An Argument for Positive Nominal Interest 1

Transcription

1 An Argument for Postve Nomnal Interest 1 Gaetano Blose 2 Herakles Polemarchaks 3 October 13, 2009 Work n progress 1 We are grateful to the Hotel of Gancolo for hosptalty. 2 Department of Economcs, Unversty of Rome III; gaetano.blose@unrom3.t 3 Department of Economcs and CRETA, Unversty of Warwck; h.polemarchaks@warwck.ac.uk

2 Abstract In a dynamc economy, such as an economy of overlappng generatons, money provdes lqudty and s domnated as a store of value. A central bank that sets the nomnal rate of nterest and dstrbutes ts proft to shareholders as dvdends s traded n the asset market. Nomnal rates of nterest that tend to zero, but do not vansh, elmnate equlbrum allocatons that do not converge to a Pareto optmal allocaton. Key words: nomnal rate of nterest; ntertemporal optmalty. JEL classfcaton numbers: D-600; E-400; E-500.

3 1 Introducton The Pareto optmalty of compettve equlbrum allocatons s a major tenet of classcal welfare economcs and the man argument n favor of compettve markets for the allocaton of resources. Devatons from the classcal paradgm sever the lnk between Pareto optmal and compettve equlbrum allocatons, wth repercussons both for the theory and practce of economc polcy. Compettve equlbrum allocatons may fall short of Pareto optmalty n two dstnct, f related, stuatons: () n economes that extend over an nfnte horzon wth a demographc structure of overlappng generatons (Gale (1971), Samuelson (1958)) and () n economes wth an operatve transactons technology wth money that provdes lqudty servces as a medum of exchange (Clower (1967)). The dynamc falure of optmalty n economes of overlappng generatons s well understood: compettve prces that attan market clearng may fal to provde consstent accountng over nfnte streams of output. Long-lved productve assets, wth streams of output that extend to the nfnte future, when traded n asset markets, guarantee that equlbrum prces provde consstent ntertemporal valuaton and restore the optmalty of compettve allocatons (Wlson (1981)). When money serves as a medum of exchange, the nomnal rate of nterest does not allow compettve prces to exhaust the statc gans from trade. Vanshng nomnal rates of nterest or, equvalently, the payment of nterest on money balances on par wth the rate of return on stores of value elmnates the suboptmalty of monetary equlbra (Fredman (1969)). The argument here s that low, but not vanshng, nomnal rates of nterest sheld the economy from ntertemporal suboptmalty at the cost of some statc neffcency. Dfferently from other arguments for a postve nomnal nterest, the argument does not appeal to nomnal rgdtes, mperfect competton or any other mperfecton or ncompleteness of fnancal markets. In an economy of overlappng generatons wth cash-n-advance constrans, a central bank ssues balances n exchange for bonds and dstrbutes ts profts, segnorage, as dvdends to shareholders (Blose, Drèze and Polemarchaks (2004)). Importantly, shares to the bank are traded n the asset market and the bank s, ntally, owned by a fnte number of ndvduals, most smply among those actve at the startng date of economc actvty. At equlbrum, the market value of the bank s at least equal to the present value of segnorage. Segnorage corresponds to the ntertemporal value of net transactons, whch s, thus, fnte. A condton of heterogenety (wthn a generaton) ensures gans to trade even at ntergeneratonal autarky, whch guarantees that, provded that the nomnal nterest s small enough, some commodtes are non-neglgbly traded over the entre nfnte horzon. As net transactons are fntely valued, so s the aggregate endowment of non-neglgbly traded commodtes. And, as a consequence, the aggregate endowment s fntely valued at equlbrum, for, otherwse, the relatve prces of neglgbly traded to nonneglgbly traded commodtes would explode across perods of trade. 1

