TECHNOLOGICAL IMPROVEMENT AND THE DECENTRALIZATION PENALTY IN A SIMPLE PRINCIPAL/AGENT MODEL. May 2, 2018

Transcription

1 TECHNOLOGICAL IMPROVEMENT AND THE DECENTRALIZATION PENALTY IN A SIMPLE PRINCIPAL/AGENT MODEL Ruochen Liang, Thomas Marschak, and Dong Wei May 2, 2018 We consider he organizer of a firm who compares a decenralized arrangemen where divisions are graned oal auonomy wih an arrangemen where perfec monioring and policing guaranee ha all divisions make he choices he organizer wans hem o make. We ask: when does improvemen in he divisions echnology srenghen he case for decenralizaion and when does i weaken i? The quesion is difficul and i is naural o sar wih a sripped-down model, where here is jus one division. In he decenralized mode, he organizer appoins a Principal who rewards a single auonomous Agen (he division manager). The Agen freely chooses an effor x. The effor need no be hidden. The Agen bears is cos. The firm hen achieves he surplus R(x) C(x), where R is revenue, is a posiive echnology parameer known o boh paries, and C(x) is he cos of he Agen s effor. When echnology improves, drops. The Agen receives a share of he revenue, namely r R(x), were 0 < r < 1. The Principal receives he residual (1 r) R(x). In he exogenous case he share r is deermined ouside he model (perhaps by Principal/Agen bargaining). In he endogenous case he Principal, who knows how he Agen responds o every possible r given he curren, chooses he r which maximizes he residual revenue. The Decenralizaion Penaly for a given equals he maximal possible surplus for ha aained under perfec monioring minus he surplus achieved in he decenralized mode. I urns ou ha here are no simple condiions on C and R which imply ha he Penaly grows (shrinks) when echnology improves. Insead, we obain a variey of resuls abou relaions beween, surplus, he Principal s generosiy (he size of he Principal s chosen share in he endogenous case), and effeciveness (he effec of a small rise in he share on he Agen s effor). * Deparmen of Mahemaics, Universiy of California, Berkeley. ** Waler A. Haas School of Business, Universiy of California, Berkeley. *** Deparmen of Economics, Universiy of California, Berkeley. 1

2 1. Inroducion Does he case for decenralizing a firm ge sronger or weaker when he producion echnology used by one or more of is divisions improves? Consider he Organizer of he firm, who seeks a good balance beween he cos of he divisions effors and he revenue which hose effors yield. One way o achieve a good balance may be inrusive bu perfec monioring and policing, which fully reveals he chosen effors and guaranees ha hey are hose he Organizer prefers. Perfec monioring/policing may be very cosly. A beer mode of organizing migh be decenralizaion, where he divisions are oally auonomous, hough heir choices may be influenced by appropriae rewards and penalies. In he decenralized mode ha we shall sudy here is a Principal who reas each division as an Agen. Each Agen freely chooses her effor and bears he effor s cos. The Principal observes he realized revenue and rewards he Agens. Each Agen s reward is a funcion of revenue, and her ne earnings are her reward minus he cos of her chosen effor. The reward funcions he Principal chooses are accepable o he Agens and are preferred by he Principal o oher possible reward funcions ha are also accepable o he Agens. The Principal pockes he residual revenue which is lef over afer he rewards have been paid. When an Agen s echnology improves, he cos of a given effor drops. The Organizer compares he decenralized Principal/Agens mode wih perfec monioring/policing. Many producion echnologies rapidly improve, bu a he same ime he coss of perfec monioring may rapidly drop as well, because of dramaic advances in monioring echniques. So he relaive meri of he wo modes requires regular reassessmen. We shall le he Organizer ake a welfare poin of view in comparing he wo modes. The Organizer s focus is he firm s surplus: he revenue earned by he divisions effors minus he cos of hose effors. Perfec monioring/policing guaranees maximal surplus. The Decenralizaion Penaly is he welfare loss due o decenralizing. I is he gap beween maximal surplus and he surplus achieved in he decenralized Principal/Agens mode; so an alernaive erm for he Penaly is surplus gap. 1 Our cenral quesion is wheher he Decenralizaion Penaly grows or shrinks when a echnical advance lowers Agens effor coss. If he Penaly subsanially grows, hen perfec monioring may now be worh wha i coss. (We will no explicily model he cos of monioring). If he Penaly shrinks, hen perfec monioring becomes less aracive even if monioring echniques have advanced. Our cenral quesion is ricky for he following reason: when he Agens echnology improves, maximal surplus rises (under weak assumpions). Maximal surplus is a moving arge. Decenralized surplus also rises, under reasonable assumpions. Bu ha does NOT mean, in general, ha as echnology improves, he rising decenralized surplus ges closer o he moving surplus arge. Our quesion appears o be very rarely asked in he abundan Principal/Agen lieraure. The cos of an Agen s effor appears in many papers and so does 1 If he firm is a regulaed monopoly, for example, hen high surplus migh be he regulaor s goal. The regulaor compares decenralizaion wih perfec monioring/policing and favors he mode ha achieves he higher surplus. Alernaively surplus may be viewed as he firm s profi. If he Organizer is also he firm s owner hen profi may be wha he seeks o maximize. 2

3 he welfare loss due o Agens second-bes choices. Bu he effec of cos improvemen on welfare loss seems o be widely negleced. Our cenral quesion is ricky for he following reason: as he Agens echnology improves, maximal surplus rises (under weak assumpions). Maximal surplus is a moving arge. Decenralized surplus also rises, under reasonable assumpions. Bu ha does NOT mean, in general, ha as echnology improves, he rising decenralized surplus ges closer o he moving surplus arge. Our quesion appears o be very rarely asked in he abundan Principal/Agen lieraure. The cos of an Agen s effor appears in many papers and so does he welfare loss due o Agens second-bes choices. Bu he effec of cos improvemen on welfare loss seems o be widely negleced. 2. The model We shall sudy a highly simplified model. There is a single effor variable x, chosen from a se Σ IR + of possible posiive effors. The se Σ may be finie or i may be a coninuum. There is no uncerainy abou he consequences of a given effor. The effor x generaes a posiive revenue R(x), where R is sricly increasing. The effor x coss C(x), where C is posiive and sricly increasing. A drop in occurs when echnology improves (or here is a fall in he price of he inpus which effor requires). For a given, we consider he surplus a he effor x, denoe W (x, ). Thus W (x, ) = R(x) C(x). In he cenralized mode perfec monioring/policing guaranees ha effor is firs-bes : i maximizes surplus. In he decenralized mode here is no direc monioring. Insead here is a self-ineresed Principal and a single self-ineresed Agen who freely chooses x Σ and bears he cos C(x). The funcions R and C, and he echnology parameer, are known o boh paries. The Principal observes he revenue R(x). Since R is sricly increasing, ha observaion also reveals he Agen s chosen x. The Principal rewards he Agen, using a reward which is a funcion of he observed revenue. We sudy an exremely simple reward scheme, namely linear revenue sharing. The Principal pays he Agen a share r (0, 1] of he revenue. So if he Agen chooses he effor x, she earns rr(x) C(x) and he ne amoun received by he Principal is he residual (1 r) R(x). We will assume ha for every (r, ) here is an effor x Σ such ha he Agen s gain rr() C(x) is nonnegaive, and his is sufficien for he Agen o be willing o paricipae. The Agen chooses o exer he effor ˆx(r, ), he smalles maximizer of rr(x) C(x) on he se Σ. We denoe he surplus when he share is r by W (r, ) (he ilde is deleed). So W (r, ) W (ˆx(r, ), ) = R(ˆx(r, )) C(ˆx(r, )). Noe ha if r = 1, hen he Agen s effor choice ˆx(1, ) is surplus-maximizing. Thus W (1, ) = W (ˆx(1, ), ) is he larges possible surplus. In he cenralized mode, perfec monioring/policing insures ha W (1, ) is achieved. 3

