Moral Hazard and Endogenous Monitoring

Transcription

1 THE PINHAS SAPIR CENTER FOR DEVELOPMENT TEL AVIV UNIVERSITY Morl Hzrd nd Endogenous Monitoring Ofer Setty 1 Discussion Pper No July 2015 I would like to thnk The Pinhs Spir Center for Development t Tel Aviv University for their finncil support. I lso thnk Yir Antler, Eddie Dekel, Zvik Neemn, Ady Puzner, Nicol Pvoni, Dotn Persitz, Michel Richter, Dn Simundz nd Dvid Weiss for very helpful comments. 1 Ofer Setty The Eitn Bergls School of Economics, Tel Aviv University. Emil: ofer.setty@gmil.com

2 Abstrct I study principl-gent problem where the principl chooses the signl's precision of the gent's ction. I use the model to study how the principl's monitoring choice depends on ech of the three properties of the gent: her disutility from performing the tsk, her probbility of succeeding in the tsk nd her outside option. JEL Clssifiction: D81; D82; J33 Keywords: Principl-gent model; Morl hzrd; Monitoring: Costly stte verifiction.

3 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 signl s precision 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 for the principl s monitoring decision between the cost of monitoring nd the cost of imposing risk on the gent. This pper uses simple model to study how the principl s monitoring choice depends on ech of three chrcteristics of the gent: her disutility from performing the tsk, her probbility of succeeding in the tsk, nd her outside option. First, how does the principl s monitoring choice depend on the disutility from performing the tsk? In principl-gent problem without monitoring n increse in tht disutility compels the principl to increse the spred between pyoffs in order to mintin the incentive-comptibility constrint tht the gent fces. Once monitoring is llowed, the principl is free to mix the two instruments tht contribute to holding the incentive-comptibility constrint: incresing the spred or incresing the precision of monitoring. I show tht n increse in the disutility from performing the tsk results in n increse in both instruments. The intuition for this result is tht the principl blnces the opertionl cost of monitoring with the cost of compensting the risk-verse gent for the spred. When the disutility from performing the tsk increses the principl continues to mintin this blnce by incresing the mrginl cost of both instruments. Second, how does the principl s monitoring choice depend on the probbility of succeeding in the tsk? The result nd the intuition for this cse is similr to the previous one. An increse in the probbility of succeeding in the tsk implies smller spred in the problem without monitoring. When monitoring is vilble the principl mintins the blnce between the two instruments by lowering both the spred nd the monitoring precision. Third, 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 show the 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. In the optiml contrct, the signl s precision increses decreses nd the dispersion of utilities fced by the 2

4 gent decreses increses with the gent s outside option if the derivtive of the inverse utility is convex concve. Observed heterogeneity in the gent s chrcteristics long the three dimensions studied in this pper rises nturlly in vrious principl-gent settings. Consider, for exmple, the problem of firm tht hires worker to do some tsk. This firm my fce individuls with different disutility levels of performing the tsk e.g., due to difference in vilbility constrints, different probbilities of succeeding in the tsk e.g., due to difference in experience nd different levels of the outside option e.g., due to difference in welth. In this context the model dicttes how the firm tht chooses both the worker s pyoff nd the qulity of monitoring should tke into ccount ll those sources of heterogeneity. Similrly, those sources of heterogeneity rise in the problem of insuring n gent ginst some dverse event. The tsk in this context is being precutious. The insurer my fce individuls with different disutility of performing the tsk e.g., due to difference in ttitudes towrds being precutious, different probbilities of succeeding in the tsk e.g., due to difference in probbilities of experiencing the dverse event, nd different outside options e.g., due to difference in ccess to lterntive insurnce schemes. 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. 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 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. 1 Thiele nd Wmbch 1999 generlize Newmn s result by llowing for ny finite number of effort levels insted of two. 2 I show tht the condition I use, Newmn s condition nd Thiele nd Wmbch s condition re equivlent. The literture on contingent monitoring systems introduces monitoring into the principl-gent problem s well. The emphsis in this literture is on how monitoring should depend on the outcome of the gent s ction rther thn the gent s chrcteristics. This question is therefore complimentry to the questions of this pper. Kim nd Suh 1992 show tht under some conditions the optiml monitoring investment is decresing in the outcome. The intuition for this result is 1 Newmn extends his model to llow for 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 Newmn s result ppers in n erly drft from