4 As long as the nomnal rate of nterest s arbtrarly low, but bounded away from zero, the statc neffcency assocated wth non-vanshng nomnal rates remans but s essentally neglgble; more mportantly, wth the stream of segnorage bounded away from zero, the bank substtutes for the long-lved productve assets that guarantee ntertemporal optmalty. In Blose and Polemarchaks (2006) we gave a smple of the argument n the specal case of The connecton between costly transactons and ntertemporal effcency was recognzed n Wess (1980); the argument there, however, was restrcted to steady-state allocatons and reled on real balances enterng drectly the utlty functons of ndvduals wth a postve margnal utlty everywhere. The argument dentfed debt wth money balances and, more mportantly, t dd not ensure dynamc effcency of non-statonary equlbrum allocaton. We organzed the development of the argument as follows: In secton 2, we gve smple examples that llustrate the argument. In secton 3, we present the argument n abstract terms, at a level of generalty that s comparable to that of Wlson (1981). Ths only requres the modfcaton of budget constrants of ndvduals that s nherted from a prmtve descrpton of sequental trades through cash-n-advance constrants. We prove the result under a hypothess of gans to trade that we show (n secton 4) to be genercally satsfed n standard statonary economes of overlappng generatons wth ntra-generatonal heterogenety. In secton 4, we descrbe a monetary economy of overlappng generatons wth cash-n-advance constrants where a central bank, whose ownershp s sequentally traded n the stock market, pegs the nomnal rate of nterest, accommodates the demand for balances and dstrbutes the segnorage to shareholders as dvdends. Not surprsngly, a canoncal ntertemporal consoldaton of sequental budget constrants reveals that relevant equlbrum restrctons of ths sequental economy are exactly those n the abstract analyss. We conclude wth some remarks. 1 2 Examples Smple, statonary economes of overlappng generatons llustrates the argument. 2 Dates or perods of trade are T = {0, 1, 2,..., t,...}. Each non-ntal generaton have a lfe span of two perods and conssts of two ndvduals, J = {a, b}. An ntally old generaton s actve at t = One commodty s exchanged and consumed at each date; the commodty s pershable. 1 A reader mght prefer to reverse the order of presentaton we chose by readng secton 4 before secton 3. Ths creates no dffculty, after a prelmnary readng of the begnnng of secton 3 for the notaton we use. 2 Mnor changes of notaton from the abstract, general argument that follows facltate the exposton 2

5 The ntertemporal utlty functon of an ndvdual s u (x, z ) = x + ln z, where x s the excess consumptons of the ndvdual when young, whle z s the consumpton when old. The endowment of an ndvdual when old s e > 0 wth quas lnear preferences, t s not necessary to specfy the endowment when the ndvdual s young, when a suffcently large endowment guarantees postve consumpton. The spot prce of the commodty s p t. Nomnal bonds, b t, of one perod maturty, serve to transfer revenue across dates. The nomnal rate of nterest s r t 0. Balances, m t, provde lqudty servces; they also serve as a store of value, but they are domnated as such by bonds. At each date, a central bank or monetary authorty ssues bonds n exchange of balances, wth 1 t b t + m t = 0, that t redeems at the followng date, wth b t + m t = 0; t earns segnorage [r t /(1+r t )]m t that t dstrbutes as dvdend to shareholders. Shares to the bank, ther number normalzed to one, are traded at each date and serve as a store of value. In the absence of uncertanty, no-arbtrage requres that the returns to bonds and shares concde, and as a consequence, the cum dvdend prce of shares, v t, satsfes f t s fnte, where v t = v 0 = r 0 0 m 0 + r t m t + 1 v t+1 ; t t t=1 1 t 1 ( r t t m t ), ( t ) = ( 0 )... ( t ), t = 1,.... The rate of nflaton s π t+1 = (p t+1 /p t ) 1, and the real rate of nterest s ρ t+1 = [( t+1 )/(1 + π t+1 )] 1; real balances are µ t = m t /p t. An ndvdual, young at t, faces the budget constrants p t x t r t b t + m t 0, p t+1 z t+1 b t + m t + p t+1 e, 3

6 and the cash n advance constrant 3 wth m t p t x t, m t 0. Equvalently, an ndvdual faces the ntertemporal budget constrant x t + r t t x t ρ t (z t+1 e ) 0, µ t = x t, the assocated holdngs of real balances. Smlarly, the cum dvdend prce of shares n real terms ϕ t, satsfes f t s fnte, where ϕ t = ϕ 0 = r 0 0 µ 0 + r t µ t + 1 ϕ t+1 ; t 1 + ρ t t= ρ t 1 ( r t t µ t ), (1 + ρ t ) = (1 + ρ 0 )... (1 + ρ t ), t = 1,.... Snce shares and bonds are perfect substtutes, t s not necessary ether to ntroduce shares explctly n the ntertemporal optmzaton of ndvduals or to dstngush between the ntal value of the bank, v 0, and debt held by the ntally old. Wth e a e b, along any equlbrum path, x a t < 0, whle x b t > 0. The soluton to the optmzaton problems of ndvduals are x a t (ρ t ) = 1 (1+ρ t)(1 θ t) ea 1 0, z a t+1(ρ t ) = (1 + ρ t )(1 θ t ), µ a t (ρ t ) = x a t, and x b t(ρ t ) = 1 (1+ρ t) eb 1 0, z b t+1(ρ t ) = (1 + ρ t ), where, θ t = (r t /( t )) < 1. Along an equlbrum path, µ b t(ρ t ) = 0, x a t + x b t + z a t + z b t = e, where e = e a + e b s the aggregate endowment of ndvduals when old. 3 x s the negatve part of x. 4

