Optimality in an Adverse Selection Insurance Economy. with Private Trading. April 2015

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1 Optmalty n an Adverse Selecton Insurance Economy wth Prvate Tradng Aprl 2015 Pamela Labade 1 Abstract An externalty s created n an adverse selecton nsurance economy because of the nteracton between prvate nformaton and the possblty of prvate tradng. The consumpton possblty set of an agent depends on the collectve decsons of all agents through the ncentve compatblty constrants, whch nclude the prces and opportuntes created by prvate tradng. When agents have prvate nformaton about the dstrbuton of ther dosyncratc endowment shocks, decentralzng an ncentve-effcent allocaton generally reures 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 prvate tradng prces that are typcally not actuarally far for any type of agent and mposng lmts on rsk sharng. Agents wll choose to under or over nsure even though full consumpton nsurance s n ther budget set, and essentally trade probablty dstrbutons, as n the rensurance market. When agents trade contngent clams at common prces, the compettve eulbra may not be prvate-tradng constraned effcent (thrd best) because of neffcent cross-subsdzaton across dfferent types of agents. Imposng a smple prce celng on the prce of nsurance wll restore effcency. JEL Classfcaton: D81, D82. Keywords: Adverse selecton, prvate-tradng arrangements, externaltes, rensurance 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 nformaton creates frctons n tradng that are mportant n understandng the structure of fnancal markets and rsk-sharng more broadly. When prvate tradng opportuntes nteract wth prvate nformaton, an externalty s created n an adverse selecton nsurance economy. The ndvdual ncentve compatblty constrants create an externalty n that the net trades of one set of agents must be ncentve compatble wth respect to the net trades of other agents. When prvate tradng opportuntes are added, the ncentve compatblty constrants are modfed to ncorporate how prvate tradng opportuntes, ncludng prces, alter the ncentves to reveal nformaton. Satsfyng the ncentve compatblty constrants wth prvate tradng prevents mplementng ncentve-effcent allocatons because the reured restrctons are not satsfed, such as separaton of markets by type, exclusvty of contracts and strct prohbton of prvate tradng. Prvate-tradng constraned effcent allocatons can be defned wthn the context of the model. Compettve eulbra are not generally prvate tradng constraned effcent because of neffcent crosssubsdzaton across dfferent types of agents. Imposng a smple prce celng on the prce of nsurance wll restore effcency. When prvate tradng cannot be prevented, agents reveal nformaton about ther type based on the value of an allocaton n the prvate tradng market. 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 eulbrum prces. In the reformulated problem, the socal planner pcks the market value of the allocaton and eulbrum prce. The condtons for exstence and unueness of compettve eulbra are establshed and the two socal plannng problems are shown to be euvalent under certan condtons. The exstence of compettve eulbra 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 eulbrum 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 uanttes 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 eulbra 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 eulbrum exsts, although t may not be unue. 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 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 eulbrum. Attar, Marott and Salane [3] provde necessary and suffcent condtons for the exstence of a pure strategy eulbrum. Whle there are mportant dfferences between the papers, they share common features such as any contract traded n eulbrum 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 eulbrum. 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 eulbra 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]. 3

4 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}, 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). 4

5 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 } f η g η c (η). (3) η 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) 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 eulbrum 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 neualty 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 5

6 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 reures 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 uanttes 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 eulbrum yelds zero expected profts because of competton among prncpals. All agents are prce takers and eulbrum 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 eulbra 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 eulbrum wth transfers. A detaled descrpton s contaned n Bsn 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 eual 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 ˆ a. The agent chooses x a = (x 1, x 2 ) to maxmze hs objectve functon subject to hs prvate-market budget constrant ˆ a1 c a1 + ˆ a2 c a2 ˆ a1 x 1 + ˆ 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 subseuent prvate market, then the prvate market-clearng prces are ˆ 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. 6

7 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 subseuent tradng opportuntes generally changes the ncentves for revealng nformaton. 2 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 η denote the prce of a contngent clam n market η {A, B} for state {1, 2}. For smplcty, normalze prces so 1 = η1 + η2 η {a, b} and assume η > 0. Markets are sad to be separate f η 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 eualzaton of prces across markets, so a = b, f an eulbrum 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 the 2 The dea subseuent 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 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 Eulbrum wth Prvate Tradng 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). For convenence, normalze prces so 1 = and assume > 0. Denote = 1 so 2 = 1 and let Q (0, 1). An agent treats as fxed the menu of contracts {c(a), c(b)} and the prce 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), ) {x R 2 + x 1 + (1 )x 2 c 1 (h) + (1 )c 2 (h)} (7) n prvate markets. Gven the menu of contracts {c(a), c(b)} and relatve prce Q, a type η agent solves subject to h {a, b} and ˆV ({c(a), c(b)}, ; η) = max {h,x 1,x 2} g η U(x ) (8) (x 1, x 2 ) ˆB(c(h), ). (9) Denote the soluton by ĥ({c(a), c(b)}, ; η) and ˆx({c(a), c(b)}, ; η) = {ˆx 1({c(a), c(b)}, ; η), ˆx 2 ({c(a), c(b)}, ; η)} for η {a, b}. Defnton 4: Gven a separatng allocaton {c(a), c(b)} I, an eulbrum n prvate markets conssts of a relatve prce Q and, for each η {a, b}, consumpton allocatons and a reported type such that (). ˆx({c(a), c(b)}, ; η) and ĥ({c(a), c(b)}, ; η) for η {a, b} solve (8) subject to h {a, b} and the budget constrant (9). (). Markets clear: f η g ηˆx ({c(a), c(b)}, ; η) η η f η g η c (ĥ({c(a), c(b)}, ; η)). (10) 8

