Optimality in an Adverse Selection Insurance Economy. with Private Trading. November 2014

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1 Optmalty n an Adverse Selecton Insurance Economy wth Prvate Tradng November 2014 Pamela Labade 1 Abstract Prvate tradng n an adverse selecton nsurance economy creates a pecunary externalty through the prce system for consumpton nsurance, generatng endogenous subsdes across hgh rsk and low rsk agents. When agents have prvate nformaton about the dstrbuton of ther dosyncratc endowment shocks, decentralzng an ncentve-effcent allocaton generally requres separaton of markets by type, exclusvty of contracts and strct prohbton of prvate tradng. If these restrctons are dropped, agents wll trade to elmnate arbtrage profts, resultng n prces that are typcally not actuarally far for any type of agent. Agents wll choose to under or over nsure even though full consumpton nsurance s n ther budget set. The condtons for exstence and unqueness of a compettve equlbrum are determned when agents trade contngent clams at common prces. The compettve equlbra are constraned effcent wth prvate tradng (thrd best). There s no Pareto-mprovng nterventon n the form of taxes or subsdes to mtgate the pecunary externalty. JEL Classfcaton: D81, D82. Keywords: Adverse selecton, prvate-tradng arrangements, optmal taxaton, pecunary externaltes 1 I wsh to thank the semnar partcpants of the Internatonal Fnance/Macroeconomcs Workshop at George Washngton Unversty, the Summer Econometrcs Socety meetng at Unversty of Southern Calforna, and the Mdwest Macroeconomcs meetngs at the Unversty of Mnnesota. I would also lke to thank Nancy Stokey and Laurence Ales for comments on an earler draft. Address: Department of Economcs, 315 Monroe Hall, George Washngton Unversty, Washngton, D.C , Phone: (202) , Emal: labade@gwu.edu. 1

2 Prvate tradng n an adverse selecton nsurance economy creates a pecunary externalty through the prce system for consumpton nsurance, generatng endogenous subsdes across hgh rsk and low rsk agents. When agents have prvate nformaton about the dstrbuton of ther dosyncratc endowment shocks, decentralzng an ncentve-effcent allocaton generally requres separaton of markets by type, exclusvty of contracts and strct prohbton of prvate tradng. If these restrctons are dropped, agents wll trade to elmnate arbtrage profts, resultng n prces that are typcally not actuarally far for any type of agent. The pecunary effect s the redstrbutve mpact of prce changes nduced by the falure to separate markets and prevent prvate tradng. Agents wll choose to under or over nsure even though full consumpton nsurance s n ther budget set. The condtons for exstence and unqueness of a compettve equlbrum are determned when agents trade contngent clams at common prces. The compettve equlbra are constraned effcent wth prvate tradng (thrd best). There s no Pareto-mprovng nterventon n the form of taxes or subsdes to mtgate the pecunary externalty created through the prce system. When prvate tradng cannot be prevented, agents reveal nformaton about ther type based on the value of an allocaton n the prvate tradng market, changng the ndvdual ncentve compatblty constrants. An agent s better off announcng he s the type provdng hm the hghest market value of an allocaton n prvate tradng, wth the result agents face the dentcal budget set n prvate markets. Ths s the bass for a convenent reformulaton of the socal planner s problem, just as n Farh, Golosov and Tsyvnsk [10]. In the orgnal problem, the socal planner chooses consumpton allocatons satsfyng the feasblty constrant and the modfed ncentve compatblty constrant based on the ndrect utlty an agent would acheve through tradng n prvate markets at equlbrum prces. In the reformulated problem, the socal planner pcks the market value of the allocaton and equlbrum prce. The condtons for exstence and unqueness of compettve equlbra are establshed and the two socal plannng problems are shown to be equvalent under certan condtons. The exstence of compettve equlbra n adverse selecton nsurance economes s often problematc. An ncentve-effcent allocaton, whch s constraned effcent (second best), s dffcult to decentralze, n part because the set of ncentve compatble allocatons s not convex, creatng a consumpton externalty; see Prescott and Townsend [14]. As dscussed by Bsn and Gottard [6], a compettve equlbrum wth exclusvty, separaton of markets and no prvate tradng may not be ncentve effcent. In these settngs, a contract s a bundle of contngent clams wth restrcted quanttes and a zero-proft condton resultng n 2

3 contracts that are actuarally far wth no cross-subsdzaton. When prvate tradng cannot be prevented, agents have an ncentve to unbundle the contngent clams n an nsurance contract to elmnate arbtrage profts and to mprove rsk-sharng. As Bsn and Gottard [5]-[6] and Rustchn and Sconolf [17] have shown, the problems of exstence of compettve equlbra are not mtgated by allowng prce-takng agents to trade standardzed contngent-clams contracts among themselves. One contrbuton of ths paper s to characterze restrctons on endowments such that an equlbrum exsts, although t may not be unque. There s an extensve lterature examnng effcent rsk sharng n the presence of prvate nformaton, where ncentve compatblty constrants often mpede effcent allocaton through compettve market mechansms because extensve restrctons on tradng are requred. An example where an ncentve-effcent allocaton s dffcult to decentralze s Atkeson and Lucas [3]. Allowng prvate tradng n these settngs often lmts rsk sharng, a pont recognzed by Jackln [10], Allen and Gale [2], and Farh, Golosov and Tsyvnsk [8], among others, because the exstence of prvate tradng opportuntes wll change an agent s ncentve to reveal nformaton. Jackln [10] and Farh, Golosov and Tsyvnsk use a varaton of a Damond-Dybvg model n whch agents can engage n prvate tradng after observng a prvate shock. Removng ths restrcton on agents actvtes results n compettve equlbra that are not ncentve effcent. Farh et al use a mechansm desgn approach to determne how a socal planner can mplement a polcy restorng ncentve effcency, whch concdes wth the frst-best allocaton. An example where unobserved tradng does not lmt rsk sharng s Cole and Kocherlakota [7], who allow agents to engage n unobserved borrowng and lendng at a rsk free rate. They fnd there s a unque effcent allocaton (frst best) that s decentralzable n compettve asset markets. Pecunary externaltes n economes wth mperfect nformaton and ncomplete markets are studed by Greenwald and Stgltz [9], who focus on when tax nterventons are Pareto mprovng. Klenthong and Townsend [11] study a model of default and endogenous collateralzed contracts n whch there s a pecunary externalty whch can be mtgated usng a market-based soluton. In the adverse selecton nsurance economy wth a pecunary externalty, I fnd there s no tax nterventon mtgatng the externalty, even though the equlbrum has the property the margnal rates of substtuton are not equalzed across agents. The non-exclusvty of contracts n adverse selecton nsurance models has been studed n many papers, recently by Ales and Mazero [1] and Attar, Marott, and Salane [3]. Nonexclusvty n these settngs mples agents can contract wth more than one prncpal and the terms of the contract are prvate nformaton. Ales 3

