Equilibrium with Mutual Organizations. in Adverse Selection Economies

Transcription

1 Equlbrum wth Mutual Organzatons n Adverse Selecton Economes Adam Blandn Arzona State Unversty John H. Boyd Unversty of Mnnesota Edward C. Prescott Arzona State Unversty and Federal Reserve Bank of Mnneapols Workng Paper 717 September 2015 Keywords: Adverse selecton equlbrum; Theory of value; Insurance; Sgnalng; Mutual organzaton; The core JE classfcaton: C62; D46; D82; G29; G22 The vews expressed heren are those of the authors and not necessarly those of the Federal Reserve Bank of Mnneapols or the Federal Reserve System. Federal Reserve Bank of Mnneapols 90 Hennepn Avenue Mnneapols, MN

2 Equlbrum wth mutual organzatons n adverse selecton economes Adam Blandn a, John H. Boyd b, Edward C. Prescott c* a Arzona State Unversty, Unted States b Unversty of Mnnesota, Unted States c Arzona State Unversty and Federal Reserve Bank of Mnneapols, Unted States Abstract We develop an equlbrum concept n the Debreu (1954) theory of value tradton for a class of adverse selecton economes whch ncludes the Spence (1973) sgnalng and Rothschld-Stgltz (1976) nsurance envronments. The equlbrum exsts and s optmal. Further, all equlbra have the same ndvdual type utlty vector. The economes are large wth a fnte number of types that maxmze expected utlty on an underlyng commodty space. An mplcaton of the analyss s that the nvsble hand works for ths class of adverse selecton economes. JE classfcaton: C62; D46; D82; G29; G22 Keywords: Adverse selecton equlbrum; Theory of value; Insurance; Sgnalng; Mutual organzaton; The core *Correspondng author: Edward C. Prescott, Edward.Prescott@asu.edu E-mal addresses: Adam.Blandn@asu.edu (A. Blandn), boydx002@umn.edu (J.H. Boyd), Edward.Prescott@asu.edu (E.C. Prescott). We would lke to thank an anonymous referee, Rob Shmer, and Manuel Amador for ther useful comments. The vews expressed heren are those of the authors and not necessarly those of the Federal Reserve Bank of Mnneapols or the Federal Reserve System.

3 1. Introducton In ths paper, socal nsurance s defned to be the centralzed provson of some form of nsurance. A queston s, why have socal nsurance? Adverse selecton s often used as justfcaton for socal nsurance, because many clam that adverse selecton can lead to the neffcent provson of nsurance by decentralzed mechansms. We fnd that f mutual organzatons are feasble and permtted, there s no falure of decentralzed arrangements n provdng nsurance or, for that matter, n dealng wth the Akerlof (1970) used-car lemons envronment or wth the Spence (1973) job market sgnalng envronment. If mutual nsurance organzatons are permtted, Adam Smth s nvsble hand works. There are no market falures due to adverse selecton n provdng nsurance. We modfy the defnton of equlbrum used by Rothschld and Stgltz (1976) n essentally one respect. We permt the agents who select nsurance arrangements to select mutual arrangements. A mutual arrangement s a set of contracts whch specfy payments contngent on both ndvdual experences and the aggregate experence of all contract holders. In addton, the set of contracts must be such that the organzaton profts are not negatve. As n Boyd-Prescott (1985), there are mutual organzatons, and a core-related equlbrum concept s developed and used. But unlke ths earler approach, the new defnton of equlbrum s smpler and more general. Exstence, unqueness (n the types utlty outcomes), and optmalty are establshed for an arbtrary fnte number of types of ndvduals. The earler Boyd-Prescott equlbrum was defned only for two types. Attempts by us and others to generalze to more than two types have faled. Wth the Rothschld-Stgltz equlbrum defnton, at the frst stage nonmutual nsurance contracts are selected. At the second stage, ndvduals choose ther optmal contract from the set of offered contracts, or choose not to nsure. In our defnton, at the frst stage mutual nsurance contracts are selected. At the second stage, ndvduals pck ther mutual arrangement optmally gven the choces of others. Thus, the second stage s a Nash equlbrum. We say that a mutual arrangement s blocked f there exsts an alternatve mutual arrangement for whch a subset of types can make themselves better off usng only ther own resources. An equlbrum s defned to be an unblocked mutual arrangement. Wth our setup, f and only f an equlbrum mutual arrangement s chosen at stage 1, no agent can offer an alternatve mutual arrangement that s proftable, gven what wll happen n the second stage. 1 We abstract from a number of features of realty that are mportant n the provson of nsurance. In our examples, no costs are assocated wth operatng an nsurance company, whereas n fact operatng an nsurance company entals large costs. Evdence that these costs 1