9 The eulbrum n prvate markets has the property an agent s allocaton s a functon of hs announced type. A compettve eulbrum wth prvate tradng s an eulbrum wth prvate markets such that the announcement-based allocaton {c(ĥ({c(a), c(b)}, ; a)), c(ĥ({c(a), c(b)}, ; b))} s feasble. Defnton 5: Gven a separatng allocaton {c(a), c(b)} F, a compettve eulbrum wth prvate tradng s a relatve prce Q and an allocaton (ˆx({c(a), c(b)}, ; a), ˆx({c(a), c(b)}, ; b)) such that () The relatve prce, allocaton (ˆx({c(a), c(b)}, ; a), ˆx({c(a), c(b)}, ; b)), and announcement ĥ({c(a), c(b)}, ; η), for η = {a, b}, are an eulbrum n prvate markets (Defnton 3); () η = ĥ({c(a), c(b)}, ; η) for η {a, b} (truth-tellng); () Feasblty: The announcement-based allocaton {c(ĥ({c(a), c(b)}, ; a)), c(ĥ({c(a), c(b)}, ; b))} F. The exstence of a compettve eulbrum s dscussed n Secton (3.2). The constraned-effcent socal plannng problem wth prvate tradng s descrbed next. 3.2 Constraned Effcency wth Prvate Tradng 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 subject to max {c(a),c(b)} Ψ η g η U(c (η)) (11) η θ f η g η c (η), (12) η g a U(c (a)) ˆV ({c(a), c(b)}, ; a), (13) g b U(c (b)) ˆV ({c(a), c(b)}, ; b). (14) The frst constrant (12) s feasblty of the allocaton. The second and thrd constrants, (13) and (14), 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 reured to provde expected 9

10 utlty at least as hgh as the type η agent can acheve by reportng he s type ĥ({c(a), c(b)}, ; η) and then tradng n prvate markets at relatve prce. 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 (8); () The ndvdual ncentve compatblty condtons are satsfed for each type and hold as strct neualtes. 3 can be convenently reformulated. Ths problem 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 (11) (14). 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, an agent s allocaton wll depend on hs announced type h. If there s an h such that c 1 (h) + (1 )c 2 (h) > c 1 (η) + (1 )c 2 (η) (15) 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), ) s ncreasng n c(h) and the ndrect utlty functon s strctly ncreasng n ts frst argument ˆV ({c(a), c(b)}, ; η) = ˆV (max[c 1 (a) + (1 )c 2 (a), c 1 (b) + (1 )c 2 (b))], ; η). Hence, f (15) holds for h, η {a, b} and h η, then h({c(a), c(b)}, ) ĥ({c(a), c(b)}, ; a) = ĥ({c(a), c(b)}, ; b) and the functon h can be used to defne the certan endowment w [0, θ] w c 1 (h({c(a), c(b)}, )) + (1 )c 2 (h({c(a), c(b)}, )). 3 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)}, ; a), ˆx({c(a), c(b)}, ; b) wll satsfy the ndvdual ncentve compatblty constrants (4). 10

11 Defne the budget constrant for an agent facng relatve prce n prvate markets by B(w, ) {x R 2 + w x 1 + (1 )x 2 }. (16) The type η agent solves V (w, ; η) = max {x B(w,)} Denote the soluton by ξ η (w, ) (ξ η1 (w, ), ξ η2 (w, )). g η U(x ). (17) Defnton 5: A compettve eulbrum n prvate markets wth market endowment w [0, θ] conssts of a relatve prce Q and consumpton allocatons {ξ a (w, ), ξ b (w, )} such that (). The demand functons ξ η (w, ) solve (17), for η {a, b}. (). Markets clear: θ η f η g η ξ η (w, ). (18) In the modfed problem, the socal planner maxmzes the Pareto-weghted ndrect utlty by pckng w [0, θ] and relatve prce Q to solve subject to max {w Θ, Q} θ η Ψ η V (w, ; η) (19) η f η g η ξ η (w, ). (20) Let (w, ) 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 (11) (14) and the modfed problem (19) (20) - 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 ). ( 1 ), horzontal ntercept ( w, 0 ), and vertcal In the frst problem (11) (14), 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 eualng c η g η1 c 1 (η) + g η2 c 2 (η), for η {a, b}. In the second problem (19) (20), the socal 11

