Global Uniqueness and Money Non-neutrality in a Walrasian Dynamics without Rational Expectations

Transcription

1 Global Unqueness and Money Non-neutralty n a Walrasan Dynamcs wthout Ratonal Expectatons Gaël Graud & Dmtros Tsomocos September 10, 2004 Abstract. We defne a non-tâtonnement dynamcs n contnuous-tme for pure-exchange economes wth outsde and nsde fat money. Traders are myopc, face a cash-n-advance constrant, and play domnant strateges n a short-run monetary strategc market game nvolvng the lmt-prce mechansm. The profts of the Bank are redstrbuted to ts prvate shareholders, but they can use them to pay ther own debts only n the next perod. Provded there s enough nsde money, monetary trade curves converge towards Pareto optmal allocatons; money has a postve value along each trade curve, except on the optmal rest-pont where t becomes a vel whle trades vansh. Moreover, genercally, gven ntal condtons, there s a pecewse globally unque trade-and-prce curve not only n real, but also n nomnal varables. Fnally, money s locally neutral n the short-run and non-neutral n the long-run. Résumé. On défnt une dynamque de non-tâtonnement en temps contnu pour les économes de consommaton, avec fat monnae nterne et externe. Les agents sont myopes, font face à une contrante de lqudté, et jouent une stratége domnante dans un jeu de marchés stratégque monétare construt autour du mécansme de prx lmtes de Mertens (2003). Les revenus de la Banque Centrale sont redstrbués à ses actonnares, mas ne peuvent être utlsés qu à la pérode suvante. S l y a suffsamment de monnae nterne, les courbes d échange monétare convergent vers une allocaton Pareto-optmale et la monnae a une valeur postve le long de ces courbes d échange, horms à la lmte, où elle devent nutle, tands que les échanges cessent. De surcroît, générquement, à dotatons ntales fxées correspond une courbe d échange et de prx unque par morceaux, non seulement en termes réels, mas encore en termes nomnaux. Enfn, la monnae est localement neutre ` court terme, mas n est jamas neutre à long-terme. CNRS umr 8095, CERMSEM, Unversté Pars-1, France. ggraud@unv-pars1.fr Saïd Busness School Unversty and St. Edmund Hall, Unversty of Oxford and Fnancal Markets Group, UK. dmtros.tsomocos@sad-busness-school.oxford.ac.uk 1

2 2 Parte C Keywords. Bank; Money; Prce-quantty Dynamcs; Lmt-prce mechansm; Insde money; Outsde money. JEL Classfcaton: D50, E40, E41, E50, E58. 1 Introducton The structure of a model reflects the practcal purposes whch drve the research n the frst place. Standard macroeconomc models reduce the aggregate economy to manageable proportons, and frequently a common smplfcaton s the representaton of each sector by agents whch behave dentcally; consequently, they are presented n representatve agent format. On the other hand, standard general equlbrum wth heterogenous agents quckly becomes ntractable: Closed form solutons typcally cannot be derved, and ther results are often not robust. The man mpedment les on the multplcty of equlbra. A second one s the statc vewpont, n effect, underlyng both standard macroeconomcs and general equlbrum theory: In most cases, theory s unable to descrbe n a sensble way what happens out of equlbrum. In the present paper, whle mantanng the basc ngredents of equlbrum analyss, namely market clearng and agent optmsaton, we offer a dynamc contnuoustme non-tâtonnement process for monetary economes wth heterogenous agents. By consderng boundedly ratonal households, we devate from the ratonal expectatons hypothess. By buldng upon the strategc market games framework, and embeddng the lmt-prce mechansm of Mertens (2003), we mantan heterogenety, and we manage to buld a model wth a monetary sector where money has non-neutal effects. In addton, short-run equlbra converge towards Pareto-optmal allocatons, and both real and nomnal varables are genercally pecewse unque. 1.1 Monetary short-run lnear economes More precsely, we consder an exchange economy whose dynamcs s drven, at each tme nstant, by a monetary short-run (lnear) economy n whch agents maxmze the frst-order approxmaton of ther current utltes subject to a cash-n-advance constrant à la Clower (1967). 1 Hence, our approach can be vewed alternatvely as the monetary counterpart of Graud (2004), whch was tself a game-theoretc rewrtng of Champsaur & Cornet (1990). We postulate a cash-n-advance constrant whereby recepts from commodty sales cannot be used contemporaneously for purchases. Therefore, traders borrow nsde money from a loan market n antcpaton of future ncome whch s used to pay back ther loans. We partly follow the monetary paradgm as set out by Dubey & Geanakoplos (1992, 2003a,b). Households are endowed wth outsde money and a government Bank njects nsde money. We depart n as much as we allow agents to send lmt-orders (and not just market orders) to the market, and profts of the Central Bank are redstrbuted to prvate shareholders (cf. Shubk & Tsomocos (1992)). Drèze & Polemarchaks (1999, 2000, 2001) n a related monetary framework assume, n addton, that the Bank dstrbutes ts profts to prvate shareholders. But, snce they are n a statc one-shot world, they can use them to pay ther own debts to 1 See also Grandmont & Younès (1972).