4 are large s that the gross output share of the U.S. nsurance carrers and related actvtes sector reported n the U.S. Natonal Accounts was 5 percent of GNP n Assocated wth the operaton of an nsurance busness are record-keepng costs, montorng costs needed to mtgate the moral hazard problems assocated wth nsurance contracts, and asset-managng costs as premums are receved pror to clams beng pad. If there were a market falure, when evaluatng socal nsurance, these costs would have to be carefully quantfed and ncorporated nto the analyss n addton to the costs that are specfc to socal nsurance. But there s no need to quantfy all these costs because the decentralzed outcome s Pareto effcent. 2. The Relatonshp to Other Equlbrum Concepts We use Debreu s (1954) defnton of an economy. In so dong, we follow Cornet, who has been a leadng contrbutor to the development of the theory of value (see, e.g., Cornet 1988). Debreu s defnton of an economy requres the specfcaton of a commodty space. The commodty space s a lnear topologcal space. There also s a set of ndvduals wth preference orderngs on subsets of the commodty space and a set of technologes, whch are subsets of the commodty space. Consequently, feasblty and Pareto optmalty are well defned. In addton, the economc statstcs of the natonal ncome accounts can be calculated and welfare analyss carred out. An assumpton s that the preference orderngs of ndvduals can be characterzed by the expected value of utlty functons, whch are contnuous real-valued functons on the underlyng consumpton set. As n Prescott and Townsend (1984), the elements of the commodty space are sgned measures on the Borel sgma-algebra of the underlyng consumpton set. The consumpton sets are sets of probablty measures n ths space. Equlbrum s not a Debreu valuaton equlbrum as t s n the Prescott and Townsend (1984) prvate nformaton economes, where households and those who pck the commodty vector n the technology sets take prces as gven. Instead, as part of the defnton of equlbrum, the blockng concept n Edgeworth s (1881) theory of the core s used. 2 But groups do not block. Rather, an ntal mutual organzaton s blocked f an actve agent wrtes a charter for a new mutual organzaton whch makes a subset of ndvduals better off whle preservng resource and ncentve feasblty. Ths s smlar to competton among nsurance frms n Rothschld and Stgltz (1976), except that the choce s a mutual arrangement and not an nsurance contract. Usng our equlbrum defnton, f mutual organzatons are prohbted and nonmutual companes permtted, all the fndngs of Rothschld and Stgltz (1976) hold. In ther classc envronment, for example, no transfers occur between ndvdual types n equlbrum, and any equlbrum allocaton must be the mnmum separatng allocaton. 2