5 tht the principl is using monitoring more intensively when the outcomes re lower becuse in those cses the gent is more likely to exert low effort. Fgrt nd Sinclir-Desggné 2007 rnk contingent monitoring systems. They extend Kim nd Suh s result by showing tht when the derivtive of the inverse utility is convex concve the principl prefers monitoring systems whose precision increses decreses with respect to the outcome. 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 V, nd normlize the vlue of low output to 0. The gent s utility is u w, where u is strictly incresing, strictly concve, three times differentible nd its inverse is three times differentible s well; w is the trnsfer to the gent; 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 c θ. 3 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 the gent s expected utility will be t lest U. 3 Since the pper focuses on the trdeoff between the spred nd the monitoring precision I mke the following ssumptions to gurntee n internl solution for θ except when the signl is non informtive: lim θ 0 c θ = 0, lim θ 1 c θ =. 4

6 3 The contrct Denote by w x the principl s trnsfer to the gent conditionl on outcome x for x {H, G, B}. Let Ĉ be the cost for principl who recommends the ction. In wht follows I ssume tht the prmeters justify creting the costly incentives for the gent to choose ction, e.g., V is high enough. Otherwise, the problem becomes trivil with recommendtion of = 0 nd full insurnce. The principl s problem is s follows: Ĉ = s.t. { min πw H + 1 π θw G + 1 π 1 θ w B + 1 π c θ + πv } w H,w G,w B,θ πu w H + 1 π θu w G + 1 π 1 θ u w B U πu w H + 1 π θu w G + 1 π 1 θ u w B u w 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 uw H = uw G > uw 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 w B must be lower thn {w H, w G } to stisfy the IC constrint see the ppendix for detils. Notice tht uw H = uw G becuse the two outcomes hve identicl informtion content high effort for sure. Rewrite the problem s follows. In the IC, substitute uw B with U nd derive u w H = U +. Using the vlues for π+1 πθ {wh, w G, w B } in the objective function nd omitting the term πv, 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: C = min {θu 1 U + } + 1 θu 1 U + 1 π c θ θ θ where π + 1 π θ. 2 5

7 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 trnsfer 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 c 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 nd cθ = 0 in Problem 2 nd get the first-best cost. 4 Optiml monitoring In this section I nlyze how optiml monitoring is ffected by ech of the three chrcteristics of the gent: the disutility from performing the tsk ā, the probbility of succeeding in the tsk π, nd the outside option U. Theorem 1 chrcterizes the contrct w.r.t. the disutility from performing the tsk ā. Theorem 1 The solution to Problem 2 hs the following chrcteristics: i the optiml signl s precision θ increses with the tsk s disutility ā; ii the utility spred increses with the tsk s disutility ā. π+1 πθ To gin intuition on Theorem 1 consider the principl-gent problem presented here except tht monitoring is unvilble. In this environment n increse in the disutility from performing the tsk compels the principl to increse the spred between pyoffs in order to mintin the incentive-comptibility constrint tht the gent fces. Once monitoring is llowed the principl 6

8 is free to mix the two instruments tht contribute to holding the incentive-comptibility constrint: incresing the spred or incresing the precision of monitoring. In this cse the principl blnces the opertionl cost of monitoring with the cost of compensting the risk-verse gent for the spred. When the disutility from performing the tsk increses the principl continues to mintin this blnce by incresing the mrginl cost of both instruments. Theorem 2 dels with the effect of the probbility of succeeding in the tsk π on optiml monitoring: Theorem 2 The solution to Problem 2 hs the following chrcteristics: i the optiml signl s precision θ decreses with the success probbility π; ii the utility spred decreses with the success probbility π. π+1 πθ The result nd the intuition for this cse is similr to the previous cse. An increse in the probbility of succeeding in the tsk implies smller spred in the problem without monitoring. When monitoring is vilble the principl decreses both the spred nd the monitoring precision mintining the blnce between the two instruments. Theorem 3 chrcterizes the contrct w.r.t. the outside option U. This requires n dditionl condition on the concvity of u 1. Theorem 3 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; iv the converse version of i iii holds when u 1 is concve. Wht is the intuition behind this Theorem? The decision mker who solves problem 2 hs utility function given by u 1. When this utility is tht of prudent individul i.e. u 1 is concve, the principl is more concerned bout spreds t low utility levels so he invests reltively more in monitoring t low levels of U to reduce the spred. 4 Since the decision mker s utility is linked to the gent s utility it is lso possible to see the intuition through the gent s perspective. 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. 4 Menezes, Geiss, nd Tressler 1980 show tht decision mker whose utility function hs positive third derivtive is downside risk verse. 7