12 planner chooses w Θ and 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, ). The budget constrant ntersects the average resource constrant at the pont S. The eulbrum allocaton {ξ a (w, ), ξ b (w, )} 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 1, 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 euals the amount collected from hm n premums. If l η (w, ) = ξ η1 (w, ) 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 1 l η(w, ) and hs expected clam s R η l η (w, ). The dstorton τ η () g η1 g [ η2 1 = g η2 R η ]. 1 The expected subsdy receved by a type η agent, eual to the amount of nsurance tmes the clams mnus payments s generally nonzero. S η (w, ) l η (w, )τ η (), 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, ) = w or 1 l η(w, ), to show θ = η [ ( f η g η1 (w + l η (w, )) + g η2 w )] 1 l η(w, ) θ w = η f η g η2 S η (w, ) = (θ 2 θ 1 )( p). The consumpton nsurance offered at common prces creates subsdes across the agent types whch do not net to zero, so feasblty reures agents receve market endowment w = Λ() such that w < θ for all Q. Before solvng the socal planner s problem, t s crtcal to establsh the condtons under whch a compettve eulbrum exsts n prvate markets. 12

13 w 1 θ 1 p Common budget set wth slope 1 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 θ p Fgure 1: Budget set and eulbrum wth prvate tradng. Pont E s the autarky endowment pont. The allocaton {c(a), c(b)} s the soluton to the problem (11) (14) and les on the common budget constrant. In the second approach, agents have endowment w, face relatve prce {c(a), c(b)}. 1 and engage n trade to consume 13

14 3.3 Eulbra 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 eulbrum can be problematc n these types of models. For a gven w [0, θ], or more generally gven a state-contngent, feasble and common endowment (θ 1, θ 2 ), an eulbrum may not exst. It s mportant to emphasze, however, the socal planner chooses (w, ) jontly and t s straghtforward to show, for any Q, there exsts a w [0, θ] such that an eulbrum exsts. Specfcally, for any Q, there exsts a level of endowment w [0, θ] such that (ξ a (w, ), ξ b (w, )) solve (17) and the allocaton s feasble,.e. satsfes (18). To develop the argument, observe the soluton ξ η (w, ) to (17) satsfes the frst-order condton U [ ] (ξ η1 (w, )) R η U (ξ η2 (w, )) =. (21) 1 Snce U s strctly concave and satsfes the Inada condtons and the budget set s convex, t follows the soluton ξ η (w, ), η {a, b} exsts and s unue. Moreover ξ η (w, ) s contnuous, contnuously dfferentable, and strctly ncreasng n w. Also ξ η1 (w, ) s strctly decreasng n. Snce both agents have dentcal budget sets B(w, ), the demand functons ξ η (w, ) are ncentve compatble for any (w, ). It follows from monotoncty the demand functons have the followng propertes: () 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, ) > ξ a1 (w, ) for all Q and w [0, θ]. () If the prce s actuarally far = g η1 for a type η agent, then ξ η (w, g η1 ) = w. Agents take on consumpton rsk f 1 R η, specfcally ξ η1 (w, ) w as 1 R η. () Under the assumptons on the utlty functon, t follows lm ξ η1(w, ) = + and lm ξ η1 (w, ) = 0. (22) 0 1 (v) The propertes of the demand functons and the assumptons on the utlty functon mply the ndrect utlty functon V (w, ; η) s uas-convex, contnuously dfferentable, and strctly ncreasng n w. 14

15 The response of V to changes n depends on whether agents take on consumpton rsk. 4 The feasblty condton after substtutng n the demand functons s θ = η f η g η ξ η (w, ). (23) The next theorem shows for any Q, there exsts a unue w [0, θ] solvng (23) and ths eulbrum s separatng. Theorem 1 To every Q, there corresponds a unue w [0, θ] solvng (23). Defne ths value by Λ(), where Λ s contnuously dfferentable n and satsfes θ = η f η g η ξ η (Λ(), ). (24) Moreover, 0 < Λ() < θ for all Q. Proof. Ths s an applcaton of the mplct functon theorem [cf Rudn pp 223]. Snce ξ η (w, ) s contnuously dfferentable n w and, and strctly ncreasng n w, the propertes of the functon Λ are establshed. The property 0 < Λ() s obvous because ξ η (w, ) s strctly ncreasng n w and Q. To demonstrate Λ() < θ, suppose to the contrary Λ() θ. If Λ() > 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() = θ, then the allocaton s clearly feasble, but there s no (common) eulbrum prce at whch agents would consume θ. To see ths, substtute the budget constrant nto (23) and rewrte, θ When w = θ, ths expresson becomes 4 The statement follows because θ (1 p)w 1 = [ f η g η2 ξ η1 (w, ) R η ]. 1 [ ] p = [ f η g η2 ξ η1 ( θ, ) R η ]. 1 1 V η(w, ) = g η2u (ξ η2 (w, )) [ξ η2 (w, ) ξ η1 (w, )]. 1 The sgn s determned by the sgn of [ξ η2 (w, ) ξ η1 (w, )], whch s postve f < g η1, 0 f = g η1, and negatve f > g η1. 15

