Optimal Taxation in an Adverse Selection Insurance Economy. September 25, 2011

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1 Optmal Taxaton n an Adverse Selecton Insurance Economy September 25, 2011 Pamela Labade 1 Abstract When agents can enter nto prvate trades and contracts are non-exclusve, the ncentve effcent allocaton cannot be mplemented n an adverse selecton nsurance economy. Optmal taxaton s studed under the assumptons that contracts are non-exclusve and prvate tradng cannot be prevented. The exstence of a compettve equlbrum, whch can be problematc n these economes, s addressed by frst dervng a set of feasble allocatons such that an equlbrum exsts and then determnng the optmal allocaton wthn ths set. The approach s based on Golosov and Tsyvnsk (2007) and Farh, Golosov, and Tsyvnsk (2009) usng the model n Labade (2009). JEL Classfcaton: D81, D82. Keywords: Adverse selecton, prvate tradng, anonymous equlbrum Frst draft - ncomplete 1 Department of Economcs, 315 Monroe Hall, George Washngton Unversty, Washngton, D.C , Phone: (301) , Fax: (301) , Emal: labade@gwu.edu. 1

2 When agents can enter nto prvate trades and contracts are non-exclusve, the ncentve effcent allocaton cannot be mplemented n an adverse selecton nsurance economy. Optmal taxaton s studed under the assumptons that contracts are non-exclusve and prvate tradng cannot be prevented. The exstence of a compettve equlbrum, whch can be problematc n these economes, s addressed by frst dervng a set of feasble allocatons such that an equlbrum exsts and then determnng the optmal allocaton wthn ths set. The approach s based on Golosov and Tsyvnsk (2007) and Farh, Golosov, and Tsyvnsk (2009) usng the model n Labade (2009). The basc model s descrbed n secton 1. The ncentve-effcent allocaton s derved n the next secton and the mplcatons of retradng and nonexclusvty are explored n secton 2. The market structure s dscussed n secton 3, n partcular developng the argument that, f there s any trade n sde markets, then all agents wll face the same contngent clams prces. The dervaton of the anonymous equlbrum s n secton 4, wth the proof of exstence n the appendx, and condtons for the exstence of an equlbrum are provded. The socal planner s problem wth retradng s descrbed next. Fnally a closed form soluton s derved for logarthmc utlty. 1 Basc Model The basc model s the adverse selecton nsurance economy of Rothschld and Stgltz [10] and Prescott and Townsend [9]. Ths s a sngle-perod pure endowment economy wth a sngle consumpton good. 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 that takes two values 0 θ 1 < θ 2, so that 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 that 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. Observe that R b > R a. For any agent, the realzaton of θ s publc 2

3 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. The followng assumpton s standard. It says that, for ether type of agent, the probablty of beng n any endowment state s strctly postve. 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 contnuous and twce contnuously dfferentable, strctly ncreasng and strctly concave. Also, as c 0, U (c) (the Inada condton). The expectaton n (2) uses an ndvdual agent s condtonal probablty of realzng θ. Snce there s no aggregate uncertanty, the only rsk an agent faces s hs dosyncratc endowment rsk. The realzaton of an agent s endowment s publc nformaton. A consumpton allocaton for a type η agent s c(η) = (c 1 (η), c 2 (η)) and satsfes a non-negatvty constrant. Let C {c(η) = (c 1 (η), c 2 (η)) c (η) 0, = 1, 2}. A par of consumpton allocatons (c(a), c(b)) C 2 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 that the set of consumpton allocatons that are feasble n the aggregate s { F (c(a), c(b)) C 2 θ η } f η g η c (η). (3) Defnton: The par of consumpton allocatons c 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. 3

