# Adverse Selection and Moral Hazard in a Model With 2 States of the World

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1 Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams, with standard convx indiffrnc curvs rprsnting risk avrs consumrs (s diagram blow, showing th risk prmium for risky prospct b). This nabls a simpl modl of markt failur du to advrs slction in comptitiv insuranc markts (Rothschild and Stiglitz, 1976), and th fficincy loss du to moral hazard rsulting from a situation with unobsrvabl ffort and a risk avrs agnt, to b constructd and illustratd graphically. (1 π)/π b U1 risk prmium First, w covr som concptual prliminaris: Assum that thr ar 2 stats of th world, a bad stat (stat 2) and a good stat (stat 1), with probability of th bad stat occurring bing π. Assum that individuals hav a utility function gw i which givs utility in stat of th world i givn walth in stat i of W i and ffort. Not that ffort is not rlvant to th advrs slction modl, and that th function g() is th sam in all stats of th world, xcpt for its input W i, thus satisfying th axioms of xpctd utility thory. Individuals can only diffr in th probability π thy fac. Suppos thr ar two typs of individual, high risk and low risk (and not that low risk individuals ar high ffort individuals in th moral hazard modl thir highr ffort rducs th chanc of th bad stat of th world occurring). Hnc th two typs hav th following xpctd utility functions, with H L : U H = H gw 2 1 H gw 1 H U L = L g W 2 1 L gw 1 L An important rsult is that th marginal rat of substitution (th gradint of th indiffrnc curvs in W 1,W 2 spac) of th two typs is thn givn by:

2 U H W MRS H 1 1,2 = U H W 2 U L W MRS L 1 1,2 = U L W 2 = 1 H H = 1 L L Along th 45º W 1 =W 2 lin, th MRS bcoms 1 H H dg dw 1 dg dw 2 dw 1 dg dw 2 dg and 1 L L rspctivly. This rflcts th important fact that th high risk typs always hav shallowr indiffrncs curv than th low risk typs. (Intuitivly, bcaus th bad stat of th world is mor likly for thm, thy do not rquir so much additional walth in th bad stat in ordr to giv up walth in th good stat.) Also, not that if ithr typ is offrd insuranc at prmium rat p, in othr words offrd th chanc to pay px in both stats of th world in ordr to gt covrag of X in th bad stat of th world thn thy ar ffctivly bing offrd an xchang rat btwn W 1 and W 2 of 1 p p. Hnc, if p= thn ach typ will fully insur by picking an optimal bundl on th 45º lin. If p thn an individual optimally chooss to partially insur by picking an optimal bundl to th right of th 45º lin (this is illustratd blow). (1 π)/π (1 p)/p U1

3 Advrs Slction Modl (Rothschild and Stiglitz, 1976) In th advrs slction modl, agnt's ffort has no impact on risk, and so for simplicity w can st H = L =0. Th two typs diffr in π in a mannr that is byond thir control. Suppos that insuranc firms play a simultanous gam in which thy offr insuranc contracts (ach possibl contract bing a singl point in (W 1,W 2 ) spac), and th diffrnt typs of individuals thn slct thir prfrrd contract. Firms aim to maximiz thir profits givn th rspons of thir customrs and th contract(s) offrd by thir comptitors. Clarly, consumrs pick th contract offrd which lis on thir highst indiffrnc curv. W first considr th situation in which th insurrs can distinguish btwn high and low risk typs. In that cas, ach typ can b offrd actuarially fair insuranc with thir prmium p tailord to thir typ. Prfct comptition btwn insuranc companis will forc prmiums to b actuarially fair in quilibrium, and will also forc firms to offr full insuranc. This is illustratd in th diagram blow. Not that th ndowmnt point lis to th right of th 45º lin bcaus without insuranc walth is highr in th good stat of th world than in th bad. Th stpr diagonal lin rprsnts th fair insuranc lin for th low risk typs, th shallowr on th fair insuranc lin for th high risk typs. UL* UH* Considr instad a situation of advrs slction whr individuals know thir typ, but firms do not. Clarly, high risk typs will now hav an incntiv to concal this information from th insurr and imitat low risk typs, unlss th insurr can dsign a contract which will lad thm to rval thir typ. This might, at first glanc, b unncssary. A pooling quilibrium would involv ach firm offring a singl contract basd on th avrag risk. Thus th fair pooling insuranc lin lis in btwn that for th two typs in th diagram blow. Unfortunatly such a pooling quilibrium is not possibl bcaus thr ar contracts which li in th shadd ara which a comptitor could offr which ar prfrrd by th low risk typs to th pooling quilibrium contract, but not by th high risk typs, and li blow th low risk fair insuranc lin, showing that thy would crat positiv profits. This mans intuitivly that othr firms would b abl to poach away th low risk typs.