4 In our sudy of he Decenralizaion Penaly we consider wo cases. In he exogenous case, he reward share is deermined ouside he model. I migh, for example, be he resul of previous bargaining beween Principal and Agen, or i migh be prescribed by law. In he endogenous case, he Principal considers all he shares in he open inerval (0, 1) and chooses a share which maximizes (1 r) R(ˆx(r, )), he residual when he Agen uses he bes-effor funcion ˆx in responding o a given share. We le r () denoe he maximizer which he Principal chooses. So in he endogenous case he Agen s effor is ˆx(r (), ) and surplus is W (ˆx(r (), ), ) = W (r (), ). The moving arge remark ha we made above suggess ha he effec of a drop in on he Decenralizaion Penaly is suble. On he oher hand, i is hard o imagine a model simpler han ours. So one migh hope ha in our simple model here are simple condiions on Σ, R, C under which he Penaly rises (falls) when drops. I urns ou, however, ha even in our model here is a sriking diversiy of resuls. There are simple examples where he Penaly rises and simple examples where i falls. Tha is he case in boh he exogenous and endogenous seings. To bring some order o his diversiy, we divide examples (Σ, R, C) ino classes. To do so, we consider he effec of a drop in on he example s bes effor ˆx and on he example s endogenouscase bes share r. A higher share simulaes he Agen o work harder, bu he srengh of he simulus depends on. Consider any pair of shares (r L, r H ), where 0 < r L < r H < 1, and suppose ha drops. In some examples he effor increase ˆx(r H, ) ˆx(r L, ) rises and in oher examples i falls. In some examples, moreover, he Principal s bes share r rises when drops, and in oher examples i falls. So we have four classes of examples. For each class we examine he effor gap he amoun by which decenralized effor falls shor of he firs-bes effor. The effor gap is ˆx(1, ) ˆx(r, ) when r is exogenous and i is ˆx(1, ) ˆx(r (), ) in he endogenous case. The effec of a drop in on he effor gap is again ricky jus as i was for he Decenralizaion Penaly (he surplus gap). Under broad condiions, boh erms of he gap rise when drops, bu he gap iself may rise or fall. The effec of a drop in on he effor gap is ineresing in iself. There are classes of examples, moreover, in which he effor gap racks he surplus gap (he Decenralizaion Penaly): when drops, he wo gaps move in he same direcion. Imagine ha firs-bes effor ˆx(1, ) has been sudied for many riples (R, C, ). Then for an impending new echnology, firs-bes effor is already known bu he welfare effecs of decenralizing remain o be discovered. If we indeed have racking, hen i suffices o observe he Agen s work o see wheher, wih he new echnology, her effor has moved closer o firs-bes effor or furher away from i. In he former case we know if we indeed have racking ha he new echnology has shrunk he Decenralizaion Penaly, so i has made perfec monioring/policing less aracive. In he laer case i has increased he Penaly. We obain resuls abou racking. We do so firs for he exogenous case 2 under he assumpion ha R and C are hrice differeniable. We consider he effeciveness of a share increase in simulaing higher effor and we classify examples according o he change in effeciveness when drops, i.e., he sign of he cross parial ˆx r (r, ). We find ha we indeed have racking, provided 2 In Theorem 4 4

5 ha he following monooniciy condiion holds: eiher ˆx r (r, ) > 0 a all (r, ) or ˆx r (r, ) < 0 a all (r, ). Tha gives us condiions 3 on he signs of C, R, C, R under which he Decenralizaion Penaly rises when echnology improves ( drops) and condiions under which he Penaly falls. As one would expec, he racking quesion is more difficul in he endogenous case. We again classify examples wih regard o he effec of echnical improvemen on effeciveness (he sign of ˆx r (r, )) bu we also classify hem wih regard o he effec of echnical improvemen on he Principal s endogenous-case generosiy, i.e., he sign of he derivaive r (). We find 4 ha we have racking if eiher of he following condiions hold: (i) for every possible (r, ), ˆx r (r, ) < 0 and r () 0; (ii) for every possible (r, ), ˆx r (r, ) > 0 and r () < 0. In conras o he exogenous case, hese resuls do no give us condiions on he he signs of he second and hird derivaives of R and C under which he Penaly falls (rises) when echnology improves. 3. Relaed lieraure A grea many Principal/Agen papers, saring wih he earlies ones, use a framework ha allows he Agen s effor o have a cos. The Agen has a uiliy funcion on her acions and rewards. In many papers Agen uiliy for he acion a and he reward y akes he form V (y) g(a). Among he early papers where his occurs are Holmsrom (1979), (1982) and Grossman and Har (1983). The acion a migh be effor and g(a) could be is cos. Welfare loss also appears very early in he lieraure. Ross (1973), for example, finds condiions under which he soluion o he Principal s problem maximizes welfare (as measured by he sum of Agen s uiliy and Principal s uiliy) and noes ha hese condiions are very srong. Bu Principal/Agen papers whose main concern is he relaion beween effor cos and welfare loss are scarce. The Principal/Agen papers closes o ours are Balmaceda, Balseiro, Correa, and Sier-Moses (2016) and Nasri, Basin, and Marcoe (2015). They sudy an Agen who has m possible effors. Each effor has a cos, which he Agen pays. There are n possible revenues. For a given effor, he probabiliy of each of he possible revenues is common knowledge, bu he revenue acually realized only becomes known afer he Agen s effor choice has been made. The Principal announces a vecor of n nonnegaive wages. For each of he n possible revenues, he vecor specifies a wage received by he Agen when ha revenue is realized. Boh Principal and Agen are risk-neural. Surplus for a given effor equals expeced revenue minus he effor s cos. The socially preferred effor maximizes surplus. In he decenralized (Principal/Agen) mode, on he oher hand, surplus is no maximal. Insead i is he surplus when he effor is he one he Principal chooses o induce. The papers sudy a fracion. Is numeraor is maximal surplus and is denominaor is wors-case Principal/Agen surplus. (When he Principal is indifferen beween several effors, he denominaor of he fracion selecs he one ha is socially wors). The fracion is a measure of he welfare loss due o decenralizing. I is shown, under sandard assumpions on he probabiliies and on he possible (revenue,effor-cos) pairs, ha he raio canno exceed m, he number of effors. Tha upper bound does no depend on he effor coss, so he papers are silen on he effec of a drop in hose coss on welfare loss. 3 Saed in a Corollary o Theorem 4 4 In Theorem 5 5

