Uncertain delivery in markets for lemons

Transcription

1 Uncertan delvery n markets for lemons João Correa-da-Slva CEMPRE and Faculdade de Economa. Unversdade do Porto. January 21st, 2009 Abstract. The noton of uncertan delvery s extended to study exchange economes n whch agents have dfferent abltes to dstngush between goods (for example a car n good condton versus a car n bad condton). In ths settng, t s useful to dstngush goods not only by ther physcal characterstcs, but also by the agent that s brngng them to the market. Equlbrum s shown to exst, and characterzed by the fact that agents always receve the cheapest bundle among those that they cannot dstngush from truthful delvery. Several examples are presented as an llustraton. Keywords: General equlbrum, Asymmetrc nformaton, Adverse selecton, Uncertan delvery, Pool, Delvery rates. JEL Classfcaton Numbers: C62, C72, D51, D82. João Correa-da-Slva (joao@fep.up.pt) s grateful to Andrés Carvajal, Carlos Hervés Beloso and Herakles Polemarchaks for useful comments, and thanks partcpants n semnars at Pars School of Economcs (Sorbonne), Insttute for Advanced Studes - U. Venna and U. Warwck. Fnancal support from CEMPRE, Fundação para a Cênca e Tecnologa and FEDER (research grant PTDC/ECO/66186/2006) s acknowledged. 1

2 1 Introducton Economc agents usually trade goods wthout havng perfect knowledge of ther characterstcs. Ths apples to frms hrng workers wth unknown productvty, to consumers buyng used cars wth unknown qualty, and to fnancal nsttutons buyng assets wth unknown return. Each trader enters the market wth specfc pror knowledge and observaton abltes concernng the characterstcs of the goods beng traded. Ths s a partcular knd of asymmetrc nformaton (adverse selecton), famous snce the semnal contrbuton of Akerlof (1970). Ths nformaton asymmetry dffers from that consdered n the models of trade wth adverse selecton of Prescott and Townsend (1984a, 1984b), Gale (1992, 1996), Bsn and Gottard (1999, 2006) and Rustchn and Sconolf (2008). In these models, agents enter the market havng prvate nformaton about ther type (endowments and preferences n each of the possble states of nature). In ths paper, each agent s prvate nformaton s descrbed by a partton of the set of commodtes, such that the agent can dstngush goods that belong to dfferent sets of the partton. Ths formalzaton follows the works of Mnell and Polemarchaks (2001) and of Meer, Mnell and Polemarchaks (2006). In these works, there are only markets for classes of goods that everyone can dstngush. If there s an agent n the economy that does not dstngush apples from oranges, then the other agents cannot trade apples (or oranges) among themselves. Ths seems very restrctve, and here an alternatve framework s consdered. The agents are allowed to buy any good. Nevertheless, an agent that buys hgh qualty goods and s not able to observe the qualty of the goods should probably expect to receve low qualty goods nstead. Frequently, the owner of a good has superor nformaton about ts characterstcs. Ths s the case that s focused n ths paper. Agents are assumed to have perfect nformaton about ther endowments, but have dfferental nformaton about the 2

3 characterstcs of the goods whch are beng offered by the other agents n the market. Consder a market n whch melons of dfferent qualty are beng offered. Some agents are able to observe the qualty of the melons, whle others are not. Now suppose that an agent who cannot dstngush the good from the bad melons, decdes, nevertheless, to buy 10 good melons. The seller may delver 10 good melons (truthful delvery), 2 good and 8 bad melons, or even 10 bad melons. Buyng 10 good melons, all that ths unnformed buyer guarantees s delvery of 10 melons (good or bad). Instrumental to the treatment of trade wth adverse selecton are the concepts of pool and delvery rate, ntroduced by Dubey, Geanakoplos and Shubk (2005). We can thnk of a set of commodtes, lke melons (good and bad), as a pool. An agent that buys 10 good melons but s unable to dstngush the good from the bad melons, may receve 2 good melons and 8 bad melons (the delvery rates are 0,2 and 0,8). An agent that can dstngush the good melons surely receves 10 good melons. An agent that buys a good may be delvered one of a set of possbltes, and takes as gven the probabltes of recevng each of the possble delveres. Ths s closely related to what was termed as uncertan delvery by Correa-da-Slva and Hervés-Beloso (2008a, 2008b and 2009), n a seres of papers that study exante trade of contngent goods, wth agents havng dfferent abltes to verfy the occurrence of the exogenous states of nature. The workngs of the economy are as follows. Goods are dstngush goods not only by ther physcal characterstcs, but also by the agent that s brngng them to the market. There are prces and rates of delvery for each of these generalzed goods, that agents take as gven. The only restrcton mposed on delvery rates s that each agent s unable to dstngush what she receves from what she bought. An equlbrum s composed by prces, delvery rates, orders and delveres, such that each agent makes an order that maxmzes the utlty of the delvered bundle, 3

