Concavity-Preserving Integration and Its Application in Principal-Agent Problems

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1 ANNALS OF ECONOMICS AND FINANCE 19-2, (2018) Cocavity-Preservig Itegratio ad Its Applicatio i Pricipal-Aget Problems Jia Xie * This paper fids the ecessary ad sufficiet coditio for a itegratio to be cocavity preservig. Usig this coditio, we ca, for the first time i the literature of the pricipal-aget problems, justify the first-order approach without requirig the cotract to be mootoic. Key Words: Cocavity-preservig itegratio; First-order approach; Pricipalaget problems. JEL Classificatio Number: D8. 1. INTRODUCTION Assume a radom variable x, whose probability desity fuctio is deoted by f(x a) idexed by a parameter a. We fid the ecessary ad sufficiet coditio for ϕ T (a) ϕ(x)f(x a)dx to be cocavity preservig, that is, ϕ T (a) is cocave i a for ay cocave fuctio ϕ(x). Jewitt (1988) ad Colo (2009) have foud sufficiet coditios for ϕ T (a) to be moo-toicity ad cocavity preservig, i.e., ϕ T (a) is odecreasig ad cocave i a for ay odecreasig ad cocave fuctio ϕ(x). Colo (2009) has also foud sufficiet coditios for ϕ T (a) to be mootoicity preservig, i.e., ϕ T (a) is odecreasig i a for ay odecreasig fuctio ϕ(x). However, there has bee o study o cocavity preservatio of ϕ T (a) as we are aware of, ad this paper teds to fill i this blak. The cocavity-preservig itegratio has importat implicatios i the pricipal-aget problem. Particularly, we are able to justify the first-order approach, as log as the optimal cotract is cocave. This is the first time * Departmet of Fiace, Mihaylo College of Busiess ad Ecoomics, Califoria State Uiversity Fullerto Nutwood Ave. Fullerto, CA 9283, USA. jxie@fullerto.edu /2018 All rights of reproductio i ay form reserved.

2 586 JIA XIE i relevat literature that the first-order approach ca be justified without requirig the cotract to be mootoic CONCAVITY-PRESERVING INTEGRATION The coditio is best represeted i the state-space formulatio. Assume that x is geerated by a parameter a ad aother radom variable s with probability desity fuctio f(s), i.e., x x( s, a). The ϕ T (a) = ϕ(x(s, a))f(s)ds. Propositio 1. ϕ T (a) is cocavity preservig if ad oly if the followig two coditios hold for all a ad all costat α: x aa (s, a)f(s)ds = 0 (1) x aa (s, a)f(s)ds 0, (2) S(α,a) where S(α, a) {s x(s, a) α}. I particular, ϕ T (a) is cocavity preservig if x(s, a) is liear i a for almost all s. Proof. We prove the sufficiecy of (1) ad (2) first. ϕ T (a) is cocave if ad oly if 2 ϕ T (a) a 0, a. We have: 2 2 ϕ T (a) a 2 = 2 ϕ(x(s, a))f(s)ds a 2 = [ϕ xx (x(s, a))x 2 a(s, a) + ϕ x (x(s, a))x aa (s, a)]f(s)ds = ϕ xx (x(s, a))x 2 a(s, a)f(s)ds + ϕ x (x(s, a))x aa (s, a)f(s)ds. (3) The first itegratio i (3) is always o-positive, as ϕ xx (x) 0 due to ϕ(x) beig cocave. The it suffices to prove that the secod itegratio is o-positive as well, that is, ϕ x (x(s, a))x aa (s, a)ds 0 if (1) ad (2) hold. First, defie I(x γ) = 1 if x γ, ad zero otherwise. ϕ x (x) is o-icreasig due to ϕ(x) beig cocave. Therefore we ca defie its iverse fuctio as follows: ϕ 1 x (β) sup{x ϕ x (x) β}. Defie: ϕ β, x (x) = β + i=1 ( ( 1 I x ϕ 1 x β + i )). (4) 1 For the literature of the first-order approach i pricipal-aget problems, see Rogerso, 1985; Jewitt, 1988; Siclaire-Desgage, 1994; Mirrlees, 1999; Colo, 2009 a,b; Jug ad Kim, 2015; ad Xie, 2017.