4 2 Incentve Effcency and Prvate Tradng When prvate tradng among agents can be prevented, the socal planner can determne the state-contngent consumpton of all agents and s able to mplement an ncentve-effcent allocaton. Defnton: The consumpton allocaton c I s ncentve effcent f there s no other par ĉ I such that g η U(ĉ η ) g η U(c η ), η = a, b, (5) wth strct nequalty for at least one type. Incentve-effcent allocatons have been studed extensvely by Prescott and Townsend, among others. As demonstrated by Prescott and Townsend, there are three types of ncentve-effcent allocatons for ths economy: () full nsurance for type a and partal nsurance for type b (separatng); () full nsurance for type b and partal nsurance for type a (separatng); () full nsurance for both such that c η = θ (poolng). Let c η denote the certan consumpton of the type wth full nsurance. For the type-h agent wth partal nsurance, h η, the consumpton allocaton satsfes U( c η ) = g h1 U(c h1 ) + g h2 U(c h2 ), (6) θ f η c η = f h [g h1 c h1 + g h2 c h2 ]. (7) Let ĉ h = (ĉ h1, ĉ h2 ) denote a soluton to (6)-(7). The soluton may requre transfers across types f c η θ η. The decentralzaton of ths constraned Pareto-effcent allocaton has been the focus of many papers, ncludng Prescott and Townsend. To decentralze an ncentve-effcent allocaton requres that markets are separated, prvate tradng s prohbted, and contracts are exclusve. Agents can enter only one market (separaton) and one contract (exclusvty), and must consume the quanttes specfed n the contract (no prvate tradng). A detaled descrpton s contaned n Bsn and Gottard [2]. Prvate Tradng and Incentve Effcency Suppose that the socal planner chooses an ncentve-effcent allocaton where only one type of agent receves full consumpton nsurance (separatng allocaton) and that prvate tradng among agents cannot be 4

5 prevented. That s, 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 rsk-sharng arrangements wth other agents. Agents may desre to enter nto rsk-sharng arrangements f t enables them to effectvely unbundle the contngent clams n a contract. For smplcty, focus on the separatng allocaton n whch the hgh-rsk agent has full consumpton nsurance at c b. Hgh-rsk agents have no ncentve to engage n further tradng wth other agents n market B. Ths s not the case for the low-rsk agents, however. The low-rsk agent purchases a contract ĉ a = (ĉ a1, ĉ a2 ) that provdes less than full nsurance. Low-rsk agents would lke to unbundle the contract by separatng ts state-contngent components and tradng. Suppose that 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 the prvate market budget constrant ˆq a1 c a1 + ˆq a2 c a2 ˆq a1 x 1 + ˆq a2 x 2, (8) where the left sde s the prvate market value of the contract purchased by a type a agent n market A. 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 would choose constant consumpton c a. At ths pont, there are no further gans from trade. Hgh rsk agents are no worse off because of ths arrangement. The dffculty ntroduced by prvate tradng s the new allocaton c a does not satsfy the orgnal ncentve compatblty constrants (4). Hence, to mplement the ncentve-effcent allocaton, t s crtcal that agents be prevented from prvate tradng. If prvate trades occur, 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 that low-rsk agents ntend to trade n prvate markets, then a hgh-rsk agent may have an ncentve to msrepresent hs type so that he too can partcpate n the prvate market. If c 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 c a s no longer feasble at the prces (g a1, g a2 ) and the self-selecton nto markets wll break down. The dea that subsequent tradng opportuntes can change the nformaton revealed by an agent s studed by Krasa [1999]. He examnes prvate nformaton exchange economes wth allocatons that cannot be mproved, n that the agent would not wsh to devate by revealng further nformaton or by the retradng of goods. An alternatve tradng mechansm s consdered next, n whch prvate markets are n 5

6 equlbrum. 3 Prvate Tradng If a separatng allocaton s chosen by the socal planner and f prvate tradng among agents cannot be prevented, then agents wll have an ncentve to engage n prvate tradng. Suppose that agents can trade n contngent clams markets and are prce-takers n those markets. Let q η denote the prce of a contngent clam n market η {A, B} for state {1, 2}. For smplcty, normalze the prces so that q η q η1 q η2 η {a, b}. Markets are sad to be separate f q η q h for η, h {a, b}, h η and there s tradng n both markets. If the relatve prce dffers across markets, then an arbtrage opportunty exsts because contracts are not exclusve, so that the elmnaton of arbtrage proft opportuntes wll result n the equalzaton of prces across markets. The market structure that wll emerge when there s prvate tradng s dscussed n secton (5). If the socal planner chooses the poolng allocaton, so that 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 assgned to each type. In the dscusson below, I assume that the socal planner wll choose a separatng allocaton, whch s a restrcton on the Pareto weghts, whch are parametrc to the model. 3.1 Constraned Effcency wth Prvate Tradng Suppose that consumers are offered a menu 2 of contracts {c(a), c(b)} F and that agents have access to a compettve market where they can trade the state-contngent components of a contract at the relatve prce q. An agent treats the menu of contracts {c(a), c(b)} and the relatve prce q n the prvate market as gven. An agent chooses hs optmal reportng strategy h {a, b} that determnes hs endowment c(h) = (c 1 (h), c 2 (h)). Hs actual after-trade consumpton (x 1, x 2 ) may dffer from hs consumpton allocaton c(h). 2 The poolng allocaton s ruled out by assumpton on the Pareto weghts. 6