16 Consder frst 1 (0, R a]. Snce 1 R a < R b t follows θ ξ a1 ( θ, ) < ξ b1 ( θ, ) so [ ] p θ > [ f η g η2 θ R η ] [ ] = 1 1 θ p, 1 whch s a contradcton. If 1 (R a, R b ], then ξ a1 ( θ, ) < θ and ξ b1 ( θ, ) θ so [ ] p θ > [ f η g η2 θ R η ] [ ] = 1 1 θ p, 1 whch s a contradcton. Fnally, for 1 (R b, ), observe ξ η1 ( θ, ) < θ for η {a, b}. It follows [ ] p θ > [ f η g η2 θ R η ] [ ] = 1 1 θ p, 1 whch agan s a contradcton. Hence Λ() < θ for Q. The mplcaton of Theorem 1 s a level of certan endowment w [0, θ] can be determned such that, gven (Λ(), ), the ξ η (Λ(), ) solve (17) and markets clear. Fgure (2) plots the functon Λ() for U(c) = ln(c) and parameter values f a = 0.6, g a1 = 0.3, and g b1 = There are three types of separatng eulbra: () 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 g a1 and g b1. Notce = 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 < 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 because Λ() < θ; f ths dd not hold, then ether w > θ or else the budget constrant ntersects the average resource constrant below the 45-degree lne, reversng the defnton of whch agent type s hgh rsk or low rsk. Specfcally, gven an eulbrum (, Λ()), f the soluton (ˆθ 1, ˆθ 2 ) to the par of euatons θ = pˆθ 1 + (1 p)ˆθ 2 (25) 5 The demand functons are ξ η1 = wg η1 Λ() = ˆθ 1 + (1 )ˆθ 2 (26) and ξ η2 = g η2w 1. Gven Q, the soluton to (28) s w = Λ( ) = θ [ fη ( g 2 η1 + g2 η2 1 )] 1. 16

17 θ For each Q, w = Λ() s the eulbrum value of the endowment such that agents are maxmzng utlty subject to the budget constrant and the allocaton s feasble. Endowment w w = Λ() 0 R a R b Relatve prce 1 Fgure 2: Eulbrum endowment as a functon of the relatve prce. For a gven relatve prce 0 < < 1, w = Λ() s the level of endowment such that agents maxmze expected utllty, markets clear and the consumpton allocaton s feasble. 17

18 has the property ˆθ 1 > ˆθ 2, then the defnton of hgh rsk and low rsk has been reversed. Solvng ths par of euatons results n ˆθ 2 ˆθ 1 = θ Λ() p. (27) Snce Λ() < θ, t follows > p to ensure type a agents are low rsk and type b agents are hgh rsk. If = p, then ths par of euatons has no soluton because Λ(p) < θ and (25) and (26) do not ntersect. For any Q, there exsts an eulbrum f w = Λ(). For any w [0, θ], there may not exst an eulbrum prce Q or there may be multple eulbra. The man result s stated n Theorem 2 but before statng that theorem, the followng lemma descrbes the relatonshp between (, Λ()). Lemma 1 ) If 0 < < p, then type a agents are hgh rsk and type b are the low rsk n the prvate tradng eulbrum (, Λ()). () Let m denote a soluton to 0 = Λ (), where there may be more than one soluton. Then g a1 < m < g b1. Proof. Part () follows from (27) because Λ() < θ for all Q so, f ˆθ 2 > ˆθ 1, where (ˆθ 1, ˆθ 2 ) solve (25) (26), then 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 Λ () = [ fη gη ] [ ξ η (w, ) ] 1 ξ η (w, ) fη gη w [ ( [ ])] [ ξη1 (w, ) ξη2 (w, ) ξ η1 (w, ) ] 1 ξ η (w, ) = fη τ η () + g η2 fη gη. 1 w The denomnator of the rght sde s strctly postve. Recall ξη1(w,) < 0. If g a1, then τ a () 0 and τ b () > 0. Also ξ a2 (w, ) ξ a1 (w, ) 0 and ξ b2 (w, ) ξ b1 (w, ) < 0 so the numerator n brackets s negatve, therefore the expresson on the rght sde s postve. Hence Λ () > 0 for g a1. If > g b1, then τ a () < 0 and τ b () 0 so the term n the numerator n brackets s postve. Also ξ a2 (w, ) ξ a1 (w, ) > 0 and ξ b2 (w, ) ξ b1 (w, ) 0 so the rght sde s negatve. Hence Λ () < 0 for g b1. It follows g a1 < m < g b1. The followng theorem summarzes the man results about the exstence of eulbra n the prvate tradng economy. 18

19 Theorem 2 () Let W = Λ(Q) denote the range of Λ. For any w W, there exsts a prvate tradng eulbrum prce where Q. () Let w m = Λ( m ), where m s the soluton to 0 = Λ (). If there s more than one soluton m, defne w m as the maxmum value or w m = max m Λ( m ). Then W (w m, θ] =, so f w (w m, θ] no prvate tradng eulbrum exsts. () If m < p for all m, then, for any w W, there exsts a unue eulbrum prce such that > p. If there s some m such that m > p, then for any w [Λ(p), w m ], there exsts at least two eulbra ( 1, 2 ) such that p < 1 < m and 2 > 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 eulbrum Q exsts for any w W ; however t may not be unue. For part (), for any Q, the eulbrum value of the market endowment satsfes w = Λ() w m, so the statement follows. For part (), the assumpton type a agents are low rsk reures p. If m < p then for any [p, 1), Λ () < 0. Hence for w (0, w m ], there exsts a unue eulbrum [p, 1). If there s some m such that p < m, then for each w [Λ(p), w m ] contnuty of Λ mples there exsts at least two eulbra 1, 2 such that 1 < m wth Λ ( 1 ) > 0 and 2 > m wth Λ ( 2 ) < 0. Defne the set P as the set of market-endowment prvate-tradng compettve eulbra, specfcally P {(, w) (0, 1) W (p, 1) and w = Λ()} 4 Constraned Effcency of Compettve Eulbra The objectves of ths secton are to establsh compettve eulbra P may not be prvate-tradng constraned Pareto effcent, to descrbe the externalty causng the neffcency, and to descrbe a smple polcy to nternalze partly the externalty causng the neffcency. Wthn the space of separatng allocatons, the socal plannng problem (19) s formulated as V(Ψ) max [ΨV (Λ(), ; a) + (1 Ψ)V (Λ(), ; b)]. (28) { Q} Denote µ η V (w, ; η). w 19