7 Wth r t = r 0, and, as a consequence, θ t = θ, an equlbrum path of real rates of nterest satsfes where ρ t+1 = e θe a e (2 θ)ρ t + θ 1, e = e a + e b < 2 s the aggregate endowment of ndvduals at the second date n ther lfe spans. If r = 0, there exst two steady-states, one wth ρ = 0 and another wth ρ = (e/2) 1 < 0; n addton, there s a contnuum of non-statonary paths ndexed by the ntal real rate of nterest, ρ 0 ( ρ, ρ ). The steady-state path wth ρ = 0, the golden rule, supports a Pareto optmal allocaton, whle all other equlbrum paths support suboptmal and Pareto ranked allocatons; ρ = (e/2) 1 < 0 support ntergeneratonal autarky. Note that, at the autarkc equlbrum, x a = (4e a e 2 )/(2e) < 0, and, as a consequence, the assocated real balances that support the equlbrum are µ = x a > 0. If r > 0, there s a steady-state equlbrum path wth 2 + e + (2 e) 2 + 4θ(e 2 θ ρ 1 θ ea ) (r) = 1 > 0. 2(2 θ) By a standard argument, there s no equlbrum path wth ρ 0 [ ρ(r), ρ (r)], where ρ(r) = [(2+e (2 e) 2 + 4θ(e (2 θ)/(1 θ)e a ))/(2(2 θ))] 1 < 0. For ρ(t) [ ρ(r), ρ (r)], real balances are bounded below by µ a ( ρ(r)) > 0 and, as a consequence, the value of the bank s well defned and, n partcular fnte, only f ρ(t) > 0. Snce ρ(t) ρ(r) < 0 f ρ(t) [ ρ(r), ρ (r)), the steady-state at ρ (r) s the unque equlbrum path. Importantly, lm r 0 ρ (r) = ρ ; as the nomnal rate of nterest tends to 0, the unque, steady-state real rate of nterest tends to the golden rule and the assocated allocaton to a Pareto optmum. The argument fals n the absence of ntrageneratonal heterogenety, when real balances need not be bounded away from zero as the economy tends to autarky. Alternatvely, Wess (1980) allows real balances to enter drectly the ntertemporal utlty functon of a representatve ndvdual, u(x, z, µ), and he wrtes the ntertemporal budget constrant as x t ρ (z t+1 e) + r t π t t µ t 0, whch follows from the hypothess that changes n the supply of balances are dstrbuted as lump-sum transfers to ndvduals when old. 5

8 At a steady-state, optmzaton requres that whle market clearng requres that the outstandng debt s u µ = r u x, r π µ = r π (z e); b = (z e) + µ. At equlbra wth debt, r = π and ρ = 0. As a consequence, the lqudty servces that balances, dstnct from debt, provde, do not sheld the economy from ntertemporal neffcency. Alternatvely, wthout debt (or, equvalently, f debt provdes lqudty servces and s not dstngushable from money), µ = z e and, wth π = 0 the real rate of nterest s necessarly postve, ρ = r > 0, whch, ndeed, guarantees ntertemporal effcency. The hypothess of non-vanshng margnal utlty for money balances plays a role smlar to that of ntrageneratonal heterogenety n our constructon, but the logc of the arguments s dfferent. 2.2 Two pershable commodtes, ndexed by l n N = {a, b}, are exchanged and consumed at each date. Indvdual only consumes commodty, but s endowed wth one unt of the other commodty,, when young and nothng when old. The ntertemporal utlty functon of an ndvdual s u ( x, z ) = x + 2z, where x and z are the consumptons of the ndvdual n commodty, respectvely, when young and when old. The prce of commodty at date t n present value terms s p t. The constant nomnal rate of nterest s r 0. An ndvdual faces the sngle budget constrant ( ) 1 p tx t + p t+1zt+1 p t, whch reflects an underlyng cash-n-advance constrant. In addton, the budget constrant of an ntally old ndvdual s p 0z0 µ ( p a t + p b ) t, whch reflects the hypothess that the ndvdual s enttled to a share µ 0 n ntertemporal segnorage, so that µ a + µ b = 1. t 6