7 An agent reportng type h faces a budget set ˆB(c(h), q) {x C qx 1 + x 2 = qc 1 (h) + c 2 (h)}. (9) Gven the menu {c(a), c(b)} and relatve prce q, a type η agent solves subject to h {a, b} and ˆV ({c(a), c(b)}, q; η) = max {h,x 1,x 2} g η U(x ) (10) (x 1, x 2 ) ˆB(c(h), q). (11) Let ˆx 1 ({c(a), c(b)}, q; η), ˆx 2 ({c(a), c(b)}, q; η) and ĥ({c(a), c(b)}, q; η) for η {a, b} denote the soluton. Defnton: An equlbrum n prvate markets, gven a menu of endowments {c(a), c(b)}, conssts of a relatve prce q > 0 and, for each η, η {a, b} consumpton allocatons (x 1 (η), x 2 (η)) and a reported type h such that (). ˆx 1 ({c(a), c(b)}, q; η), ˆx 2 ({c(a), c(b)}, q; η) and ĥ({c(a), c(b)}, q; η) solve (10) subject to h {a, b} and the budget constrant (11). (). Markets clear: f η g ηˆx ({c(a), c(b)}, q; η) f η g η c (ĥ({c(a), c(b)}, q; η)). η η The exstence of an equlbrum n adverse selecton nsurance economes can be problematc and s the topc of dscusson n secton (3.2). The constraned effcent allocaton wth prvate markets and unobservable consumpton, called the SP 3 program, or the thrd best program, s studed next. The socal planner pcks {c(a), c(b)} F that maxmze the Pareto-weghted expected utlty of agents subject to the feasblty condton, the ncentve compatblty constrants and the possblty that agents may trade n prvate markets. Let Ψ η denote a Pareto weght for a type η agent such that Ψ η > 0 and 1 = Ψ a + Ψ b. The constraned effcent socal plannng problem wth prvate tradng (SP 3 ) s Ψ η g η U(c (η)) (12) subject to max {c(a),c(b)} η θ η 7 f η g η c (η), (13)

8 g a U(c (a)) ˆV ({c(a), c(b)}, q; a), (14) g b U(c (b)) ˆV ({c(a), c(b)}, q; b). (15) The frst constrant (13) s feasblty of the allocaton. The second and thrd constrants requre the consumpton allocaton to result n expected utlty that s at least as hgh as the agent type η can acheve by reportng he s type ĥ({c(a), c(b)}, q; η) and then tradng n prvate markets at relatve prce q. It s straghtforward to show that, f for some h {a, b}, qc 1 (h) + c 2 (h) > qc 1 (η) + c 2 (η) η, h {a, b} h η, then every agent wll announce that he s type h or h = ĥ({c(a), c(b)}, q; a)ĥ({c(a), c(b)}, q; b). Ths follows because the ndrect utlty functon ˆV s strctly ncreasng n the market value of the consumpton contract and all agents face the dentcal relatve prce n prvate markets. As a result, all agents wll then face the dentcal budget set ˆB(c(h), q). It follows that the consumpton x ˆB(c(h), q) wll be ncentve compatble. Defne the value of the consumpton allocaton c(h) n prvate markets as (1 + q)w qc 1 (a) + c 2 (a) = qc 1 (b) + c 2 (b), where w s the certan consumpton endowment, so that the endowment s (w, w). 3 I now show the condtons under whch choosng the consumpton allocaton to solve (SP 3) s equvalent to the socal planner choosng (w, q). If a type η agent has the endowment w and faces relatve prce q n prvate markets, hs budget set s B(w, q) {x C (1 + q)w = qx 1 + x 2 }. The agent solves 3 Alternatvely one can defne ŵ as V (w, q; η) = max {x B(w,q)} g η U(x ) (16) ŵ = qc 1 (h) + c 2 (h) so that ŵ = (1 + q)w. Specfyng (ŵ, q) s equvalent to specfyng (w(h), q).the problem s posed usng w because the opportunty set for w s straghtforward. 8