6 Noe ha welfare loss is also defined as a fracion in a larger lieraure, iniially developed by compuer scieniss. Typically he objec of sudy is a game. The fracion sudied is ofen called he price of anarchy. Is numeraor is he payoff sum in he socially wors equilibrium of he game. Is denominaor aainable when he players cooperae is he larges possible payoff sum. 5 In our seing i is naural o use he surplus gap raher han a raio in defining he Penaly (welfare loss) due o decenralizing. Perfec monioring (cenralizaion) would eliminae he gap bu i would be expensive. If is cos exceeds he gap hen decenralizaion is he preferred mode. If we allow more han one Agen, hen pars of he large lieraure on he design of organizaions become relevan. The designer has a goal, say surplus (profi) maximizaion, and can choose beween a srucure where a single member commands he choices made by all he ohers, and a srucure where everyone is auonomous. The laer srucure migh be modeled as a game. A raher small piece of he design lieraure sudies he communicaion and compuaion coss of each srucure and he rade-off beween hose coss and some measure of gross performance (e.g., gross expeced surplus, before he coss are subraced). The problem is far more complex han he one we consider here and he resuls remain scarce and specialized Plan of he remainder of he paper. In Secion 5 we examine five examples where he se of possible effors and he se of possible values of are no finie. The examples provide a preview of our general resuls. In Secion 6 we develop basic resuls, presened in wo heorems. They do no require differeniabiliy, so finie examples are covered. The resuls concern he exogenous case in he firs heorem and he endogenous case in he second. Secion 7 presens wo exogenous-case heorems which require differeniabiliy. Secion 8 presens wo endogenous-case heorems which again require differeniabiliy. Secion 9 considers he shape of he funcion which relaes r o he Principal s gain. A concave shape implies a proposiion abou he negoiaion se when he wo paries bargain abou he size of r. Secion 10 explores an alernaive model, ofen sudied in he Principal/Agen lieraure. Revenue for a given effor is now uncerain, hough he probabiliies are common knowledge. he Agen s reward is no longer a fixed share of revenue. Raher, he Principal chooses a vecor of wages, one for each of he possible realized revenues. Boh paries are risk-neural. We ask wheher we again have one of our basic resuls: he Agen never works less when echnology improves, We hen consider he effec of improved echnology on he Decenralizaion Penaly. Secion 11 provides concluding remarks abou exensions and variaions of he model. 5. Some examples 5 A variey of social siuaions are sudied from his poin of view. One of hem concerns opimal versus selfish rouing in ransporaion (Roughgarden (2005)). Ohers are found in Nissan, Roughgarden, Tardos, and Vazirani (eds.) (2007). Many of hese sudies develop bounds on he price of anarchy. Several of hem (e.g., Babaioff, Feldman, and Nissan (2009)) consider a Principal/Agen seing. 6 Surveys of he design lieraure wih communicaion and compuaion coss are found in Garicano and Pra (2013) and Marschak (2006). A model in which revenue is shared by a group of game-playing Agens is sudied in Courney and Marschak (2009). Each player chooses effor and bears is cos. Equilibria of he game are compared wih he welfare-maximizing effors. The paper finds condiions under which he welfare loss drops (rises) when effor coss shif down. 6

7 Our model, simple as i is, urns ou o have quie diverse resuls. To illusrae he diversiy, we now consider five examples. In all five of hem he effor se and he se of possible values of he echnology parameer are coninua and calculus mehods are used o sudy hem. In he simples finie example, on he oher hand, here would be jus wo values of and wo values of effor. We can consruc simple finie examples yielding a variey of answers o he quesions we jus lised, bu hey are no presened here. In each of our five non-finie examples we provide some saemens ha we shall subsequenly generalize. 5.1 A Classic monopoly example where marginal revenue falls and marginal cos is fla. For convenien reference we shall call his our Classic monopoly example or, for breviy, our Classic example. I is suggesed by he inroducory monopoly diagram in he ypical ex, where marginal revenue drops and marginal cos is fla or rising. We may hink of he Principal as a monopolis who delegaes he choice of produc quaniy o he Agen. Quaniy will be our effor. A effor x, price is A Bx, where A > 0, B > 0. Cos is C(x) = x and revenue is R(x) = Ax Bx 2. Marginal revenue becomes negaive a x = A. To keep price and marginal revenue posiive, our se of possible effors will be Σ = ( 0, A ). 2B We consider a se Γ of pairs (r, ), where (i) every r (0, 1) belongs o one and only one pair, and (ii) for every pair (r, ) Γ here is a posiive effor ˆx(r, ) which belongs o Σ and is he unique maximizer of r R(x) C(x). Tha is he case for 2B Γ {(r, ) : 0 < r < 1; 0 < < Ar} and ˆx(r, ) = A 2B 2Br. For a given, he Agen s bes response o he share r if (r, ) Γ is he effor ˆx(r, ), which belongs o Σ. In every example ha we sudy we will specify a similar se Γ, having he properies (i) and (ii). Noe ha in he Classic example, Γ is he inerior of a riangle. In a diagram wih r on he horizonal axis and on he verical axis he riangle has verices a (0, 0), (1, 0) and (1, A). In oher examples Γ migh be a recangle, as in he example which follows (in 5.2). In sill oher examples one of he boundaries of Γ migh have curvaure. We shall le Γ denoe he se of values of which we consider. Thus In our Classic example, Γ = (0, A). Γ { : (r, ) Γ for some r (0, 1)}. Now for every (r, ) Γ consider he derivaive of ˆx wih respec o r, he derivaive wih respec o, and he cross derivaive. They will be denoed by ˆx r, ˆx and ˆx r. We have he following resuls. Some of hem will be generalized o wider classes of examples. 7