3 Global Unqueness and Money Non-neutralty 3 the Bank. As a consequence, there s no outsde money n ther model, and nomnal ndetermnacy of the statc equlbra obtans. 2 Here, when a shareholder receves dvdends at the end of tme t, she can only use them as cash (outsde money) at tme t + dt. The consequence s that, at varance wth Dubey & Geanakoplos (2003a,b), the ext of the quantty, m, of outsde money s ncpent because ths money (beng equal, along each equlbrum trade curve, to the Bank s nstantaneous proft) s renjected n the economy one nano-second later. But, unlke Drèze & Polemarchaks (2001, strong ndetermnacy) and Dubey & Geanakoplos (2003b, generc local unqueness), we get global unqueness of the outcome. Ths dynamc process of nfntesmal trades contnues tll the resultng system of dfferental equaton converges to a rest-pont. If there s enough nsde money, ths restpont s a Pareto-optmal pont. Otherwse, the economy remans stuck at a non-effcent pont. As Graud (2004) shows, ths lmt-prce process nduces a dscontnuous vector feld. Ths dscontnuty s unavodable, and can be traced back to the fundamental lack of contnuty that characterzes both strategc market games and double-auctons (Mertens (2003) lmt-prce mechansm s an extenson of both, and therefore nherts the dscontnuty). We thus nvoke Flppov s soluton concept to dfferental equatons wth a dscontnuous rght-hand sde n order to establsh exstence of monetary trade curves and convergence of the state of the economy to a long-run (non-monetary,.e., Walrasan) prce equlbrum. The advantage of the formulaton we adopt n ths paper s that t enables us to ntroduce a Central Bank provdng nsde money, keeps a cash-n-advance constrant, and formulate all these ngredents wthn a dynamcs whch s drven va a full-blown local game assocated to each short-run economy. Intraperod nomnal nterest rates are endogenously determned at each nstant, and serve as market-clearng prces between bonds and nsde money. The feasblty constrant on each money-sellng order s tantamount to a cash-n-advance constrant. Hahn s (1965) famous puzzle s solved because agents who borrow M nsde money from the Central Bank at nterest rate m M(t) r(t) > 0 must return (1 + r(t))m(t) > M(t). If r(t) = and ths wll turn out to be the case along each equlbrum monetary trade curve, all the outsde money m wll ext the economy at tme t along wth nsde money (before beng renjected at tme t + dt). Fsher s quantty theory of money takes the followng form n each short-run economy (except at a rest-pont of the dynamcs where trades collapse): m + M(t) = p c (t)qc(t, p c ), (1) where m (resp. M(t)) s the aggregate amount of outsde (resp. nsde) money put on the market at tme t, p c (t) s the (endogenous) current prce of commodty c, and q c(t, p c ) s the (endogenous as well) current quantty of commodty c put up for sale at p c. m s not ndexed by t because t wll turn out to reman constant across tme, whatever beng the dynamcs of nsde money. We show that the classcal dchotomy holds n the short-run, provded there s enough nsde money. Indeed, provded there are 2 The excepton s when the government budget constrant s volated, n whch case Drèze & Polemarchaks model would also nduce nomnal determnacy. Volaton of the government budget constrant s equvalent to exstence of outsde money. Notce that, n the present paper, the governement volates ts budget constrant only durng a nano-second, snce the profts earned by the Central Bank at tme t are redstrbuted (as outsde money) at tme t + dt. Ths suffces to recover determnacy. c

4 4 Parte C enough gans to trade,.e., M(t) > m γ(e(t)),3, the old quantty theory of money, defended by the neo-classcal school holds water: One can separate the real and nomnal sdes of the economy, solvng the real sde for relatve prces, and fxng ther levels by the stock of nomnal money. But ths holds only n the short-run. Indeed, the amount of money M(t) must change over tme n order to keep the precedng nequalty. How t changes wll then necessarly affect both nomnal and real varables along every trade curve. Moreover, the (mplct) prce of money (to be dstngushed both from the nterest rate) s always equal to 1, and ts velocty s a decreasng functon of tme (and of r(t)), bounded from above by 1. Fnally, not only does money have value n our model, ts value s determnate: For generc economes, the dynamcs of nterest rates, prce levels and commodty allocatons are unquely defned (wthn a certan tme nterval). Monetary polcy beng non-neutral n the long-run, ts effects can n prncple be tracked because of the global unqueness of soluton paths to our dynamcs A central clearng house for nsde and outsde money To elaborate our own defnton of a short-run outcome, we recall n secton 3.2. nfra the ngredents of Dubey & Geanakoplos (2003a) defnton of a monetary equlbrum. As every Walrasan-lke equlbrum concept, ths monetary equlbrum does not guarantee, n general, global unqueness of ts correspondng allocaton, even when restrcted to lnear short-run economes. On the other hand, t heavly rests on the perfect foresght assumpton. Indeed, each perod t s dvded n three subperods t α, t β and t γ. At tme t α, agents borrow money from a Central Bank by sellng bonds; at t β, they exchange commodtes for money; at t γ, they repay ther loans to the Central Bank. When sellng bonds n the frst subperod t α, households need to perfectly antcpate both t β -commodty prces and nterest rates (the latter beng charged at tme t γ ) n order not to default. And from a game-theoretc vewpont, ths no-default constrant also nduces the use of a generalzed game, and not a full-blown normal-form game, or the ntroducton of penalty rules n order to guarantee that, at a Nash equlbrum, nobody wll go bankrupt. In ths paper, we keep the sprt (orgnatng n Shubk & Wlson (1977)) of ntroducng money n general equlbrum theory va a central Bank. However, we drop the ratonal expectatons hypothess, whch, though commonly made n the general equlbrum lterature, would be at odds wth the postulated myopa of our boundedly ratonal agents. The key nsght of our analyss conssts n extendng (along the lnes of Mertens (2003)) the paradgm ntroduced by Dubey & Geanakoplos (2003a) to lmtprce orders.e., orders that are condtonal on the realzaton of prces (or nterest rates). Ths enables to relax the ratonal expectatons hypothess, to end-up wth a classcal one-shot normal-form game, and to recover the unqueness of the short-run outcome assocated to each state of the economy (equvalently, to get a vector feld, and not just a cone feld). For ths purpose, we equp each ndvdual wth a certan amount of bonds wth whch she starts afresh at each tme t. Before enterng the market for commodtes, 3 Where γ(e(t)) s a (short-run verson of the) measure of gans to trade ntroduced n Dubey & Geanakoplos (1992). 4 The exploraton of ths fascnatng topc s left for further research.