20 The frst-order condton for (28) s 0 = Λ () [Ψµ a + (1 Ψ)µ b ] + Ψµ a [ξ a2 (w, ) ξ a1 (w, )] + (1 Ψ)µ b [ξ b2 (w, ) ξ b1 (w, )] where Roy s Identty as been used, whch takes the form n ths applcaton. V η (w, ) = µ η [ξ η1 (w, ) ξ η2 (w, )] Theorem 3 The soluton to (19)-(20)) has the followng propertes. (). If m p, then satsfes (p, g b1 ). () If m < p, then the soluton to (29) s unue. () If p < m, then there exsts two solutons whch cannot be Pareto ordered. The optmal soluton can be mplemented by settng a celng Proof. If m < p, then Λ () < 0 for all > p. Snce ξ η2 (w, ) ξ η1 (w, ) = w ξ η1(w, ), 1 the frst-order condton can be expressed as [ ] [ ] [ Λ Ψµ a ξa1 (w, ) w (1 Ψ)µ b () = + Ψµ a + (1 Ψ)µ b 1 Ψµ a + (1 Ψ)µ b If g b1, then ξ η1 (w, ) w > 0 1 < R b on the relatve prce of nsurance. ] [ ξb1 (w, ) w 1 ]. (29) for η = a, b and the rght sde of (29) s postve and the left sde s negatve, whch s a contradcton, hence < g b1. The unueness of when m < p follows from the property Λ s decreasng n for > m. Let p < 1 < m < 2 be two eulbra such that w = Λ( 1 ) = Λ( 2 ). By constructon, the budget constrant B( 1, w) ntersects B( 2, w) at w and B( 1, w) les above B( 2, w) for c 1 > w and c 2 < w and les below for c 1 > w, c 2 < w. Snce ξ η1 (, w) > w for < g b1, the type b consumpton bundle ξ b ( 1, w) cannot be purchased when = 2, hence type b agents strctly prefer B( 1, w). A smlar argument can be used to demonstrate type a agents strctly prefer the consumpton bundle ξ a ( 2, w) to ξ a ( 1, w). Hence the eulbra cannot be Pareto ranked. 20

21 4.1 Ineffcent Cross-Subsdzaton Theorem (3) establshes the optmal soluton to the socal plannng problem (19)-(20) satsfes (p, g b1 ). Compettve eulbra (, Λ()) such that g b1 are not prvate-tradng constraned effcent. The reason s neffcent cross subsdzaton. Recall θ w = f η g η2 [ξ η1 (w, ) w][r η 1 ] When > g b1, then R η 1 < 0 for both types of agents. The expected clam of an agent s R η(ξ η1 (w, ) w) whle the payment s 1 (ξ η1(w, ) w). For both types of agents the expected benefts are less than the payments. When p < < g b1, the expected clam for a type bagentexceedsthecost, whletheoppostestrueforatypea agent. As a result, the prvate tradng constraned effcent outcome has the property type a agents are subsdzng type b agents. 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 reure 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 not separated by type, contracts are not exclusve, and agents can enter nto prvate-tradng arrangements. The resultng eulbrum 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. Essentally agents are tradng probablty dstrbutons, as n the rensurance lterature. The result that some states are under-nsured whle others are over-nsured for an agent. A prce celng on the prce of nsurance s a smple polcy to elmnate neffcent cross-subsdzaton across dfferent types of agents wth the result compettve eulbra 21

22 are prvate-tradng constraned effcent. 22

23 Appendx A To demonstrate not all eulbra n P are constraned-effcent, I show there may be two eulbra wth budget constrants ntersectng at the same common endowment (θ 1, θ 2 ) and these eulbra can be Pareto ranked. Defne the set of of common feasble endowments Θ as Θ {θ (θ 1, θ 2 ) R 2 + pθ 1 + (1 p)θ 2 and θ 2 > θ 1 }. For any θ Θ, defne X(θ) = {x η = (x η1, x η2 ) θ + x η 0}. and defne B(, θ) = {x X(θ) x 1 + (1 )x 2 = 0} Let x η (, θ) = arg max x B(,θ) { } g η U(θ + x η ) Defnton 1 A common endowment compettve eulbrum s a prce p < < 1, endowment θ Θ, and net trades (x e a, x e b ) such that x e η = x η (, θ) and the allocaton s feasble θ = η f η g η [θ + x η (, θ)]. To determne f a common endowment compettve eulbrum exsts, let Q η (θ) denote the (unue) autarky prce at whch an agent of type η has zero demand, U (θ 1 ) Q η (θ) R η U, η = a, b. (θ 2 ) Snce θ 1 < θ 2, t follows Q η (θ) > R η, η = a, b and snce R b > R a t follows Q b (θ) > Q a (θ) for all θ Θ. Defne a η(θ) Q η(θ) 1 + Q η (θ) 23