9 Denote the soluton as ξ η (w, q) (ξ η1 (w, q), ξ η2 (w, q)). The socal plannng problem can now be restated as follows. The socal planner pcks 0 < w θ and relatve prce q > 0 to solve subject to Let (w, q ) denote the soluton. max {w,q} θ η Ψ η V (w, q; η) (17) η f η g η ξ η (w, q). (18) I now prove the equvalence between the soluton to (17) and the soluton to (SP3). Lemma 1 Let (w, q ) be the soluton to (17) and ξ η (w, q ) for η {a, b} be solutons to (16). Then c (η) = ξ η (w, q ) for η {a, b} and {1, 2} (19) s a soluton to (SP3). Proof. The frst step s to show the soluton to SP3, denoted (c (a), c (b)) can be mplemented for some w and q satsfyng the feasblty condton for (18) n that (c (a), c (b)) would solve (17). The second step s to take (w, q ) solvng (17) and show the (c(a), c(b)) gven by c (η) = ξ η (w, q ; η) are feasble and solve (SP3). Take any soluton {c(a), c(b)} to SP3 and let q be the equlbrum relatve prce for prvate markets gven these allocatons. Note that the ncentve compatblty constrant for SP 3 can be wrtten g η U(c (η) V (qc 1 (h) + c 2 (h), q; η) for h, η meanng that type η doesn t get more expected utlty from the endowment by pretendng to be type h and tradng. By (17) V (qc 1 (η) + c 2 (η), q; η) V (qc 1 (h) + c 2 (h), q; η) 9

10 Snce V s strctly ncreasng n ts frst argument, ths s equvalent to qc 1 (η) + c 2 (η) = max h {a,b} [qc 1(h) + c 2 (h)] The next step s to show that for (w, q) and the gven allocaton {c(a), c(b)}, wll solve (17) for any η and ths mples (18) f x η (w, q) = c (η) for all η and because {c(a), c(b)} satsfes (12).But {c(a), c(b))} s a soluton to (10) and s stll a soluton under the addtonal constrant η = h. Ths proves that any soluton {c(a), c(b)} to (12) may be mplemented through the approprate choce of w and q. Moreover the values of the maxmand n (12) and (17) are equal for these parameter values; hence the maxmum n (17) s at least as large as the one n (12). Suppose now that (w, q ) solve problem (17); 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 (16) for (w, q ). The next step s to check that {c(a), c(b)} s feasble for (17). so t satsfes (13) (15). Clearly (13) follows from (18). Snce the constrant to (17) s bndng so that q c 1 (a) + c 2 (a) = q c 1 (b) + c 2 (b) = (1 + q )w ˆV ({c(a), c(b)}, q ; η) = V (w, q ; η) for all η {a, b}. To be completed The two approaches to solvng for the constraned effcent allocaton wth prvate markets are llustrated n Fgure (1). The socal planner can choose consumpton allocatons {c(a), c(b)} such that c(η) B(w, q). In Fgure (1), the average feasblty constrant s the lne through the pont θ wth slope R, wth a horzontal ntercept at θ p 1 and vertcal ntercept at θ p 2. The budget constrant for an agent s the lne through the pont w on the 45-degree lne wth slope q. The budget constrant ntersects the resource constrant at the pont S where S = (ˆθ 1, ˆθ 2 ). 4 The endowment pont s E = (θ 1, θ 2 ) (the autarky pont). The consumpton allocatons c(a) and c(b) are n the budget set and satsfy feasblty. The socal plannng problem SP3 specfes the 4 The pont S satsfes θ = p 1ˆθ 1 + p 2ˆθ 2 (1 + q)w = qˆθ 1 + ˆθ 2. 10