5 Global Unqueness and Money Non-neutralty 5 traders can trade ther bonds aganst nsde money. In stage t β, they then play a domnant strategy n the lmt-prce mechansm, takng nto account the fact that, afterwards, they wll have to fully delver on ther bonds. Fat money n our model corresponds to the paper used as cash n everyday lfe. All cash of course looks the same, regardless of whether t was orgnally njected as nsde money or as outsde money. Nevertheless, n our model as n Dubey & Geanakoplos (2003a), the orgn of the money plays a crtcal role n determnng the endogenous real and monetary varables. Hence, and ths s of course specfc to ths paper, t s pvotal n the determnaton of the whole dynamcs. Snce t s the nterplay between today s outsde and nsde money (or, equvalently, yesterday s profts of the Central Bank and today s nsde money) that s responsble for the results obtaned here, a serous justfcaton s needed for the dstncton between both types of money. When the State njects money nto the prvate sector n exchange for assets promsng the future delvery of money, ts arrval foreshadows ts departure, and we call t nsde money. Money njected by the State as a transfer, or n exchange for real assets, commodtes or labor (whch gves no clam on future repayment) s called outsde money. In the absence of a Central Bank provdng lqudty, there would be no nsde money n our model whch at least contradcts the fact that fat money s a creature of the State, and mmedately prompts the queston as to where outsde money comes from. On the other hand, n a model wth just outsde money used as a medum of exchange, nomnal prces would be ndetermnate. To realze ths, just consder Mertens (2003) lmt-prce mechansm assocated to a graph of trades that s star-shaped wth respect to (and only to) some worthless numérare (called outsde money). Then, nomnal prces would be ndetermnate (as they are n standard GET). By contrast, n a stuaton nvolvng only nsde money, our model reduces to the Walrasan (.e., non-monetary) one ntroduced n Graud (2004a). As a consequence, real trades are determnate, but there s nomnal ndetermnacy n prces (and r 0) Fnally, co-exstence of nsde and outsde money does not suffce per se, however, to drve our results. Suppose, ndeed, that at each tme t, nsde money can be exchanged aganst outsde money, but that outsde nfntesmal trades are performed thanks to the classcal Shapley-Shubk (1977) model of tradng-posts, and that there are separate tradng-posts for each type of money. The varable r s the relatve prce of outsde to nsde money. Then, as soon as M > m, all the ndvduals wll sell ther whole endowment of outsde money aganst nsde money, and perform all ther trades n commodtes solely wth outsde money. Player wll end up wth Mm h m = m h m M, and wll spend ths amount of cash to buy commodtes. The quantty theory of money wll then look lke M = p c ẋ c, (2) c and the endogenous varables p c of the real sector of the economy wll agan be ndetermnate (and depend exclusvely on M). Only r wll depend both on M and m. Consequently, a slght change n M, suffcently small to keep M > m, wll not affect the real terms of trade, so that money wll agan be essentally neutral. Thus, the last ngredent for our recpe to hold water s to organze trades not accordng to the decentralzed tradng-posts à la Shapley-Shubk (called TP-mechansm n the sequel),

6 6 Parte C but accordng to Shapley wndows model. 5 The next secton presents the model n detals. Secton 3 s devoted to our man results. A last secton offers some concludng remarks. 2 The model 2.1 The fundamentals Commodtes The long-run economy E s defned by C 1 consumpton goods c {1,..., C} and I 1 types of households = 1,..., N, characterzed by (u, ω ). 6 Each type s represented by a contnuum of clones, say [0, 1], together wth the (restrcton of the) Lebesgue measure. For each, R C + represents type s tradng set. The functon u : R C + R s the (long-term) utlty functon of type. The vector ω R C + \ {0}, s her/hs ntal endowment. Defnton () An allocaton s an ntegrable, type-symmetrc map x : [0, 1] N R C+1 belongng to the feasble set τ: τ := { x L 1 ([0, 1] N, R C ++) : 1 0 x d = ω := 1 0 ω d = 1 ω }. N () An allocaton x s ndvdually ratonal whenever u (x ) u (ω ) for a.e. [0, 1] N. Because of type-symmetry of allocatons, we dentfy τ wth: τ := {x ( R C ) N + (x ω ) = 0 }. For every household, ˆX (resp. ˆX ) s the subset of consumpton bundles x >> 0 such that x ω := h ω h (resp. the subset of ponts x n ˆX such that u (x ) u (ω )). Assumpton (C)() For each, the restrcton of u to ˆX u (x )> 0, and s quas-concave. s C 1, verfes () For each and every x X ˆX, u (x ) x > 0. Moreover, the restrcton of u to ˆX s strctly quas-concave. Ths assumpton s weak n the sense that t makes no use of any boundary condton n order to keep away the dynamcs from the boundary τ of the feasble set. Smlarly, preferences are not assumed to be strctly monotone. A map x : [0, 1] N R C s sad to be type-symmetrc whenever each restrcton, x [0,1], on the th element of the Cartesan product [0, 1] N s a.e. constant for all = 5 See Graud (2003) for the dstncton between both market games. 6 Throughout the paper, N C desgnates the fnte set {1,..., C}, u(x) s the gradent of u at x.

7 Global Unqueness and Money Non-neutralty 7 1,..., N. We denote by x R C, the equvalence class of ths restrcton. Hence, we use the notaton x n three dfferent senses: As the vector n R C, whch s the common ndvdual allocaton attrbuted to each household of type, as the constant functon whch maps each agent n [0, 1] to the vector x, and descrbes the symmetrc allocatonselecton of households of type, and as the ntegral of ths constant functon on the unt nterval [0, 1], whch gves the aggregate allocaton of agents of type. Ths sense wll be clear from the context. Money Fat money s present n the economy: At tme t = 0, each type has a prvate endowment of outsde money m (0) 0 and of bonds b > 0 and a stock M = M(0) > 0 of money (nsde money) s held by the Central Bank. Outsde money s owned by households free and clear of debt. Insde money s always accompaned by debt when t comes nto households hands. The quanttes m = (m (0)), b = (b ) and M are exogenously fxed, and m = m (0) (resp. b = b ) represents the aggregate stock of outsde money (resp. bonds) held by the agents at the begnnng of tme. (Snce b wll reman constant across tme, there s no need for ndexng t wth respect to tme.) The monetary long-run economy s defned by (E, m, b, M) and the prvate sector s defned by (E, m) (u, ω, m, b ). Let p c (t) > 0 denote the prce of good c n terms of money and r(t) 0 denote the money rate of nterest on the Bank loan. The vector (p(t), r(t)) R C + R + denotes the market prce at tme t. The confguraton space of our dynamcs s the set of states,.e. of feasble allocatons n commodtes and stocks of money (x, m, M) τ R N+1 + wth m = m. Trades occur n contnuous tme. The stock of bonds b = (b ) s constant across tme because we make the assumpton that, at each tme t, each ndvdual starts afresh wth the same stock b (that wll be solely used n order to borrow nsde money n stage t α, and s gven back to household as she repays her loan at the end of tme t). At each tme t, the profts of the Central Bank wll be equal to r(t)m(t) (where r(t) s the current ntra-perod nterest rate). It s dstrbuted to ts prvate shareholders accordng to some fxed ownershp structure ν : ν [0, 1] such that ν = 1. However, shareholder can use only at tme t + dt, the cash receved as dvdend at tme t. There s no loss of generalty n postulatng that, for every, ν > 0. Indeed, n our strpped-down model, myopc households make no expectatons about the future, hence have no precautonary demand for money. As a consequence, they spend ther whole cash at tme t n order to repay ther loans. Thus, an agent recevng no dvdend from the Central Bank at tme t, wll enter the markets at tme t + dt wth no outsde money. Such an agent won t be able to trade any more, and could be dsregarded as well The short-run economy At each tme t, the state s (x(t), m(t), M(t), b); the short-run economy T x(t),m(t),m(t),b E s a monetary lnear economy defned by the same characterstcs as E except that: 7 Actually, we could add a precautonary demand for money wthout alterng our results.