24 The demand functons x η, η = a, b have the followng propertes. () An agent of type η nsures (does not nsure, takes on rsk) f and only f 1 > Q η(θ)). That s x η1 (, θ) 0 as 1 Q η(θ). () An agent of type η nsures partally (fully, more than fully) f and only f 1 < R η). That s θ 1 + x η1 (, θ) θ 2 + x η2 (, θ) An mmedate conseuence of () and () s as 1 R η. 1 < Q η(θ) ( 1 = Q η(θ), 1 > R η ( 1 = R η, () If Q a (θ) < R b, then full consumpton nsurance for hgh rsk (type b) agents mples that low rsk type a agents take on rsk. That s, Q a (θ) < R b = x a1 (g b1, θ) < 0. There are two cases, Case I: R a < R b < Q a (θ) < Q b (θ), (30) Case II: R a < Q a (θ) R b < Q b (θ). (31) The results for compettve eulbra startng from a common endowment θ Θ are studed n Theorem 1 of Labade [2009], whch s summarzed below. Theorem 4 Under assumptons (1) (2), () f Q a (θ) > R b (Case I), there exsts at least one eulbrum wth 1 (Q a(θ), Q b (θ)), and there are no eulbra wth () f R b = Q a (θ), then 1 = R b s an eulbrum; () f Q a (θ) < R b (Case II), then n any eulbrum The proof s omtted. See Labade [2009]. 1 [R b, Q a (θ)]; 1 ( R, Q a (θ)) (R b, Q b (θ)). 1 ( R, R b ) and at least one wth I now establsh the connecton between the common endowment eulbra and the set of compettve eulbra P. Assocated wth each (, w) P s an endowment pont (θ 1, θ 2 ) on the resource constrant, 24

25 subject to some restrctons. Let θ (θ 1, θ 2 ) Θ be the soluton θ = pθ 1 + (1 p)θ 2 (32) To ensure θ 1, θ 2 ) Θ reures w 1 > Λ() = θ 1 + (1 )θ 2 (33) θ 1 p and w < θ p.6. The budget sets B(θ, ) and B(w, ) then ntersect at the pont (θ 1, θ 2 ). The next step s establsh the values of θ Θ satsfyng Case I and Case II n Theorem (3). Ths reures determnng the allocaton θ Θ satsfyng the condton n part () of Theorem (3). Let θ a (θ a 1, θ a 2) solve the par of euatons θ = pθ 1 + (1 p)θ 2 (34) R b = Q a (θ) = R a U (θ 1 ) U (θ 2 ). (35) Then = g b1 s an eulbrum when the endowment s θ a. Defne the set Θ I as Θ 1 = {(θ 1, θ 2 ) Θ θ 1 θ a 1 and θ 2 θ a 2} and defne the set Θ II Θ2 = {(θ1, θ2) Θ θ1 > θ a 1 and θ 2 < θ a 2} Case I n (30) corresponds to (θ 1, θ 2 ) Θ 1 whle Case II n (31) corresponds to (θ 1, θ 2 ) Θ 2. Set = g b1 and defne w = Λ(g b1 ). The common endowment pont correspondng to ths eulbrum s the soluton to the par of euatons θ = pθ 1 + (1 p)θ 2 (36) Λ(g b1 ) = g b1 θ 1 + g b2 θ 2 (37) Denote the soluton as ˆθ = ( ˆθ 1, ˆθ 2 ). Proposton 1 The soluton θ a to (34)-(35) solves the system (36)-(37). Proof. Defne w a g b1 θ1 a + g b2 θ2. a Then ξ a1 (w a, g b1 ) = θ1, a ξ a2 (w a, g b1 ) = θ2 a and ξ b1 (w a, g b1 ) = ξ b2 (w a, g b1 ). Ths allocaton satsfes the market-clearng condton (20) settng Λ(g b1 ) = w a. Hence θ a = ˆθ. 6 Prove ths 25

