Module 4: Moral Hazard - Linear Contracts

Transcription

1 Module 4: Mol Hzd - Line Contts Infomtion Eonomis (E 55) Geoge Geogidis A pinipl employs n gent. Timing:. The pinipl o es line ontt of the fom w (q) = + q. is the sly, is the bonus te.. The gent hooses whethe the ept o ejet the ontt. If the gent epts it, then goto t =3. If the gent ejets it, then he eeives his outside option U, the pinipl eeives pofit 0, nd the gme ends. 3. The gent hooses tion / e ot A [0, ]. 4. Output q = + " is elized, whee " N(0, ) 5. The pinipl pys the gent, nd the pties pyo s e elized. The pinipl is isk neutl. His pofit funtion is E [q w (q)] The gent is isk vese. His utility funtion is U (w, ) =E e (w(q) ()) with () = Rtionlity ssumptions:. Upon obseving the ontt w ( ), the gent hooses his tion to mximize his expeted utility.

2 . The pinipl, ntiipting (), hooses the ontt w ( ) to mximize his expeted pofit. Fist Best Benhmk: Suppose the pinipl ould hoose the tion. We ll this benhmk the fist best o the e ient outome. Equivlent to sy tht the gent s tion is veifible o onttible. Pinipl solves: mx,w(q) E [ + w (q)] s.t. E e (w(q) ()) U Individul Rtionlity (IR) Solution ppoh: Jensen s inequlity =) E x [ e x ] pple e Ex[x] Beuse the pinipl hooses the tion, optiml wge must be independent of q; i.e., w (q) = Beuse highe w (q) deeses the pinipl s pofit nd ineses the gent s pyo, (IR) must bind. So: e ( ()) = U =) = () ln ( U) The lst eqution pins down the wge s funtion of the tion. We now substitute into the objetive funtion. We hve: mx pple ln ( U) Fist ode ondition: = 0 Optiml solution: = nd hene w (q) = ln ( U) +

3 Notes: Intuitively, beuse the gent is isk vese nd he does not hoose the tion, it is suboptiml to expose him to isk. In genel, (IR) will bind t the optimum. Othewise, the pinipl is leving money on the tble. Mol Hzd Now suppose tht the pinipl nnot hoose the gent s tion. Tde-o s:. Beuse the gent is isk vese nd the pinipl is isk neutl, the pinipl wnts to insue the gent.. Beuse the pinipl nnot enfoe ptiul tion, she must povide inentives to the gent. Exteme ses: Full insune (but no inentives): Py flt wge; i.e., w (q) =. Full inentives (but no insune): Agents pys flt fee nd buys the output; i.e., w (q) = + q. Solution Appoh Fist, solve the gent s mximiztion poblem fo bity w (q): mx U = mx = mx = mx = mx = mx E e E n [w(q) ()] h i e + (+") h i e + )E e " h i e + )e e ( + ) o 3

4 Theefoe, the gent s poblem edues to mx + The fist-ode ondition fo the gent s optiml e ot hoie is: ( )= Unless, in equilibium, e ot is less thn fist best. The pinipl will then mximize mx,, E [ + ( + )] = ( ) s.t. = + u Fist eqution is the inentive omptibility onstint (IC) nd the seond is the individul tionlity (IR) with u = ln U. The pinipl will hoose = u (s.t. IR binds). Substituting into the pinipl s objetive funtion: ( ) mx + u Solution: nd = u = + () ( + ), Beuse negtive slies e llowed, the IR onstint is binding. The equilibium level of e ot is = ( + ) 4

5 whih is lwys lowe thn the fist-best level of e ot, fb =. Comptive Sttis Inentives e lowe poweed ; i.e., = + is lowe when: the gent is moe isk-vese; i.e., if is lge e ot is moe ostly; i.e., if is lge thee is gete unetinty; i.e., if is lge. Is line ontt optiml (mong ll possible ontts)? NO! Milees s shoot-the-gent ontt is optiml hee: ( q (x) = w H if x q 0 othewise w L whee w H >w L. By hoosing w H, w L nd q 0 ppopitely, it is possible to implement fist best (ppoximtely). Agent eeives w H lmost suely, yet hs inentives fom fe of w L. But this esult depends uilly on the ssumption N (0, ). Wht to mke of line ontts Even if line ontts e not optiml hee, they e tttive fo thei simpliity nd fo being esy to hteize nd intepet. Nonline models e often vey sensitive to the ptiul ssumptions of the model (e.g., the distibution funtion of ). Nonline ontts e lso pone to gming. Conside Milees shoot-the-gent ontt in dynmi wold. Afte output hs ehed q 0, the gent hs no inentive to exet e ot. 5

6 Refeenes Bod S., (0), Letue Notes. Bolton nd Dewtipont, (005), Contt Theoy, MITPess. Edee F., (0), Letue Notes. L ont J-J. nd Mtimont D., (00), The Theoy of Inentives: The Pinipl-Agent Model, Pineton Univesity Pess. Segl nd Tdelis, (00), Letues on Contt Theoy, Stnfod Univesity (online link). 6