9 The signl s cost does not depend on the outside option. When u 1 is convex, 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 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 wu w u w 2 3 3: There is convex function h : R R + such tht 1 u w 4: 1 u u 1 u u = h u w. Condition 1 is the one used in this pper. Condition 2 is used in Thiele nd Wmbch Condition 3 is used in Newmn Condition 4 implies tht u is more risk verse thn 1 u the sense of Prtt 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 wu w u w 2 2, mking their condition weker thn his. However, s Proposition 1 shows, Newmn s condition is equivlent to theirs. 5 in 5 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 in u 2 income is stricter condition thn wht Newmn requires. In fct 1 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 8

10 References FAGART, M.-C., AND B. SINCLAIR-DESGAGNÉ 2007: Rnking Contingent Monitoring Systems, Mngement Science, 539, KIM, S. K., AND Y. S. SUH 1992: Conditionl Monitoring Policy Under Morl Hzrd, Mngement Science, 388, MENEZES, C., C. GEISS, AND J. TRESSLER 1980: Incresing Downside Risk, Americn Economic Review, 705, MILGROM, P., AND C. SHANNON 1994: Monotone Comprtive Sttics, Econometric, 621, NEWMAN, A. F. 2007: Risk-bering nd Entrepreneurship, Journl of Economic Theory, 1371, PRATT, J. W. 1964: Risk Aversion in the Smll nd in the Lrge, Econometric, 32, THIELE, H., AND A. WAMBACH 1999: Welth Effects in the Principl Agent Model, Journl of Economic Theory, 892,

11 APPENDIX Proof of clim 1 Lemm 1 In the optiml solution either w H > w B or w G > w B, or both. Proof. Rewrite the IC s: πu w H + 1 π θu w G [π + 1 π θ] u w B +. 3 Since > 0 nd since the sum of the coefficients of { u w H, u w G} is equl to the coefficient of u w B nd positive, if both w H w B nd w G w 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 w 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 w H = w G. Proof. Assume tht w H > w G. The optiml solution cn be improved s follows. Decrese w H by ε nd increse w G πε by. By construction this chnge does not ffect the objective function 1 πθ becuse π w H ε + 1 π θ w G + = πw H + 1 π θw G. To study the effect on the π 1 πθ IR, consider the prt of the IR composed of πu w H + 1 π θu w 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 w H ε + 1 π θu 1 π θ u w G + επ 1 π θ w G + Divide both sides by ε nd rerrnge to get: u w H u w H ε ε επ > πu w H + 1 π θu w G 1 π θ u w G > π u w H u w H ε 4 < u w G + επ u w G 1 πθ επ 1 πθ In the limit this is u w H < u w G, which is true by the negtion ssumption tht w H > w G. Thus the IR nd similrly the IC become slck, which is contrdiction to Lemm 2. Therefore it is impossible for w H to be strictly greter thn w G in the optiml solution. The sme line of proof elimintes the possibility of w G > w H by showing tht incresing w H by ε nd decresing w G by πε 1 πθ violtes Lemm

12 Lemm 4 In the optiml solution the IC holds with equlity. Proof. By Lemmt 1 nd 3 w H > w B. If the IC is slck then the objective function cn be improved. Decrese w H by ε nd increse w 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 w H + 1 π 1 θ u w 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 w H ε + 1 π 1 θ u w B επ + > πu w H + 1 π 1 θ u w B 1 π 1 θ π u w H u w H ε < 1 π 1 θ u w B επ + u w B. 6 1 π 1 θ Divide both sides by ε nd rerrnge to get: u w H u w H ε u w B + < ε επ 1 π1 θ επ 1 π1 θ u w B. 7 In the limit this is u w H < u w B, which is true becuse w B < w H. Now decrese w H by δ in order to improve the objective function without dmging ny of the constrints. Lemm 5 In the optiml solution u w 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 w B = U. Clim 1 is then combintion of Lemmt 1, 3, nd 5. Proof of Theorem 1 Theorem 1 The solution to Problem 2 hs the following chrcteristics: i the optiml signl s precision θ increses with the tsk s disutility ā; ii the utility spred increses with the tsk s disutility ā. π+1 πθ Proof. The two prts of the theorem re proved sequentilly. Proof of i The proof is bsed on monotone comprtive sttics Milgrom nd Shnnon, w = u 1 U + 2 w 1 π = u 1 θ 2 U + <