4 wth the (correctly antcpated) delveres beng feasble. The man result of ths paper s the exstence of equlbrum (Secton 2). A byproduct of the exstence proof s the fndng of an equlbrum n whch agents receve the cheapest bundle among those that they cannot dstngush from truthful delvery. Ths s what should be expected from optmzng agents that set delvery rates for the goods they offer. To llustrate the man ntutons offered by the model, the examples presented by Meer, Mnell and Polemarchaks (2006) are explaned and solved (Secton 3). It s left for future work the study of the case n whch agents set, smultaneously, the prces of the goods they offer and the respectve delvery rates. The relevance of ths s made evdent n the work of Wlson (1980). 2 The model We consder an economy n whch a fnte number of agents, I = {1,..., I}, trade a fnte number of goods, l L = {1,..., L}. To capture the usual context n whch the seller has superor nformaton about the qualty of the goods that she brngs to the market, t s useful to consder a generalzed noton of a good, ncorporatng n ts descrpton the name of the agent that s endowed wth the good. Ths allows us to study markets n whch agents may not have the ablty to dstngush good cars from bad cars n general, but are able to observe the qualty of ther own cars. Such reformulaton mples the need to adapt endowments and preferences to ths new setup. We refer to good l that s n the ntal endowment of agent as the generalzed good (l, ). Endowments n terms of these generalzed commodtes, f IR LI +, relate to the 4

5 usual defnton of endowments, e IR L +, as follows: f IR IL +, wth f (l,) = e l and f (l,j) = 0, l,, j. Smlarly, the utlty functons n terms of these generalzed commodtes, V, can be obtaned from the usual utlty functons, U : IR L + IR, as follows: V : IR LI + IR; V (x ) = U(z ), where z l = j x (l,j). Agents wsh to maxmze ther utlty functons, V (x ), whch are contnuous, concave and strctly ncreasng 1. Each agent has specfc abltes to dstngush the dfferent goods that are traded n the market. These observaton abltes are descrbed by a partton of the set of generalzed goods, P, such that (l, j ) P (l, j) f and only f agent cannot dstngush good (l, j ) from good (l, j). The nablty to dstngush between two goods, (l, j) and (l, j ), mples that an agent that buys certan quanttes of (l, j) and (l, j ), say y (l,j) and y (l,j ), may receve dfferent quanttes, x (l,j) and x (l,j ), such that: x (l,j) + x (l,j ) = y (l,j) + y (l,j ). More generally, when buyng y = (y (1,1),..., y (1,I) receve x = (x (1,1),..., x (1,I), x (2,1) x (1,1) x (1,I) x (2,1). x (L,I) =, y (2,1),..., x (L,I) ) such that: k (1,1),(1,1) k (1,1),(1,I) k (1,1),(2,1) k (1,1),(L,I)..... k (1,I),(1,1) k (1,I),(1,I) k (1,I),(2,1) k (1,I),(L,I) k (2,1),(1,1) k (2,1),(1,I) k (2,1),(2,1) k (2,1),(L,I) k (L,I),(1,1) k (L,I),(1,I) k (L,I),(2,1) k (L,I),(L,I),..., y (L,I) ), agent wll y (1,1). y (1,I) y (2,1). y (L,I), 1 By strctly ncreasng, t s meant that an ncrease n consumpton of any of the goods s strctly desred by the agents: x x and x x mples that V (x ) > V (x ). 5