8 ˆx r (r, ) = 2Br 2, which is posiive. For a given, increasing he share evokes more effor. We shall show 7 ha in any example, finie or nonfinie, increasing he share never evokes less effor. ˆx (r, ) = 1, which is negaive. When r is fixed and echnology improves (when 2Br drops), he Agen works harder. We shall show 8 ha in any example, finie or nonfinie, he Agen never works less when drops. We have ˆx r (r, ) = 1 1 > 0. So echnology improvemen (a drop in ) diminishes 2B r 2 effeciveness (he effor increase evoked by a small rise in r). When he Agen uses he bes effor ˆx(r, ), he receives r R(ˆx(r, ) ˆx(r, ). In our Classic example, he derivaive of ha expression wih respec o urns ou o be negaive. 9 So in he exogenous case, echnology improvemen is good news for he Agen. We shall provide 10 a simple proof ha his saemen holds in any example, finie or nonfinie. We find ha surplus = W (r, ) = R(ˆx(r, )) C(ˆx(r, )) = 1 [(Ar ) (BAr + B 2Br)]. 4B 2 r2 The derivaive wih respec o of he expression in square brackes is 2BAr 2 2B + 4Br. Our requiremen ha < Ar implies ha his is negaive. 11 Thus, for a fixed r < 1, decenralized exogenous-case surplus rises when echnology improves ( drops). We shall see 12 ha his always holds, for boh finie and infinie message ses, as long as R and C are sricly increasing. For all Γ, we have 13 W (1, ) < 0. Maximal surplus 14 rises when echnology improves ( drops). As we shall see 15, a rivial argumen shows ha his always holds in boh finie and non-finie examples. (1 r) We have W r (r, ) = > 0. Br 3 So W r (r, ) and ˆx r (r, ) have he same sign. Tha implies, as we shall see, ha he exogenous Decenralizaion Penaly (surplus gap) W (1, ) W (r, ) and he exogenous effor gap ˆx(1, ) 7 In Par (a) of Theorem 1. 8 In Par (b) of Theorem 1. 9 The derivaive is ˆx (r, ) [rr (ˆx(r, )) C (ˆx(r, )] C(ˆx(r, )). Tha is negaive, since 0 < r < 1 and ˆx(r, ) saisfies he firs-order condiion 0 = rr C. 10 In Par (f) of Theorem The derivaive is negaive if Ar 2 > (2r 1). Tha is he case a r = 0 and a r = 1 (since < A). A all r (0, 1) our requiremen < Ar implies ha 2Ar, he derivaive of he lef side of he inequaliy wih respec o r exceeds 2, he derivaive of he righ side. So a all (r, ) Γ he inequaliy holds. 12 In Par (g) of Theorem For derivaives, we shall use he symbols W r, W, W r, which are analogous o our symbols ˆx r, ˆx, ˆx r. 14 In he monopoly seing our general erm surplus should no be confused wih consumers surplus. Our Surplus is he same as he monopoliss profi. 15 In Par (d) of Theorem 1 8

9 ˆx(r, ) move in he same direcion when echnology improves, i.e., he exogenous surplus gap racks he exogenous effor gap. There are finie examples where ha is no he case. Bu we shall show 16 ha if R and C are hrice differeniable hen i mus be he case, because, as we shall prove, W r ˆx r 0. We now urn o he endogenous case. The Principal chooses a bes share r, bu excludes r = 1, which would give he Agen all of he revenue. To sudy he consequence of choosing r = 0, we would have o specify how he Agen responds o r = 0. The naural answer is zero effor, bu here would hen be some echnical difficulies in our analysis of cerain examples. So, as already noed, we confine aenion o he case where he Principal chooses a share in he open inerval (0, 1) and he Agen s response is a posiive effor. We will show 17 ha under simple condiions (which are saisfied in he Classic example), he Principal s gain (1 r) (R(ˆx(r, )) is posiive for all r (0, 1) and is concave on (0, 1). Tha implies ha here is a share in (0, 1) which solves he firs-order condiion 0 = d dr [(1 r) R(ˆx(r, )] = R(ˆx(r, )) + (1 r) R (ˆx(r, )) ˆx r (r, ) and maximizes he Principal s gain on he se (0, 1). Tha share is our r () (0, 1). In our Classic example he Principal s firs-order condiion urns ou o be he cubic equaion 0 = A 2 r 3 + r When we graph he implici funcion r (), we obain he following figure for he case A = 2, B = 3: FIGURE 1 HERE The graph reveals ha r is increasing in. The share-choosing Principal becomes less generous when echnology improves. Bu we can esablish his fac for a very wide class of examples ha includes he Classic example wihou any graphing. We shall show 18 ha i mus hold whenever R < 0 and ˆx r < 0 (as in he Classic example). Thus a drop in makes he Principal less generous if i makes effeciveness drop and in addiion he Agen s increased effor (in response o he lower ) makes marginal revenue drop. Nex consider he endogenous effor ˆx(r (), ). Figure 2 shows ha in he Classic example he endogenous effor rises when echnology improves ( drops). We shall show 19 ha his mus happen, for he endogenous case, in every example, finie or nonfinie. FIGURE 2 HERE The endogenous decenralizaion Penaly (surplus gap) is W (1, ) W (r (), ). Tha can be graphed as a funcion of, even wihou an explici expression for r (). We do so in Figure 3 16 In Theorem In Theorem In Theorem In Par (b) of Theorem 2. 9

10 Fig. 1 graph of r () for he Classic example wih A = 2, B = 3

11 Fig. 2 graph of ˆx(r (), ) for he Classic example wih A = 2, B = 3

12 for he Classic example. We see ha for large furher echnical improvemen (a furher drop in ) raises he Penaly, bu for small furher improvemen lowers i. FIGURE 3 HERE We now urn o he racking quesion. For he Classic example, Figure 4 shows boh he Penaly (surplus gap) and he effor gap ˆx(1, ) ˆx(r (), ). When increases each gap firs rises and hen falls and for each he gaps move in he same direcion, so we indeed have racking. FIGURE 4 HERE 5.2 A Cubic-revenue example, where, jus as in he Classic example, echnology improvemen diminishes effeciveness and generosiy, bu now we do no have racking. In his example: R(x) = x 3 x 2, C(x) = x, and he se of possible (r, ) pairs is Γ = {(r, ) : r (0, 1); r}, so he se of possible values of is Γ = (0, 1). We find jus as in he Classic example ha for all (r, ) in Γ, effeciveness diminishes when drops, i.e., ˆx r (r, ) > Now consider r (), he implici funcion which solves he Principal s firs-order equaion for (0, 1). Tha funcion urns ou o saisfy a cumbersome polynomial equaion. 21 Figure 5 graphs he implici funcion r. 20 We have ˆx(r, ) = 1 ( 1 ) 1/2. 3 r Nex we obain ˆx r (r, ) = d dr [ ( 1 ) ] 1/2 = 1 ( r ) 1/2 r r 2. We hen have: 4 3 ˆx r (r, ) = d d [ ( 1 ) ] 1/2 r r 2 = ( 1 ) 1/2 ( ) ( 1 r r 2 + r d d [ ( 1 ) ]) 1/2. r Bu So we indeed have ˆx r (r, ) > The equaion is d d [ ( 1 ) ] 1/2 = 1 ( r 2 1 ) 3/2 r r 2 < 0. 0 = 16[r ()] [r ()] [r ()] [r ()] r ()

13 W (r (), ) W (1, ) Figure 3: W (1, ) and W (r (), ) for he Classic case, wih A = 2, B = 3

14 surplus gap effor gap Figure 4: The wo gaps: surplus gap (W (1, ) W (r (), )) and effor gap (ˆx(1, ) ˆx(r (), )) for he Classic case, wih A = 2, B = 3