4 UL UH Considr now whthr a sparating quilibrium could xist, and its fficincy proprtis. This would involv two contracts aimd at th high and low risk typs, such that thy will slf slct, thus rvaling thir typ. In ordr to prvnt th high risk typs from imitating th low risk typs and taking th wrong contract, th low risk typs would nd to b offrd partial insuranc, at point b L, rathr than full insuranc as in th cas with prfct information. Th low risk typs will thrfor b on a lowr indiffrnc curv than with prfct information, and thus this outcom would b Parto infficint. Although infficint, such an quilibrium will xist if th poold insuranc lin passs undrnath th indiffrnc curv of th low risk typ, lablld U L ' in th diagram blow. bh bl UL* UH' UL'

5 Howvr, if th poold fair insuranc lin lis abov th indiffrnc curv for th low risk typs, as will occur if thr is a larg proportion of low risk typs, and is illustratd blow, thn th sparating quilibrium dos not xist ithr, bcaus both typs could b poachd away by a comptitor offring a pooling contract in th shadd ara. In fact no stabl quilibrium thn xists, and th modl prdicts unstabl comptitiv insuranc markts. UH' UL' Policy consquncs This classic modl suggsts that advrs slction will rndr comptitiv insuranc markts at bst infficint, and at worst unstabl. Signalling or scrning might allviat th problm, by nabling asir sparation of high and low risk typs, but probably still with a dadwight loss. Also, thr may b an quity problm with th sparating quilibrium (both undr prfct information and undr asymmtric information). For xampl, it may not b morally accptabl to say that in halthcar, for instanc, popl with pr xisting conditions should pay highr prmiums, or that mn should pay highr automobil insuranc (du to highr risk of crashing) or that womn should gt lss gnrous pnsions for a givn contribution rcord (bcaus thy ar at highr risk, from th insurr's prspctiv, of living longr). If thr ar both quity and fficincy problms with comptitiv insuranc markts, this provids a justification in conomic thory for various forms of social insuranc (.g. UK citizns ar ffctivly forcd to purchas halth insuranc in a social pooling contract by th National Halth Srvic). Moral Hazard du to Unobsrvabl Effort A scond ky application of this framwork would b th dadwight loss to socity du to th agncy cost of inducing an agnt to produc an ffort lvl whn th agnt is risk avrs. In a moral hazard modl, thr is a singl agnt, who must dcid whthr to bcom low risk by xrting high ffort, or high risk by xrting low ffort. Howvr, th utility functions w alrady dfind will dscrib th situation providd that H L and H L.

6 Th modl would fit a numbr of storis. For instanc, w might hav th standard txtbook cas of a workr (th agnt) bing contractd to prform som projct for a privat company (th principal) in which th outcom is risky but dpnds partly and unobsrvably on th ffort of th workr. On th othr hand, w might hav th xampl of moral hazard in a social insuranc situation. For xampl, whthr somon is unmployd or not dpnds partly but not fully on th ffort thy put in to finding and kping a job. In th bad stat of th world, th individual is unmployd and hnc has a lowr walth lvl than in th good stat of th world. If thr was no moral hazard problm (i.. job sarch/rtntion ffort was obsrvabl) thn, assuming a risk avrs individual, th fficint outcom would b for th govrnmnt to fully insur th individual by paying unmploymnt bnfit qual to post tax arnings whn mployd. With unobsrvabl ffort, on th othr hand, th optimal scond bst outcom is to mak walth whn unmployd lowr in ordr to provid an incntiv for high job sarch ffort. Hnc scond bst optimal unmploymnt bnfits will not fully rplac post tax arnings whn mployd. Similarly, th scond bst outcom in th cas of workr bing contractd is that rmunration is to som dgr prformanc rlatd, vn though this imposs risk on th workr, for which th contractor must compnsat thm. In what follows, w assum that th optimal scond bst outcom (and thrfor th profitmaximizing outcom for th contractor/mployr) would rquir th agnt to xrt high ffort. (Th sam would thn b tru of th Parto fficint outcom, sinc rmoving th agncy cost would mak high ffort vn mor socially bnficial.) W will xplain using th txtbook cas of a privat company contracting a workr to prform a task. Th diagram blow shows th crtain wag that would b rquird in ordr to induc th agnt to tak on th task and put in high ffort (/bcom low risk ) (point a) and low ffort (/b high risk ) (point b). With obsrvabl ffort, th contractor would offr point a (and rquir high ffort in th contract), and th agnt would thus hav utility lvl U H '. With unobsrvabl ffort, th contractor could no longr offr point a, bcaus th agnt would thn choos to put in low ffort, and thus th contractor would not b maximizing profits. By instad offring point c, th agnt would thn wakly prfr to put in high ffort. Howvr, this rsults in a highr xpctd wag bing paid, th diffrnc bing th agncy cost, which concptually is th risk prmium rquird to compnsat th agnt for th gratr risk thy now fac. Sinc th agnt is on th sam indiffrnc curv as with obsrvabl ffort, but th principal is now wors off, this is clarly Parto infficint. b a c UH' UL' agncy cost

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