4 and Mazero, for example, address non-exclusvty assumng nsurance s sold through an ntermedary, there s free entry, and agents can sgn multple contracts wth ntermedares. They determne condtons for a separatng equlbrum. Attar, Marott and Salane [3] provde necessary and suffcent condtons for the exstence of a pure strategy equlbrum. Whle there are mportant dfferences between the papers, they share common features such as any contract traded n equlbrum yelds zero profts, so there are no cross subsdzatons across types and tradng takes place among agents specalzng n ether buyng or sellng of nsurance. I assume agents drectly trade among themselves through standardzed contracts and cross subsdzaton across types s a feature of an equlbrum. The basc model s descrbed n Secton 1 and the ncentve-effcent allocaton s descrbed n Secton 2, and the mplcatons of prvate tradng and non-exclusvty of contracts are dscussed. The problem of constraned effcency wth prvate tradng s stated n Secton 3 along wth a convenent reformulaton of the problem. The exstence of equlbra s problematc and ths ssue s addressed n n Secton 3.2. The soluton to the constraned effcent problem wth prvate tradng s derved n Secton 4. Implementaton of the optmal allocaton s dscussed n Secton 4.1. The market structure and elmnaton of arbtrage proft opportuntes s dscussed n the Appendx. 1 Basc Model The basc model s the adverse selecton nsurance economy of Rothschld and Stgltz [15], Prescott and Townsend [14], Bsn and Gottard [5], and Labade [13]. Ths s a sngle-perod pure endowment economy wth a consumpton good that s tradeable and dvsble. There s a contnuum of agents ndexed over the unt nterval and two types of agents, a and b. A fracton f a of agents are type a and f b = 1 f a are type b. An agent s type s prvate nformaton. The endowment θ s a dscrete random varable takng two values 0 θ 1 < θ 2, so θ 1 s the bad state and θ 2 s the good state. The random varable θ s ndependently dstrbuted across agents. For a type η agent, the probablty of drawng θ s g η, η {a, b} and {1, 2}. Let g a2 > g b2 so type a agents are low rsk and type b are hgh rsk. Denote R η g η1 g η2 for η {a, b}, 4

5 as a measure of type rsk, where R b > R a. The realzaton of θ s publc nformaton. Let θ η = g η1 θ 1 + g η2 θ 2 η = a, b denote the expected endowment for type η, where θ a > θ b, and let θ = f a θa + f b θb (1) denote average endowment. Defne the (uncondtonal) probablty of realzng θ 1 as p η f η g η1 and let R p 1 p. The followng assumpton s standard and states the probablty of beng n any endowment state s strctly postve for ether type of agent. Assumpton 1 g η > 0, η = a, b, = 1, 2. A type-η agent has preferences g η U(c ). (2) Assumpton 2 The functon U s twce contnuously dfferentable, strctly ncreasng and strctly concave. As c 0, U (c) (the Inada condton). The expectaton n (2) uses an agent s condtonal probablty of realzng θ. Snce there s no aggregate uncertanty, the only rsk an agent faces s hs dosyncratc endowment rsk. A consumpton allocaton for a type η agent s c(η) = (c 1 (η), c 2 (η)), where c(η) R 2 +. A par of consumpton allocatons (c(a), c(b)) R 4 + s feasble n the aggregate f the economy-wde resource constrant holds. Snce there s a contnuum of agents, the Law of Large Numbers mples the set of consumpton allocatons feasble n the aggregate s { F (c(a), c(b)) R 4 + θ η } f η g η c (η). (3) Defnton 1: The par of consumpton allocatons (c(a), c(b)) F s ncentve compatble f g η U(c (η)) g η U(c (h)), h η, h, η = a, b. (4) 5

6 Let I F denote the set of feasble consumpton allocatons that are ncentve compatble. The set I s generally not convex and, as dscussed by Prescott and Townsend [14], ths non-convexty makes the applcaton of standard general equlbrum analyss problematc. There s a consumpton externalty created by the constrant (4) because the net trades of an agent n one market must be ndvdually ncentve compatble wth respect to the net trades of an agent n the other market. 2 Incentve Effcency and Prvate Tradng When prvate tradng among agents s prohbted and enforced, the socal planner determnes the statecontngent consumpton of all agents and s able to mplement an ncentve-effcent allocaton. Defnton 2: The consumpton allocaton (c(a), c(b)) I s ncentve effcent f there s no other par ĉ I such that g η U(ĉ (η)) wth strct nequalty for at least one type. g η U(c (η)), η = a, b, (5) Incentve-effcent allocatons have been studed extensvely by Prescott and Townsend, among others. As they have proven, there are three types of ncentve-effcent allocatons for ths economy: ()-() full nsurance for type η and partal nsurance for type h, η h and η, h {a, b} (separatng); () full nsurance for both c (η) = θ (poolng). To decentralze an ncentve-effcent allocaton requres separaton of markets by type, prohbton of prvate tradng, and exclusvty of contracts. Agents can enter only one market - A or B (separaton), one contract (exclusvty), and must consume the quanttes specfed n the contract (no prvate tradng). Agents are offered a menu of contracts by a prncpal, typcally an nsurance company, and agents can enter nto one contract. Any contract traded n equlbrum yelds zero expected profts because of competton among prncpals. All agents are prce takers and equlbrum prces are actuarally far. As a result, there s no cross-subsdzaton across types. A poolng allocaton that s ncentve effcent cannot be decentralzed. Whle not all compettve equlbra are ncentve effcent, mplyng the frst welfare theorem may not hold, a constraned verson of the second welfare theorem does hold: Any ncentve-effcent separatng allocaton can be decentralzed as a compettve equlbrum wth transfers. A detaled descrpton s contaned n Bsn 6

7 and Gottard [6]. Suppose the socal planner chooses a separatng allocaton { c(a), c(b)} that s ncentve-effcent, but prvate tradng among agents cannot be prevented. After an agent self-selects nto a market and purchases an nsurance contract, but before observng the realzaton of hs endowment, agents can enter nto rsksharng arrangements wth other agents. For smplcty, focus on the separatng allocaton n whch the hgh-rsk agent self-selects nto market B and has full consumpton nsurance equal to c. Hgh-rsk agents have no ncentve to engage n further tradng wth other agents n market B. The low-rsk agent self-selects the contract c(a) ( c a1, c a2 ) provdng less than full nsurance. Low-rsk agents would lke to unbundle the contract by separatng ts state-contngent components and then tradng among themselves. Suppose a prvate market opens n whch clams can be traded at prces ˆq a. The agent chooses x a = (x 1, x 2 ) to maxmze hs objectve functon subject to hs prvate-market budget constrant ˆq a1 c a1 + ˆq a2 c a2 ˆq a1 x 1 + ˆq a2 x 2, (6) where the left sde s the prvate market value of the contract selected by a type a agent. If only type a agents enter the orgnal market A and the subsequent prvate market, then the prvate market-clearng prces are ˆq a = g a and the low-rsk agent chooses constant consumpton ĉ a g a1 c a1 + g a2 c a2. At ths pont, there are no further gans from trade. The result of prvate tradng s low-rsk agents are better off whle hgh-rsk agents are no worse off. But ths creates the followng dffculty: If hgh-rsk agents know low-rsk agents ntend to trade n prvate markets, then a hgh-rsk agent may have an ncentve to msrepresent hs type so he too can partcpate n the prvate market. If ĉ a > c b, then a type b agent wll msrepresent hs type to enter market A. But f all agents self-select nto market A, then ĉ a s no longer feasble at the prces (g a1, g a2 ) and the self-selecton nto separate markets breaks down. The dffculty ntroduced by prvate tradng s the new allocaton ĉ a does not satsfy the orgnal ncentve compatblty constrants (4). The exstence of subsequent tradng opportuntes generally changes the ncentves for revealng nformaton. 2 2 The dea subsequent tradng opportuntes can change the nformaton revealed by an agent s studed by Krasa [12], who examnes prvate nformaton exchange economes wth allocatons that cannot be mproved, n the agent would not wsh to devate by revealng further nformaton or by the retradng of goods. 7