6 where k (l,j ),(l,j) denotes the number of unts of good (l, j ) that agent receves for each unt of good (l, j) that she buys. The delvery matrx, k, s endogenous (equlbratng varable). The delvery matrces that are compatble wth the abltes of agent to dstngush commodtes are such that, for each (l, j): (l,j ) P (l,j) k (l,j ),(l,j) = 1 and k (l,j ),(l,j) = 0, (l, j ) / P (l, j). The set of matrces that satsfy these condtons s denoted K, and K = I =1 K. It should be clear that f agent s able to dstngush all the commodtes, that s, f P (l, j) = {(l, j)}, (l, j), then K has a sngle element (the dentty matrx). In ths case, agent s sure of recevng exactly the bundle that she buys. To smplfy our problem, we wll always assume that agents have perfect nformaton about ther endowments. Assumpton 1 (Perfect nformaton about own endowments). (, l) : P (l, ) = {(l, )}. Only goods that exst are prced and traded n the market. those for whch f (l,j) > 0, a condton whch s equvalent to f (l,j) j Such goods are > 0. Ths restrcton mples straghtforward modfcatons of the spaces n whch f, k, V and P are defned. We wll denote the set of goods that exst by M L I, and the number of goods that exst by M L I. When the classcal nterorty assumpton holds (e 0, ), all the goods are traded n the market and therefore M = L I and M = L I. Takng prces, p M, and delvery rates, k K, as gven, agent trades ts ntal endowments, f IR M +, for a bundle, y IR M +, that maxmzes utlty, 6

7 V (k y ), among those that satsfy the budget restrcton, y B (p): B (p) = { y IR M + : p y p f }. Notce that Assumpton 1 guarantees ndvdual ratonalty of partcpatng n the market. The agent can always buy ts own endowments. Defnton 1 (Equlbrum). An equlbrum of the economy E {f, V, P } I =1 s composed by a prce system, p M +, ndvdual choces, y = (y 1,..., y I ) IRIM +, delvery rates, k = (k 1,..., k I ) K, and the resultng allocaton, x = (x 1,..., x I ) IRIM +, whch satsfy: y arg max V (k y B (p y ), [ndvdual optmalty]; ) x = k y, [delvery]; x f [feasblty]. Theorem 1 (Exstence of equlbrum). There exsts an equlbrum of the economy. Proof: Consder, for now, bounded choce sets. For choces n the upper bound to mply aggregate excess delvery, let E = (l,j) M f (l,j) j convex and bounded choce sets: + 1 and defne the followng Y = {y IR M + : y (l,j) E, (l, j)}. The budget set of agent, n ths bounded economy, s: B (p) = { y Y : p y p f }. Let ψ (y, p, k) = arg max {V (k z )}. z B (p) 7

8 The utlty functon, V (k y ), s contnuous wth respect to both k and y. As long as p f > 0 (whch s always the case for p 0), the correspondence B (p) = {y Y : p y p f } s contnuous wth non-empty compact values. When ths s the case, we know, from Berge s Maxmum Theorem 2, that the demand correspondence, ψ (y, k, p), s upper hemcontnuous wth nonempty compact values. It s also convex-valued, because V s concave and k s constant. Let M ɛ = { p M : p ɛ }. We wll start by fndng a fxed pont wth strctly postve prces, on M ɛ, and then let ɛ 0 to obtan a sequence of fxed ponts. { Let ψp(y, ɛ p, k) = arg max q } (k y f ). q M ɛ And let ψ k (y, p, k) = arg mn d K {p d y }. All these correspondences (ψ, ψ ɛ p and ψ k ) are upper hemcontnuous wth nonempty compact and convex values. Therefore, the product correspondence, ψ ɛ = I =1 ψ ψ ɛ p I =1 ψ k, also s. Applyng the Theorem of Kakutan, we fnd that there exsts a fxed pont of ψ ɛ, that we denote by (y ɛ, p ɛ, k ɛ ). Consderng a sequence, {ɛ n } n IN, that converges to zero, we obtan a sequence of fxed ponts, {(y n, p n, k n )} n IN. The sequence s contaned n a compact set, therefore a subsequence converges to (y, p, k ). We want to consder ths subsequence and verfy that ts lmt s an equlbrum. Suppose that the sequence of prces n the nteror of the smplex, {p n }, converges to a prce on the border of the smplex. There s at least one agent whose ncome does not tend to zero (p f > 0), and therefore whose demand (whch s u.h.c.) s drven to the bound of the choce set (recall that utlty s strctly ncreasng). Ths mples that, for suffcently large n and n the lmt, there s aggregate excess delvery of at least some good: (l, j) : x (l,j) = (l,j ) k (l,j),(l,j ) y (l,j ) > f (l,j). The budget restrctons mply that p n yn p n f, and the defnton 2 See, for example, Alprants and Border (2006). 8