15 FIGURE 5 HERE Jus as in he Classic example, r rises a every possible value of (all (0, 1)). Finally, we plo he surplus gap and he effor gap. FIGURE 6 HERE We find ha for in he inerval (.48,.63), he effor gap rises bu he surplus gap falls. Unlike he Classic example, we do no have racking. 5.3 A Price-aker example where marginal cos rises and marginal revenue is fla In his example we may hink of he Principal as a price-aker (wih price equal o one). He delegaes quaniy choice o he Agen and he Agen s cos funcion is quadraic. Price-aker is a convenien label for his example. The example is defined as follows: The se of possible effors is Σ = IR +. R(x) = x. C(x) = 1 2 (x 1)2. The se of possible pairs (r, ) is he recangle Γ = {(r, ) : 0 < r < 1; 0 < < 1}. We find he following: Given r, he Agen chooses ˆx(r, ) = r + 1. We hen have ˆx r(r, ) = 1 2 rises when echnology improves (when drops). < 0. Effeciveness When he Agen uses he bes effor ˆx(r, ), he receives r+ r2. The derivaive wih respec o 2 is r2 < 0. So, jus as in he Classic example echnology improvemen is good news for he 2 2 Agen in he exogenous case. As we already noed, we will provide a simple (calculus-free) proof ha his good news saemen holds, for he exogenous case, in any example, finie or nonfinie. Exogenous surplus is R(ˆx(r, )) C(ˆx(r, )) = r + 1 r2 2. Surplus-maximizing effor is and maximal surplus is The exogenous surplus gap (he Penaly) is 1 r + r2. Is derivaive wih respec o is 2 2 negaive. The exogenous effor gap is 1 r, which also has a negaive derivaive. So he exogenous surplus gap racks he exogenous effor gap, jus as in he Classic example. As already noed, his will be proved o hold, in he exogenous case, for any differeniable example. We now urn o he endogenous case. We find ha: 11

16

17 effor gaps surplus gap Fig. 6 Surplus and effor gaps in he Cubic-revenue example. For (.48,.63), ses bu surplus effor gap rises falls. bu surplus gap falls.

18 The soluion o he Principal s firs order condiion 0 = d dr [R(ˆx(r, )) C(ˆx(r, ))] is r () = 1 2. So we have r () < 0 a every possible. (Recall ha < 1). So, in sharp conras o he Classic monopoly example, he Principal becomes more generous when echnology improves. We find ha jus as in he exogenous case a drop in is good news from he welfare poin of view. This mus be he case 22 whenever as in he Price-aker example and he Rising Marginals example which we consider nex he Principal becomes more generous (or says jus as generous) when drops. 23 The endogenous Penaly (he endogenous surplus gap) is Is derivaive wih respec o is ( 2 1), which is negaive, since < 1. The Penaly rises when 8 2 echnology improves. Again, noe he conras wih he Classic example, where he Penaly drops when echnology improves, once has dropped below a criical value. The endogenous effor gap is 1+ r ()+ = Tha also has a negaive derivaive. 2 2 So he endogenous effor gap racks he exogenous surplus gap. Bu ha is NOT implied, as we shall see, by he fac ha ˆx r < 0 and r () < A Cubic-cos example, where, jus as in he Price-aker example, echnology improvemen increases boh effeciveness and generosiy, bu now we have opposie direcions raher han racking. In his example R(x) = 1 2 x2 and C(x) = 1 3 x3 + a 2 x2 ɛx, where ɛ > 0 and a > 0. The numbers a, ɛ and he se Σ of possible effors will be chosen as we proceed. The riple (a, ɛ, Σ) will have he propery ha C(x) > 0 for all x Σ. The Agen s firs-order condiion for given r, is 22 Shown in Par (d) of Theorem The argumen is as follows: We have rx = (x 2 + ax ɛ). d d [R(ˆx(r (), )) C(ˆx(r (), ))] = [ˆx r r + ˆx ] (R C ) C. We have R C > 0 (because of he firs-order condiion rr C = 0, where 0 < r < 1). Since ˆx < 0 and r 0, we conclude ha he derivaive is negaive, so we indeed have good news from he welfare poin of view. 12

19 This is solved by Noe ha ˆx(r, ) = (a ) r 2 ( ) + 4ɛ a r 2 > 0. ˆx r = 1 2 ( = [ ( a r ) ] 2 1/2 + 4ɛ 2 ( a r ) ) a r (a ) r 2 + 4ɛ (a ) = 1 r 2 ( ) 2 + 4ɛ a r (a ). r 2 + 4ɛ Our se Γ of possible (r, ) pairs will be { Γ = (r, ) ; Now assume ha 1 a ɛ < 3 4 a2. ( 1 a, 2 a2 + 4ɛ ) } ; 0 < r < 1. Then 1 a < 2, so Γ is no empy. Moreover a r/ 0 for all r (0, 1). Under hese a2 + 4ɛ assumpions we have ˆx r (r, ) < 0. Effeciveness increases when echnology improves Consider he fracion Muliply numeraor and denominaor by ˆx r = (a ) r 2 ( ) + 4ɛ a r ) 2 + 4ɛ. (a 2 r ( a r ) 2 ( + 4ɛ + a r ). The new fracion simplifies o 4ɛ [ (a ) ]. 2 r 2 (a ) + r 2 + 4ɛ Since a > r, he denominaor is sricly increasing in. Hence he whole fracion is sricly decreasing in. So, as claimed, we have ˆx r (r, ) < 0. 13

20 In he endogenous case he firs-order condiion saisfied by he Principal s chosen share r saisfies he firs-order condiion r = 1 R R ˆx r = ˆx2 ˆxˆx r = 1 ˆx 2ˆx r = 1 1 q (a r ) 2 +4ɛ (a r ) 2 q. r (a ) 2 +4ɛ (a r ) r (a ) 2 1/2 +4ɛ So, if r () is a maximizer of (1 r) R(ˆx(r, )), i saisfies ( ) 2 a r () (+) r + 4ɛ () = 1. 2 ( ) We can show he following for every in our se Γ 1 = a, 2 of possible values of : a2 + 4ɛ (1) There is a unique value of r, denoed r (), such ha for every Γ, r () saisfies he firs-order condiion and hence r () is he unique maximizer of (1 r) R(ˆx(r, )) on he inerval (0, 1). (2) r () < 0 (he Principal becomes more generous when echnology improves) We firs prove (2). To do so, we use a very general resul, which holds even wihou differeniabiliy. I will be proved below, in Par (a) of Theorem 2. I says ha if H > L, hen r ( H ), where r () is any H r ( L ) L maximizer of (1 r) R(ˆx(r, )) on (0, 1). So when rises, he righ side of (+) falls, i.e., r () < 0. Nex rewrie he firs-order condiion saisfied by he Principal s chosen share as (a r)2 + 4ɛ 0 = f(r, ) 1 r 2. 2 Then f(0, ) < 0, i.e., 2 > a + 4ɛ for all of he possible values of. Tha is he case since 2 = 2 a2 + 4ɛ a 2 + 4ɛ 2 and <. We also have f(1, ) < 0. Hence, by he Inermediae Value Theorem, for any ( a2 + 4ɛ ) 2 0,, we have f(r (), ) = 0 for some r () (0, 1). Moreover, since r () < 0, ha r () is a2 + 4ɛ he only soluion o f(r, ) = 0 in (0, 1). Tha esablishes (1). 14