3 CONCAVITY-PRESERVING INTEGRATION AND ITS APPLICATION 587 The ϕ β, x (x) approximates ϕ β x(x) max(β, ϕ x (x)) uiformly from above. Figure 1 shows ϕ β, x (x) ad ϕ x (x), as fuctios of x. Figure Figure 1: 1: ϕ β, x β, (x) x (x) ad ad ϕ x (x) (x) as as fuctios fuctios of of x The we have: The we have: ϕ β,x (x(s, a))x aa (s, a)f(s)ds x [ ( ( = β ϕ β, + I x ϕ 1 x β + x (x(s, a))x aa (s, a)f(s)ds)) ] i x aa (s, a)f(s)ds [ i=1 ( 1 = β + I x ϕ 1 x (β + i ) ] = β x aa (s, a)f(s)ds + 1 ( ( I ) x x aa ϕ 1 x(s, a)f(s)ds β + i )) x aa (s, a)f(s)ds i=1 =β x aa (s, a)f(s)ds + 1 ( I x ϕ 1 x (β + i ) = 1 ( ( I x ϕ 1 x β + i )) x aa (s, a)f(s)ds ) x aa (s, a)f(s)ds = ( I x ϕ 1 x (β + i ) = 1 x aa (s, a)f(s)ds ) x aa (s, a)f(s)ds 0, (5) S(ϕ 1 x (β+ i = 1 ),a) where the first equality follows S(ϕ 1 x (β+ i ), a) x aa(s, a)f(s)ds from substitutig 0, (4) i, the third equality (5) follows from (1), ad the iequality at the ed follows from (2). The we have: where the first equality follows from substitutig (4) i, the third equality follows from (1), ϕ β x(x(s, a))x aa (s, a)f(s)ds = lim ϕ β, x (x(s, a))x aa (s, a)f(s)ds 0, ad the iequality at the ed follows from (2). The we have: (6) where the iequality follows from (5). The accordig to the mootoe covergece theorem, we have: ϕ β x(x(s, a))x aa (s, a)f(s)ds = lim ϕ β, x (x(s, a))x aa (s, a)f(s)ds 0, (6) ϕ x (x(s, a))x aa (s, a)f(s)ds = ϕ β x(x(s, a))x aa (s, a)f(s)ds 0, lim β where the iequality follows from (5). The accordig to the mootoe covergece theorem, where the iequality follows from (6). This completes the proof of sufficiecy. 4

4 588 JIA XIE To proof the ecessity of (1) ad (2), we eed to costruct cocave fuctios ϕ(x) such that, if (1) or (2) does ot hold, the ϕ T (a) is ot cocave i a. First, assume (1) does ot hold, i.e., there is a costat λ 0 such that x aa (s, a)f(s)ds = λ. Let ϕ(x) = λx. The ϕ(x) is cocave i x, with ϕ x (x) = λ ad ϕ xx (x) = 0. The by substitutig the last two equalities ito (3), we have 2 ϕ T (a)/a 2 = λ x aa (s, a)f(s)ds = λ 2 > 0. Therefore, ϕ T (a) is ot cocave i a. Secod, assume (2) does ot hold, i.e, there is a costat α such that S(α,a) x aa(s, a)f(s)ds = µ > 0. Let S c (α, a) be the complemet set of S(α, a), i.e., S c (α, a) {s x(s, a) > α}. The accordig to (1), S c (α,a) x aa(s, a)f(s)ds = S(α,a) x aa(s, a)f(s)ds = µ < 0. The we costruct ϕ(x) as follows: { µx if x α, ϕ(x) = µx if x > α. The ϕ(x) is cocave i x, with ϕ xx (x) = 0 ad ϕ x (x) = { µ if x α, µ if x > α. (7) Equatio (7) is equivalet to ϕ x (x(s, a)) = { µ if s S(α, a), µ if s S c (α, a). (8) The accordig to (3), we have 2 ϕ T (a) a 2 = ϕ x (x(s, a))x aa (s, a)f(s)ds = ϕ x (x(s, a))x aa (s, a)f(s)ds + = µ S(α,a) S(α,a) S c (α,a) x aa (s, a)f(s)ds µ x aa (s, a)f(s)ds. S c (α,a) ϕ x (x(s, a))x aa (s, a)f(s)ds = µ 2 + µ 2 = 2µ 2 > 0, (9) where the secod equality follows from the fact that S(α, a) ad S c (α, a) are complemet sets, the third equality follows from (8), the fourth equality follows from the assumptio that S(α,a) x aa(s, a)f(s)ds = µ ad that S c (α,a) x aa(s, a)f(s)ds = µ. (9) implies ϕ T (a) is ot cocave i a. We have thus prove that (2) is ecessary for ϕ T to be cocavity preservig. Fially, if x(s, a) is liear i a, the x aa (s, a) = 0 ad thus both (1) ad (2) hold.