8 8 Parte C The set of nfntesmal trades of agent n T x(t),m(t) E s the shfted cone (x (t) + m (t)) + R C+1 + Intal endowments of consumpton goods (resp. outsde money) are replaced by 0. Current allocatons (x (t)) (resp. current endowment (m (t)) ) play the role of short-sale constrants. household s short-run preferences are gven by the lnear utltes over commodtes nduced by ther current gradents. 8 In other words, agent s short-run utlty v : R C R, s: v (ẋ (t)) := u (x (t)) u (x ) ẋ (t) := g (x (t)) ẋ (t). (3) At each state, the moton of E s dctated by some strategy-proof equlbrum of a strategc market game G[T x(t),m(t),b(t),m(t) E] assocated to the short-run economy T x(t),m(t),b(t),m(t) E. Ths captures the myopa of consumers: They are not sophstcated enough to solve an ntertemporal optmzaton programme (nvolvng the heroc soluton of, say, the assocated Hamlton-Jacob-Bellmann partal dfferental equaton). Rather they try to trade n the drecton of the steepest ncrease of ther current utlty. 2.3 The monetary short-run game Each perod t R + s dvded nto three subperods. At the frst subperod t α, agents borrow money from the Bank by sellng bonds ; at tme t β, they sell commodtes for money and buy goods wth money ; eventually, at tme t γ, Bank s loans are repad wth money. Agents enter the commodty market wth the money they got from the loan market plus ther ntal holdng n outsde money. In stage t β, agents meet n a monetary verson of the strategc market game nduced by Merten s (2003) lmt prce mechansm. Ths mechansm tself can be vewed alternatvely as the mult-tem extenson of double auctons, or as the extenson of Shapley s wndows model (see Sah & Yao (1989)) to lmt-prce orders. 9 Money s denoted by m and bonds by b. The tme perod t s fxed. Market orders In order to descrbe the acton of a player, let us begn wth market orders. The vector q of s market order has 2C + 1 components: q bm s the quantty of bonds sold by to the Bank for money at tme t α ; q mc s the money spent by to purchase c at tme t β, c = 1,..., C; q cc s the quantty of c sold by for commodty c at tme t β, c c = 1,..., C. Alternatvely, a trader s sgnal has two components: the frst one conssts n q bm and s sent n stage t α, the second s a (C + 1) (C + 1) matrx whose k, l-entry q kl 8 Money and bonds are of course worthless. 9 See Graud (2003) for an ntroducton to strategc market games, and a dscusson of ths pont.

9 Global Unqueness and Money Non-neutralty 9 ndcates the amount of tem k s/he s offerng n exchange for tem l. Prces are then computed accordng to the followng set of equatons: r = q bm M, (4) C+1 ( N ) C+1 qkl p ( N k = p l k=1 =1 k=1 =1 q lk ) l = 1,..., C + 1 (5) where the prce of money s normalzed to 1. Fnally, commodtes are redstrbuted n such a way that s fnal allocaton s x l := ω l + 1 C+1 p l k=1 C+1 p k qkl k=1 q lk. (6) In other words, prces form so as to clear all markets; all of nsde money M s dsbursed to households n proporton to ther bonds; the nterest rate r forms as a market-clearng prce on the market for bonds aganst nsde money; and at each commodty-money market, all the money (or, commodty) receved s dsbursed to households n proporton to the commodty (or, money) sent by them. Gven prces (p, r), the (compettve) budget set B(p, r, x (t), m ) of household s the set of all market orders and fnal allocatons (q, ẋ (t)) R 2C+1 R C + that satsfy the constrants below for all c C and all t 0: C c=1 q mc m + qbm 1 + r (7) q cm x c (t) (8) q bm m + qbm 1 + r C c=1 q mc + C c=1 p c q cm (9) ẋ c (t) q cm x c (t) + qmc (10) p c (7) s the cash-n-advance constrant faced by each trader every perod. A common crtcsm s that such cash-n-advance constrants are ad hoc, and do not adequately capture lqudty. However, n any strategc market game or n a monetary economy, they emerge naturally. Ther man ntuton s that the dfferent nstruments and commodtes of the economy are not equally lqud. As long as there exst dfferent lqudty parameters whch are less than 1 for the endowment vector, money demand s postve to brdge the gap between expendtures and recepts. Otherwse, the budget constrants collapse to the standard Arrow-Debreu constrants.

10 10 Parte C (8) precludes commodty short sales, and (9) specfes loan repayment n the fnal subperod from money carred over from the frst subperod and recepts from commodty sales. Fnally, (10) descrbes the fnal allocatons. Lmt-orders We now supplement the precedng market structure by allowng traders to send lmt-prce orders to the market. For varous reasons (that are spelled out n Mertens (2003)) only sellng orders are allowed but ths mples no loss of generalty: If a player wants to buy a commodty, s/he just has to sell some money! Defnton A lmt-order to sell tem l n exchange for tem c 10 gves a quantty q lc to be sold, and a relatve prce p+ l. The order s to sell up to q p + lc unts of tem l c n exchange for tem c f the relatve prce p l p c p + l p + c = 0, one gets a famlar market order. s greater than, or equal to, p+ l. When p + c Remark. A lmt-order to sell l aganst m at relatve prces p + m = 0, p + l > 0 s, n fact, an order not to buy money. Every trader n T x(t),m(t),b(t),m(t) E s allowed to send as many (market- and/or lmt-)orders as s/he wants. Nevertheless, due to the fact that a short-run economy s lnear, we shall see that every player has at her dsposal a unque domnant strategy on the commodty market, whch n addton nvolves at most C + 1 lmt-prce orders (whose lmt-prces wll exactly concde wth ths agent s current margnal rates of substtuton among commodtes and money). 11 In order to prepare for the (next) defnton of a monetary order book, observe that, gven some collecton of orders, one can defne a fcttous lnear monetary economy as follows: Defnton A fcttous lnear monetary economy L = I, I, µ, b, e, M, s defned by a postve, bounded measure space (I, I, µ) of traders, and measurable functons b, e : I R C+3 +, e beng ntegrable. In such a lnear economy L, there are C + 1 objects of exchange: C consumpton commodtes and money. Every trader s I consumpton set s R+ C+1. Trader s utlty for x s b x, and e = (e 1,..., e C+2, e C+1 ) s her/hs ntal endowment, wth ts last component beng s current holdng of money. We desgnate the set of agents of a lnear economy by an abstract measure space (I, I, µ) because we wll need later on to nterpret t as a set of lmt orders. Fortunately, I wll rapdly turn out to be equal to [0, 1] N (equpped wth the product of Lebesgue measures) n most of the stuatons of nterest for us. I s, now, the set of orders; For each fcttous agent (.e., for each order), ts lnear utlty s gven by b := (p + 1,..., p+ C, 0) ; Its ntal endowment s defned as e := (q 1,..., q C, q m). 10 Here, an tem may desgnate a consumpton commodty as well as money or a bond. 11 See Graud (2004a) for detals.