8 3 Prvate Tradng As dscussed n the prevous secton, the exstence of prvate-tradng arrangements mples the agents can choose to consume a convex combnaton of the allocaton offered by a socal planner by engagng n prvate tradng. Ths process of unbundlng the state-contngent components of any allocaton s the key property of prvate tradng and motvates the market structure descrbed n ths secton. Suppose agents can trade contngent clams and are prce takers, as n Bsn and Gottard [5]. Let q η denote the prce of a contngent clam n market η {A, B} for state {1, 2}. For smplcty, normalze prces so 1 = q η1 + q η2 η {a, b} and assume q η > 0. Markets are sad to be separate f q η q h for η, h {a, b}, h η, {1, 2} and there s tradng n both markets. If the relatve prce dffers across markets, then an arbtrage opportunty exsts because contracts are not exclusve, and elmnaton of arbtrage proft opportuntes wll result n the equalzaton of prces across markets, so q a = q b, f an equlbrum exsts. The elmnaton of arbtrage profts and the emergng market structure are dscussed n Part B of the Appendx. If the socal planner chooses the poolng allocaton, so c η = θ for all η and, then there s no ncentve to enter nto prvate-tradng arrangements because agents are dentcal across states and types. Whether or not the socal planner chooses the poolng allocaton wll depend on the Pareto weghts. In the dscusson below, I assume the socal planner chooses a separatng allocaton. 3.1 Compettve Equlbrum n the Retradng Market Agents are offered a separatng allocaton n the form of a menu of contracts {c(a), c(b)} I. An agent makes an announcement of hs type and t s assumed all agents of the same type make the same announcement (symmetry). Defne φ η as the fracton of the populaton announcng type η {a, b} such that φ a +φ b = 1 and φ η 0. 3 Followng Klenthong and Townscent, defne the market fundamental as z = {c(a), c(b)}, (φ a, φ b ). 3 Under the assumpton of symmetry, the par φ a, φ b satsfes (φ a, φ b ) {(1, 0), (f a, f b ), (0, 1)}, so all agents announce they are type a, theannouncementstruthful, orallagentsannouncetheyaretypeb, where φ a = f b and φ b = f a has been ruled out under the assumpton the menu of contracts {c(a), c(b)} s ndvdually ncentve compatble. Ths 8

9 The agent receves hs announcement-based state-contngent allocaton and then has access to a compettve market where he can enter nto prvate-tradng arrangements at prces {q 1, q 2 }. For convenence, normalze prces so 1 = q 1 + q 2 and assume q > 0. Denote q = q 1 so q 2 = 1 q and let q Q (0, 1). The market-clearng prce q Q s the soluton to where ˆτ η (c(η), q) = (τ η1 (c(η), q), τ η2 (c(η), q)) s the soluton to subject to f η g ηˆτ η (c(η), q) 0. (7) η max τ η g η U(c η + τ η ) (8) 0 qτ η1 + (1 q)τ η2. (9) Defnton 3: Gven a separatng allocaton {c(a), c(b)} I, an equlbrum n prvate markets conssts of a relatve prce q Q and, for each η {a, b}, consumpton allocatons such that (). ˆτ η (c(η), q) for η {a, b} solve (8) subject to the budget constrant (9). (). The market clearng condton (7) s satsfed. The exstence of an equlbrum s studed n Bsn and Gottard () and Labade () for a common ntal endowment θ 1, θ 2. The exstence of an equlbrum for the more general case s dscussed n Appendx A. 3.2 Constraned Effcency wth Prvate Tradng An agent treats as fxed the menu of contracts {c(a), c(b)} offered by the socal planner and the prce q n the prvate market. An agent chooses hs optmal reportng strategy h {a, b}, determnng hs allocaton c(h) = (c 1 (h), c 2 (h)). Hs after-trade consumpton (x 1, x 2 ) may dffer from hs allocaton c(h) f he chooses to engage n prvate tradng. An agent reportng type h faces a budget set ˆB(c(h), q) {x R+ 2 qx 1 + (1 q)x 2 qc 1 (h) + (1 q)c 2 (h)} (10) n prvate markets. Gven the menu of contracts {c(a), c(b)} and relatve prce q Q, a type η agent solves pont s dscussed below. ˆV ({c(a), c(b)}, q; η) = max {h,x 1,x 2} g η U(x ) (11) 9

10 subject to h {a, b} and (x 1, x 2 ) ˆB(c(h), q). (12) Denote the soluton by ĥ({c(a), c(b)}, q; η) and ˆx({c(a), c(b)}, q; η) = {ˆx 1({c(a), c(b)}, q; η), ˆx 2 ({c(a), c(b)}, q; η)} for η {a, b}. Defnton 4: Gven a separatng allocaton {c(a), c(b)} F, an equlbrum n prvate markets conssts of a relatve prce q Q and, for each η {a, b}, consumpton allocatons and a reported type such that (). ˆx({c(a), c(b)}, q; η) and ĥ({c(a), c(b)}, q; η) for η {a, b} solve (11) subject to h {a, b} and the budget constrant (12). (). Markets clear: f η g ηˆx ({c(a), c(b)}, q; η) η η f η g η c (ĥ({c(a), c(b)}, q; η)). (13) The equlbrum n prvate markets has the property an agent s allocaton s a functon of hs announced type. A compettve equlbrum wth prvate tradng s an equlbrum wth prvate markets such that the announcement-based allocaton {c(ĥ({c(a), c(b)}, q; a)), c(ĥ({c(a), c(b)}, q; b))} s feasble. Defnton 5: Gven a separatng allocaton {c(a), c(b)} F, a compettve equlbrum wth prvate tradng s a relatve prce q Q and an allocaton (ˆx({c(a), c(b)}, q; a), ˆx({c(a), c(b)}, q; b)) such that () The relatve prce q, allocaton (ˆx({c(a), c(b)}, q; a), ˆx({c(a), c(b)}, q; b)), and announcement ĥ({c(a), c(b)}, q; η), for η = {a, b}, are an equlbrum n prvate markets (Defnton 3); () η = ĥ({c(a), c(b)}, q; η) for η {a, b} (truth-tellng); () Feasblty: The announcement-based allocaton {c(ĥ({c(a), c(b)}, q; a)), c(ĥ({c(a), c(b)}, q; b))} F. The exstence of a compettve equlbrum s dscussed n Secton (3.2). The constraned-effcent socal plannng problem wth prvate tradng s descrbed next. The socal planner offers a contract maxmzng the Pareto-weghted expected utltes. Let Ψ η denote a Pareto weght for a type η agent, wth Ψ η > 0 and 1 = Ψ a + Ψ b. The constraned-effcent allocaton wth prvate tradng s the soluton to max {c(a),c(b)} Ψ η g η U(c (η)) (14) η 10