21 Now consider he case where a = 1 and ɛ = 0.6. Tha mees our requiremen ɛ < 3 4 a2. Define our se of possible effors o be Σ = (1, ]. Then Γ = (1, 1.084) and C(x) > 0 for every x Σ. Figure 7 graphs he surplus gap, which falls when echnology improves ( drops). Figure 8 graphs he effor gap, which rises when echnology improves. We have opposie direcions raher han racking. FIGURES 7 AND 8 HERE 5.5 An example where marginal revenue rises bu marginal cos rises faser. I will be convenien o call his he Rising Marginals example. We have: The se of possible effors is Σ = IR +. R(x) = x a, C(x) = x b, 0 < a < b. The se of possible pairs (r, ) is Γ = {(r, ) : 0 < r < 1; > 0}. We obain he following: ˆx(r, ) = ( ) 1 b a b. ra ˆx r (r, ) = 1 ( ) 1/(a b) b (a b) 2 1/(a b) 1 r 1/(b a) 1. Tha is negaive. So when echnology a improves, effeciveness increases. In he endogenous case he Principal chooses he share r () = a. The Principal s generosiy b remains unchanged when echnology changes The firs-order condiion saisfied by r can be wrien r = 1 R(ˆx(r, )) R (ˆx(r, )) ˆx r (r, ). In he example we obain: R(x) R (x) = x a, and ˆx r = ( ) 1/(a b) b 1 a b a r1 1/(b a), ˆx ˆx r = r (b a). So he firs-order condiion is r = 1 r (b a)r a. Tha is solved by r = a b. 15

22 W(1,) W(r (),) Fig. 7: Surplus gap in he Cubic-cos example.

23 Fig. 8: Effor gap in he Cubic-cos example.

24 Jus as in he Price-aker example, Improvemen in echnology is good news for he sharechoosing Principal. Tha is he case because r = 0. Even hough we have an explici expression for r, compuing he derivaive of endogenous effor gap wih respec o and he derivaive of endogenous surplus gap (Penaly) wih respec o is cumbersome. I urns ou ha boh are negaive. So he endogenous surplus gap racks he endogenous effor gap. This is rue as well in he Classic and Price-aker examples (bu no in he Cubic-revenue example). The fac ha i is rue in he Classic and Price-aker examples does no follow from he signs of ˆx r and r in hose examples. In conras, we shall show 27 ha, in he endogenous case, he surplus gap racks he effor gap whenever (as in he Rising Marginals example) ˆx r < 0 and r Basic resuls ha do no require differeniabiliy. We now sae wo heorems which are proved wihou requiring differeniabiliy of R or C. Theorem 1 concerns he exogenous case and Theorem 2 concerns he endogenous case. Boh heorems hold for all examples in which R and C are sricly increasing. An example is defined by a se Σ of possible posiive effors, he funcions R and C, and a se Γ of possible pairs (r, ). Recall ha for every r (0, 1), Γ conains some pair (r, ). The se of values of such ha (r, ) Γ for some r is again denoed Γ. Recall ha he funcion ˆx has he propery ha for all (r, ) Γ, we have ˆx(r, ) Σ and r R(ˆx(r, )) C(ˆx(r, )) r R(x) C(x) for all x Σ. The exogenous-case Theorem 1 has eigh pars. Par (a) says ha he Agen never works less when he share r rises (while remaining less han one), and sricly prefers he higher share. Par (b) says ha he Agen never works less hard when drops (echnology improves) Par (c) says ha he surplus-maximizing effor canno fall when drops. Par (d) says ha maximal surplus mus rise when drops. Par (e) says ha a drop in is never bad news for he Principal and Par (f) says ha i mus be good news for he Agen. Par (g) says ha a drop in is never bad news from he welfare poin of view. Par (h) says ha a rises in he share r is never bad news from he welfare poin of view and is good news if and only if he Agen s effor changes afer he drop. Thus, in he exogenous case, i is in he social ineres for he Principal o be more generous. Theorem 1 Le R and C be sricly increasing on Σ. Then: (a) ˆx(r H, ) ˆx(r L, ) and R(ˆx(r H, )) C(ˆx(r H, )) > R(ˆx(r L, )) C(ˆx(r L, )) whenever (r L, ) Γ, (r H, ) Γ, and 0 < r L < r H < 1. (b) ˆx(r, L ) ˆx(r, H ) whenever (r, L ) Γ, (r, H ) Γ, and 0 < L < H. (c) ˆx(1, L ) ˆx(1, H ) whenever L, H Γ, and 0 < L < H. (d) W (1, L ) > W (1, H ) whenever L, H Γ and 0 < L < H. 27 In Par (b) of Theorem 5. 16

25 (e) (1 r) R(ˆx(r, L )) (1 r) R(ˆx(r, H )) whenever (r, L ) Γ, (r, H ) Γ, and 0 < L < H. (f) rr(ˆx(r, L )) L C(ˆx(r, L )) > rr(ˆx(r, H )) H C(ˆx(r, H )) whenever (r, L ) Γ, (r, H ) Γ, and 0 < L < H. (g) W (r, L ) > W (r, H ) whenever (r, L ) Γ, (r, H ) Γ, and 0 < L < H. (h) W (r H, ) W (r L, ) whenever (r H, ) Γ, (r L, ) Γ, and 0 < r L < r H < 1s. The inequaliy is sric if and only if ˆx(r H, ) ˆx(r L, ). The proof of Theorem 1, like all he subsequen proofs, is found in he Appendix. In proving Pars (e),(f),(g), (h) we use he simple observaion ha when drops or r rises, he Agen could coninue o use he same effor as before he change. In proving Pars (a),(b),(c),(d) we use a basic proposiion from monoone comparaive saics. 28 Theorem 2 concerns he endogenous case. Par (a) says ha he raio of he Principal s chosen share o he echnology parameer canno fall when drops. Bu, as we have already seen in he examples, he chosen share iself may rise or fall or say he same. Neverheless Par (b) says ha in he endogenous case he Agen never works less hard when drops. Par (c) says ha in he endogenous case a drop in is never bad news for he Principal. (Tha is he endogenous counerpar of Par (h) of heorem 1). Par (d) says ha in he endogenous case a drop in mus be good news from he welfare poin of view. Theorem 2 Le R and C be sricly increasing on Σ. Le r () denoe a maximizer of (1 r) R(ˆx(r, )) on he inerval (0, 1). Then (a) r ( L ) L r ( H ) H whenever L, H Γ and 0 < L < H. (b) ˆx(r ( L ), L ) ˆx(r ( H ), H ) whenever L, H Γ and 0 < L < H. (c) (1 r ( L )) R(ˆx(r ( L ), L ) (1 r ( H )) R(ˆx(r ( H ), H ) whenever L, H 0 < L < H. Γ and (d) W (r ( L ), L ) > W (r ( H ), H ) whenever L Γ, H Γ, and 0 < L < H. While Par (c) of Theorem 2 ells us ha in he endogenous case echnical improvemen can never be bad news for he Principal, he siuaion is differen for he Agen. Figure 9 is a graph 28 See, for example, Sundaram (1996). The proposiion concerns a funcion h : IR IR IR which displays sricly increasing differences. [For such a funcion we have h(u H, v H ) h(u L, v H ) > h(u H, v L ) h(u L, v L ) whenever u H > u L, v H > v L ]. The proposiion is as follows: If a funcion h(u, v) displays sricly increasing differences, and if u H maximizes h(u, v L ), hen u H u L if v H > v L. maximizes h(u, v H ) while u L 17