9 Market clearng smply requres that x t + z t = 1. At equlbrum, sequental Walras Law mples p t + p t+1zt+1 = p tz t. Ths completes the descrpton of the economy. Let ɛ = r () 1 for r 0. Renterpretng terms, one mght suppose that every ndvdual wth only 1 ɛ unts of commodty when young and nothng when old. In addton, a real productve asset, ntally owned by old ndvduals, delver ɛ unts of commodty at every date. We consder equlbra n two dstnct cases. Frst, r = 0. From the budget constrants of ntally old ndvduals, t follows that z0 = 0 and, so, explotng sequental Walras Law, that x t = 1 and zt = 0 for every t. Ths requres p t+1 2p t for every t. The equlbrum allocaton clearly fals Pareto optmalty. Alternatvely, r > 0. From the budget constrants of ntally old ndvduals, t follows that p 0z0 = µ ( p a t + p b ) t, and, as a consequence, that ( t p a t + pt) b s fnte. By a canoncal argument, the equlbrum allocaton acheves Pareto optmalty. We show that a steady state equlbrum exsts under an equal dstrbuton of segnorage, µ a = µ b. Assume that x t = 0 and zt = 1 for every t. To obtan equlbrum prces, observe that, from the budget constrants of young ndvduals, ( ) 1 p t+1 = p t, whle, from the budget constrants of ntally old ndvduals, p 0 = p a 0 = p b 0; t follows that ( ) t 1 p t = p a t = p b t = p 0, at every date t. For an arbtrary dstrbuton of segnorage, a steady state equlbrum mght not exst. To verfy ths, observe that, at a statonary equlbrum, x t = x and zt = z for every t, wth x + z = 1. If x > 0, by utlty maxmzaton, p t+1 2p t, whch would volate the fact that ( t p a t + pt) b s fnte. Hence, x = 0, whch mples, by the budget constrant of a young ndvdual and utlty maxmzaton, ( ) 1 2p t p t+1 = p t. t 7

10 In addton, an ntal condton requres ( r p 0 = µ ) From both condtons, t follows that ( ) 1 2µ b µ a and 2µ a t ( ) 1 µ b. ( p a t + p b ) t. Hence, a statonary equlbrum mght not exst for an arbtrary dstrbuton of shares well known for (statonary) economes of overlappng generatons wth multple ndvduals n each generaton and multple commodtes. Ths example s desgned to delver an extremely clear concluson about effcency at equlbrum. In partcular, a smplfyng assumpton, that each ndvdual s endowed only wth the commodty that he does not consume, elmnates prce dstortons due to cash-n-advance constrants, whch only operates through pure wealth effects. Wthn each generaton, there are actually nfnte gans to trade, as young ndvduals are clearly better off by exchangng ther endowments. Intergeneratonal trade allows for a further ncrease n welfare. 3 The Abstract Argument There s a countable set of ndvduals, I = {...,,...}, a countable set of perods of trade, T = {0, 1, 2,..., t,...}, and a fnte set of physcal commodtes n every perod of trade, N. The commodty space s L = R L, where L = T N. 4 The consumpton space of an ndvdual s L +, the postve cone of the commodty space, and an element, x, of the consumpton space s a consumpton plan. An ndvdual s characterzed by a preference relaton,, on the consumpton space and an endowment, e, of commodtes, an element of the consumpton space tself. He s also enttled to a share µ 0 of aggregate revenue, so that, across ndvduals, µ = 1. 4 The set of all real valued maps on L s L = R L. An element x of L s sad to be postve f x (l) 0 for every l n L; neglgble f x (l) = 0 for all but fntely many l n L. For an element x of L, x + and x are, respectvely, ts postve part and ts negatve part, so that x = x + x and x = x + +x. The postve cone, L +, of L conssts of all postve elements of L. Also, L 0 s the vector space consstng of all neglgble elements of L. Fnally, for every element x of L, L (x) = {v L : v λ x, for some λ > 0} s a prncpal deal of L. Unless otherwse stated, every topologcal property on L refers to the tradtonal product topology. We remark that, throughout the paper, the term postve s used to mean greater than or equal to zero. 8

11 Fundamentals, (..., (, e, µ ),... ), are restrcted by canoncal assumptons, so that every sngle ndvdual s neglgble. The aggregate endowment s e, whch s understood to be a lmt n the product topology. Assumpton 1 (Preferences) The preference relatons of ndvduals are convex, contnuous, weakly monotone and locally non-satated. Assumpton 2 (Endowments) The endowments of ndvduals are postve, neglgble elements of the commodty space. Assumpton 3 (Aggregate Endowment) The aggregate endowment s a postve element of the commodty space. An allocaton, x = (..., x,... ), s a collecton of consumpton plans. It s balanced whenever x = e. It s feasble whenever x e. It s ndvdually ratonal whenever, for every ndvdual, x e. For a feasble allocaton x, aggregate consumpton, x, s an element of L (e), where e = e s the aggregate endowment. Trade occurs ntertemporally subject to transacton costs. In an abstract formulaton, t smplfes matters to assume that ndvduals can only trade f they delver a value that s proportonal to the value of ther net transactons. Such revenues from transactons accrue to a central authorty that redstrbutes them to ndvduals as lump-sum transfers, accordng to gven shares. Ths abstracton corresponds to the descrpton of a sequental monetary economy under a complete asset market and a central bank that pegs a constant nomnal rate of nterest and accommodates the demand for balances. In addton, the central bank, whose ownershp s sequentally trade on the asset market, redstrbutes ts proft (segnorage) as dvdends to shareholders. Prces of commodtes p are also an element of L +. These are, n a sense, dscounted or Arrow-Debreu prces. The dualty operaton on L + L + s defned by p v = sup {p v 0 : v 0 [0, v] L 0 }, that may be nfnte. The budget constrant of an ndvdual s p (x e ) ( + p x e ) µ w, where µ 0 s the share of the ndvdual n the aggregate postve transfer w. For gven a postve nomnal rate of nterest, r, an (abstract) r-equlbrum conssts of a balanced allocaton, x, prces, p, and a aggregate postve transfer, w, such that p ( x e ) w 9