5 Gven expected utlty maxmzaton, the utlty functon of each ndvdual type s lnear, and the set of ncentve-compatble allocatons s a convex affne cone. The ntersecton of the resource constrant set and the ncentve-compatble allocaton set s a convex subset wth dmenson one less than the number of types. Each person ether selects the mutual organzaton that s best gven the choces of other people or chooses not to nsure. Thus, the second stage s a Nash equlbrum. The economy s large wth an atomstc measure of each of the fnte number of types of ndvduals. Type s prvate nformaton. 3. The Formal Analyss The economes are large wth a fnte number of ndvdual types I. The measures of n type are λ. The underlyng consumpton sets are C, a closed and bounded subset of R. The commodty space s the space of sgned measures on the Borel sgma-algebra of C. Each ndvdual s consumpton possbltes set s X, the set of probablty measures on the Borel sgma-algebra of C. Preferences of a type ndvdual are ordered by the expected value of a contnuous functon v : (1) u( x) = v( c) x( dc). An allocaton s an n-tuple { x }. The ncentve compatblty constrants (ICs) are j (2) u ( x ) u ( x ) for all, j I. Type must weakly prefer x to j x. The sngle resource constrant (RC) has the form (3) λ( r( x ) π) 0, I where the r : X R functons are lnear and the π are parameters. In the Rothschld Stgltz (1976) nsurance case the resource constrant s the condton that a mutual nsurance organzaton cannot dstrbute more n clams than t takes n as premums. In the Spence (1973) sgnalng envronment, t s the condton that wage payments to members of a mutual organzaton cannot exceed the value of producton of that organzaton. 3

6 An allocaton s feasble f t satsfes the ncentve constrants, (2), and the resource constrant, (3). An allocaton x s a Pareto optmum f for no other feasble allocaton y, u ( y ) u ( x ) for all wth strct nequalty for at least one. Workng n the Utlty Space We transform the problem nto the utlty space as the proofs of exstence and optmalty of unblocked mutual arrangements are carred out n ths space. Assocated wth each allocaton x s a utlty vector ux ( ) = { u( x)}. The set of utlty vectors that satsfy the ncentve constrants s denoted by IC. Ths set s a convex affne cone. The set of allocatons that yeld a gven pont n IC s convex. The mappng from utlty vectors n the IC set to allocatons that we use s denoted by xu ( ). If the nverse mage of uxs ( ) not a pont, then the mean allocaton of all allocatons that yeld utlty vector u s the value of the functon xu ( ). Ths mean allocaton s n the set of allocatons that yeld u, because the nverse mage of the functon ux ( ) s a convex set. The aggregate net transfer made by type s (4) t ( u) = λ ( r( x ( u) π ). The resource constrant s (5) t ( u ) = 0. The set of utlty vectors that satsfy ths constrant s a hyper-plane wth dmenson one less than the cardnalty of I. Defnton: A feasble utlty vector u s blocked by feasble better under u ', denoted by B, t ( u ) 0. B Some Results I u ' f for the set of types that do Result 1: The Pareto optmum utlty set, P, s a closed, convex subset of the hyper-plane that satsfes (5). Result 2: Any non-pareto optmal u s blocked by the set of types I and a Pareto superor feasble utlty vector. Result 3: Any unblocked u s a Pareto optmum. Result 4: Utlty vector u Ps blocked f and only f there s a u' P that blocks t. 4

7 Therefore, the problem s to establsh the exstence of an unblocked u P. Proposton: An equlbrum exsts and s unque n the utlty space. Proof: et P be the Pareto utlty set. (). Pck a drecton n the Pareto set. Movng n that drecton makes one set of types better off and the other set worse off. Thus a drecton parttons the types nto two sets. Movng n the gven drecton makes one set of types better off and the complment set of types worse off. (). A pont u n P and a drecton defne a lne. The ntersecton of a lne and P s a lne segment. (). Gven a lne segment n a subset n P, that subset can be parttoned nto lne segments parallel to ths lne segment. Along any of these parttons, utltes and transfers vary lnearly. A necessary condton for an unblocked utlty pont to be on the lne s that transfers by the beneftng group for the gven drecton mnmzes the absolute value of the transfers by that group n that partton. The set of all utlty ponts that satsfy ths property for some partton s a convex set wth dmensonalty less than that of the subset. All unblocked utlty ponts le n that set. (v). Gven that the dmensonalty of P s fnte, begnnng wth subset P, by nducton a set of dmenson zero can be found that contans the unblocked utlty ponts, u *. Comment: The allocaton that yelds u * may not be unque. If not unque, we select the allocaton that s the average of the set that yelds ths utlty outcome. Gven the lnearty of the mappng from feasble arrangements to utltes, the nverse mage of any pont n the utlty possblty set s a convex set. Thus, ths average s a pont n the set. 4. Applcaton to the Adverse Selecton Insurance Envronment The envronment has measures λ of people of type {1, 2,..., I} = I. The same symbol s used for both the cardnalty of the set and the set tself. An ndvdual s type s prvate nformaton. A person has a random endowment subsequent to contractng ej {1, 2,..., J} = J. The probablty of a type havng endowment e j s π j. 5