9 of ψ k mples that p n kn y n p n yn (the dentty matrx s enough for equalty). In the lmt: p x = p k y p f. (1) From (1), we know that f there s some good (l, j) wth strctly postve prce (p (l,j) > 0), for whch x(l,j) > f (l,j) (excess delvery), there must be another, (l, j ), for whch,j ) x(l < f (l,j ) (excess supply). To understand the dea of the proof, start by supposng that, n the lmt, there s a sngle good, (l, j), wth maxmal excess delvery. From the defnton of ψ p, the prce of ths good tends to 1, and the remanng goods have vanshng prces. Therefore, the aggregate budget restrcton becomes: y (l,j) f (l,j). Gven the defnton of the ψ k : x (l,j) = (l,j ) k (l,j),(l,j ) y (l,j ) = k (l,j),(l,j) y (l,j) y (l,j). Whch mples that x(l,j) f (l,j) (no excess delvery). Contradcton. Now let s move on to the general case. Suppose that, n the lmt, there s a set of goods, G, ted for the maxmal excess delvery: (l, j) G : x (l,j) = (l,j ) k (l,j),(l,j ) y (l,j ) > f (l,j). (2) In the lmt, the prces of these goods must add to 1, whle the prces of the remanng goods become null. Aggregatng the budget restrctons: (l,j) G p (l,j) y (l,j) (l,j) G p (l,j) f (l,j). Consder the set of goods wth hghest prce, G 1 G. From the defnton of the ψ k, we know that buyng one unt of a good n G 1 mples delvery of quanttes 9

10 of goods n G 1 that add to 1 unt or less, because buyng goods that do not belong to G 1 mples no delvery of goods n G 1. Then: x (l,j) (l,j) G 1 y (l,j). (l,j) G 1 In fact, consderng the set of goods wth prce hgher or equal to some threshold, t > 0, denoted G t G, we have (from an analogous reasonng): x (l,j) (l,j) G t y (l,j). (l,j) G t Observng that y domnates x n the sense of Lorenz (for prce nequalty among goods nstead of ncome nequalty among agents), we obtan: { p } (l,j) x (l,j) { p } (l,j) y (l,j) (l,j) G (l,j) G p (l,j) x (l,j) p (l,j) y (l,j) p (l,j) f (l,j). (l,j) G Whch contradcts (2). (l,j) G (l,j) G There s not excess delvery of any good, therefore, x s feasble and p 0. Exstence of equlbrum n the bounded economy s establshed. To check that ths s an equlbrum when the bounds on the choce sets are removed, we must verfy that ndvdual choces reman unaltered. Observe that the bound on the choce sets s large enough for the ndvdual choces, y, to be n the nteror of Y (otherwse we would not have feasblty). Snce preferences are convex, we are sure that the bounds are not bndng. If there were a strctly better choce outsde Y, then there would also be a strctly better choce n the fronter of Y. QED Under general condtons, equlbrum exsts. Furthermore, the equlbrum that was found has a nce property: delvered bundles are as cheap as possble (to see ths, look at the defnton of ψ k n the proof of exstence). Although delvery 10

11 rates were taken as gven by the agents, they are concde wth those that would be expected from ther optmzng behavor. Ths property nduces a natural refnement of the equlbrum concept. 3 Examples In ths secton, some examples presented as an llustraton. 3 Here the notaton s smpler than n the prevous secton, because t wll not be necessary to deal wth generalzed goods n a formal way. In the frst example, an agent that does not dstngush two goods s not able to consume the hgh qualty good, n spte of beng wllng to pay any prce for a small quantty of ths good. The second example, two partal substtute goods are traded at the same equlbrum prce. The agent that does not dstngush the two goods receves both goods, wth delvery rates determned by the leftovers from the choces of the nformed agents. Fnally, an example of a job market wth two frms: one that can observe the productvty of workers, and another that cannot. Wages end up reflectng productvty, wth the nformed frm hrng the more productve workers. 3.1 Cherry pckng Three ndvduals, I = {1, 2, 3}, trade three commodtes, L = {0, r, g}, that we can thnk of as money, red cherres and green cherres. 3 The examples were adapted from those presented by Meer, Mnell and Polemarchaks (2006). The equlbrum concept used here leads to qualtatvely dfferent solutons. 11