11 Global Unqueness and Money Non-neutralty 11 Monetary order books The tmng of trades n each short-run economy s such that, n a thrd stage, households stll have to delver on ther bonds. As shown by Dubey & Geanakoplos (2003a, Lemma 1), the multple constrants of each stage t α, t β can be summarzed n a unque (non-lnear) constrant where revenues from sales are dscounted by the nterest rate. In a sense, the bankng system extracts (nsde) money every tme a household purchases beyond ts outsde fnancal wealth m. 12 We capture ths property n our short-run game G[T x,m,b,m E] as follows. Suppose frst that r = 0. Any order on the commodty-money market of stage t β, consstng of a lnear utlty u, together wth an endowment e, s equvalent to a set of C + 1 separate orders, the c th of them sellng an amount e c (resp. e m ) of commodty c (resp. money) wth the utlty u. Therefore, we concentrate on sell-orders of a sngle tem say c 0. Snce money s worthless, every truth-tellng sell-order wll nvolve a zero utlty for money. But, beng neglgble, t s a domnant strategy for each player to reveal the truth when sendng sell-orders to the market. Hence, a utlty functon wll typcally have the form: v (ẋ) = C g c (x )ẋ c, (11) c=1 where g c(x ) s the personal relatve prce of c for agent (cf. (3)). Ths corresponds to a sell-order of commodty c 0 aganst any other commodty accordng to whch one wll yeld the most value n terms of the personal relatve prce system (g 1(x ),..., g C(x )). Formally, f p s the emergent prce vector, ths order wll be executed as an order to sell e c 0 aganst ẋ c := p c p c0 e c 0 for c n Argmax {v (ẋ c ) c = 1,..., C}, provded that there s at least some commodty c such that pc 0 p c g c(x. If ths last ) condton s not satsfed, then the order s nexecuted (and automatcally dsappears from the order book). If ths last condton s satsfed at most as an equalty, then the order may be only partally executed (and the nexecuted part automatcally dsappears from the order book). If ths condton s satsfed as a strct nequalty, then the order s fully executed. gc0 (x ) Next, f r > 0, the same logc apples, except that, when a player s sellng commodtes, her revenue s rescaled at the rato 1 1+r n order to take account of the dsspaton of money n the system whch s but the cost to pay for the fact that nsde money facltates trades. (Remember, nevertheless, that the money dsspated at tme t s mmedately renjected nto the system at tme t + dt through the dvdends.) Ths s equvalent, to requrng that, when a player announces (11) to the market and sells commodty c 0 (aganst money), then an outcome wll be computed for the modfed economy where (11) has been replaced by: 12 Ths vewpont s developed by Dubey & Geanakoplos (1992), and s consstent wth Dubey & Shubk s (1978) semnal approach.

12 12 Parte C v (ẋ) = (1 + r)g c 0 (x ) + C c=1,c c 0 g c (x )ẋ c. (12) On the other hand, when she s sendng an order to sell money (.e., to buy some commodty), ths player s announcement s not modfed. That such a modfed order s equvalent to a standard one when due account s taken from the dscount factor r can be best vewed as follows. As long as a player s usng her own outsde money n order to fnance her purchase, she ncurs no dscount rate. Ths s reflected by the fact that an order to sell money (aganst commodty) stays unmodfed. On the contrary, as soon as a player spends some nsde money n order to fnance addtonal purchases, then she has to ncorporate ths cost n her budget constrant. Everythng then goes as f the prce at whch she wll be ready to sell endowment n commodty c 0 was not p c0 but 1 1+r p c 0 < p c0. Thus, the correspondng order should reman nexecuted as long as 1 p c0 1 + r p c < gc0 (x ) g c(x ). But ths s equvalent to modfyng the utlty functon assocated wth the correspondng lmt-prce order accordng to (12). Let us call r-monetary order such a lmt-prce order where revenues from sales are dscounted by the nterest rate r accordng to (12). Observe that the dscount mposed by the market makers s smlar to some transacton cost that ntroduced a wedge between buyng and sellng prces. Even more, snce t affects only revenues from sales, t works lke a bd-ask spread. Gven some r > 0, a r-monetary order book n G(T x,m,b,m E) s a fcttous lnear monetary economy O = ( I, I, µ, b, e, M ) such that each agent s utlty verfes (??). 2.4 Monetary nfntesmal trades We now defne the short-term outcome that wll be nduced by a collecton of r-monetary order books sent by players n the short-term economy T x(t),m(t),b(t),m(t) E. Such an outcome wll provde the drecton n whch the state of the underlyng economy E moves (.e., the nfntesmal trades, (ẋ(t), ṁ(t), ḃ(t)), occurrng) at tme t. Consder a fcttous lnear monetary economy L = ( I, I, µ, b, e, M ). Defnton A pseudo-outcome of L s a prce system p R C+1 + \ {0} and an nfntesmal trade ẋ verfyng p m = 1, 13 and () For every agent I, p b = 0 mples ẋ = 0 R C+1. () For every I, ẋ maxmzes b ẋ subject to the (nfntesmal) budget constrants: ẋ (x, m ), p ẋ 0 (13) 13 That s, the prce of money s normalzed to 1. and ( p k = 0 ẋ k = 0).