8 The underlyng consumpton space s C = { x R J : x c for all j}. The c s suffcently large that ncreasng t does not ncrease the set of feasble allocatons. Ths s possble because the resource constrant, necessary for feasblty, s bounded. The commodty pont s x = { x j( dc)} j J. The consumpton possblty set X s a vector of J probablty measures, and the utlty functons are. u () x = π v() c x ( dc) I j J j j The contnuous functon v: C R s strctly ncreasng and concave. The Rothschld-Stgltz Two-Type Adverse-Selecton Envronment + There are two types and two possble postve endowments: 0 < eb < eg. The subscrpts denote bad ( b ) and good ( g ). The two types are low-rsk ( ) and hgh-rsk ( H ) ndvduals. Thus, π b < π. We normalze the populaton sze to one so that Hb j λ H + λ = 1. Type average endowments are e = π b+ π g for I. b g Dependng on the parameter values, the equlbrum wll fall nto one of two categores: equlbra wth no transfers between types, or equlbra wth postve transfers from type to type H ndvduals. The unblocked utlty vector s the one that maxmzes type- utlty on the utlty possblty fronter. If the measure of low-rsk people s suffcently large, the set of Pareto optma s a pont wth everyone consumng the average populaton endowment x( λ ). Ths average s an ncreasng functon n λ. For the unblocked contract, the probablty measures for an type condtonal on that ndvdual endowment realzaton place all probablty on a sngle pont: c b for x b and c b for x g. Ths follows from the assumed propertes of the utlty functon v. We denote the 6

9 consumpton of a hgh-rsk ndvdual, whch s the same for both possble endowments, by c H. Thus, each Pareto optmum s characterzed by a trplet of real numbers ( cb, cg, c H). The utlty functons n terms of these three varables are denoted by U ( c ) = vc ( ) and U( c, c) = π vc ( ) + π vc ( ). b g b b g g H H H To fnd the Pareto set, we frst fnd the utlty-maxmzng contract for type gven an average transfer, T. These transfers are constraned to the nterval T [0, ( e e ) λ ]. H H The upper end of the nterval results n the average consumpton of the two types beng equal. The transfer for whch the utlty of type s maxmal s the equlbrum T *. We proceed n two steps. Frst, we consder the problem of maxmzng the utlty of a type gven T. The value of the soluton s denoted by uˆ ( T ). Ths program s uˆ ( T) = max U ( c, c ) ch, cc b g b g subject to the followng constrants: (). The H have enough resources to fnance ther consumpton: c e + Tλ / λ ; H H H (). The have enough resources to fnance ther consumpton: π c + π c e T ; b b g g (). The H weakly prefer ther contract to the contract: U ( c, c ) U ( c, c ); H H H H b g (v). The weakly prefer ther contract to the H contract: U ( c, c ) U ( c, c ). b g H H sep The curves for all fractons λ go through the separatng utlty pont u, for whch T = 0. To fnd the Pareto optmum utlty sets, the utlty of type s maxmzed gven the utlty level of type H. Ths functon s denoted by uˆ ( u ). The nature of ths functon depends on the fracton of low-rsk types. If ths fracton s suffcently small, the curve s as depcted n 7 H