12 and, for every ndvdual, p (x e ) ( + p x e ) µ w and z x = p (z e ) ( + p z e ) > µ w. An (abstract) r-equlbrum nvolves a speculatve bubble f b = w p ( x e ) > 0. Notce that an (abstract) 0-equlbrum s what the lterature tradtonally refers to as an equlbrum wth (possbly) postve outsde money, or wth (possbly) a postve speculatve bubble. Lemma 1 The value of net transacton, p ( x e ), s fnte at every r-equlbrum wth r > 0. Proof. Obvous. Q.E.D. Allocaton z Pareto domnates allocaton x f, for every ndvdual, z x wth z x for some. Allocaton z Malnvaud domnates allocaton x f z Pareto domnates x, whle z = x for all but fntely many ndvduals (Malnvaud (1953)). For a gven postve nomnal rate of nterest, r, an allocaton, x, s Pareto (Malnvaud) r-undomnated f t s not Pareto (Malnvaud) domnated by an alternatve allocaton, z, that satsfes ( z e ) + z ( x e ) + x. Evdently, a Pareto (Malnvaud) 0-undomnated allocaton concdes wth a standard Pareto (Malnvaud) effcent allocaton. Lemma 2 Every r-equlbrum allocaton s a Malnvaud r-undomnated allocaton. Proof. If not, there s an allocaton z that Malnvaud domnates allocaton x and satsfes ( z e ) + z ( x e ) + x. Thus, for every ndvdual, ( r ) p (z e ) + p z 10 p (x e ) + p x,

13 wth at least one strct nequalty. Snce the allocaton z concdes wth the allocaton x for all but fntely many ndvduals, aggregaton across ndvduals yelds a contradcton. Q.E.D. An allocaton, x, nvolves unform trade f there s a decomposton L f L b of the (reduced) commodty space L (e), wth { L f v L : v λ } ( x e ), for some λ > 0, and an allocaton v such that v belongs to L (e) and, for some λ > 0 small enough, x λx b + v f x for every ndvdual. Ths requres that the set of commodtes can be parttoned nto commodtes that are traded n some unformly strctly postve amount and commodtes that are not, n such a way that all ndvduals can ncrease ther welfare by a large enough ncrease n consumpton n the former set of commodtes, even when consumpton n the latter set of commodtes s slghtly reduced. Assumpton 4 (Gans to Trade) Every ndvdually ratonal balanced Malnvaud r-undomnated allocaton, wth r > 0 suffcently small, nvolves unform trade. The gans to trade hypothess extends the condton Blose, Drèze and Polemarchaks (2004) and Dubey and Geanakoplos (2005). It has as consequence that the value of the aggregate endowment s fnte at equlbrum. Lemma 3 The value of the aggregate endowment, p e, s fnte at every r-equlbrum wth r > 0 suffcently small. Proof. Consder the decomposton of the (reduced) commodty space L f L b = L (e) L n the hypothess of a unform trade. Clearly, p defnes a postve σ-addtve lnear functonal on L f. Thus, p e s unbounded only f p e b s unbounded and, hence, only f p x b s unbounded. Also, p v f s fnte, where v s the allocaton mentoned n the defnton of unform trade. For every ndvdual, z x mples p (z e ) ( ) r + p z p (x e ) + p x. Snce ( z x ) ( + x e ) ( z e ), t follows that p (z x ) ( ) + 1 p (z x ). As x λx b + v f x for some λ > 0 suffcently small, usng the prevous argument, wth z = x λx b + v f mples that ( ) 1 p v f λp x b. 11