26 ( For each θ Θ 1, there are at least two eulbra such that 1 (p, g b1 ) and 2 Qa(θ) ( ) (p, g b1 ) and 2 Qa(θ) 1+Q, Q b (θ) a(θ) 1+Q b (θ). Let 1+Q a(θ), Q b (θ) 1+Q b (θ) ). 1 w 1 = 1 θ 1 + (1 1 )θ 2 w 2 = 2 θ 1 + (1 2 )θ 2 It follows w 1 > w 2 because 1 < 2 and θ 2 > θ 1. The two budget constrants ntersect at the pont θ 1, θ 2. For 1, the type b, agent has net demands such that x b1 ( 1, θ) > 0 and x a2 ( 1, θ) < 0 such that θ 1 + x b1 ( 1, θ) > θ 2 + x b2 ( 1, θ), so hs consumpton s below the 45-degree lne on the budget constrant.the type b agent has net demands such that x b1 ( 2, θ) > 0 and x a2 ( 2, θ) < 0 such that θ 1 + x b1 ( 2, θ) < θ 2 + x b2 ( 2, θ), so hs consumpton s on the budget constrant on the segment (θ 1, θ 2 ) and (w 2, w 2 ). For the type b agent, hs consumpton (θ 1 + x b1 ( 2, θ), θ 2 + x b2 ( 2, θ) s contaned n hs budget set ˆB( 1, θ) so he s better off wth consumpton(θ 1 + x b1 ( 1, θ), θ 2 + x b2 ( 1, θ)). For 1, the net demands for a type a agent satsfy x a1 ( 1, θ) > 0 and x a2 ( 1, θ) < 0 such that θ 1 + x a1 ( 1, θ) < θ 2 + x a2 ( 1, θ). Hence the type a agent s consumpton les along the budget constrant on the lne segment between the pont (θ 1, θ 2 ) and (w 1, w 1 ). For the eulbrum wth prce 2, observe the net demands for a type a agent satsfy x a1 ( 2, θ) < 0 and x a2 ( 2, θ) > 0 such that θ 1 +x a1 ( 2, θ) < θ 2 +x a2 ( 2, θ). Hence the type a agent s consumpton les along the budget constrant on the lne segment above the pont (θ 1, θ 2 ). For the type a agent, defne the expected utlty of hs consumpton n eulbrum ( 1, w 1 ) as Ū a = g a1 U(θ 1 + x a1 ( 1, θ) + g 2a U(θ 2 + x a2 ( 1, θ)) Defne the compensated demand curve h a ( 2, Ūa) (h a1 ( 2, Ūa), h a2 ( 2, Ūa). Observe 2 h a1 ( 2, Ūa) + (1 2 )h a2 ( 2, Ūa) > w 2, hence the type a s better off n eulbrum 1, w 1. Hence the compettve eulbrum 1, w 1 Pareto domnates 2, w 2. A smlar argument can be constructed for a common endowment θ Θ 2 for whch there are at least two ( ) Q eulbra such that 1 (p, a(θ) 1+Q ) and Q a(θ) 2 R b, b (θ) 1+Q b (θ).hence attenton wll be focused on (p, g b1 ) for all θ Θ. Appendx B The euvalence between the orgnal problem (11)-(14) and the modfed problem (19)-(20) s summarzed n Theorem 3; the proof follows the proof of Lemma 1 n FGT. 26

27 Theorem 5 Let (w, ) be a soluton to (19)-(20) and let {ξ a (w, ), ξ b (w, )} be a soluton to (17), gven (w, ), where (w, ) s a prvate tradng compettve eulbrum. Then {c(a), c(b)} defned by c(η) = (ξ η1 (w, ), ξ η2 (w, )) for all η {a, b} (38) s a soluton to (11) (14). Conversely, let {c(a), c(b)} be a soluton to (11) (14) wth the property t s a separatng allocaton. Then there exsts a Q and w = Λ() Θ solvng (19) subject to (20) f ξ η (w, ) = c(η) such that {ξ a (w, ), ξ b (w, )} solve (17). 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 (11) (14), denoted {c (a), c (b)} can be mplemented for some w and satsfyng (20), n that {c (a), c (b)} would solve (17). Start wth a soluton {c(a), c(b)} to (11) that s a separatng allocaton and let Q be the assocated eulbrum prce, as n Defnton 2. The ncentve compatblty constrants (13)-(14) can be wrtten g η U(c (η)) ˆV (c 1 (h) + (1 )c 2 (h), ; η) for all h {a, b} and each η. From the defnton of ˆV n (8) t follows ˆV (c 1 (ĥ({c(a), c(b)}, ; η))+(1 )c 2(ĥ({c(a), c(b)}, ; η)), ; η) ˆV (c 1 (h)+(1 )c 2 (h), ; η) for all h {a, b} whch s euvalent to ˆV ({c(a), c(b)}, ; η) ˆV (max[c 1 (a) + (1 )c 2 (a), c 1 (b) + (1 )c 2 (b)], ; η) (39) Snce ˆV s strctly ncreasng n ts frst argument, t follows or c 1 (η) + (1 )c 2 (η) = max [c 1(h) + (1 )c 2 (h)] h {a,b} c 1 (a) + (1 )c 2 (a) = c 1 (b) + (1 )c 2 (b). (40) 27