13 Global Unqueness and Money Non-neutralty 13 () For every tem c, p c = 0 mples that, for µ-a.e., ( ) p e > 0 b (c) = 0. P (L) wll denote the set of pseudo-outcome prces, and for all p P (L), X p (L) the correspondng set of pseudo-outcome allocatons. We are now n a poston to defne monetary short-term outcomes. Defnton (Mertens (2003)) () A pseudo-outcome s proportonal f all buyers who quoted the market prce as lmt prce should get ther orders executed n the same proporton, and smlarly for sellers: For every par of tems (c, c ) N C+1, wth non-zero prces, there exsts m cc 0 s.t. a) m cc + m c c > 0; b) m c1 c 2 m c2 c 3 m c3 c 4 = m c1 c 3 m c3 c 2 m c2 c 1 (consstency); c) all agents wth non-zero utlty whose demand set δ p {c, c } receve them n quanttes proportonal to m cc and m c c, where the demand set of at prce p s } δp := {l N C+1 p l r(b, l, k)p k, k N C+1, wth r(b, l, k) := b l denotng the margnal rate of substtuton between l and k (wth b k the conventon b 0 := 0). () A short-term outcome of L s defned by the followng algorthm: pck any proportonal pseudo-outcome, settle the correspondng trades, and repeat the procedure wth the lnear sub-economy L restrcted to the commodtes that had zero prce. Untl the algorthm ends. 2.5 Strategy-proof trade curves Our short-run strategc market game s feasble: at the start of perod t, the Bank holds M(t) and households hold m(t) of money. Money market clearng (4) n stage t α guarantees that the Bank stock M(t) flows to traders at the end of t α. When sendng orders to the central clearng house n stage t β, everythng goes as f players would not use nsde money, but only outsde money. The use of outsde money s mplct n the fact that they can send orders to sell commodtes aganst commodtes (but then wth the specfc cost (12) descrbed n the precedng secton). Thus, the commodtes to be traded n stage t β are the consumpton commodtes plus outsde money. Commodty market clearng n stage t β s guaranteed by the fact that Mertens (2003) mechansm s balanced (cf. (6), see also Lemma 1 (a), p. 448 f needed). As a consequence, the total stock of commodtes and outsde money s conserved and redstrbuted among the households durng the second stage. At the end of the two frst stages, all of (M +m)(t) s wth households. The no-default constrant (9) s always satsfed because of (12), and mples that the total bonds sold by households do not exceed (M + m)(t). At the end of stage t γ, the Bank holds (1 + r(t))m(t) M(t) + m(t), and households hold the balance m(t) r(t)m(t). The proft of the Bank s therefore r(t)m(t), and t s

14 14 Parte C redstrbuted to ts shareholders at the begnnng of perod t + dt as a new endowment n outsde money. Consequently, no money dsappears from the system. The ntal endowment, m (t+dt) = m (t)+ṁ (t), n outsde money of household at the begnnng of tme t+dt wll be the amount of outsde money she was left at the end of t γ (.e., the dfference between the rght-hand sde and the left-hand sde of (9)) plus her dvdend: (p(t)) m (t) + ṁ (t) = m (t) + qbm 1+r C + C c=1 p c (t)q cm c=1 qmc (p(t)) (p(t)) q bm (p(t)) + ν r(t)m(t). A strategy s of player n the game G(T x,m,b,m E) conssts n sendng an order to buy nsde money n subperod t α and a lmt-prce order for each par (k, k ) of tems n perod t β. Players have no memory, and cannot condton ther current behavor the past. Let us denote by ϕ (s) the fnal allocaton receved by player whenever the strategy profle s := (s h ) h s played. Havng defned the rules of the game, t remans to characterze the players behavor. We shall consder only weakly domnant strateges. Defnton A strategy-proof profle s a strategy profle s such that a.e. player plays a weakly domnant strategy n the short-run game, takng m (t) as hs/her current ntal endowment n outsde money,.e., for a.e. player τ [0, 1] N one has: g (x ) ϕ (s) g (x ) ϕ (s ) where s s the strategy profle obtaned by replacng s wth some arbtrary strategy. We are eventually ready to defne the dynamcs of the Lmt-Prce exchange Process (LPP). For every strategy profle s, we denote by π(s) R C+1 ++ (resp. ẋ(s)) the set of short-term outcome prces (resp. trades) nduced by s n T x(t),m(t),b(t),m(t) E. Defnton. A monetary strategy-proof trajectory s a soluton of the followng dfferental ncluson: (x(0), m(0)) = (ω, m(0)) and ( [ ]) ( [ ]) ẏ(t) = ϕ s G[T x,m,b,m E] and p(t) Π s G[T x,m,b,m E]. (14) [ ] where t 0, s G[T x,m,b,m E] s a strategy-proof profle of G[T x,m,b,m E]. Here soluton has to be understood as folllows. Let ẋ(t) f(x(t)), (15) where f : R m R m s a possbly dscontnuous generalzed vector feld. Defnton (Flppov (1988)) A Flppov soluton of (15), s an absolutely contnuous trajectory φ : [a, b) τ such that, for a.e. t [a, b), φ(t) G F (φ(t)) := ε>0 A N co { y d(y, f(φ(t)))) < ε, y / A }. (16)

15 Global Unqueness and Money Non-neutralty 15 where N := famly of sets A R m of (Lebesgue) measure zero. 3 Unqueness and non-neutralty 3.1 Global nomnal unqueness Defnton. (Mertens (2003)) A market order to sell commodty k for commodty j s nexecutable f there exsts a partton of N L nto A B such that j A, k B, and for every agent, (α) ether e a = 0 a A, (β) or b b = 0 b B. Defnton () A lnear economy L = ( I, I, µ, b, e ) s weakly reducble f there exsts a partton A B = N L such that for each agent, ether b b = 0 b B, or e a = 0 a A, and there exsts some trple ( 0, b, a) wth e 0 b > 0, b 0 b = 0 and b 0 a > 0. () L s weakly rreducble f t s not weakly reducble,.e., f t admts no nexecutable order. We shall need the followng, farly weak assumpton: 14 Assumpton (I) E s dynamcally weakly rreducble, that s, for every x τ, the short-term economy T x,b,m,m E s weakly rreducble. A strct trade ẋ n some lnear economy L s such that ether ẋ 0 or b (ẋ e ) > 0 for a.e. agent. Mertens (2003) proves that every short-run outcome s Pareto optmal when optmalty s checked only wth respect to strct trades. A feasble allocaton (x, m) s nfntesmally Pareto-optmal f there does not exst any path φ : [a, b) τ such that φ(a) = (x, m) and u (x ) φ (x) 0 for every, wth at least one strct nequalty. It s nfntesmally optmal n strct trades f there does not exst any path φ : [a, b) τ such that φ(a) = (x, m), φ (x) nvolves only strct trades n T x,m,b,m E, and u (x ) φ (x) 0 for every, wth at least one strct nequalty. We denote by θ (resp. Θ) the set of such nfntesmally optmal allocatons (resp. n strct trades). Clearly, (x, m) θ (resp. Θ) ff (x, m) s Pareto-optmal (resp. Pareto-optmal when only strct trades are allowed) n T x,m,b,m E. Fnally, θ s the relatve nteror of θ. Fnally, for a gven r 0, a trade ẋ s r-nfntesmally optmal n strct trades f there does not exst any path φ : [a, b) τ such that φ(a) = (x, m), φ (x) nvolves only strct trades n T x,b,m E, and ũ r (x ) φ (x) 0 for every, wth at least one strct nequalty, where ũ r s the modfed utlty functon defned as follows.15 Let z R C be a trade vector of (wth postve components representng purchases and negatve ones representng sales). For any scalar γ > 1, defne: τ c(γ) := mn { z c, zc(γ) entals a dmnuton of purchases n z by the fracton 1 concave) utlty functon ũ r ( ) s gven by: 14 See Graud (2004) for a dscusson of ths assumpton. 15 See Dubey & Geanakoplos (2003a) for detals. 1 } 1 + γ z c. (17) 1+γ. The (contnuous and