11 14 12 w(1+q) 10 θ p2 State 2 consumpton 8 6 E S c(a) θ w 4 w c(b) Far odds lne Slope R 2 Budget constrant Slope -q 0 θ w (1+q)w State 1 consumpton θ q p2 Fgure 1: Budget Set wth Prvate Tradng 11

12 consumpton allocatons c(a), c(b). Alternatvely, the socal planner can be modeled as specfyng a pont w on the 45-degree lne and the slope q. 5 Before solvng the socal planner s problem, t s crtcal to establsh the condtons under whch a compettve equlbrum n prvate markets exsts. As dscussed n Bsn and Gottard [2006] and Labade [2009], the exstence of a compettve equlbrum can be problematc. 3.2 Equlbra n Prvate Tradng Condtons that ensure the exstence of an equlbrum are dscussed n ths secton when the socal planner pcks w W (0, θ). The soluton ξ η (w, q) to (16) satsfes the frst-order condton U (ξ η1 (w, q)) R η = q. (20) U (ξ η2 (w, q)) Snce U s strctly concave and satsfes the Inada condtons, t s mmedate that the soluton ξ η (w, q), η = a, b exsts and s unque. Moreover ξ η (w, q) s contnuous and strctly ncreasng n w and 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 that 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 (q, w) > ξ a1 (q, w) for all q > 0 and 0 < w < θ () A type η agent nsures partally (fully, more than fully) as q > R η (q = R η, q < R η ) ξ η1 (w, q) w as q R η Defnton: A prvate tradng equlbrum s an ntal endowment w W, a prce q e > 0 and consumpton demands (x e a, xe b ) F, such that (). Agents solve (16), so that x e η = ξ η (w, q e ) for η = a, b. 5 The socal planner can also choose to reallocate the endowment by specfyng ˆθ = (ˆθ 1, ˆθ 2 ) and the prce q. The dscusson below focuses on the equvalence between the soluton to SP 3 and the socal plannng problem wth decson varables (w, q), dscussng the equvalence of other approaches wth the SP3 n an appendx (to be wrtten). 12

13 (). The equlbrum condton s satsfed at q e θ = η f η g η ξ η (w, q e ). The exstence of an equlbrum can be establshed under certan condtons as follows: For any q > 0 and w W, the equlbrum condton can be expressed as θ = η f η {g η1 ξ η1 (w, q) + g η2 [(1 + q)w qξ η1 (w, q)]} (21) or θ (1 + q)p 2 w = η f η g η2 ξ η1 (w, q)[r η q]. (22) Defne H(w, q) θ p 2 (1 + q)w, and defne Ξ η : W R + R by Ξ η (w, q) g η2 ξ η1 (w, q)[r η q] and let Ξ(w, q) η f η Ξ η (w, q). The key equaton (21) can then be expressed as H(w, q) Ξ(w, q) = 0. (23) The propertes of the functons H and Ξ are descrbed next. The functon H has the followng propertes. (). H s decreasng n q, w. (). At q = 0, H(w, 0) = θ p 2 w. (). As q ncreases, lm q H(w, q) =. (v) At w = θ, H( θ, q) = θp 2 [ R q]. (v). There exsts a value of q gven w W such that H(w, q) = 0. Defne q : W R + as [ ] θ p2 w q(w). (24) p 2 w 13

14 Observe that q s decreasng n w. At w = θ, q( θ) = R. Also lm q(w) =. w 0 The functon H can be rewrtten as [ ] θ H(w, q) = p 2 (1 + q)w. p 2 In fgure (1), the functon H s proportonal to vertcal dstance between the ntercept of the resource constrant θ p 2 and the vertcal ntercept of the budget constrant (1 + q)w. Next, the propertes of the functon Ξ depend on the demand functons ξ. Observe that It follows mmedately that and lm ξ η1(w, q) = + and lm ξ η2 (w, q) = 0, η = a, b, q 0 q 0 lm ξ η1(w, q) = 0 and lm ξ η2(w, q) = +, η = a, b. q + q + lm Ξ(w, q) = + q 0 lm Ξ(w, q) = 0. q + Lemma 2 Gven w W, there exsts a R < q < R b such that Ξ(w, q) = 0. Proof. At q = R, Ξ(w, R) > 0. Ths follows because Ξ(w, q) = f a g a2 ξ a1 (w, q)[r a R] + f b g b2 ξ b1 (w, q)[r b R] > f a g a2 ξ a1 (w, q)[r a R] + f b g b2 w[r b R] > f a g a2 w[r a R] + f b g b2 w[r b R] = 0 where the frst nequalty follows because ξ b1 (w, R) > w and the second nequalty follows because ξ a1 (w, R) < w. Hence Ξ(w, R) > 0. 14