10 Fgure 1(a). The equlbrum utlty vector s denoted by u *. At ths pont, no transfers between types take place. The Pareto set, denoted n red, s the decreasng porton of ths concave functon. Fg. 1 uˆ ( uh) n the Rothschld-Stgltz envronment: a) small λ ; b) bg λ. The greater the populaton fracton of the low-rsk people, the hgher the ntersecton wth the 45 degree lne, where all consume the mean endowment ndependent of type and ndvdual endowment realzaton. For a suffcently hgh fracton of the low-rsk type, the Pareto set s a pont set wth everyone gettng utlty veλ ( ( )) wth certanty. Relaton to Rothschld-Stgltz Equlbrum sep Only for the fractons λ for whch the slope of the curve s negatve at u, as n Fgure 1(a), does the Rothschld-Stgltz (R-S) equlbrum exst. In these cases, the R-S equlbrum and ours are the same. But our equlbrum always exsts and s unque. General Case The equlbrum n the more general case can be found by solvng the problem of mnmzng the 1 norm of the transfer vector over the set of Pareto optma. The mnmum exsts gven that t s a convex mnmzaton problem wth a compact constrant set and a 8

11 contnuous objectve functon. In some cases, the optmum wll have nondegenerate lotteres condtonal on the endowment, namely when dfferent degrees of rsk averson can be exploted to separate the types. 5. Applcaton to the Spence Sgnalng Envronment We turn now to the smple Spence (1973) sgnalng equlbrum, whch does not requre lotteres to convexfy the economy. A pont n the commodty space s x= (,) cs where c s 2 consumpton and s s the sgnal and the commodty space s R. 2 max Types are denoted by I = {1, 2}. The consumpton set s X = { x R+ : x x }. Indvduals of type 2 are the hgh-productvty ndvduals ( π2 > π1 > 0 ) and have the lower dsutlty of sgnalng (0 < θ2 < θ1). The utlty functons are u = c θ s for I. max The x s suffcently large that all resource feasble allocatons are n ths max max consumpton set. A suffcently large c s π 2. A suffcent large s s π2 / θ 2. Aggregate transfers from the hgh-productvty type to the low-productvty type are t 2 ( u) = π 2 c 2 ( u). The equlbrum utlty vector s u *. Fgure 2(a) specfes the set of utlty allocatons that are ncentve feasble. The set s a convex cone. Fgure 2(b) specfes the feasble set of allocatons and (n red) the Pareto set. For ths example, the lne defned by the resource constrant beng bndng has a negatve slope, whch requres the fracton of hgh-productvty types to be below a crtcal value. If t s not below ths crtcal value, the lne has a weakly postve slope, and the Pareto set s a pont set wth everyone consumng the populaton average productvty and havng a zero sgnal. The nterestng case s the one n whch the Pareto set s a downward-slopng lne. Fgure 2(c) specfes the sgn of the transfers from the hgh-productvty to low-productvty people along the Pareto set. The utlty vector for the equlbrum s the pont u *. 9

12 Fg. 2 The utlty spaces n the Spence sgnalng envronment: a) the IC set; b) the Pareto set; c) transfers along the Pareto set. 10

13 Relaton to Spence s Valuaton Equlbrum Spence used a valuaton equlbrum wth externaltes. The equlbrum depends on the commodty space. The commodtes were the set of sgnals permtted, S, and the consumpton good. et c be the numerare and w s the wages of the varous types of labor. A Spence equlbrum requres that all ndvduals select sgnals that maxmze ther utlty gven the prces and, for every s that was chosen by some type, the average productvty of those who choose that s s equal to w s. Any feasble allocaton s an equlbrum for an approprately selected S, as was shown n Prescott and Townsend (1981). For each sgnal, the assocated wage s the average productvty of people who choose that sgnal. 6. Concludng Comments The equlbrum concept developed requres there be a sngle resource constrant and that ndvduals maxmze expected utlty. Ths s an mportant class of envronments that have receved a great deal of attenton n the economcs lterature. Equlbrum concepts that are useful n some envronments are not useful n others. We see ths equlbrum as expandng the set of envronments for whch there s a useful equlbrum concept. 11

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