16 16 Parte C ũ r (x) = u (e + (x e )(r)). (18) Needless to say, f r = 0, a feasble allocaton x s r-nfntesmally optmal n strct trades f, and only f, t belongs to Θ. We denote by Θ r the subset of r-nfntesmally optmal allocatons n strct trades. Of course, due to the transacton cost nduced by r, the short-term outcome of a short-run economy need not be Pareto-optmal n the correspondng lnear economy, even when optmalty s checked wth respect to strct allocatons. (It would be so for sure f M = +,.e., r = 0.) Nevertheless, the next theorem says that the possblty of retradng n contnuous tme ensures that all r-gans to strct trades wll be exhausted at the end of a monetary trade curve. Theorem 1. Under (C)(), for b > m + M, these three parameters beng fxed, () every short-run economy L = T x,m,b,m E admts a unque short-term outcome (ẋ, ṁ, p, r). () Moreover, r = m M. Except when x Θ r, the correspondng short-term prce π = (p, r) s unque,.e. P (L) reduces to a sngleton. () Monetary trade curves exst. () Provded M, m > 0 and, f n addton, (C)() and (I) hold, all monetary trade curves converge to θ r. Proof. () and () If b s suffcently large (e.g., b(t) > m(t) + M(t)), we can be sure that the feasblty constrant on the market for bonds wll never be bndng: Indeed, each player must return qb (t) to the Bank n stage t γ, and there s at most m(t) + M(t) unts of money n the economy. Snce players are neglgble and play a domnant strategy, there s no loss of generalty n assumng that, at the end of the thrd stage t γ, after repayng the Bank, no player wll be left wth unowed cash. Otherwse, she should have spent more money earler n order to purchase commodtes, or else curtaled her sale of commodtes, mprovng her (strctly monotone) short-run utlty. Hence, exactly m(t) + M(t) s owed to the Bank, so that (1 + r(t))m(t) = M(t) + m(t),.e., r(t) = m(t) M(t). As a consequence, exactly r(t)m(t) = m(t) = m s redstrbuted to the households at the end of tme t, so that wll start at tme t + dt wth ν m unts of outsde money n her pocket. From now on, we therefore consder the quantty m (t) of outsde money held by household as a constant (ndependent from t for every t > 0). On the other hand, even f they are nformed of the r-manpulaton operated by the market-makers on ther commodty-sell orders, neglgble players have no nterest to manpulate ther preferences. Manpulatng ther announcements has no effect on the emergng sort-run prce p, whle the very defnton of a short-term outcome mples that the nduced outcome wll solve: Max u (x ) ẋ (r), under the constrants: p c = 0 ẋ = 0 and p ẋ 0 and ẋ (x, m ), on the economy restrcted to the commodtes belongng to the support of p.

17 Global Unqueness and Money Non-neutralty 17 Thus, when analyzng a local game G[T x,m,m,b E], one can restrct attenton to the r- monetary lnear economy T x,m,r E obtaned from T x,m,m,b E by the same transformaton as the one used to go from (11) to (12), and wth r := m M. Let us denote by ( π(t x,m,r E), ẋ(t x,m,r E) ) the unque short-term outcome assocated to T x,m,b,m E). Exstence and () now follow from the exstence and unqueness results of optmal allocatons (called short-term outcomes here) obtaned n Mertens (2003) for general lnear economes. There, unqueness n prce s understood up to a normalzaton. Here, as we mpose that money s prce be equal to 1, prces are automatcally normalzed (hence unque n nomnal terms). () Exstence of monetary trade curves then follows from standard arguments (see Graud (2004) for detals): We denote by V : τ T τ R C+1 the cone feld assocatng to each state the set of nfntesmal trades n commodtes and money (ẋ, ṁ) nduced by our dynamcs. Except on the Θ r, ths cone feld reduces to a vector feld. As s clear from Graud (2004a), ths vector feld s dscontnuous n general. Flppov (1988) then ensures that the dfferental equaton wth dscontnuous rght-hand assocated to ths vector feld can be translated nto a dfferental ncluson that s upper sem-contnuous, non-empty-,convex- and compact-valued. Exstence of monetary trade curves then follows from standard exstence results for dfferental nclusons, see, e.g., Aubn & Cellna (1984) see Graud (2004, Theorem ). (v) We frst remark that, under (C)(), every ndvdually ratonal trade ẋ n every short-run monetary economy T x,b,m,m E at some state such that x τ, must be strct. Indeed, u (x ) ẋ u (x ) x > 0. Thus, Θ r reduces to θ r. The rest of the proof follows the standard Lyapounov argument, see Graud (2004, Theorem 4.2.1). The next result states that, gven aggregate ntal endowments (ω, m, M), and for a dense subclass of monetary economes of partcular nterest, namely fntely subanalytc (see Graud (2004a,b)) economes, the vector feld assocated to our dynamcs s smooth on an open and dense subset of the feasble set. Thanks to the Cauchy- Lpschtz theory of smooth dfferental equatons, ths mples that, when restrcted to ths generc subset, the Cauchy problem nduced by our dynamcs admts a (pecewse) unque soluton path not only n real, but also n nomnal terms. Proposton 1. For any fntely subanalytc economy E satsfyng (C)() and (), then, for every fxed r, the feasble set τ can be parttoned as: τ = R C where both R and C are fntely subanalytc subsets, the latter beng closed, of dmenson strctly less than CN C =dmr, and contanng θ r. Moreover, the restrcton of V to the (open and dense) subset R s a real-analytc, hence smooth, vector feld. Proof. We frst need to prove that θ r s of measure zero n τ. It follows from the standard argument nvolvng the strct quas-concavty of preferences 16 and from Dubey & Geanakoplos (2003a, Lemma 2) that a pont x τ r belongs to θ r ff t s 16 See Graud (2004, Lemma ).