15 At q = R b, Ξ(w, R b ) = f a g a2 ξ a1 (w, R b )[R a R b ] < 0. Hence, because Ξ s contnuous, there exsts a q such that R < q < R b satsfyng Ξ(w, q) = 0. Gven w W, defne the soluton to ths equaton or Ξ(w, q) = 0. as ˆq(w), There are three cases: () ˆq(w) < q(w), () ˆq(w) > q(w), and () ˆq(w) = q(w). Lemma 3 Let w W. If ˆq(w) < q(w), then there exsts a q satsfyng (23) such that q e (0, ˆq(w)) ( q(w), ). Proof. It follows from assumptons (1)-(2) that Ξ(w, q) s contnuous and contnuously dfferentable n q. At q = 0, Ξ(w, 0) > H(w, 0). Snce Ξ(w, ˆq(w)) < H(w, ˆq(w)), t follows that there s a q ˆq(w) such that (23) holds because H(w, ˆq(w)) > H(w, q(w)) = 0. Snce Ξ(w, q(w)) < H(w, q(w)) = 0, and lm q Ξ(w, q) = 0 whle lm q H(w, q) =, t follows that there s some q > q(w) such that (23) holds. At q e = q(w), Ξ(w, q(w)) < 0 = H(w, q(w)). Snce lm q H(w, q) = and lm q Ξ(w, q) = 0, t follow that there s some q e > q(w) such that (23) holds. An example s provded n Fgure 2. Lemma 4 If q(w) R b, then w w(r b ) where w(r b ) θ p 2 (1 + R b ). 15

16 10 8 φ(w, q) 6 4 H(w,q) 2 q(w) q 2 0 q 1 R a ˆq(w) R b relatve prce q Fgure 2: Determnaton of the equlbrum prce Ξ(w,q) H(w,q) 2 0 R R b R a Fgure 3: Example where an equlbrum does not exst 16

17 Proof. By defnton q(w(r b )) = R b and q(w) s strctly decreasng n w. An example where an equlbrum does not exst s llustrated n the next fgure. Lemma 5 If ˆq(w) = q(w), then w > w(r b ). Proof. Ths follows because R < ˆq(w) < R b and for any 0 < w < w(r b ), q(w) R b. Snce q(w) s strctly decreasng n w, t follows that w > w(r b ). The par (w, q) such that ˆq(w) = q(w) wll satsfy the par of equatons Ξ(w, q) = 0 H(w, q) = 0 or q s the soluton to ( ) θ Ξ p 2 (1 + q), q The functon Ξ s not monotonc n w or q. If a soluton exsts, denote t as q and defne w = = 0 θ p 2 (1 + q ). Lemma 6 If ˆq(w) = q(w), then there s an equlbrum prce q ( R, R b ). Proof. The lower bound on the value of q(w) s at R where w = θ. If w < θ then (q)(w) > R. Snce ˆq(w) < R b, the nequalty follows. The results for the exstence of an equlbrum are summarzed n the followng theorem. Theorem 1 () If 0 < w w(r b ), then there exts a q e solvng (23) such that q e (0, ˆq(w)) ( q(w), ). () If w w(r b ) and ˆq < q(w), then there exsts an equlbrum q e such that q e (0, ˆq(w)) ( q(w), ). () If w exsts and w > w w(r b ) and ˆq(w) = q(w), then q e = ˆq(w) = q(w) s an equlbrum. (v) If θ > w > w such that ˆq > q(w), then f an equlbrum exsts, q e > ˆq(w). Proof to be completed. 17

