Morl Hzrd nd Endogenous Monitoring Ofer Setty * Tel Aviv University This Drft: Februry 2014 Abstrct How do principl s incentives to monitor depend on n gent s outside option? The principl cn incentivize the risk-verse gent by both incresing monitoring precision nd spreding out the consumption of the risk-verse gent. I show tht the principl s optiml choice of the signl s precision wekly increses decreses with the gent s outside option if the derivtive of the inverse of utility is convex concve. JEL Clssifiction: D81; D82; J33. Keywords: Principl-gent model; Morl hzrd; Monitoring; Costly stte verifiction. * I thnk Yir Antler, Eddie Dekel, Zvik Neemn, Ady Pusner, Nicol Pvoni, Dotn Persitz, Michel Richter, Dn Simundz nd Dvid Weiss for very helpful comments. Correspondence: Ofer Setty, Deprtment of Economics, Tel Aviv University. E-mil: ofer.setty@gmil.com.
1 Introduction In the cnonicl principl-gent model, risk-neutrl principl provides to risk-verse gent trnsfer tht depends on noisy signl of the gent s ction. Becuse of the gent s risk version, the spred in possible trnsfers for given ction implies cost to the principl. I contribute to this literture by llowing the principl to choose the precision of the noisy signl of the gent s ction. More precise signls llow for reduction of the risk ssocited with the trnsfers, nd hence reduce the verge trnsfer to the gent. Since the precision of the signl is costly, this frmework posits trde-off between the cost of monitoring nd the cost of imposing risk on the gent. How does the principl s monitoring choice depend on the gent s outside option? The signl s cost is ssumed to be independent of the outside option. The signl s benefit, however, depends on the outside option in nontrivil wy. I present condition under which the cost of spreding out utilities is incresing in the outside option. When this is the cse, the signl s vlue increses with the outside option, leding to the choice of more precise monitoring. The min finding of this pper is tht in the optiml contrct, the signl s precision increses decreses nd the dispersion of utilities fced by the gent decreses increses with the gent s outside option if the derivtive of the inverse utility is convex concve. The model presented cn be pplied in principle-gent setting with gents tht re heterogenous in their outside option. Consider, for exmple, the problem of optiml unemployment insurnce. In this problem the government is risk-neutrl principl tht insures risk-verse worker ginst unemployment shocks. The desired outcome for the government is employment, but this requires the worker to exert costly job-serch effort. Some sttes in the United Sttes provide higher unemployment insurnce benefits, thereby implying higher outside option. How should the government trde off monitoring unemployed workers to ensure tht they exert effort in their job serch, versus punishing unemployed workers who fil to find employment? Does this trdeoff depend on the generosity of unemployment insurnce nd hence on the gent s outside option? This pper s nlysis cn be pplied to this question, mong others. A few ppers study welth effects in principl-gent model. Newmn 2007 looks t how occuptionl choice depends on welth given two occuptions tht differ in the mount of risk borne. 1 In his model, workers who differ in their initil welth level choose between entrepreneurship tht entils risky pyoff nd being worker with risk-free pyoff. Using condition on 1 An erly drft by Newmn tht studies tht problem dtes 1995. 2
the inverse of mrginl utility, Newmn concludes tht there is threshold welth level such tht workers with t most tht welth level choose the riskier occuption nd vice vers. Thiele nd Wmbch 1999 generlize Newmn s result by llowing for ny finite number of effort levels insted of two. I show tht the condition I use, Newmn s condition nd Thiele nd Wmbch s condition re equivlent. To the best of my knowledge, endogenous investment in monitoring with generl utility hs not been studied in principl-gent model. The closest study is n extension in Newmn s pper bove in which he dds monitoring. He shows tht the ssignment of workers to monitoring technologies follow their outside option. His result, however, is restricted to log utility nd he leves the extension to generl utilities for future reserch. 