14 Aggregaton across ndvduals yelds a contradcton. Q.E.D. As the aggregate endowment s fntely valued at equlbrum, canoncal conclusons about effcency and the absence of speculatve bubbles can be drawn. Proposton 1 (Almost Pareto Optmalty) No r-equlbrum allocaton, x, wth r > 0 suffcently small, s Pareto domnated by an alternatve allocaton, z, that satsfes ( ) 1 z e. Proof. As the aggregate endowment s fntely valued, t s clear that every r-equlbrum allocaton, wth r > 0 small enough, s a Pareto r-undomnated allocaton (the proof s just an adaptaton of the proof of lemma 2). So, n order to prove that the statement n the proposton holds true, suppose not. It follows that x s Pareto domnated by an alternatve allocaton z that satsfes ( z e ) + z e + z e ( x e ) + x. Ths contradcts Pareto r-undomnaton. Q.E.D. Proposton 2 (No Speculatve Bubbles) No r-equlbrum, wth r > 0 suffcently small, nvolves a speculatve bubble. Proof. As the aggregate endowment s fntely valued, the result follows from the aggregaton of budget constrants across ndvduals. Q.E.D. It remans to understand the restrctons mpled by the gans to trade hypothess (assumpton 4). 4 Gans to Trade n a Statonary Economy The hypothess on gans to trade (assumpton 4) s genercally satsfed n a standard statonary economy of dentcal overlappng generatons of heterogenous ndvduals. We shall smply provde the core argument, as detals are straghtforward but heavy n terms of notaton. The set of ndvduals s I = J T, where T = {0, 1, 2,..., t,...} are dates or perods of trade and J s a fnte set of ndvduals wthn a generaton: for every t n T, I t = {(j, t) : j J } s generaton t. All generatons I t+1 are 12

15 dentcal and have lfe spans T t+1 = {t, t + 1} T. The ntal generaton I 0 has lfe span T 0 = {0} T. Preferences are strctly monotone over the lfe span of an ndvdual: for an ndvdual n generaton t n T, preferences are strctly monotone on the postve cone of L t = R Lt R L = L, where L t = T t N. Endow the (reduced) commodty space L (e) wth the supremum norm v = sup {λ > 0 : v λe}. As the economy s statonary, ths nvolves no loss of generalty. Suppose that there s ɛ > 0 such that, for every ndvdually ratonal, balanced, Malnvaud effcent allocaton, x, the aggregate net trade of every generaton t n T s ɛ-bounded away from autarky, that s, ( x e ) + I t In a statonary economy of dentcal overlappng generatons, ths s a rather weak requrement when there are at least two ndvduals n each generaton. 5 It follows that there s ɛ > 0 such that, provded that r > 0 s small enough, for every ndvdually ratonal, balanced, Malnvaud r-undomnated allocaton, x, the aggregate net trade of every generaton t n T s ɛ-bounded away from autarky. Ths s evdent. Let e l be the aggregate endowment of commodty l n L (regarded as an element of the commodty space L). For a generaton t n T, let g (t) n L be a commodty such that e g(t) 1 ( x e ) +. ɛ I t Such a commodty exsts because net trades are unformly bounded away (n the sup norm) from zero by ɛ > 0. Decompose the aggregate endowment as e = e f + e b, where e f = e l, and e b = l g(t ) l g(t ) 5 As the allocaton s Malnvaud effcent, t s Pareto effcent wthn every generaton. As the allocaton s ndvdually ratonal and preferences are strctly monotone, f postve net trades vansh wthn a generaton, so do negatve net trades. Thus, usng the fact that all generatons are dentcal, ɛ > 0 above does not exst only f no-trade s a Pareto effcent allocaton wthn a typcal generaton. Ths does not occur genercally n preferences and endowments. e l. > ɛ. 13

16 Clearly, L f = L (e f ) and L b = L (e b ) are such that L (e) = L f L b. In addton, e f t T e g(t) 1 ɛ t T I t ( x e ) + = 1 ɛ ( x e ) + = 1 ɛ ( x e ), so that L f { v L : v λ ( x e ), for some λ > 0 }. For an ndvdual n generaton t n T, let v = v f = e g(t). Takng nto account multplctes and usng the fact that generatons overlap for at most two perods, t s easly verfed that v = e g(t) (#J ) e g(t) 2 (#J ) e l = 2 (#J ) e f. t T I t t T l g(t ) Usng statonarty hypotheses and the strct monotoncty of preferences over relevant consumpton spaces, t s smple to show that there s 1 > λ > 0 such that, for every ndvdual, x λx b + v f x. The gans to trade hypothess (assumpton 4) s satsfed. 5 Sequental Trade The abstract framework accommodates a sequental economy of overlappng generatons. We here present the classcal arguments for the consoldaton of budget constrants that are mpled by a sequentally complete asset market. 5.1 Prces and Markets In every perod of trade, there are markets for commodtes, balances and assets. Balances are the numérare at every date. A constant postve nomnal rate of nterest, r, s pegged by the monetary authorty. 6 The asset structure conssts of a one-perod nomnally rsk-free bond and an nfntely-lved securty that pays off nomnal dvdends n every perod. Short 6 As far as ndvduals and commodtes are concerned, notaton s as n secton 3. In partcular, an element x of L = R T N decomposes, across perods of trade, as x = (x 0,..., x t 1, x t, x t+1,...), where each x t s an element of R N ; an element x of E = R T decomposes, across perods of trade, as x = (x 0,..., x t 1, x t, x t+1,...), where each x t s an element of R. 14