12 Agent 1 s endowed wth money, agent 2 wth red cherres and agent 3 wth green cherres. e 1 = {12, 0, 0}; e 2 = {0, 12, 0}; e 3 = {0, 0, 12}. All agents prefer green cherres to red cherres. Agent 2 does not lke red cherres at all. Preferences are descrbed by the followng utlty functons: U 1 (x 1 ) = ln(x 0 1) + ln(x r 1) + 2ln(x g 1); 4 U 2 (x 2 ) = ln(x 0 2) + 2ln(x g 2); U 3 (x 3 ) = ln(x 0 3) + ln(x r 3) + 2ln(x g 3). There s asymmetrc nformaton because agent 1 cannot dstngush red cherres from green cherres whle agents 2 and 3 are able to dstngush the three commodtes. P 1 = {{0}, {r, g}}; P 2 = {{0}, {r}, {g}}; P 3 = {{0}, {r}, {g}}. Agents 2 and 3 are perfectly nformed, and therefore receve exactly what they buy. On the other hand, agent 1 can buy ether red cherres or green cherres, but receves whatever knd of cherres s delvered (because agent 1 does not dstngush the two goods). 5 5 The group of commodtes desgnated as cherres can be seen as a lst contanng two alternatves: red cherres and green cherres. In the orgnal context of an economy wth uncertan delvery (see Correa-da-Slva and Hervés-Beloso (2008a, 2008b and 2009) for a detaled descrpton), whch was of ex-ante contractng for future contngent delvery (agents could not dstngush states of nature, but could dstngush commodtes), delvery had to be ether of red cherres or green cherres. Here, addtonal complexty arses from the fact that there s an nfnte number of possbltes for the delvery of 10 cherres (10 red and 0 green, 5 red and 5 green, 7 red and 3 green, etc.). 12

13 Snce agent 2 wll not buy red cherres, her budget restrcton s (notce that the prce of money s normalzed to p 0 = 1): p 0 x p r x r 2 + p g x g 2 = p r e r 2 x p g x g 2 = 12p r. The optmalty condton mples equalty between the ratos between margnal utlty and prce, for each good demanded: x 0 2 = 0.5p g x g 2. From the budget restrcton and the optmalty condton, we fnd the demand of agent 2: (x 0 2, x r 2, x g 2) = (4p r, 0, 8 pr p g ). Smlarly, we can obtan the demand functon of agent 3: x p r x r 3 + p g x g 3 = 12p g (x 0 3, x r 3, x g 3) = (3p g, 3 pg x 0 3 = p r x r 3 = 0.5p g x g p, 6). r 3 Lookng at the demand of agents 2 and 3 for green cherres, we fnd that p r < p g, otherwse there would be excess demand. More precsely: 8 pr p g pr 0.75p g. Suppose that agent 1 buys a quantty x rg 1 of cherres (guarantees delvery of red cherres and green cherres such that x r 1 + x g 1 = x rg 1 ). If red cherres are cheaper than green cherres, then the agent should receve only red cherres, and no green cherres. If the agent receved some green cherres, then one could wonder why someone s delverng these green cherres, nstead of tradng them n the market for red cherres plus money and delverng the red cherres whle keepng the money (there would be an arbtrage opportunty). 13