2 The model A risk-neutrl principl contrcts with risk-verse gent. The gent s ction {0, }, > 0 is her privte informtion. This ction determines output o {H, L} owned by the principl s follows: p H = π, p H 0 = 0. Denote the vlue for the principl from high output by ω, nd normlize the vlue of low output to 0. The gent s utility is u c, where u is strictly incresing, strictly concve, three times differentible nd its inverse is differentible; c is consumption; nd is normlized such tht it is the gent s utility cost for exerting effort. The principl cn cquire binry signl s {G, B} on the gent s ction, representing good nd bd outcomes, respectively. The good signl outcome cn hppen only if the gent s ction is =, i.e., p G 0 = 0. This mens tht the ccurcy of the signl is determined by p G. If p G = 0, the signl crries no informtion; if p G = 1, the signl perfectly revels the gent s ction. Denote p G by θ nd let θ be choice vrible of the principl. The signl s cost is strictly incresing convex function κ θ. The signl nd output probbilities, conditionl on effort, re independent. In generl, the contrct should include precision choice for ny level of output. However, since cquiring n informtive signl is costly, the principl will lwys set θ = 0 for n gent with outcome H. This is becuse p H 0 = 0 implies tht outcome H revels the gent s ction. This simplifies the contrct s there re only three possible outcomes in equilibrium: {H, G, B}. The contrct specifies recommendtion on ction, the precision choice of the monitoring technology when low output is relized, nd trnsfer to the gent for ny outcome. The ction recommendtion must be incentive comptible. In ddition, the contrct requires tht, on verge, the gent 3
will receive t lest U. 3 The optiml contrct Denote by c x the principl s choice of consumption conditionl on outcome x for x {H, G, B}. Let Ĉ U be the cost for principl who recommends the ction for n gent with outside option U. In wht follows I ssume tht the prmeters justify creting the costly incentives for the gent to choose ction, e.g., ω is high enough. Otherwise, the problem becomes trivil with recommendtion of = 0 nd full insurnce. The principl s problem is s follows: ĈU = s.t. { min πc H + 1 π θc G + 1 π 1 θ c B + 1 π κ θ + πω } c H,c G,c B,θ πu c H + 1 π θu c G + 1 π 1 θ u c B U πu c H + 1 π θu c G + 1 π 1 θ u c B u c B 1 The first constrint is the individul-rtionlity IR constrint. The second constrint is the incentive-comptibility IC constrint. The left-hnd side of the two constrints is the expected utility for the gent conditionl on =. The right-hnd side of the IC constrint is the utility for the gent conditionl on = 0. Notice tht since the IC constrint holds, the objective function ssumes the probbilities given ction. The following clim determines the rnking of the trnsfers. Clim 1 In the optiml solution uc H = uc G > uc B = U. All proofs re relegted to the ppendix. This clim is bsed on severl properties of the problem: both the IR nd the IC constrints re tight, nd c B must be lower thn {c H, c G } to stisfy the IC constrint. It is strightforwrd to see tht the IC nd IR re tight see the ppendix for detils. In the IC, substitute uc B with U nd derive u c H = U +. Using the vlues for π+1 πθ {ch, c G, c B } in the objective function nd omitting the term πw, which is independent of the choice vrible, leds to the following convex optimiztion problem, whose solution for θ is identicl to tht of Problem 1: CU = min {u 1 U + } + 1 u 1 U + 1 π κ θ θ 4 2
where = π + 1 π θ. To understnd the role of monitoring precision in this problem, consider the solution to the first best. In the first best, the principl observes the gent s effort, so no monitoring is required. The first best lloction is then fixed consumption level independent of output tht is equl to u 1 U +. In this cse the principl compenstes the gent only for her effort. The principl s cost in 2 differs from the principl s cost in the first best in two spects. First, in the constrined problem, monitoring my be used upon low output with cost of 1 π κ θ. Second, in the constrined problem, the principl is required to crete spred in trnsfers conditionl on outcomes. Therefore, the principl delivers to the gent utility s lottery between u 1 U + with probbility nd u 1 U with probbility 1. I refer to the difference between the two utilities U +, U s the spred. The verge utility delivered through the lottery is U +. This is, by construction, equl to the utility delivered in the first best. Therefore, the only role of the signl in this problem is to reduce the risk ssocited with the spred nd thus reduce the cost of delivering utility s lottery rther thn s certinty equivlent. Indeed, if the signl ws without cost, the principl would set θ = 1, nd both the lloction nd the principl s cost would be identicl to those of the first best. To see this, substitute θ = 1 in Problem 2 nd get the first best cost. The following