18 18 Parte C Pareto-optmal (n the usual sense) for the auxlary economy E r defned as follows. Each household s utlty u s changed nto ũ r as defned above by (18) (wth the help of (17)). Snce u s strctly quas-concave and ncreasng, so s ũ r. Thus, the set of Pareto ponts n E r s homeomorphc to the (N 1) unt smplex (cf. Mas-Colell (1985), Prop , p. 155). So s therefore θ r. As a consequence, t s neglgble n τ. Thus, we can perturb our generalzed vector feld n a way analogous to the one followed n Graud (2004, Theorem 4.3.1) n order to be able to apply Flppov s theory. From there, the proof follows verbatm Theorem n Graud (2004). Mathematcally, the proof looks the same as n Graud (2004a). From an economc vewpont, there s a bg dfference however n the way prces have been normalzed. In Graud (2004a), prces are normalzed a pror n the unt smplex, because the whole real dynamcs s homogeneous of degree zero wth respect to prces. Here, prces are endogenously normalzed by equaton (2). Of course, ths generc global unqueness result has to be contrasted wth the generc local unqueness of monetary equlbra obtaned n Dubey & Geanakoplos (2003a, Theorem 3). The set C of crtcal economes beng fntely subanalytc, t s the fnte, dsjont unon of smooth submanfolds, all of them of dmenson less than CN C. The pcture that can be derved from the prevous theorem s therefore the followng: τ can be parttoned nto fntely many open, dsjont subsets, separated by smooth submanfolds, such that the unon of these open subsets (=R) s dense n the feasble set, and the restrcton of our vector feld to each open subset s smooth. 3.2 Long-run non-neutralty of money Is money neutral n our model? It s clear that f both m and M are multpled by some constant λ > 0, then nothng changes n the analyss. Ths means that there s no money lluson. However, f ether m or M s changed separately, then there wll be, n general, a change n the long-run real varables characterzng the monetary trade curves of the economy. We show n ths subsecton how to characterze the short-run and long-run mpact of such a monetary change on the real sector. 17 Unless otherwse specfed, and n order to avod trvaltes, we assume throughout from now on that m > 0. Let us start by stressng that a short-term outcome of the short-run economy T x,m,b,m E does not concde, n general, wth a monetary equlbrum (wth ratonal expectatons) n the sense gven to ths word by Dubey & Geanakoplos (2003a). For ths purpose, we frst recall the authors defnton. Gven prces (p, r), the (compettve) budget set B(p, r, x (t), m ) of type s the set of all market orders and fnal allocatons (q, ẋ (t)) R 2C+1 R C + that satsfy the constrants below for all c C and all t 0: C q mc m + qbm 1 + r c=1 17 The quanttatve comparatve dynamcs of the mpact of money n the long-run can be done, n prncple, snce our whole dynamcs s computable (see Graud (2004a). However, ths pont s left for further research. (19)

19 Global Unqueness and Money Non-neutralty 19 q cm x c (t) (20) q bm m + qbm 1 + r C c=1 q mc + C c=1 p c q cm (21) ẋ c (t) q cm x c (t) + qmc (22) p c Agents have zero endowment n bonds, and therefore no feasblty constrant s put on bond-sellng market-orders. Ths s at varance wth Mertens (2003) non-monetary mechansm, but n accordance wth the tradtonal modellng of fnancal assets n perfectly compettve general equlbrum theory and wth, e.g., Peck & Shell (1991). Consequently, there s only a no-default constrant (21). Ths (nfntesmal budget) constrant can be wrtten n a more compact way: q bm (22) + c p c q cm, where (22) s the dfference between the rght-and the left-hand sde of (19). Due to ths constrant, we do not end up wth a full-blown game, on account of the fact that no player can default. But, as already remarked by Dubey & Geanakoplos (2003a, footnote 14), ths s not a real ssue. By addng suffcently harsh default penaltes, one could get a classcal game, and stll guarantee that, at least at (strategc) equlbrum, nobody goes bankrupt. A vector (p(t), r(t), (q, ẋ (t)) ) R C ++ R + (R 2C+1 + R C +) I s a monetary equlbrum (n the sense of Dubey & Geanakoplos (2003a) adapted to our lnear/short-run settng) of T x(t),m(t),b,m(t) E f all agents market orders are n ther compettve budget sets: (q, x (t)) B(p(t), r(t), x (t), m ), (23) demand equals supply for the loan market and for all good markets: q bm = M(t) (24) 1 + r(t) q mc p c (t) = q cm, c C (25) and each agent optmzes

20 20 Parte C v (ẋ (t)) v (ẋ (t)) for all (q, ẋ (t)) B(p(t), r(t), x (t), m ). (26) In order to compare short-term outcomes wth monetary equlbra n ths sense, we need to recall the defnton of the measure of gans to trade γ(x), as ntroduced by Dubey & Geanakoplos (1992). For any γ 0, we say that x = (x ) ( R C ++) N s not γ-pareto-optmal f there exst feasble trades z = (z ) ( R C) N such that z = 0; x + z >> 0 and u (x [γz ]) > u (x ) for all, 18 where x c [γz ] := x c +mn{ z c, z c } 1+γ for every c = 1,..., C. Thus, the feasble trades contemplated to γ-pareto mprove x nvolve a tax of on trades. If, at the allocaton x, we can fnd some prce vector p R C + such that p z 0 u (x [γz ]) u (x ), then, x s γ-pareto optmal. Of course, 0-Pareto optmalty concdes wth the standard noton of Pareto optmalty. Fnally, the gans to trade γ(x) at x τ s defned as the supremum of all γ for whch x s not γ-pareto optmal. γ 1+γ Proposton 2. () Under (C), for b suffcently large, f t M(t) grows suffcently rapdly, so that M(t) > m γ(x(t)), (27) for every t, then every short-run outcome s a monetary equlbrum of the correspondng short-run economy. Provded that each u s strctly quasconcave, every monetary trade curve converges to some pont x Θ. () On the contrary, f at some pont x, γ(x) < m M, then the short-run outcome of T x,m,b,m E concdes wth no-trade, and x s a rest-pont of the dynamcs. Proof. () In vew of Theorem 1 and Proposton 2, t suffces to show that, f M(t) > m γ(x(t)), then T x(t),m(t),b,m(t) E admts a monetary equlbrum whch s also ts unque short-run outcome. For ths purpose, observe that all the assumptons of Theorem 2 n Dubey & Geanakoplos (2003a) are verfed by T x(t),m(t),b,m(t) E. Take therefore a monetary equlbrum of T x(t),m(t),b,m(t) E. By Lemma 1 of Dubey & Geanakoplos (2003a), t s such that every agent s maxmzng her short-run utlty over the non-lnear budget set nduced by the condton (see also Fgure 1 below): 18 Snce preferences are strctly ncreasng, there s no need to dstngush weak from strct γ-pareto optmalty.