propoition 4. Suppoe Aumption 2 hold. Then the concluion in Propoition 3 hold. Furthermore, the ignal i not miued at the olution to [BP], that i, x (0, ) x(, q i )for all [, 0), and x(, q i ) x (0, ) for all (0, ], for i = 1, 2,...,n. 26 The derivation of thi neceary condition i preented in the proof of Propoition 2 in the Appendix. 27 Notice that Z ji (0) = 0 becaue c(0, q i ) = c(0, q j ) for all q i, q j Q. Therefore, becaue output ditortion when = 0 would not limit the upplier incentive to mirepreent the prevailing ignal quality, no uch ditortion are implemented, that i, x(0, q i ) = x (0, ) for all q i Q at the olution to [BP]. 28 When Aumption 2 hold, Z ji () = 1 [ q j qi ], which i independent of. It i alo readily verified that Z φi qi ji ()i independent of a long a c(, q i ) = c 0 + S() Q(q i ), where S(0) = 0, S () > 0 for all [, ], and Q(q j ) > Q(q i ) > 0 for all q j > q i. Indeed, the enuing analyi hold under thi generalization of Aumption 2, a the proof of Propoition 4 reveal.

14 CHU AND SAPPINGTON / 257 One additional feature of the olution to [BP] warrant emphai. If the higher q realization are ufficiently likely, the buyer will decline to link induced output to the realization of the public ignal when the mallet ignal qualitie are reported. By etting x(, q i ) = x (0, ) for all [, ] for the mallet q i realization (ee Figure 4), the buyer can enure that the upplier earn no rent when he claim the ignal quality to be one of the mallet q i realization. 29 Thu, ignoring the public ignal here play a role imilar to the role that hutdown (i.e., inducing no output) play in baic advere election problem. The difference here i that urplu continue to be generated when the ignal i ignored (becaue output i trictly poitive), although the urplu i le than could be achieved by tailoring output to the upplier realized unit cot. 30 Becaue ignoring the public ignal in thi manner eliminate rent without eliminating the entire urplu, the public ignal may optimally be ignored for an extenive range of quality level. Corollary 3 illutrate thi more general concluion. Corollary 3. Suppoe Aumption 2 hold, φ i = 1, and q n i = q i 1 + for all i = 1,...,n, where q 0 = 0 and n = 2 ñ, where ñ 1 i an integer. Then x(, q i ) = x (0, ) for all i = 1,...,ñ at the olution to [BP], o induced output i not linked to the public ignal for one half of the quality realization. 31 Corollary 3 report that when the ignal quality effectively ha a uniform ditribution, the induced output doe not vary with the public ignal (o x(, q i ) = x (0, )) for the mallet half of the quality realization. The region in which the public ignal i o ignored can be even more pronounced when the mallet q realization are relatively unlikely wherea the larget q realization are relatively likely. In thi cae, ignoring the public ignal for a large portion of the maller q realization can reduce the upplier expected rent ubtantially without reducing expected total urplu unduly Renegotiation Before concluding, we briefly conider the effect of renegotiation in our model. To thi point, we have aumed that the buyer can credibly commit to the term of the contract that he deign before oberving the realization of the public ignal. Now uppoe, in contrat, that mutually advantageou renegotiation of the original contract can take place after the ignal realization i oberved. We will demontrate that the poibility of renegotiation need not be contraining for the buyer in thi etting, even though it often i in dynamic advere election model. Conider the impact of cotle renegotiation initiated by the buyer in the binary etting of Propoition 1. Suppoe the buyer in thi etting initially offer to the upplier the {T (, q 1 ), x(, q 1 )} contract that arie at the olution to [BP]. Recall that thi contract induce output below the firt-bet level (x(, q 1 ) < x (, q 1 ) < x (, q 2 )) when < 0 and above the firtbet level (x(, q 1 ) > x (, q 1 ) > x (, q 2 )) when > 0. Thi contract alo generate nonnegative expected profit for the upplier for both realization of the ignal quality, and o the upplier will alway accept thi initial contract. After the realization of the ignal i oberved publicly, the buyer will propoe a new contract that maximize her expected utility while enuring the upplier at leat the profit he could ecure under the initial contract. At thi time, the buyer belief about the prevailing ignal quality 29 When x(, q i ) doe not vary with, (q i q j ) (q i q i ) = x(, q i ) [c(, q i) c(, q j )] f () d = It i readily hown that if Aumption 2 hold and the public ignal i ignored at the olution to [BP] for ome q j, then the ignal i ignored for all q i < q j. 31 ñ If n = 2 ñ + 1, then the ignal i not ued for the fraction of the quality realization. 2ñ+1 32 Correponding concluion hold when q ha a continuou, rather than a dicrete, ditribution. In particular, uppoe the cumulative ditribution function for q i G(q) = ( qq )γ for q [0, q], where γ>0. Further uppoe c(, q) = c 0 + S()q and S() f () d = 0. Then the buyer optimally et x(, q) = x (0, ) for all q [0, q 0 ], where q 0 = q[1 + γ ] γ 1. Thu, a γ increae from 0 to, q0q increae from to 1. e