26 of he Agen s endogenous-case ne earnings r () R(ˆx(r ()), ) C(ˆx(r ()), ) in he Classic example. Once has dropped o a criical value ha is close o 0.5, a furher drop is Bad news for he Agen. Informally: in he endogenous case, he Principal is never he enemy of echnical progress bu he Agen migh be. FIGURE 9 HERE 7. Two exogenous-case heorems which require differeniabiliy. 7.1 Effeciveness and he effor gap move in he same direcion when changes. Theorem 3 Le Γ be an open se in IR 2+. Suppose ha he funcions R and C are hrice differeniable. Suppose ha he following monooniciy condiion is me: we eiher have ˆx r > 0 for all (r, ) Γ or ˆx r < 0 for all (r, ) Γ. Suppose, in addiion, ˆx is coninuous wih respec o r a all poins in (0, 1]. Then ˆx r (r, ) > 0 (< 0) a every (r, ) Γ if and only if d d [ˆx(1, ) ˆx(r, )] > (< 0) a every (r, ) Γ. The assumpions of he heorem are me in all he examples we have presened. Noe ha he pair (r (), ) belongs o Γ, so he heorem applies, in paricular, o ˆx r (r (), ) and he endogenous effor gap ˆx(1, ) ˆx(r (), ). The proof (in he Appendix) is very simple A heorem abou he effor gap and he surplus gap. We firs formally define exogenous racking in examples where R and C are wice differeniable on he effor se Σ and, for fixed r (0, 1], he Agen s effor choice ˆx(r, ) Σ solves he he firs-order condiion 0 = rr (x) C (x). Definiion Noe ha we could sae a more general definiion, no requiring differeniabiliy. There we would say ha he example has he exogenous racking propery if we have {ˆx(1, L ) ˆx(r, (+) L ) > ˆx(1, H ) ˆx(r, H ) ( ˆx(1, L ) ˆx(r, L ) < ˆx(1, H ) ˆx(r, H ) ) whenever (r, L ), (r, H ) Γ and 0 < L < H if and only if we also have { W (1, L ) W (r, (++) L ) > W (1, H ) W (r, H ) ( W (1, L ) W (r, L ) < W (1, H ) W (r, H ) ) whenever (r, L )(r, H ) Γ and 0 < L < H. For opposie direcions he appropriae inequaliies are reversed. Using his definiion, one could explore he racking quesion for finie examples. 18

27 Fig. 579 The Agen s ne earnings for he Classic example wih A = 2, B = 3

28 An example (R, C, Γ, Σ), wih R and C hrice differeniable, has he exogenous racking (opposie direcions) propery if d d [ˆx(1, ) ˆx(r, )] d d [W (1, ) W (r, )] > 0 (< 0) a all (r, ) Γ. The nex heorem concerns exogenous racking in Inerior examples. In an Inerior example he Agen s bes effor is he unique soluion o a firs-order equaion and he same is rue for he Principal s chosen share. Before providing he definiion, we recall ha for every (r, ) Γ we have 0 < r < 1. Recall also ha Γ denoes he se of possible values of. ( Γ = { : (r, ) Γ for some r}). Definiion 2 An example (Σ, R, C, Γ) is Inerior if Σ IR +, and Γ IR 2+, are open ses. R, C are hrice differeniable on Σ and R > 0, C > 0. There exiss a wice differeniable funcion ˆx : (0, 1] Γ Σ such ha for r (0, 1], ˆx(r, ) saisfies he firs-order condiion 0 = rr (x) C (x) and is he unique maximizer of rr(x) C(x) on Σ. For every Γ, here exiss a share r () (0, 1) which saisfies he firs-order condiion 0 = d [(1 r) R(ˆx(r, ))] and is he unique maximizer of (1 r) R(ˆx(r, )) on (0, 1). dr All he examples we have discussed saisfy hese condiions. 30 Theorem 4 An inerior example has he exogenous racking propery if he effor se is Σ = (0, J), where J > 0, and he monooniciy condiion of Theorem 2 holds (we eiher have ˆx r > 0 for all (r, ) Γ or ˆx r < 0 for all (r, ) Γ). Sraighforward calculaion yields he following Corollary, proved (ogeher wih heorem 4) in he Appendix. Corollary The following hold for an inerior example in which he monooniciy condiion of Theorem 2 is saisfied, he effor se is Σ = (0, J) (where J > 0), and ˆx r (r, ) > 0, ˆx (r, ) < 0 for all (r, ) Γ: (i) he Decenralizaion Penaly (surplus gap) is decreasing in (so he Penaly grows when echnology improves) if a every effor x (0, J) we have R (x) 0, R (x) = C (x) = Consider he condiion we imposed in Theorem 3. We require ha he funcion ˆx is coninuous wih respec o r a every r (0, 1]. The hird iem in our Inerior Example definiion insures ha his is indeed he case. 19

29 (ii) he Decenralizaion Penaly (surplus gap) is increasing in (so he Penaly shrinks when echnology improves) if a every effor x (0, J) we have R (x) < 0, C (x) = 0, R (x) 0. Even hough we are in he relaively sraighforward exogenous case, he Corollary s sufficien condiions for he Penaly o grow (shrink) when echnology improves are resricive bu simple. When we urn o he endogenous case, we find no similarly simple condiions on R and C which ell us, all by hemselves, he direcion in which he Penaly moves when echnology improves. 8. Endogenous-case resuls which require differeniabiliy. Classifying he endogenous-case resuls. We have seen ha in he endogenous case here are examples where he Decenralizaion Penaly (surplus gap) rises when echnology improves and here are examples where i falls. There are examples where we have endogenous racking (surplus and effor gaps move in he same direcion when changes), bu here oher examples where ha is no rue. There is no endogenous analog of Theorem 4 in which we again have racking under very general assumpions. Here is our racking definiion for he endogenous case. Definiion 3 An inerior example (R, C, Γ, Σ) has he endogenous racking (opposie direcions) propery if d d [ˆx(1, ) ˆx(r (), )] d d [W (1, ) W (r (), )] > 0 (< 0) a all Γ = { : (r, ) Γ for some r}. An insrucive way o bring order o he rich variey of endogenous resuls is o characerize he way ha (a) effeciveness ˆx r (r, ) and (b) he Principal s chosen share r move when changes. For a drop in, we consider four combinaions: (a) and (b) boh rise; hey boh fall; (a) rises and (b) falls; (a) falls and (b) rises. In paricular we find in Theorem 5 ha when one rises and he oher falls, hen we indeed have endogenous racking. In Theorem 6 we find ha if R < 0 hen a drop in canno lead o more effeciveness and greaer Principal s generosiy. Before proceeding o hese heorems we presen a four-box endogenous-case able which serves as a guide o hose heorems and heir relaion o he examples we have considered. 20