18 3.3 Soluton of the Constraned Problem wth Prvate Tradng The frst-order condtons are Φη V (w, q; η) Φη V (w, q; η) q = λ f η = λ f η g η ξ η (w, q) g η ξ η (w, q) q Solve (26) for λ and substtute nto (25) and rewrte ( ) ( ) V (w, q; η) ( ξ η (w, q) ) ( ) V (w, q; η) ξ η (w, q) Φη fη g η = Φη fη g η q q Roy s dentty, whch s V η (w, q) q = V η(w, q) ξ η1 (w, q), can be substtuted nto (28) to obtan ( Φη ) ( ) V (w, q; η) ( ξ η (w, q) fη g η = V η (w, q) Φ η ξ η1 (w, q) q The Slutsky-Hcks equatons are h η1 (q, U η) q h η2 (q, U η ) q = ξ η1(w, q) q = ξ η2(w, q) q + ξ η1(w, q) ξ η1 (w, q) + ξ η2(w, q) ξ η1 (w, q) ) ( fη (25) (26) (27) ) ξ η (w, q) g η (28) where h η (q, Uη ) s the Hcksan or compensated demand curve and U η ) s a gven level of utlty. Solve the Slutsky-Hcks equaton for ξη(w,q) q and substtute nto (28) to obtan ( Φη ) ( ) V (w, q; η) h η (q, Uη) fη g η q ( ) V (w, q; η) = Φη fη ξ η1(w, q) ξ η (w, q) g η 4 Logarthmc Example When U(c) = ln c, the demand functons are ξ η1 = ŵg η1 q 18 ( Φη ξ η1(w, q) (29) ) V (w, q; η) ξ η (w, q) fη g η

19 c η2 = g η2 ŵ where ŵ (1 + q)w. The socal planner pcks ŵ, q to maxmze ( ) ] gη1 ŵ Φ η [g η1 ln + g η2 ln (g η2 ŵ) q subject to the resource constrant η η θ η f η [ g 2 η1 ŵ q + g 2 η2ŵ Let µ denote the multpler for the resource constrant. The frst-order condtons wth respect to ŵ, q are [ ( )] 1 g 2 Φ η ŵ = µ η1 f η q + g2 η2 (31) η Φ η g η1 q η = µŵ η Solve (31) for µ, substtute nto (32, and solve for q [ fη q gη1 2 = [1 ] Ph η g η1 ] Φη g η1 fη gη2 2 The value of w s determned from w = ( ) [ ( )] 1 θ g 2 η1 fη 1 + q q + g2 η2 ] (30) f η g 2 η1 q 2 (32) Fgure 3 llustrates the optmal soluton for the followng parameter values: f a = 0.6, g a1 = 0.3, g b1 = 0.6 and Φ a = The endowment pont (autarky) s E = [2, 8] and θ = The equlbrum prce s q e = 0.92, whch les between R = 0.72 and R b = 1.5, and the optmal level of ncome s w = In ths example, the socal planner can acheve the optmal consumpton allocaton c(a) = [3.11, 6.70] and c(b) = [6.23, 3.83] by changng the ntal endowment to S = [0.60, 9.01] or by offerng the menu of consumpton {c(a), c(b)}, or by settng w = The functons H and Ξ for the optmal level of ncome w are llustrated n the Fgure (5). (33) 4.1 Other Numercal Examples In the example llustrated n Fgure (6), Φ a = f a, g a1 = 0.3, g b1 = 0.7, R = 0.72, θ = The soluton to the socal planner s problem s q e = 0.76 and w = Observe that the budget constrant does not ntersect the average feasblty constrant (far odds lne). 19

20 state 2 consumpton S E 8 6 c(a) 4 c(b) w θ state 1 consumpton 5 Market Structure Fgure 4: Socal planer s soluton for logarthmc utlty There are two possbltes for the organzaton of markets: there s a sngle market n whch all clams contngent on realzaton {1, 2} trade for the dentcal prce q or else there are separate markets A and B where contngent clams trade at prces q η. The socal planner offers a consumpton vector c I. An agent who announces he s type η receves c η. If the agent trades n market h, h {A, B}, he faces a budget constrant q h1 c η1 + q h2 c η2 = q h1 x η1 + q h2 x η2 h {a, b} Nonexclusvty of contracts mples that agents can enter nto multple trades and that 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 nonexclusvty 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. 20