14 Followng ths reasonng, snce p r < p g, then x g 1 = 0. Assumng that agent 1 s aware of ths (delvery rates are antcpated and taken as gven), we can fnd her demand functon. The quantty x r 1 and the prce p r can also be nterpreted as the quantty and prce of (undstngushed, or pooled) cherres, x rg 1 and p rg. 6 x p r x r 1 = 12 x 0 1 = p r x r 1 x x 0 1 = 12 x 0 1 = 6; p r x r 1 + p r x r 1 = 12 x r 1 = 6. p r For demand to equal supply: x x x 0 3 = 12 4p r + 3p g = 6; x r 1 + x r 2 + x r 3 = pg = 12; p r p r x g 1 + x g 2 + x g 3 = 12 8 pr = 6 p r = 0.75p g. p g These equatons allow the determnaton of equlbrum prces: p = (p 0, p r, p g ) = (1; 0.75; 1). The allocaton s, therefore (y = x yelds truthful delveres): x 1 = (6, 6, 0) = (6, 8, 0); p r x 2 = (4p r, 0, 8 pr ) = (3, 0, 6); p g x 3 = (3p g, 3 pg, 6) = (3, 4, 6). p r 3.2 Modfed cherry pckng Is t always the case that agent 1 does not consume green cherres? In ths modfed example, we fnd that agent 1 can consume green cherres f the ncentves for agents 2 and 3 to delver only red cherres dsappear. 6 Agent 1 prefers any nteror bundle (x 1 0) to a bundle that s n the fronter of the consumpton set. But she cannot get any green cherres and, therefore, her utlty s nfntely negatve. Recall that we are assumng that, among bundles wth x g 1 = 0 (n the fronter of the consumpton set), the preferences of agent 1 are descrbed by v 1 (x 1 ) = ln(x 0 1) + ln(x r 1). 14

15 Ths occurs f green cherres become much more abundant than red cherres. Let s double the endowments of agent 3 (green cherres): e 1 = {12, 0, 0}; e 2 = {0, 12, 0}; e 3 = {0, 0, 24}. Demand of agent 2 remans unaltered: x p g x g 2 = 12p r (x 0 2, x r 2, x g 2) = (4p r, 0, 8 pr x 0 2 = 0.5p g x g p ). g 2 Whle demand of agent 3 doubles: x p r x r 3 + p g x g 3 = 24p g x 0 3 = x r 3p r = 0.5x g 3p g x 3 = (6p g, 6 pg p r, 12). Now the demand of agents 2 and 3 for green cherres does not exceed supply as long as p r 1.5p g : x g 2 + x g 3 = 8 pr p g pr 1.5p g. There are three possbltes: (a) the prce of green cherres s hgher than the prce of red cherres and thus agent 1 only consumes red cherres (as n the prevous example); (b) the prce of red cherres s hgher than the prce of green cherres and thus agent 1 only consumes green cherres ; (c) the prces of green cherres and red cherres concde. In case (a), there would be excess supply of green cherres, as aggregate consumpton s lower than 20: x g 1 + x g 2 + x g 3 = pr + 12 < 20. pg In case (b), there would be excess supply of red cherres, as aggregate consumpton s lower than 6: x r 1 + x r 2 + x r 3 = pg p r < 6. 15

16 Thus, n equlbrum we must have case (c): p r = p g = p rg. Thus: x 2 = (4p rg, 0, 8) x 2 + x 3 = (10p rg, 6, 20). x 3 = (6p rg, 6, 12) The only canddate for an equlbrum allocaton gves agent 1 the followng consumpton bundle: x 1 = (12 10p rg, 6, 4). To check whether ths s an equlbrum, we need to fnd the demand of agent 1. A problem that we face s that agent 1, through her demand, may nfluence the qualty of the cherres (the proporton between red and green cherres). It s assumed that she takes the proportons of delvered red and green cherres as gven. In equlbrum, ths proporton must be fulflled (otherwse t would not be an equlbrum). Snce we already have a sngle canddate for the equlbrum consumpton of agent 1, x 1 = (12 10p rg, 6, 4), we must assume that agent 1 expects to receve 60% red cherres and 40% green cherres. The utlty and the maxmzaton condton of agent 1 are (wth x rg 1 = x r 1 + x g 1): u 1 (x 1 ) = lnx ln(0.6x rg 1 ) + 2ln(0.4x rg 1 ) x 0 1 = 1 3 prg x rg 1. Puttng ths together wth the budget restrcton, demand s obtaned: x p rg x rg 1 = 12; x 0 1 = 1 3 prg x rg 1 (x 0 1, x r 1, x g 1) = (3, p 9, 0.4 rg p rg ). For demand of money to equal supply: x x x 0 3 = p rg + 6p rg = 12 p rg = 0.9. Equlbrum prces are, therefore, p = (1; 0.9; 0.9), and the allocaton s: x 1 = (3; 6; 4); x 2 = (3.6; 0; 8); x 3 = (5.4; 6; 12). 16