13 According to the monotone comprtive sttics theorem, θ wekly increses with if 2 w θ 0.6 Proof of ii Inspection of the spred = shows tht the effect of n increse in both nd θ π+1 πθ on the spred is mbiguous. A further inspection into the FOC shows, however, tht this is not the cse. The FOC is: c θ = ā u 1 U + u 1 U + u 1 U 9 As θ increses the LHS increses becuse c is convex in θ. By differentiting the RHS w.r.t. the spred it cn be verified tht the RHS is incresing in the spred. Therefore n increse in θ must be ccompnied by n increse in the spred. Proof of Theorem 2 Theorem 2 The solution to Problem 2 hs the following chrcteristics: i the optiml signl s precision θ decreses with the success probbility π; ii the utility spred decreses with the success probbility π. π+1 πθ Proof. The two prts of the theorem re proved sequentilly. Proof of i The proof follows the sme line of proof s in theorem 1. w u 1 = 1 θ U + ā u 1 π 2 w π θ = 1 π 1 θ 2 3 u 1 U + > 0, U + u 1 U cθ 10 where the derivtion of 2 w uses the fct tht the FOC w.r.t. θ is zero t the optimum. This π θ estblishes by the monotone comprtive sttics theorem tht s the prmeter π increses, the monitoring precision θ decreses. Proof of ii Inspection of the spred s denomintor π + 1 πθ shows tht the effect of decrese in both π nd θ on the spred is mbiguous. However, s directly given by the proof of Theorem 1 6 In the cse of mximiztion problem supermodulrity is required between {, θ}. Here the sign of the cross derivtive is opposite becuse it is minimiztion problem s min w = mx w. 12

14 ii decrese in θ must be ccompnied by decrese in the spred to keep the FOC. Proof of Theorem 3 Theorem 3 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; iv the converse version of i iii holds when u 1 is concve. Proof. The three prts of the theorem re proved sequentilly. Proof of i The proof follows the sme line of proof s in Theorem 1 i. w u U = 1 U + u 1 U + u 1 U 11 2 w 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 w 0. This is stisfied if U θ u 1 is convex. To see this, rewrite 2 w 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 w 0. U θ, so the spred de- π+1 πθ Proof of ii By i, θ increses with U if u 1 is convex. The spred is creses s θ increses. Proof of iii The principl s cost in problem 2 is composed out of the cost of providing trnsfer, equl to: u 1 U u 1 U nd the monitoring cost 1 π c θ. 13

15 The first-best cost for the principl is u 1 U +. Therefore, the difference between the principl s cost of providing trnsfer in the first best nd in 2 is result of the requirement of spreding out utilities. Define the cost of spreding out utilities { U, U + } t utility U 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 u 1 U } u 1 U Since under Condition 1 u 1 is convex, Jensen s inequlity implies tht: u 1 U u 1 U > u 1 { U + } + 1 U = u 1 {U + } D U > 0 Proof of iv The proof follows the sme rguments s in the proof for prts i iv bove for u 1 concve. Proof of Proposition 1 Proposition 1 The following conditions re equivlent: 1: u 1 is convex 2: u wu w u w 2 3 3: There is convex function h : R R + such tht 1 u w 4: 1 u u 1 u u = h u w. 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 14

16 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 15 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 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 w u w u w Condition 2 Condition 1 Using the implicit function theorem nd differentiting both sides of w = u 1 uw gives the equlity u 1 u w = 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 17 15

17 To prove tht Condition 2 Condition 1, show tht not Condition 1 not Condition 2. Not condition 1 w s.t. u 1 u w = 1 is concve: By 17 nd using tht u w is positive: u 3u w 2 u w u w < 0, u w u w > 3u w 2 u w u w u w 2 > 3 16