21 10 8 Ξ(w,q) 6 Excess Demand 4 2 H(w,q) 0 R a R q q R b relatve prce q Fgure 5: The functons H and Ξ 21

22 E State 2 consumpton 8 6 c(a) 4 c(b) w State 1 consumpton θ Fgure 6: An optmal allocaton for whch there s no dentcal endowment pont 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 that provdes the hghest level of certan consumpton, so ncentve compatblty requres 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 that c h1 c h2, h η. Assume that c η θ. If only type h agents self select nto market h, then equlbrum prces satsfy q h = g h and type h agents wll trade such that they acheve full consumpton nsurance equal 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 that, f 22

23 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 equlbrum prce cannot satsfy q h = g h. 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. 6 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 0 c 1 [q a1 q b1 ] + c 2 [q a2 q b2 ] and an arbtrage opportunty exsts. Tradng across markets wll elmnate the arbtrage opportunty untl q η = q h. IIn 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, q 1 c a1 + q 2 c a2 q 1 c b1 + q 2 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 equal to θ, there wll be a sngle market for contngent clams. Moreover, f the allocaton c I has the property that nether type has full nsurance, then the consumpton allocaton for each type must have the same value n sde markets. Specfcally, defne q q 1 q 2 then the requrement that allocatons have the dentcal market value requres w qc a1 + c a2 = qc b1 + c b2 (34) Defne B(c η, q) to be the set of consumpton trades that satsfy the budget constrant (wth equalty) for relatve prce q, and endowment c η B(c η, q) {x C qx 1 + x 2 = ac η1 + c η2 }. (35) 6 Assume that f an agent s ndfferent, then the agent wll truthfully reveal hs type. 23

24 Hence the opportunty to trade n sde markets mposes an addtonal restrcton on consumpton allocatons besdes ncentve compatblty, namely that allocatons must also have the same market value. 6 Concluson An exchange economy wth adverse selecton and prvate nformaton s studed under the assumpton that 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 sde arrangements. The result s the compettve equlbrum wth anonymty. Anonymty means that all agents face the same budget set n net trades. The allocaton s ndvdually ncentve compatble, although t s not ncentve-effcent. Snce agents face the same budget constrant n net trades, 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. The equlbrum has the property that the margnal rate of substtuton s equalzed across states n the contngent clams market. The anonymous equlbrum s derved and shown to exst, although t may not be unque. A closed-form soluton s derved for a set of parameter values and plotted n Fgure 3. A comparson of expected utlty for varous allocatons s provded n Table I. One drecton for future work s to determne the condtons under whch the anonymous mechansm s coaltonally ncentve compatble and to determne whether the smple market mechansm studed here s an organzatonal structure that wll emerge endogenously usng coalton theory, as n Boyd, Prescott and Smth [2]. 24

25 References [1] A. Bsn, P. Gottard, Compettve Equlbra wth Asymmetrc Informaton, J. Econ. Theory 87, [2] A. Bsn, P. Gottard, Effcent Compettve Equlbra wth Adverse Selecton, J. Polt. Economy 114 (2006), [3] E. Farh, M. Golosov, and A. Tsyvnsk, A Theory of Lqudty and Regulaton of Fnancal Intermedaton, Revew of Economc Studes 76(3), [4] M. Golosov, A Tsyvnsk, Optmal Taxaton wth Endogenous Insurance Markets, Q. Journal of Economcs,May (2007), [5] P. J. Hammond, Markets as Constrants: Multlateral Incentve Compatblty n a Contnuum Economy, Rev. Econ. Stud. 54 (1987), [6] S. Krasa, Unmprovable Allocatons n Economes wth Incomplete Informaton, Econ. Theory 87 (1999), [7] P. Labade, Anonymty and Indvdual Rsk, J. Econ. Theory 144 (2009), [8] A. Mas-Colell, Whnston and Green, Mcroeconomc Theory, Oxford Unversty Press (1995). [9] E. Prescott, R. Townsend, Pareto Optma and Compettve Equlbra wth Adverse Selecton and Moral Hazard, Econometrca 52 (1984), [10] M. Rothschld, J. Stgltz, Equlbrum n Compettve Insurance Markets: An Essay on the Economcs of Imperfect Informaton, Quart. J. Econ. 90 (1976),