28 Denote ths value as ŵ. The next step s to use ŵ and n (17) and show {c(a), c(b)} solve (17). Snce the eualty n (40) s a property of the soluton to (8), then ths mples (19) f ξ(w, ; η) = c(η). The allocaton {c(a), c(b)} satsfes the socal planner s feasblty constrant (12) and s stll a soluton under the restrcton ĥ({c(a), c(b)}, ; a) = ĥ({c(a), c(b)}, ; b) whch s mplct n (17). Snce {c(a), c(b)} s a soluton to (11), t follows {c(a), c(b)} can be mplemented by the approprate choce of w,. Moreover ˆV ({c(a), c(b)}, ; η) = V (w, ; η) where ŵ s defned n (40). The second step s to take (w, ) solvng (19) (20) and show the {c(a), c(b)} gven by c(η) = (ξ η1 (w, ), ξ η2 (w, )), where ξ η (w, ) solve (17), are feasble and solve (11). Gven (w, ), the allocaton {c(a), c(b)} wll solve (17) for any η and, because {c(a), c(b)} satsfes (12) ths mples (20) f ξ η (w, ) = c (η) for all η and. By assumpton {c(a), c(b)} s a soluton to (8) and s stll a soluton under the addtonal constrant ĥ({c(a), c(b)}, ; a) = ĥ({c(a), c(b)}, ; b). Hence any soluton {c(a), c(b)} to (11) that s a separatng allocaton may be mplemented through the approprate choce of w and. Moreover the values of the maxmand n (11) and (19) are eual for these parameter values; hence the maxmum n (19) s at least as large as the one n (11). Suppose (w, ) solve problem (19) and let {c(a), c(b)} be gven by c (η) = ξ η (w, ), where {ξ a (w, ), ξ b (w, )} s the soluton to (17) gven (w, ). The next step s to determne f {c(a), c(b)} s feasble for (11), so t satsfes (12). If (20) s satsfed by {ξ a (w, ), (ξ b (w, )} and c(η) = ξ η (w, ) then clearly (12) follows from (20). Snce the budget constrant n problem (19) s bndng, t follows c 1 (a) + c 2 (a) = c 1 (b) + c 2 (b) = w so ˆV ({c(a), c(b)}, ; η) = V (w, ; η) for all η {a, b}, 28

29 because wth certan endowment w, gven ths relatve prce and prvate markets, a consumer does as well reportng truthfully η = ĥ({c(a), c(b)}, ; η) as he does by lyng, h = ĥ({c(a), c(b)}, ; η) h η. The ncentve compatblty constrants (13) (14) follow from the property {ξ a (w, ), ξ b (w, )} solve (17). Hence, f (w, ) solve (19), then the nduced allocatons are feasble for problem (11). Also, the maxmands are eual, and the maxmum n (11) s at least as large as the one n (19). Hence, f the allocaton s separatng, the maxmums n (11) and (19) concde. Any separatng allocaton {c(a), c(b)} solvng (11) nduces a (w, ) solvng (19). Appendx C: 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 h, 1 h, or else () there are separate markets A and B, where contngent clams trade at prces {( a, 1 ), ( b, 1 b )} and a b. The socal planner offers a consumpton vector c I. An agent who announces he s type η receves c(η). If the agent subseuently trades n market h, h {A, B}, he faces a budget constrant h c η1 + (1 h )c η2 = h x η1 + (1 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

30 If an agent of type η self selects nto market η, then by symmetry all agents of type η self select nto market η. The mplcatons of separate markets are derved for the three cases. Case : Both types have full nsurance at consumpton levels c a, c b. If c a c b, then both agents wll announce the type provdng the hgher level of certan consumpton, so ncentve compatblty reures c a = c b, whch s feasble f c η = θ (f less than θ there are unused resources). If the socal planner offers each agent θ for both types and states, then agents are dentcal across all states and types so there s no dosyncratc rsk, no tradng and hence no sde markets are formed. Case : One type has full nsurance at c η whle the other does not, so c h1 c h2, h η. Assume c η θ. If only type h agents self select nto market h, then the eulbrum prce satsfes h = g h1 and type h agents wll trade such that they acheve full consumpton nsurance eual to ĉ h = g h1 c h1 + g h2 c h2. If ĉ h ĉ η, then the fnal consumpton allocaton s not ncentve compatble. If, as assumed, a type h agent self-selects nto market h, then ĉ h ĉ η, otherwse he would have announced type η. It follows, f type h agents obtan ĉ h > ĉ η, then type η agents would also announce type h n antcpaton of the hgher consumpton after trade and the eulbrum prce cannot satsfy h = g h1. If c η = θ, then after tradng wth other agents of type h η, a type h agent has fnal consumpton ĉ h = θ. Type η agents have no ncentve to announce they are type h and type h agents are ndfferent between announcng type η and type h. Case : Nether agent has full nsurance. There are two sub-cases. If c c η = c h, as would be the case f agents receve dentcal contngent endowments (θ 1, θ 2 ), and f markets were separate, then all agents wll engage n arbtrage tradng because, for any (c 1, c 2 ), 0 c 1 [ a b ] and an arbtrage opportunty exsts. Tradng across markets wll elmnate the arbtrage opportunty untl η = h. In the other case, f agents receve dfferent endowments, c a c b, then agents wll engage n trade to elmnate arbtrage profts. Moreover, f the value of the consumpton allocatons c I n the sde markets s dfferent, c a1 + (1 )c a2 c b1 + (1 )c b2, then an agent wll announce the type wth the hghest value of consumpton n sde markets. To summarze, unless one or both types have full consumpton nsurance, where certan consumpton s eual to θ, there wll be a sngle market for contngent clams. Moreover, f the allocaton c I has the 30