11 subject to θ f η g η c (η), (15) η g a U(c (a)) ˆV ({c(a), c(b)}, q; a), (16) g b U(c (b)) ˆV ({c(a), c(b)}, q; b). (17) The frst constrant (15) s feasblty of the allocaton. The second and thrd constrants, (16) and (17), replace the ndvdual ncentve compatblty constrants (4), where the rght sde s the ndrect utlty functon of an agent tradng n prvate markets. The consumpton allocaton s requred to provde expected utlty at least as hgh as the type η agent can acheve by reportng he s type ĥ({c(a), c(b)}, q; η) and then tradng n prvate markets at relatve prce q. The allocaton has three key propertes: () each agent reports hs type truthfully; () each agent has a zero net trade poston n prvate markets n the sense c(η) s the soluton to (11); () The ndvdual ncentve compatblty condtons are satsfed for each type and hold as strct nequaltes. 4 Ths problem can be convenently reformulated. An Alternate Specfcaton Farh, Golosov and Tsyvnsk (FGT) study a Damond-Dybvg model wth retradng and prove an agent wll base hs announcement of type on the prvate-market value of an allocaton. A smlar argument can be appled n ths adverse selecton settng to reformulate the problem descrbed n (14) (17). In ths alternate specfcaton, the socal planner chooses the market value of an allocaton nstead of the ndvdual components to solve the constraned effcent problem wth prvate tradng. For a gven contract {c(a), c(b)} that s separatng and a prce q Q, an agent s allocaton wll depend on hs announced type h. If there s an h such that qc 1 (h) + (1 q)c 2 (h) > qc 1 (η) + (1 q)c 2 (η) (18) 4 As wll be shown below, each agent wll announce the type wth the hghest market value n prvate markets and wll, therefore, face the same budget set. From the defnton of ˆV, the allocaton ˆx({c(a), c(b)}, q; a), ˆx({c(a), c(b)}, q; b) wll satsfy the ndvdual ncentve compatblty constrants (4). 11

12 for η, h {a, b} and η h, then each agent wll announce he s a type h, regardless of hs true type. Ths follows because the budget set ˆB(c(h), q) s ncreasng n c(h) and the ndrect utlty functon s strctly ncreasng n ts frst argument ˆV ({c(a), c(b)}, q; η) = ˆV (max[qc 1 (a) + (1 q)c 2 (a), qc 1 (b) + (1 q)c 2 (b))], q; η). Hence, f (18) holds for h, η {a, b} and h η, then h({c(a), c(b)}, q) ĥ({c(a), c(b)}, q; a) = ĥ({c(a), c(b)}, q; b) and the functon h can be used to defne the certan endowment w Θ [0, θ] w qc 1 (h({c(a), c(b)}, q)) + (1 q)c 2 (h({c(a), c(b)}, q)). Defne the budget constrant for an agent facng relatve prce q n prvate markets by The type η agent solves B(w, q) {x R 2 + w qx 1 + (1 q)x 2 }. (19) V (w, q; η) = max {x B(w,q)} Denote the soluton by ξ η (w, q) (ξ η1 (w, q), ξ η2 (w, q)). g η U(x ). (20) Defnton 5: A compettve equlbrum n prvate markets wth market endowment w Θ conssts of a relatve prce q Q and consumpton allocatons {ξ a (w, q), ξ b (w, q)} such that (). The demand functons ξ η (w, q) solve (20), for η {a, b}. (). Markets clear: θ η f η g η ξ η (w, q). (21) In the modfed problem, the socal planner maxmzes the Pareto-weghted ndrect utlty by pckng w Θ and relatve prce q Q to solve subject to max {w Θ,q Q} θ η Ψ η V (w, q; η) (22) η f η g η ξ η (w, q). (23) 12

13 Let (w, q ) denote the soluton; the exstence of a soluton s dscussed n Secton (3.2). The two approaches to determnng the constraned-effcent allocaton wth prvate tradng - the orgnal problem (14) (17) and the modfed problem (22) (23) - are llustrated n Fgure (1). The common endowment pont n autarky s E = (θ 1, θ 2 ). The average feasblty constrant s the lne through the pont ( θ, θ) wth slope R, ( ) ( ) horzontal ntercept θ p, 0 θ and vertcal ntercept 0, 1 p. The budget constrant for an agent s the lne through the pont (w, w) wth slope ntercept (0, w 1 q ). ( q 1 q ), horzontal ntercept ( w q, 0 ), and vertcal In the frst problem (14) (17), the socal planner chooses consumpton allocatons {c(a), c(b)}, located on the common budget set n Fgure 1. The expected consumpton of a type η agent s a pont on the 45-degree lne equalng c η g η1 c 1 (η) + g η2 c 2 (η), for η {a, b}. In the second problem (22) (23), the socal planner chooses w Θ and q Q. Each agent, regardless of type, engages n tradng subject to the budget constrant and a type η agent wll pck consumpton c(η) = ξ η (w, q). The budget constrant ntersects the average resource constrant at the pont S. The equlbrum allocaton {ξ a (w, q), ξ b (w, q)} corresponds to a par of contngent clams contracts that are typcally not actuarally far. The rate of exchange between state 1 and state 2 consumpton, the slope of the budget constrant q 1 q, can be thought of as measurng the cost of nsurance n state 1. Far nsurance means the amount expected to be pad out by an agent equals the amount collected from hm n premums. If l η (w, q) = ξ η1 (w, q) w s the amount of consumpton shfted to state 1 (nsurance), then the payments out of state 2 consumpton are per unt of nsurance for a type η agent s q 1 q l η(w, q) and hs expected clam s R η l η (w, q). The dstorton τ η (q) R η q 1 q. The expected subsdy receved by a type η agent, equal to the amount of nsurance tmes the clams mnus payments s generally nonzero. S η (w, q) l η (w, q)τ η (q), The endogenous cross-subsdes S η (w, q) are generated by a pecunary externalty, specfcally the redstrbutve effects of prce changes nduced by the falure to separate markets and the exstence of prvate tradng. 13