17 sales are allowed on bonds, but not on the securty. Prces of the securty q are a postve element of E = R T. These are spot prces. Dvdends of the securty y are a postve element of E. Ths securty s n postve net supply and, to smplfy, the supply s normalzed to the unty. Dscount factors, a, a postve element of E, are obtaned by settng a t = ( ) t 1. No arbtrage, jontly wth the fact that the securty cannot be domnated by bonds at equlbrum, mples that, n every perod of trade, a t q t = a t y t + a t+1 q t+1. Ths condton reflects the nnocuous assumpton that the securty s prced cum dvdend. As far as the ntertemporal transfer of wealth s concerned, bonds and the securty are perfect substtutes under ths no-arbtrage prcng. A standard argument mples that, n every perod of trade, q t 1 a s y s. a t s t That s, the prce of the securty s at equal to or greater than ts fundamental value. The dsplacement of the market value of the securty from ts fundamental value s the speculatve bubble. Prces of commodtes p are a postve element of L. To avod an excess of notaton, we nterpret p as present value prces of commodtes. So, current (or spot) prces of commodtes are ( 1 p 0,..., a 0 1 a t 1 p t 1, 1 a t p t, 5.2 Sequental Budget Constrants ) 1 p t+1,.... a t+1 Sequental constrants are canoncal. Indvdual formulates a consumpton plan, x, a postve element of L, and a fnancal plan, ( m, z, v ), consstng of holdngs of balances, m, a postve element of E, of the securty, z, a postve element of E, and of short-term bonds, v, an element of L. Indvdual enters perod of trade t wth some accumulated nomnal wealth, wt; he trades n assets and balances accordng to the budget constrant ( ) 1 m t + (q t y t ) zt + vt w t; he uses balances for the purchase of commodtes, as prescrbed by a cash-nadvance constrant, 1 p t (x t e ) + t m a t ; t 15

18 receves balances from the sale of commodtes and he enters the followng perod of trade t + 1 wth nomnal wealth w t+1 = m t + q t+1 z t + v t 1 a t p t (x t e t). In addton, a wealth constrant of the form 1 p s e s wt+1 a t+1 s t+1 s mposed n order to avod Ponz schemes. Fnally, the ntal nomnal wealth s gven by the ntal prce of the securty, w 0 = µ q 0, where µ 0 s the ntal share of ndvdual nto the securty. If a consumpton plan, x, and a fnancal plan, ( m, z, v ), satsfy all the above descrbed restrctons at all perods of trade, we say that fnancal plan ( m, z, v ) fnances consumpton plan x (equvalently, consumpton plan x s fnanced by fnancal plan ( m, z, v ) ). The sequental budget constrant of ndvdual s the set of all consumpton plans, x, that are fnanced by some fnancal plan. Lterally nterpreted, our sequental budget constrant mght appear contradctng the hypothess of overlappng generatons of ndvduals. Indeed, t can be argued that an ndvdual mght not be actve at some date and, so, t s meanngless to assume that consumptons and wealth accumulaton of such an ndvdual are restrcted by the entre sequence of constrans. Observe, however, that an ndvdual should be regarded as not beng actve at some date only f he has no endowment of commodtes and hs utlty s unaffected by the consumpton of commodtes at that date. These are jont assumptons of preferences and endowments. Lettng the ndvdual trade when he should be regarded as not beng actve adds redundant constrants wthout alterng the substance. A skeptcal reader mght assume that an ndvdual s characterzed by a tme horzon T T of consecutve perods of trade. Both the consumpton plan and the fnancal plan can be assumed to be zero out of the gven tme horzon. In the same sprt of the above observaton, one mght be wllng to assume that the ntal share nto the securty s strctly postve only for ndvduals that are actve n the ntal perod of trade. 5.3 Intertemporal Budget Constrants By a canoncal consoldaton, provded that there are no arbtrage opportuntes, sequental budget constrant reduces to a sngle ntertemporal budget constrant of the form p t (x t e ) t + p t (x t e t) µ q 0. t The underlyng demand of balances satsfes, n every perod of trade t, m t 1 a t p t (x t e t) +, t 16