Theorem chrcterizes the contrct w.r.t. the outside option U under condition on u 1. Theorem 1 The solution to Problem 2 hs the following chrcteristics if u 1 is convex: i the optiml signl s precision θ increses with the outside option U; ii the utility spred decreses with the outside option U; π+1 πθ iii the cost of spreding out utility increses with the outside option U. The converse version of the theorem holds when u 1 is concve. Wht is the intuition behind this Theorem? For ny level of outside option utility, the principl weighs the cost of the signl ginst its benefit of reducing the risk ssocited with the spred. The signl s cost does not depend on the outside option. When the conditions bove hold, the cost of spreding out utilities increses with the outside option. In this cse, the vlue of monitoring increses nd the principl increses her investment in the signl. This, in turn, results in smller spred between utilities. 5
4 An equivlence result The condition tht u 1 is convex is relted to other conditions tht cn be found in the literture. I mke the following observtion: Proposition 1 The following conditions re equivlent: 1: u 1 is convex 2: u cu c u c 2 3 3: There is convex function h : R R + such tht 1 u c 4: 1 u u 1 u u = h u c. Condition 1 is the one used in this pper. Condition 2 is used in Thiele nd Wmbch 1999. Condition 3 is used in Newmn 2007. Condition 4 implies tht u is more risk verse thn 1 u the sense of Prtt 1964. Exmples of utility functions tht stisfy those conditions re IARA, CARA, nd CRRA with coefficient of reltive risk version of t lest 1 2. As explined in the introduction, Thiele nd Wmbch generlize Newmn s result by llowing ny finite number of effort levels insted of two. They lso interpret Newmn s condition s implying tht u cu c u c 2 2, mking their condition weker thn his. However, s Proposition 1 shows, Newmn s condition is equivlent to theirs. 2 in References NEWMAN, A. F. 2007: Risk-bering nd Entrepreneurship, Journl of Economic Theory, 1371, 11 26. PRATT, J. W. 1964: Risk Aversion in the Smll nd in the Lrge, Econometric, 32, 122 136. THIELE, H., AND A. WAMBACH 1999: Welth Effects in the Principl Agent Model, Journl of Economic Theory, 892, 247 260. 2 Thiele nd Wmbch describe Newmn s condition s requir[ing] tht the inverse of mrginl utility is convex in income., which indeed implies tht u u 2. However, requiring the inverse of mrginl utility to be convex u 2 1 in income is stronger condition thn wht Newmn requires. In fct u my even be concve s long s it is more convex thn ux, s is the cse of CRRA with coefficient of risk version in [ 1 2, 1]. Insted, Newmn s condition is tht 1 u is convex in utility rther thn in income. This condition is equivlent, s Proposition 1 shows, to u u 3. u 2 6
APPENDIX Proof of clim 1 Lemm 1 In the optiml solution either c H > c B or c G > c B, or both. Proof. Rewrite the IC s: πu c H + 1 π θu c G [π + 1 π θ] u c B +. 3 Since > 0 nd since the sum of the coefficients of { u c H, u c G} is equl to the coefficient of u c B nd positive, if both c H c B nd c G c B then the IC cnnot hold. Lemm 2 In the optiml solution the IR holds with equlity. Proof. The solution with slck IR cn be improved by decresing c B by ε. For smll ε, the IR is still slck. The IC remins slck or becomes slck see 3. The objective function increses by ε, which is contrdiction to the solution being optiml. Lemm 3 In the optiml solution c H = c G. Proof. Assume tht c H > c G. The optiml solution cn be improved s follows. Decrese c H by ε nd increse c G πε by. By construction this chnge does not ffect the objective function 1 πθ becuse π c H ε + 1 π θ c G + = πc H + 1 π θc G. To study the effect on the π 1 πθ IR, consider the prt of the IR composed of πu c H + 1 π θu c G. The chnge mkes the IR slck becuse it is lottery with the sme certinty equivlent but with less risk. Formlly, the clim is tht: πu c H ε + 1 π θu 1 π θ u c G + επ 1 π θ c G + Divide both sides by ε nd rerrnge to get: u c H u c H ε ε επ > πu c H + 1 π θu c G 1 π θ u c G > π u c H u c H ε 4 < u c G + επ u c G 1 πθ επ 1 πθ In the limit this is u c H < u c G, which is true by the negtion ssumption tht c H > c G. The IC is unffected see 3. Thus the IR becomes slck, which is contrdiction to Lemm 2. Therefore it is impossible for c H to be strictly greter thn c G in the optiml solution. The sme line of proof elimintes the possibility of c G > c H by showing tht incresing c H by ε nd decresing c G by πε 1 πθ violtes Lemm 2. 5 7