14 Ths subsdy can be used to explan the dfference between average endowment θ and the market endowment w. Rewrte the resource constrant wth the demand functons usng the defnton of l η, where ξ η2 (w, q) = w or q 1 q l η(w, q), to show θ = η ( [g η1 (w + l η (w, q)) + g η2 w q )] 1 q l η(w, q) θ w = η f η g η2 S η (w, q). The msprced consumpton nsurance creates subsdes across the agent types whch do not net to zero, so feasblty requres agents receve certan endowment w = Λ(q) such that w < θ for all q Q. Before solvng the socal planner s problem, t s crtcal to establsh the condtons under whch a compettve equlbrum exsts n prvate markets. 3.3 Equlbra wth Prvate Tradng As dscussed n Bsn and Gottard [5] [6], Prescott and Townsend [14], and Rustchn and Sconolf [17], among others, the exstence of a compettve equlbrum can be problematc n these types of models. For a gven w Θ, or more generally gven a state-contngent, feasble and common endowment (θ 1, θ 2 ), an equlbrum may not exst. It s mportant to emphasze, however, the socal planner chooses (w, q) jontly and t s straghtforward to show, for any q Q, there exsts a w Θ such that an equlbrum exsts. Specfcally, for any q Q, there exsts a level of endowment w Θ such that (ξ a (w, q), ξ b (w, q)) solve (20) and the allocaton s feasble,.e. satsfes (21). To develop the argument, observe the soluton ξ η (w, q) to (20) satsfes the frst-order condton U [ ] (ξ η1 (w, q)) q R η U (ξ η2 (w, q)) =. (24) 1 q Snce U s strctly concave and satsfes the Inada condtons and the budget set s convex, t follows the soluton ξ η (w, q), η {a, b} exsts and s unque. Moreover ξ η (w, q) s contnuous, contnuously dfferentable, and strctly ncreasng n w. Also ξ η1 (w, q) s strctly decreasng n q. Snce both agents have dentcal budget sets B(w, q), the demand functons ξ η (w, q) are ncentve compatble for any (w, q). It follows from monotoncty the demand functons have the followng propertes: 14

15 w 1 q θ 1 p Common budget set wth slope q 1 q State 2 Consumpton E S c(a) w c b w θ c θ a a θ b c(b) Average Resource Constrant wth Slope R 0 State 1 Consumpton w q θ p Fgure 1: Budget set and equlbrum wth prvate tradng. Pont E s the autarky endowment pont. The allocaton {c(a), c(b)} s the soluton to the problem (14) (17) and les on the common budget constrant. In the second approach, agents have endowment w, face relatve prce {c(a), c(b)}. q 1 q and engage n trade to consume 15

16 () Snce R b > R a, hgh-rsk agents (type b) purchase more consumpton n state 1 than low-rsk agents (type a), ξ b1 (w, q) > ξ a1 (w, q) for all q Q and w Θ. () If the prce s actuarally far q = g η1 for a type η agent, then ξ η (w, g η1 ) = w. consumpton rsk f q 1 q R η, specfcally Agents take on ξ η1 (w, q) w as q 1 q R η. () Under the assumptons on the utlty functon, t follows lm ξ η1(w, q) = + and lm ξ η1 (w, q) = 0. (25) q 0 q 1 (v) The propertes of the demand functons and the assumptons on the utlty functon mply the ndrect utlty functon V (w, q; η) s quas-convex, contnuous, contnuously dfferentable, and strctly ncreasng n w. The response of V to changes n q depends on whether agents take on consumpton rsk. 5 The feasblty condton after substtutng n the demand functons s θ = η f η g η ξ η (w, q). (26) The next theorem shows for any q Q, there exsts a unque w Θ solvng (26) and ths equlbrum s separatng. Theorem 1 To every q Q, there corresponds a unque w Θ solvng (26). Defne ths value by Λ(q), where Λ s contnuous and contnuously dfferentable n q and satsfes θ = η f η g η ξ η (Λ(q), q). (27) Moreover, 0 < Λ(q) < θ for all q Q. 5 The statement follows because V η(w, q) q = g η2u (ξ η2 (w, q)) [ξ η2 (w, q) ξ η1 (w, q)]. 1 q The sgn s determned by the sgn of [ξ η2 (w, q) ξ η1 (w, q)], whch s postve f q < g η1, 0 f q = g η1, and negatve f q > g η1. 16

17 Proof. Ths s an applcaton of the mplct functon theorem [cf Rudn pp 223]. Snce ξ η (w, q) s contnuous, contnuously dfferentable n w and q, and strctly ncreasng n w, the propertes of the functon Λ are establshed. The property 0 < Λ(q) s obvous because ξ η (w, q) s strctly ncreasng n w and q Q. To demonstrate Λ(q) < θ, suppose to the contrary Λ(q) θ. If Λ(q) > θ, then the consumpton allocaton (w, w) les on the budget set of both types of agents, yet the allocaton sn t feasble. If Λ(q) = θ, then the allocaton s clearly feasble, but there s no (common) equlbrum prce at whch agents would consume θ. To see ths, substtute the budget constrant nto (26) and rewrte, θ (1 p)w 1 q When w = θ, ths expresson becomes [ ] p q θ 1 q = f η g η2 ξ η1 (w, q) = [ f η g η2 ξ η1 ( θ, q) R η [ R η q ]. 1 q q ]. 1 q Consder frst q q 1 q (0, R a]. Snce 1 q R a < R b t follows θ ξ a1 ( θ, q) < ξ b1 ( θ, q) so [ ] p q θ > [ f η g η2 θ R η q ] [ ] = 1 q 1 q θ p q, 1 q whch s a contradcton. If q 1 q (R a, R b ], then ξ a1 ( θ, q) < θ and ξ b1 ( θ, q) θ so [ ] p q θ > [ f η g η2 θ R η q ] [ ] = 1 q 1 q θ p q, 1 q q whch s a contradcton. Fnally, for 1 q (R b, ), observe ξ η1 ( θ, q) < θ for η {a, b}. It follows [ ] p q θ > [ f η g η2 θ R η q ] [ ] = 1 q 1 q θ p q, 1 q whch agan s a contradcton. Hence Λ(q) < θ for q Q. The mplcaton of Theorem 1 s a level of certan endowment w Θ can be determned such that, gven (Λ(q), q), the ξ η (Λ(q), q) solve (20) and markets clear. Fgure (2) plots the functon Λ(q) for U(c) = ln(c) and parameter values f a = 0.6, g a1 = 0.3, and g b1 = The demand functons are ξ η1 = wg η1 q and ξ η2 = g η2w 1 q. Gven q Q, the soluton to (33) s w = Λ(q ) = θ [ ( )] g 2 1 η1 fη q + g2 η2 1 q. 17