5 CONCAVITY-PRESERVING INTEGRATION AND ITS APPLICATION THE FIRST-ORDER APPROACH IN PRINCIPAL-AGENT MODEL The pricipal-aget model ca be described as follows. There is a pricipal ad a aget. The aget exerts a effort a R + that stochastically geerates a sigal, deoted by a radom variable x. Let f(x a) deote the desity fuctio of x. The outcome is determied by the fuctio π(x), ad the paymet to the aget is specified by s(x). The pricipal is risk eutral, with her welfare beig π(x) s(x), while the aget is risk averse, with a utility fuctio u(s(x)). I additio, the aget has a icreasig ad covex cost fuctio c(a). The pricipal s problem is to choose a target actio a ad a cotract s ( ) that solve the follow program: max [π(x) s (x)]f(x a )dx (10) a,s ( ) s.t. u(s (x))f(x a )dx c(a ) 0, ad (11) a = argmax a u(s (x))f(x a)dx c(a). (12) Costrait (11) is the participatio costrait, which states that the aget s expected utility must be o less tha his outside reservatio utility, which is ormalized to zero. Costrait (12) is the icetive compatibility costrait, which states that, give the paymet schedule s, a maximizes the aget s expected utility. A techical challege is the ifiite umber of costraits imposed by (12). A commo solutio is to use the first-order approach, which replaces (12) with the first-order ecessary coditio that u(s (x))f a (x a )dx c a (a ) = 0, (13) where the subscript deotes a partial derivative i a. The first-order approach is valid if u(s (x))f(x a)dx is cocave i a. (14) Propositio 2. If s (x) is cocave ad Coditios (1) ad (2) are satisfied, the (14) holds ad the first-order approach is valid. Proof. Sice the utility fuctio u( ) is icreasig ad cocave, if s (x) is cocave i x, so is u(s (x)). The accordig to Propositio 1, u(s (x))f(x a)dx is cocave i a. A importat implicatio of Propositio

6 590 JIA XIE 2 is that, for the first time i the pricipal-aget model literature, the cotract does ot have to be odecreasig for the first-order approach to be valid. REFERENCES Colo, J. R., Two New Coditios Supportig the First-Order Approach to Multisigal Pricipal-Aget Problems. Ecoometrica 77, Colo, J. R., Supplemet to Two New Coditios Supportig the First-Order Approach to Multisigal Pricipal-Aget Problems. Ecoometrica Supplemetary Material, 77, proofs.pdf. Jewitt, I., Justifyig the first-order Approach to Pricipal-Aget Problems. Ecoometrica 56, Jug, J.Y. ad S. K. Kim, Iformatio space coditios for the first-order approach i agecy problems. Joural of Ecoomic Theory 160, Mirrlees, J. A., The Theory of Moral Hazard ad Uobservable Behavior: Part I. Review of Ecoomic Studies 66, Rogerso, W. P., The First-Order Approach to Pricipal-Aget Problems. Ecoometrica 53, Siclair-Desgagé, B., The First-Order Approach to Multi-Sigal Pricipal- Aget Problems. Ecoometrica 62, Xie, Jia, Iformatio, Risk Sharig ad Icetives i Agecy Problems. Iteratioal Ecoomic Review 2017,