Lemm 4 In the optiml solution the IC holds with equlity. Proof. By Lemmt 1 nd 3 c H > c B. If the IC is slck then the objective function cn be improved. Decrese c H by ε nd increse c B επ by. By construction this chnge does 1 π1 θ not ffect the objective function. The IC still holds. Consider the prt of the IR composed of πu c H + 1 π 1 θ u c B. Those chnges mke the IR slck becuse it is lottery with the sme certinty equivlent but with less risk. Formlly, the clim is tht: πu c H ε + 1 π 1 θ u c B επ + > πu c H + 1 π 1 θ u c B 1 π 1 θ π u c H u c H ε < 1 π 1 θ u c B επ + u c B. 6 1 π 1 θ Divide both sides by ε nd rerrnge to get: u c H u c H ε u c B + < ε επ 1 π1 θ επ 1 π1 θ u c B. 7 In the limit this is u c H < u c B, which is true becuse c B < c H. Now decrese c H by δ in order to improve the objective function without dmging ny of the constrints. Lemm 5 In the optiml solution u c B = U. Proof. Since both the IR nd the IC re tight, nd since the LHS of both constrints is identicl, the RHS of both constrints is equl nd u c B = U. Clim 1 is then combintion of Lemmt 1, 3, nd 5. Proof of Proposition 1 Proposition 1 The following conditions re equivlent: 1: u 1 is convex 2: u cu c u c 2 3 3: There is convex function h : R R + such tht 1 u c 4: 1 u u 1 u u = h u c. Proof. The proof goes s follows: Condition 1 Condition 3 Condition 4 Condition 2 Condition 1. Condition 1 Condition 3 Let h be u 1. Then hux = u 1 1 ux = = 1. By Condition 1 uu 1 ux u x u 1 1 is convex, nd therefore there exists convex function h such tht: = h u x. u x 8
Condition 3 Condition 4 Denote: f 1, g u. By Condition 3 h such tht f = h g. Then: u x f = h g g f = h g 2 + g h = h g f h + f g g f f = h g h + g g 8 u is incresing g > 0, h is incresing becuse h = f g h is convex h > 0, h g 0 h f f g. g = u u > 0 h > 0, 1 u Condition 4 Condition 2 Using tht u 1 = u 2 u nd tht u 1 1 u 2 s u 2u. By Condition 4 1 1 u u u u so we get tht: 1 u u u 2 u 1 u 2 u u u u u 2 u 1 u 2 u u 2 u u 3 u 2 = 2 u 3 u 2 u 2 u rewrite u c u c u c 2 3 9 Condition 2 Condition 1 Using the implicit function theorem nd differentiting both sides of c = u 1 uc gives the equlity u 1 u c = 1. Differentite 1 twice with respect to utility gives: u u du 1 du d 2 u 1 du 2 = = u 2 u dc du = u 3 u d u 3 u = 3 u 4 u 2 dc du du u 3 u dc du = 3 u 5 u 2 u 4 u 10 9
To prove tht Condition 2 Condition 1, show tht not Condition 1 not Condition 2. Not condition 1 c s.t. u 1 u c = 1 is concve: By 10 nd using tht u c is positive: u 3u c 2 u c u c < 0, u c u c > 3u c 2 u c u c u c 2 > 3 Proof of Theorem 1 Theorem 1 The solution to Problem 2 hs the following chrcteristics if u 1 is convex: i the optiml signl s precision θ increses with the outside option U; ii the utility spred decreses with the outside option U; π+1 πθ iii the cost of spreding out utility increses with the outside option U. The three prts of the theorem re proved sequentilly. Proof of i The proof is bsed on monotone comprtive sttics notice tht C is twice differentible. C u U = 1 U + u 1 U + u 1 U 11 2 C U θ = 1 π u 1 U + u 1 1 π U u 1 U + According to the monotone comprtive sttics theorem, θ wekly increses with U if 2 C U θ 0.3 This is stisfied if u 1 is convex. To see this, rewrite 2 C s: U θ { 1 π u 1 U + u 1 U u 1 U + }, 12 nd notice tht u 1 U+ u 1 U is the verge slope of u 1 between { U, U + }, where > 0, nd u 1 U + is the slope of u 1 t U +. If u 1 is convex then the slope of u 1 t { U + is higher thn the verge slope t U, U + } 12 is negtive 2 C 0. U θ Proof of ii 3 In the cse of mximiztion problem supermodulrity is required between {U, θ}. Here the sign of the cross derivtive is opposite becuse it is minimiztion problem s min C = mx C. 10
By i, θ increses with U if u 1 is convex. The spred is, so the spred decreses s θ π+1 πθ increses. Proof of iii The principl s cost in problem 2 is composed out of the cost of providing consumption, equl to: u 1 U + + 1 u 1 U nd the monitoring cost 1 π κ θ. The first best cost for the principl is u 1 U +. Therefore, the difference between the principl s cost of providing consumption in the first best nd in 2 is result of the requirement of spreding out utilities. Define the cost of spreding out utilities s the difference between the costs: DU = {u 1 U + } + 1 u 1 U u 1 U +, 13 nd notice tht the curly brckets include lottery with prizes { U +, U} with probbilities {, 1 }, whose expected prize is U +. This mens tht D U is the difference between lottery nd certinty equivlent U +, vlued by the function of u 1. Since u is concve D U > 0 u. The dependence of this cost on U is the following derivtive: D U = { u 1 U + + 1 u 1 U } u 1 U +. 14 Since under Condition 1 u 1 is convex, Jensen s inequlity implies tht: u 1 U + + 1 u 1 U > u 1 { U + } + 1 U = u 1 {U + } D U > 0 11