18 There are three types of separatng equlbra: () Type a has full consumpton nsurance, consumng w = Λ(g a1 ) and type b agents have partal consumpton nsurance; () Type b agents have full consumpton nsurance w = Λ(g b1 ) and type a agents have partal consumpton nsurance; and () Nether type has full consumpton nsurance, so q g a1 and q g b1. Notce q = p and w = Λ(p) s an example of case () where agents face a budget constrant parallel to and strctly below the average resource constrant. Ths formulaton of the problem doesn t dstngush between hgh-rsk and low-rsk agents. To llustrate ths, suppose q < g a1 n Fgure 1. Then the vertcal ntercept of the resource constrant s greater than the vertcal ntercept of the budget constrant or θ 1 p > w 1 q because Λ(q) < θ. If ths budget constrant ntersects the average resource constrant, t wll do so below the 45-degree lne, reversng the defnton of whch agent type s hgh rsk or low rsk. Specfcally, gven an equlbrum (q, Λ(q)), f the soluton (ˆθ 1, ˆθ 2 ) to the par of equatons θ = pˆθ 1 + (1 p)ˆθ 2 (28) Λ(q) = qˆθ 1 + (1 q)ˆθ 2 (29) has the property ˆθ 1 > ˆθ 2, then the defnton of hgh rsk and low rsk has been reversed. Solvng ths par of equatons results n ˆθ 2 ˆθ 1 = θ Λ(q) q p. (30) Snce Λ(q) < θ, t follows q > p to ensure type a agents are low rsk and type b agents are hgh rsk. If q = p, then ths par of equatons has no soluton because Λ(p) < θ and (28) and (29) do not ntersect. For any q Q, there exsts an equlbrum f w = Λ(q). For any w Θ, there may not exst an equlbrum prce q Q or there may be multple equlbra. The man result s stated n Theorem 2 but before statng that theorem, the followng lemma descrbes the relatonshp between q, Λ(q). Lemma 1 ) If 0 < q < p, then type a agents are hgh rsk and type b are the low rsk n the prvate tradng equlbrum (q, Λ(q)). () Let q m denote a soluton to 0 = Λ (q), where there may be more than one soluton. Then g a1 < q m < g b1. Proof. Part () follows from (30) because Λ(q) < θ for all q Q so, f ˆθ 2 > ˆθ 1, where (ˆθ 1, ˆθ 2 ) solve (28) (29), then 18

19 θ For each q Q, w = Λ(q) s the equlbrum value of the endowment such that agents are maxmzng utlty subject to the budget constrant and the allocaton s feasble. Endowment w w = Λ(q) 0 R a R b Relatve prce q 1 q Fgure 2: Equlbrum endowment as a functon of the relatve prce. For a gven relatve prce 0 < q < 1, w = Λ(q) s the level of endowment such that agents maxmze expected utllty, markets clear and the consumpton allocaton s feasble. 19

20 q p. By assumpton g a1 < g b1 so, f ˆθ 2 < ˆθ 1, the defnton of hgh rsk and low rsk has been reversed. For part (), the functon Λ s contnuous and dfferentable. Dfferentatng Λ wth respect to q Λ (q) = [ fη gη = [ fη g η2 ] [ ξ η (w, q) ] 1 ξ η (w, q) fη gη q w ( ξη1 (w, q) The denomnator of the rght sde s strctly postve. Recall ξη1(w,q) q q τ η (q) + ξ )] [ η2(w, q) ξ η1 (w, q) ] 1 ξ η (w, q) fη gη. 1 q w < 0. If q g a1, then τ a (q) 0 and τ b (q) > 0. Also ξ a2 (w, q) ξ a1 (w, q) 0 and ξ b2 (w, q) ξ b1 (w, q) < 0 so the numerator n brackets s negatve, so the expresson on the rght sde s postve. Hence Λ (q) > 0 for q g a1. If q > g b1, then τ a (q) < 0 and τ b (q) 0 so the term n the numerator n brackets s postve. Also ξ a2 (w, q) ξ a1 (w, q) > 0 and ξ b2 (w, q) ξ b1 (w, q) 0 so the rght sde s negatve. Hence Λ (q) < 0 for q g b1. It follows g a1 < q m < g b1. The followng theorem summarzes the man results about the exstence of equlbra n the prvate tradng economy. Theorem 2 () Let W = Λ(Q) denote the range of Λ. For any w W, there exsts a prvate tradng equlbrum prce q where q Q. () Let w m = Λ(q m ) where q m s the soluton to 0 = Λ (q). If there s more than one soluton q m, defne w m as the maxmum value or w m = max q m Λ(q m ). Then W (w m, θ] =, so f w (w m, θ] no prvate tradng equlbrum exsts. () If q m < p for all q m, then, for any w W, there exsts a unque equlbrum prce q such that q > p. If there s some q m such that q m > p, then for any w [Λ(p), w m ], there exsts at least two equlbra (q 1, q 2 ) such that p < q 1 < q m and q 2 > q m. Proof. For part (), t follows from the propertes of the functon Λ establshed n Theorem 1 the range W = Λ(Q) s well-defned and an equlbrum q Q exsts for any w W ; however t may not be unque. For part (), for any q Q, the equlbrum value of the market endowment satsfes w = Λ(q) w m, so the statement follows. For part (), the assumpton type a agents are low rsk requres q p. If q m < p then for any q [p, 1), Λ (q) < 0. Hence for w (0, w m ], there exsts a unque equlbrum q [p, 1). If there s some q m 20

21 such that p < q m, then for each w [Λ(p), w m ] contnuty of Λ mples there exsts at least two equlbra q 1, q 2 such that q 1 < q m wth Λ (q 1 ) > 0 and q 2 > q m wth Λ (q 2 ) < 0. For any feasble exogenous endowment (θ 1, θ 2 ) satsfyng (28) (29) such that θ 2 > θ 1, the equlbrum prce s the fxed pont of the mappng T : Q Q defned by q = T q = θ 2 Λ(q) θ 2 θ 1 (31) There may be no fxed pont or multple fxed ponts to (31). The operator T s not monotonc because the rght sde s decreasng n q f q < g a1 whle t s ncreasng f q > g b1. 4 Soluton to the Socal Plannng Problem An optmal allocaton s ether poolng or separatng. If the soluton s poolng, then all agents consume θ. If t s separatng, then prvate tradng alters the socal plannng problem specfed n (14) (17). The equvalence between the orgnal problem (14)-(17) and the modfed problem (22)-(23) s summarzed n Theorem 3; the proof follows the proof of Lemma 1 n FGT and therefore s contaned n the Appendx Part A. Theorem 3 Let (w, q ) be a soluton to (22)-(23) and let {ξ a (w, q ), ξ b (w, q )} be a soluton to (20), gven (w, q ), where (w, q ) s a prvate tradng compettve equlbrum. Then {c(a), c(b)} defned by c(η) = (ξ η1 (w, q ), ξ η2 (w, q )) for all η {a, b} (32) s a soluton to (14) (17). Conversely, let {c(a), c(b)} be a soluton to (14) (17) wth the property t s a separatng allocaton. Then there exsts a q Q and w = Λ(q) Θ solvng (22) subject to (23) f ξ η (w, q ) = c(η) such that {ξ a (w, q ), ξ b (w, q )} solve (20). Snce the proof s based on the proof n FGT, t s contaned n Appendx A. Theorem 2 summarzed the condtons for the exstence and unqueness of an equlbrum. Theorem 3 summarzes the results for the socal plannng problem when a separatng allocaton s chosen and an equlbrum exsts n prvate markets. A socal planner wll choose q Q and w = Λ(q) to solve (20) 21