15 258 / THE RAND JOURNAL OF ECONOMICS coincide with her belief when he deign a nonrenegotiable contract in problem [BP] (becaue the upplier alway accept the initial contract in the preent etting). When the realized ignal i negative, the buyer know that the upplier unit cot i maller when q = q 2 than when q = q 1. Conequently, the buyer optimally preent the upplier with the option of increaing output from x(, q 1 )tox (, q 2 ) in return for an additional payment of c(, q 2 )[x (, q 2 ) x(, q 1 )]. The upplier will find thi additional option to be at leat a profitable a the relevant option in the original contract if and only if q = q 2. The buyer will not offer an additional option that expand output toward x (, q 1 ) and increae payment at a rate that would be compenatory for the upplier when q = q 1. Doing o would provide exceive rent to the upplier when q = q When the realized ignal i poitive, the buyer know that the upplier unit cot i higher when q = q 2 than when q = q 1. Conequently, the buyer optimally preent the upplier with the option of reducing output from x(, q 1 )tox (, q 2 ) in return for a reduction in payment of c(, q 2 )[x(, q 1 ) x (, q 2 )]. Becaue thi payment reduction exceed the cot aving the upplier would ecure under the new option when q = q 1, the upplier elect the new option only when q = q 2. Thu, by initially offering the {T (, q 1 ), x(, q 1 )} contract that arie at the olution to [BP], the buyer enure that thi contract govern her interaction with the upplier when q = q 1.The correponding contract when q = q 2 induce firt-bet output and leave the upplier with the ame rent that he ecure under the initial contract. Conequently, the upplier ex ante expected profit i preciely hi expected profit at the olution to [BP], and induced output i exactly the output induced at the olution to [BP]. Therefore, the buyer expected utility in thi binary etting with renegotiation coincide with her expected utility at the olution to [BP]. 6. Concluion and extenion We have analyzed optimal procurement contract in a etting where a upplier ha privileged knowledge of the quality of a public ignal about hi production cot. In ome pecial etting, the optimal procurement contract exhibit tandard feature in thi etting. In particular, local downward incentive compatibility contraint bind and the upplier rent increae a the ignal quality increae. To limit thi rent, the buyer underue the public ignal. She may even ignore the public ignal altogether for a pronounced range of the maller quality realization. More generally, optimal procurement contract in the preent etting can differ qualitatively from the correponding contract that arie in baic advere election problem. For intance, the upplier may not be able to command rent from hi private information. Furthermore, when the agent i able to ecure rent, the rent may not vary monotonically with hi private information. In addition, the buyer may miue the ignal and thereby create incentive for the upplier to exaggerate the prevailing ignal quality. Conequently, incentive compatibility contraint may bind upward for ome realization of the ignal quality and bind downward for other realization. The equilibrium pattern of binding contraint reflect in part the potential for reducing the upplier rent by ditorting output. Becaue thi potential can vary nonytematically with both the ignal quality and the ignal realization, the region of upward and downward binding incentive compatibility contraint need not be neatly eparated, a they often are in advere election problem with countervailing incentive. 34 Thu, varied pattern of binding participation 33 The buyer alo will not offer an additional option that reduce output below x(, q 1 ) and that i compenatory for the upplier when q = q 1. The output reduction would reduce total urplu becaue x(, q 1 ) < x (, q 1 )when < 0. Furthermore, the new option would not reduce the upplier rent when q = q 2 becaue the original option (with the larger output) remain available to the upplier. 34 For example, conider the etting in which countervailing incentive arie becaue the agent unit cot of producing output for the principal (c) varie inverely with hi reervation utility level. In thi etting, the agent will be tempted to: (i) exaggerate hi unit cot when c i relatively mall, and o induced output i relatively large; and (ii) undertate hi unit cot (and thereby exaggerate hi reervation utility) when c i relatively large, and o induced output i relatively mall. Thee conideration are complicated in our analyi by the fact that the nature and the extent of the