17 In ths case, agent 1 consumes both red cherres and green cherres, whch are traded at the same prce. Agents 2 and 3 optmze by delverng 3.6 unts of red cherres and 5.4 unts of green cherres to agent 1. The quanttes of red and green cherres that agent 1 buys (y1 r and y1) g are rrelevant, gven that they add to 10. If y = x, there s truthful delvery. In any case, the delvery rates adjust to be such that delvery s surely that calculated above: x r 1 = 6 and x g 1 = A job market Consder an economy wth two frms, A and B, that have an ntal endowment of money and hre labor. There are two types of labor, 1 and 2 (type 2 s more productve than type 1). Money s desgnated as good 0. The frms have the same preferences and endowments (8 unts of money ): e A = {8, 0, 0} u A (x A ) = x A0 + x A1 + 4x A2 ; and e B = {8, 0, 0} u B (x B ) = x B0 + x B1 + 4x B2. The prce of money s normalzed to 1, and the wages of each type of labor are denoted w 1 and w 2. Two workers supply labor, at the expense of ther tme of lesure. Ther endowments are 4 unts of tme. e 1 = {0, 4, 0} e 2 = {0, 0, 4} and u 1 (x 1 ) = x 10 1 x 11 ; u 2 (x 2 ) = x 20 1 x 22. There s asymmetrc nformaton because frm A can dstngush between the two types of labor, but frm B cannot. P A = {{0}, {1}, {2}}; P B = {{0}, {1, 2}}. 17

18 For demand of labor by frm A to be fnte, t s necessary that w 1 1 and w 2 4. In fact, t s clear that n equlbrum we wll have w 2 = 4, otherwse there would be no demand for labor of type 2. Notce that, at ths wage, frm A s wllng to hre any quantty of labor of type 2. Optmzaton by the workers yelds demand for lesure. Agent 2 dedcates half of the tme to work and the other half to lesure. w 2 x 2 22 = 1 x 22 = w 1/2 2 = 2. If w 1 = 4, then the worker of type 1 would also wsh to sell 2 unts of labor. Only frm B could use ths labor (frm A would not fnd t proftable to hre labor of type 1 at w 1 = 4). But, clearly, frm B would not be wllng to pay w 1 = 4, knowng that the labor would not be 100% of type 2. Thus: w 1 < 4 and all the labor hred by frm B s of type 1 (workers of type 2 are not be wllng to work for a wage lower than 4, whch s what they receve from frm A). Frm B s aware of ths reasonng, and knows, therefore, that the labor hred s 100% of type 1. For frm B to demand a fnte amount of labor, t s necessary that w 1 = 1. In sum, these consderatons mply that equlbrum prces are w 1 = 1 and w 2 = 4. Agan, optmzaton by the worker yelds agent 1 s demand for lesure. w 1 x 2 11 = 1 x 11 = w 1/2 1 = 1. From the budget restrctons, we obtan the money ncome of each agent: x 10 + w 1 x 11 = w 1 e 11 x 10 = w 1 (4 x 11 ) x 10 = 3; x 20 + w 2 x 22 = w 2 e 22 x 20 = w 2 (4 2) x 20 = 8. The correspondng money ncome and lesure tme of each agent are: x 1 = (1, 3, 0); x 2 = (8, 0, 2). 18