22 and a soluton exsts. Snce the allocaton {ξ a (w, q ), ξ b (w, q )} solves the socal plannng problem (14) (17) so the compettve equlbrum (w, q ) s constraned effcent wth prvate tradng. Any nterventon shftng (q, Λ(q)) wll change the allocaton but s not Pareto mprovng. In the absence of exclusvty of contracts, preventon of prvate tradng, and enforced separaton of markets, there s no nterventon by the socal planner that can elmnate the pecunary externalty and move the economy to an ncentve-effcent allocaton. 7 Wthn the space of separatng allocatons, the socal plannng problem (22) s formulated as V(Ψ) max [ΨV (Λ(q), q; a) + (1 Ψ)V (Λ(q), q; b)]. (33) {q Q} For each 0 < Ψ < 1, the optmal allocaton (separatng or poolng) s the soluton to W (Ψ a, Ψ b ) = max [ U( θ), V(Ψ a, Ψ b ) ] (34) Denote The frst-order condton for (33) s µ η V (w, q; η). w 0 = Λ (q) [Ψµ a + (1 Ψ)µ b ] + Ψµ a [ξ a2 (w, q) ξ a1 (w, q)] + (1 Ψ)µ b [ξ b2 (w, q) ξ b1 (w, q)] where Roy s Identty as been used, whch takes the form n ths applcaton. 8 V η (w, q) q = µ η [ξ η1 (w, q) ξ η2 (w, q)] 7 Let c(a), c(b) be a soluton to (14) (17) wth equlbrum prce q and w = Λ(q) and assume to the contrary ĉ(a), ĉ(b) s a feasble allocaton such that one type of agent s better off whle the other s no worse off. Agents face dentcal relatve prces otherwse an arbtrage proft opportunty exsts. If the market value of one allocaton exceeds the other, then all agents wll announce the same type. If ĉ(a), ĉ(b) les on the budget set B(w, q), then the agent who s better off would have chosen that allocaton, contradctng the assumpton c(a), c(b) solves (14) (17). If the allocaton ĉ(a), ĉ(b) les above the budget set, then the allocaton s not feasble. 8 If q m > p and q = q m, then the Pareto weght Ψ satsfes Ψ = µ b [ξ b1 (Λ(q m ), q m ) ξ b2 (Λ(q m ), q m )) µ a(ξ a2 (Λ(q m ), q m ) ξ a1 (Λ(q m ), q m )) + µ b (ξ b1 (Λ(q m ), q m ) ξ b2 Λ(q m ), q m ) so the Pareto weght depends on the share of consumpton nsurance, where l η(w, q) = ξ η1 (w, q) ξ η2 (w, q). 22

23 S State 2 Consumpton c(a) E Budget constrant θ c a V a w c b w c(b) Average Resource Constrant V b 0 State 1 Consumpton Fgure 3: Example of optmal separatng allocaton 23

24 Fgure (3) llustrates the optmal soluton, 9 whch s a separatng allocaton when the parameters are f a = 0.4, g a1 = 0.2, and g b1 = 0.7. The endowment pont (autarky) s E = [2, 8] and θ = 5. The equlbrum prce s q e = 0.68, whch les between g a1 and g b1, and the optmal level of ncome s w = Notce U( θ) = 1.61 and the separaton allocaton results n Pareto-weghted expected utlty The ndfference curves are V a and V b. 4.1 Implementaton Startng from the common endowment (θ 1, θ 2 ), the ex ante problem solved by a type η agent s subject to max c η g η θ qc η1 + (1 q)c η2 + g η U(c η ) (35) g η1 [θ w] (36) The socal planner can choose a q (0, 1) and determne rghts to trade n prvate markets equal to θ w, whch s ncentve compatble. The optmal allocaton {c (a), c (b)}, the soluton to (14) (17), and the assocated equlbrum prce q can be mplemented n several ways. Clearly, the socal planner can drectly allocate {c (a), c (b)}, an allocaton from whch there s no ncentve to engage n prvate tradng. A second approach s for the socal planner to allocate w = Λ(q ), where w satsfes w = q c 1 (a) + (1 q )c 2 (a) = q c 1 (b) + (1 q )c 2 (b), whch results n the equlbrum {c (a), c (b)}. Ths allocaton can be mplemented by a set of taxes faced by all agents regardless of type {ˆτ 1, ˆτ 2 }, where ˆτ = w θ. A thrd way to mplement the allocaton s for the 9 A closed form soluton can be computed when U(c) = log(c). The soluton s where The value of w s q = K 1 + K [ fηg η1 2 K = [1 ] Ψ ηg η1 ] Ψηgη1 fηg η2 2. w = Λ(q ) = θ [ ( )] g 2 1 η1 fη q + g2 η2 1 q. 24

25 socal planner to choose a state-contngent allocaton that s dentcal for all agents, specfcally an allocaton (ˆθ 1, ˆθ 2 ) solvng θ = p 1 ˆθ1 + p 2 ˆθ2 (37) Λ(q ) = q ˆθ1 + ˆθ 2. (38) Tradng wll occur at prce q such that agents wll consume {c (a), c (b)}. Pont S n Fgure 1 llustrates a soluton to the par of equatons (37)-(38). 10 As mentoned earler, not every optmal allocaton can be mplemented by the approprate choce of a common state-contngent endowment (ˆθ 1, ˆθ 2 ) solvng (37)-(38). Consder the followng counterexample: Fx q = p and determne w = Λ(p), whch s an equlbrum. The budget constrant Λ(p) px 1 + (1 p)x 2 does not ntersect the average feasblty constrant at any pont. Hence, there exst optmal allocatons mplementable by a choce of (q, w) that cannot be acheved by choosng a common endowment (ˆθ 1, ˆθ 2 ) along the feasblty constrant. 5 Concluson Prvate nformaton creates frctons n tradng that are mportant n understandng the structure of fnancal markets and rsk sharng more broadly. Incentve-effcent allocatons n the optmal contractng lterature generally requre extensve restrctons on the actons of agents, because the ndvdual ncentve compatblty constrants create a consumpton externalty n that the net trades of one agent must be ndvdually ncentve compatble wth the net trades of other agents. These extensve restrctons on agents actvtes may be mplausble n decentralzed fnancal markets n whch agents have an ncentve to elmnate arbtrage proft opportuntes and mprove rsk sharng. An exchange economy wth adverse selecton and prvate nformaton s studed under the assumpton rsk averse agents trade drectly n a contngent clams market. Markets are 10 To see ths, defne z η (w, q ) ξ η (w, q ) ˆθ and observe 0 = f η gη z η (w, q ) so at q, the z η (w, q ) for η = a, b and = 1, 2 are the equlbrum net trades. 25