30 THE EFFECT OF IMPROVED TECHNOLOGY (A DROP IN ) IN FOUR GROUPS OF INTERIOR EXAMPLES WHEN DROPS, EFFECTIVE- NESS OF A SHARE INCREASE WHEN DROPS, EFFEC- TIVENESS OF A SHARE IN- FALLS. HENCE SO DOES THE EX- CREASE RISES. HENCE SO OGENOUS EFFORT GAP (SEE THEOREM 3). ˆx r > 0 and hence [ˆx(1, ) ˆx(r, )] > 0 d d DOES THE EXOGENOUS EFFORT GAP (SEE THEO- REM 3). ˆx r < 0 and hence [ˆx(1, ) ˆx(r, )] < 0 d d WHEN DROPS, PRINCI- 1 SEE CLASSIC AND 2 SEE RISING PAL BECOMES LESS GEN- CUBIC-REVENUE EXAMPLES. WE MARGINALS EXAMPLE. EV- EROUS OR GENEROSITY STAYS THE SAME. r 0 HAVE TRACKING IN THE CLASSIC EXAM- PLE BUT IN THE CUBIC-REVENUE EXAMPLE WE HAVE OPPOSITE DIRECTIONS (IF THE SET OF POSSIBLE VALUES OF IS PROP- ERLY CHOSEN). ERY EXAMPLE THAT LIES IN THIS BOX HAS THE TRACKING PROP- ERTY. (See Theorem 5, Par (a)). WHEN DROPS, PRINCI- 3 SEE EXPLODING 4 SEE THE PAL BECOMES MORE GEN- MARGINALS EXAMPLE. EVERY EX- PRICE-TAKER EXAMPLE, EROUS. r < 0 AMPLE THAT LIES IN THIS BOX HAS THE TRACKING PROPERTY. (See Theorem 5, Par (b)). AN EXAMPLE WITH R < 0 CANNOT BE IN THIS BOX. (See Theorem 6). WHERE WE HAVE TRACKING AND THE CUBIC-COST EXAMPLE, WHERE WE HAVE OPPOSITE DIRECTIONS. The example in Box 3, which we call he Exploding Marginals example, has funcions R and C such ha C > R and boh C and R are very large. I appears difficul o consruc Box-3 examples where ha is no he case. 31 The Exploding Marginals example is as follows: Σ = (0, 1). Γ = {(r, ) : 0 < r < 1; r (e, ee )} (e is he base of he naural logarihms). R(x) = e x2. C(x) = x [ 0 2e e p e p p ] dp. The proof ha he Exploding Marginals example indeed lies in Box 3 is provided in he Appendix. We now have Theorem 5, a wo-par heorem, which concerns Box 2 and Box One can prove, for example, ha if R = 1 2 x2, so ha R = 1, hen we canno be in Box 3. 21

31 Theorem 5 Consider an inerior example (Σ, Γ, R, C). (a). Suppose he following holds: for every Γ we have r () 0 and for every (r, ) Γ we have ˆx r (r, ) < 0. Then we have endogenous racking. (b). Suppose he following holds: for every Γ we have r () < 0 and for every (r, ) Γ we have ˆx r (r, ) > 0. Then we have endogenous racking. The nex heorem does no direcly concern he wo gaps. Bu i implies ha if marginal revenue is decreasing or consan (R 0) in an inerior example and he Principal has a unique bes share, hen he example canno be in Box 3. Theorem 6 Suppose ha in he inerior example (Σ, Γ, R, C) we have: R (x) 0 a every x Σ. ˆx r (r, ) 0, ˆx (r, ) < 0 and ˆx r (r, ) > 0 a every (r, ) Γ. r () is he unique maximizer of (1 r) R(ˆx(r, )) on (0, 1), Then r () 0 for all Γ. I is difficul o give a clear inuiion for Theorems 4 and 5. Tha is a lile easier for Theorem 6, which says ha if marginal revenue is decreasing, and effeciveness drops when echnology improves, hen when echnology improves, he Principal does no become more generous (r () 0), i.e., we canno be in Box 3. Inuiively one migh say: when drops, increasing he share above is previous level would damage he Principal, because he exra revenue due o exra effor has dropped (marginal revenue has declined) and a he same ime he exra effor evoked by a share increase has dropped as well. 8.2 A summary. Reurn o our original puzzle: when does echnical improvemen lower he Penaly and when does i raise he Penaly? Here is a summary of wha Theorems 5 and 6 have old us abou he puzzle in he endogenous case. 22

32 Consider any inerior example. If, in ha example, echnical improvemen makes he Principal less generous or keeps his generosiy unchanged, while a he same ime i raises he effeciveness of a share increase (r () 0 and ˆx r (r, ) < 0), hen he improvemen raises he Penaly or keeps i unchanged. If echnical improvemen makes he Principal more generous, while a he same ime i decreases he effeciveness of a share increase or leaves i unchanged (r () < 0 and ˆx r (r, ) 0) which canno happen if marginal revenue is nonincreasing hen he improvemen lowers he Penaly or keeps i unchanged. Unforunaely here are no simple condiions on R and C, similar o hose in he Corollary o he exogenous-case Theorem 4, which imply, all by hemselves, ha he Penaly rises (falls) when echnology improves. 9. Finding he Principal s bes share for a given. The funcion r () may be increasing on he se Γ of possible values of. I may also be decreasing or consan. We have discussed he implicaions of each case. Bu we have no ye sudied, in a general way, he shape of he Principal s gain as a funcion of r (0, 1) when is fixed. The gain for fixed is (1 r) R(ˆx(r, )). The graph of he non-negaive values of (1 r) R(ˆx(r, )), wih r on he horizonal axis, sars a zero and ends a zero. The graph coincides wih he Principal s gain curve excep a r = 0 and r = 1, since he Principal confines aenion o he open inerval (0, 1). I would be paricularly helpful if he gain curve rises and hen falls, achieving is maximum heigh a r (). More generally, he curve could rise unil r = r () and could hen be fla for an inerval before descending. Le us call such a gain curve single-peaked. As long as he gain is posiive a some r (0, 1) he curve is single-peaked if i is concave on (0, 1). The following heorem provides condiions under which he gain is indeed concave. The heorem has wo pars. The firs par does no require differeniabiliy wih respec o r, bu he second par does. Informally, he second par says ha we have concaviy if marginal revenue drops (R < 0) and in addiion he effeciveness of a share increase drops when he share increases (ˆx rr < 0). Theorem 7 (a) If, for a fixed, R(ˆx(r, )) is concave on (0, 1), hen he Principal s gain (1 r) R(ˆx(r, ) is also concave on (0, 1). (b) Consider an inerior example (Σ, Γ, R, C) where Σ = (0, J), wih J > 0. Then R is concave on (0, J) if for all x (0, J) we have R (x) < 0, and for all (r, ) Γ we have ˆx rr (r, ) < 0. If R (x) < 0, hen a sufficien condiion for ˆx rr < 0 is r R (x) C (x) 0. Suppose ha he Principal s gain curve is indeed single-peaked, and suppose ha he share he Principal uses is deermined by bargaining beween he Principal and he Agen. In he 23