19 Ignorng the ndetermnacy on the allocaton of labor of type 1 (frm B s assumed to hre all the supply), we fnd: x A = (x A0, x A1, x A2 ) = (4, 0, 2); x B = (x B0, x B1, x B2 ) = (7, 1, 0). 4 Concludng remarks In an economy n whch agents trade goods wth uncertan qualty, the ablty to observe the qualty of the good s very useful. To study markets such as the used car market (Akerlof, 1970), t s natural to assume that agents know the qualty of the goods that they brng to the market, but not the qualty of the goods brought by the other agents. To model ths knd of nformaton asymmetry, we have consdered a generalzed noton of a good, ncorporatng n ts descrpton the agent that s endowed wth ths good. Ths allowed us to study economes n whch agents may not have the ablty to dstngush good cars from bad cars, but are able to observe the qualty of ther own cars. Equlbrum s shown to exst, and characterzed by the fact that agents always receve the cheapest delvery that s consstent wth ther observaton abltes (that s, that they cannot dstngush from truthful delvery) In ths model, the prce of the same good may vary across sellers (observe that each seller faces buyers wth dfferent nformaton), but prce dscrmnaton s not allowed (the prce of a good does not depend on the buyer). Notce also that f two agents sell the same good, wth equal delvery rates, they must sell t at the same prce. Snce the prces of goods depend on the agent that brngs them to the market, assumng prce-takng behavor s questonable. It stll provdes a natural bench- 19

20 mark, but ts comparson wth alternatve assumptons would be nterestng, as well as the study of a replcated economy. There could be objectons to the assumpton of agents takng own delvery rates as gven, but t can be verfed that the equlbrum delvery rates concde wth those that would be expected from optmzng agents. Anyway, the message of the work of Wlson (1980) suggests the study of the case n whch agents set, smultaneously, the prce of the goods they brng to the market, and the correspondng delvery rates. Ths s left for future work. References Alprants, C.D. and K.C. Border (2006), Infnte Dmensonal Analyss, 3rd edton, Sprnger-Verlag, Berln. Akerlof, G.A. (1970), Market for Lemons : Qualty Uncertanty and the Market Mechansm, Quarterly Journal of Economcs, 84 (3), pp Arrow, K.J. and G. Debreu (1954), Exstence of an Equlbrum for a Compettve Economy, Econometrca, 22 (3), pp Bsn, A. and P. Gottard (1999), Compettve Equlbra wth Asymmetrc Informaton, Journal of Economc Theory, 87 (1), pp Bsn, A. and P. Gottard (2006), Effcent Compettve Equlbra wth Adverse Selecton, Journal of Poltcal Economy, 114 (3), pp Correa-da-Slva, J. and C. Hervés-Beloso (2008a), Subjectve Expectatons Equlbrum n Economes wth Uncertan Delvery, Journal of Mathematcal Economcs, 44 (7-8), pp Correa-da-Slva, J. and C. Hervés-Beloso (2008b), General Equlbrum wth Prvate State Verfcaton, FEP Workng Papers,

21 Correa-da-Slva, J. and C. Hervés-Beloso (2009), Prudent Expectatons Equlbrum n Economes wth Uncertan Delvery, Economc Theory, 39 (1), pp Dubey, P., J. Geanakoplos and M. Shubk (2005), Default and Punshment n General Equlbrum, Econometrca, 73 (1), pp Gale, D. (1992), A Walrasan Theory of Markets wth Adverse Selecton, Revew of Economc Studes, 59, pp Gale, D. (1996), Equlbra and Pareto Optma of Markets wth Adverse Selecton, Economc Theory, 7, pp McKenze, L.W. (1981), The Classcal Theorem on Exstence of Compettve Equlbrum, Econometrca, 49 (4), pp Meer, M., E. Mnell and H.M. Polemarchaks (2006), Nash-Walras Equlbra wth Prvate Informaton on Both Sdes, mmeo, October Mnell, E. and H.M. Polemarchaks (2001), Nash-Walras Equlbra of a Large Economy, Proceedngs of the Natonal Academy of Scences of the Unted States of Amerca, 97 (10), pp Prescott, E.C. and R.M. Townsend (1984a), Pareto Optma and Compettve Equlbra wth Adverse Selecton and Moral Hazard, Econometrca, 52 (1), pp Prescott, E.C. and R.M. Townsend (1984b), General Compettve Analyss n an Economy wth Prvate Informaton, Internatonal Economc Revew, 25 (1), pp Rustchn, A. and P. Sconolf (2008), General Equlbrum n Economes wth Adverse Selecton, Economc Theory, 37 (1), pp Wlson, C. (1980), The nature of equlbrum n markets wth adverse selecton, The Bell Journal of Economcs, 11 (1), pp