26 not separated by type, contracts are not exclusve, and agents can enter nto prvate-tradng arrangements. The resultng equlbrum allocaton s ndvdually ncentve compatble, although t s not ncentve-effcent. Snce agents face the same budget constrant n prvate-tradng markets, but face dfferent endowment dstrbuton rsk, agents wll execute dfferent net trades dependng on type. The result that some states are under-nsured whle others are over-nsured for an agent. There s a pecunary externalty created through the prce system, rasng the queston whether mplementaton of a tax-subsdy scheme can mtgate the mpact of the externalty. As long as prvate tradng arrangements cannot be prevented, the compettve equlbrum s constraned effcent (thrd best) and nterventons wll not be Pareto mprovng. 26

27 Appendx Secton A: Exstence of a Compettve Equlbrum For any ntal feasble and ncentve compatble allocaton {c(a), c(b)}, defne Q(c(η)) = R η U (c η1 ) U (c η2 ) () Snce R b > R a, hgh-rsk agents (type b) purchase more consumpton n state 1 than low-rsk agents (type a), ξ b1 (w, q) > ξ a1 (w, q) for all q Q and w Θ. () If the prce s actuarally far q = g η1 for a type η agent, then ξ η (w, g η1 ) = w. consumpton rsk f q 1 q R η, specfcally ξ η1 (w, q) w as q 1 q R η. Agents take on Secton B: The proof of Theorem 1 s a straghtforward applcaton of the proof to Lemma 1 n FGT. Proof. The frst step s to show the soluton to (14) (17), denoted {c (a), c (b)} can be mplemented for some w and q satsfyng (23), n that {c (a), c (b)} would solve (20). Start wth a soluton {c(a), c(b)} to (14) that s a separatng allocaton and let q Q be the assocated equlbrum prce, as n Defnton 2. The ncentve compatblty constrants (16)-(17) can be wrtten g η U(c (η)) ˆV (qc 1 (h) + (1 q)c 2 (h), q; η) for all h {a, b} and each η. From the defnton of ˆV n (11) t follows ˆV (qc 1 (ĥ({c(a), c(b)}, q; η))+(1 q)c2(ĥ({c(a), c(b)}, q; η)), q; η) ˆV (qc 1 (h)+(1 q)c 2 (h), q; η) for all h {a, b} whch s equvalent to ˆV ({c(a), c(b)}, q; η) ˆV (max[qc 1 (a) + (1 q)c 2 (a), qc 1 (b) + (1 q)c 2 (b)], q; η) (39) Snce ˆV s strctly ncreasng n ts frst argument, t follows qc 1 (η) + (1 q)c 2 (η) = max [qc 1(h) + (1 q)c 2 (h)] h {a,b} 27

28 or qc 1 (a) + (1 q)c 2 (a) = qc 1 (b) + (1 q)c 2 (b). (40) Denote ths value as ŵ. The next step s to use ŵ and q n (20) and show {c(a), c(b)} solve (20). Snce the equalty n (40) s a property of the soluton to (11), then ths mples (22) f ξ(w, q; η) = c(η). The allocaton {c(a), c(b)} satsfes the socal planner s feasblty constrant (15) and s stll a soluton under the restrcton ĥ({c(a), c(b)}, q; a) = ĥ({c(a), c(b)}, q; b) whch s mplct n (20). Snce {c(a), c(b)} s a soluton to (14), t follows {c(a), c(b)} can be mplemented by the approprate choce of w, q. Moreover ˆV ({c(a), c(b)}, q; η) = V (w, q; η) where ŵ s defned n (40). The second step s to take (w, q ) solvng (22) (23) and show the {c(a), c(b)} gven by c(η) = (ξ η1 (w, q ), ξ η2 (w, q )), where ξ η (w, q ) solve (20), are feasble and solve (14). Gven (w, q), the allocaton {c(a), c(b)} wll solve (20) for any η and, because {c(a), c(b)} satsfes (15) ths mples (23) f ξ η (w, q) = c (η) for all η and. By assumpton {c(a), c(b)} s a soluton to (11) and s stll a soluton under the addtonal constrant ĥ({c(a), c(b)}, q; a) = ĥ({c(a), c(b)}, q; b). Hence any soluton {c(a), c(b)} to (14) that s a separatng allocaton may be mplemented through the approprate choce of w and q. Moreover the values of the maxmand n (14) and (22) are equal for these parameter values; hence the maxmum n (22) s at least as large as the one n (14). Suppose (w, q ) solve problem (22) and let {c(a), c(b)} be gven by c (η) = ξ η (w, q ), where {ξ a (w, q ), ξ b (w, q )} s the soluton to (20) gven (w, q ). The next step s to determne f {c(a), c(b)} s feasble for (14), so t satsfes (15). If (23) s satsfed by {ξ a (w, q ), (ξ b (w, q )} and c(η) = ξ η (w, q ) then clearly (15) follows from (23). Snce the budget constrant n problem (22) s bndng, t follows q c 1 (a) + c 2 (a) = q c 1 (b) + c 2 (b) = w so ˆV ({c(a), c(b)}, q ; η) = V (w, q ; η) for all η {a, b}, 28

29 because wth certan endowment w, gven ths relatve prce q and prvate markets, a consumer does as well reportng truthfully η = ĥ({c(a), c(b)}, q; η) as he does by lyng, h = ĥ({c(a), c(b)}, q; η) h η. The ncentve compatblty constrants (16) (17) follow from the property {ξ a (w, q ), ξ b (w, q )} solve (20). Hence, f (w, q ) solve (22), then the nduced allocatons are feasble for problem (14). Also, the maxmands are equal, and the maxmum n (14) s at least as large as the one n (22). Hence, f the allocaton s separatng, the maxmums n (14) and (22) concde. Any separatng allocaton {c(a), c(b)} solvng (14) nduces a (w, q ) solvng (22). Secton B: Dscusson of Market Structure The market structure assumed n Secton 3 s dscussed. Agents trade drectly among themselves after recevng the announcement-based endowment and t s assumed there s no fnancal ntermedary. There are two possbltes for the organzaton of compettve markets when agents trade drectly among themselves: () There s ether a sngle market n whch all clams contngent on realzaton {1, 2} trade for the dentcal prce vector q h, 1 q h, or else () there are separate markets A and B, where contngent clams trade at prces {(q a, 1 q q ), (q b, 1 q b )} and q a q b. The socal planner offers a consumpton vector c I. An agent who announces he s type η receves c(η). If the agent subsequently trades n market h, h {A, B}, he faces a budget constrant q h c η1 + (1 q h )c η2 = q h x η1 + (1 q h )x η2 η, h {a, b}. Non-exclusvty of contracts mples agents can enter nto multple trades and current trades are not condtonal on prevous trades. The frst ssue s whether markets can be separate when there s prvate nformaton about type and non-exclusvty n contracts. An agent can enter nto contngent clams markets multple tmes as long as the budget constrant s satsfed. There are three possbltes for c I: () both types have full nsurance; () one type has full nsurance whle the other does not; () nether type has full nsurance. To start, suppose there are separate markets. 29