19 wth the equalty whenever r > 0. Also, the holdng of bonds and of the securty, wtch are perfect substtutes as far as ntertemporal transfers of wealth are concerned, can be assumed to satsfy, n every perod of trade t, 1 m t + q t+1 zt + vt ( = x s e 1 s) + p s (x s e a s). t a t s t+1 As a matter of mere fact, usng a more compact notaton, a consumpton plan, x, s restrct by a sngle ntertemporal budget constrant of the form p (x e ) ( + p x e ) µ q 0. The fnancal plan, ( m, z, v ), that fnances an ntertemporally budget feasble consumpton plan, x, can be recovered, up to an ntrnsc multplcty due to redundant assets. 5.4 The Monetary Authorty The securty s backed by the ownershp of a central bank, whch ssues balances aganst bonds and dstrbutes ts proft as a dvded to shareholders. A plan, (m, v, y), of the monetary authorty conssts of a supply of balances, m, a postve element of E, a demand of short-term bonds, v, an element of E, and dvdends to shareholders, y, a postve element of E. A sequental budget constrant mposes ( ) 1 m v = y. The monetary authorty accommodates the demand for balances (that s, m = m ) and runs balanced accounts (that s, m = v), so that y = m. 5.5 Sequental Equlbrum Equlbrum requres market clearng only for commodtes and assets, as the demand of balances s accommodated by the monetary authorty. Gven a postve nomnal rate of nterest, r, a sequental r-equlbrum conssts of a collecton of plans for ndvduals, (..., ( x, ( m, z, v )),... ), s t a plan for the monetary authorty, (m, v, y), prces, p, and securty prces, q, such that the followng condtons are satsfed. (a) For every ndvdual, consumpton plan x s -optmal, subject to sequental budget constrant, and s fnanced by fnancal plan ( m, z, v ). 17

20 (b) The monetary authorty accommodates the demand for balances and runs a balanced budget or m = m, v = m, y = m. (c) Markets for commodtes and assets clear or x = e, z = 1, v = v. Clearly, at a sequental equlbrum, securty prces nvolve no arbtrage opportuntes and, n addton, the securty s not domnated by bonds. 5.6 Abstracton At equlbrum, ( ) r 1 m t = p t ( ) ( x a t e + r 1 t) = p t ( x t a t e t) t and, as a consequence, p t ( x t e t) = a t y t q 0. t t Thus, usng consoldaton of sequental budget constrants, at a (sequental) r-equlbrum, t follows that p ( x e ) q0 and, for every ndvdual, p (x e ) ( + p x e ) µ q 0 and z x mples p (z e ) ( + p z e ) > µ q 0. These are the only substantal equlbrum restrctons, as market clearng for bonds and the securty can be verfed to hold. As a concluson, a sequental r-equlbrum concdes wth an abstract r-equlbrum. 18

21 6 Concludng Remarks In the abstract formulaton, every ndvdual s subject to a sngle budget constrant of the form p (x e ) ( + p x e ) w, where w would be nterpreted, dependng on the partcular nsttutonal framework, as the value of ntal asset holdngs plus possbly transfers n present value terms. Thus, Walras Law mposes f + b = p (x e ) + p (x e ) = w = w, where w, f and b are understood to be (possbly non-fnte) lmts. 7 The argument for almost Pareto optmalty moves from the observaton that the value of net transactons s fnte at equlbrum. As long as nomnal rate of nterest s strctly postve, r > 0, ths occurs whenever f s fnte. In addton, by local non-sataton of preferences, w s fnte f at least one ndvdual s enttled to a postve share of t (that s, w = α w, wth α > 0, for some ndvdual ). If w s fnte, then w p ( x e ) = f suffces to argue that f s fnte. Incdentally, the above nequalty rules out a negatve speculatve bubble, w f = b 0, but the crucal pont s only that t guarantees a fnte value of f. Sequental trades and, n partcular, a central bank quoted on the stock market serve to nterpret w as the ntal market value of the central bank and f as the ntal fundamental value of the central bank. Thus, w f, wth w fnte, s nherted by a prmtve descrpton of sequental trades under the assumpton of free dsposal on long-term securtes, so as to rule out a negatve market value of the central bank. Could the same concluson be drawn n other nsttutonal frameworks? In Blose, Drèze and Polemarchaks (2004), a central bank trades balances for bonds and runs a balanced account by redstrbutng ts proft to shareholders. Ths bascally requres f = w, whch by tself does not ensure a fnte value of f. However, f ths redstrbuton of the proft s nterpreted as occurrng ntertemporally (that s, shares are nto the ntertemporal value of segnorage w), w would be fnte and conclusons would be equvalent. Alternatvely, n the sprt of the fscal theory of prce determnaton (Woodford (1994)), one nterprets w as a gven stock on publc debt, whch s, thus, fnte. A pror, t does not follow that w f, whch ncdentally shows that the prce level mght stll be ndetermnate (n that context, f = w mples an 7 The dscusson here s only suggestve, so that we avod detals on condtons for welldefned, though not fnte, lmts. 19

22 ntertemporally balanced publc budget). However, f one assumes that publc debt cannot be negatve, wth ambguous mplcatons for sequental publc budget constrans, then b 0 and, so, w f, thus leadng to analogous conclusons. References [1] Blose, G., J. H. Drèze and H. M. Polemarchaks (2004), Monetary equlbra over an nfnte horzon, Economc Theory, 25, [2] Blose, G. and H. M. Polemarchaks (2006),, Internatonal Journal of Economc Theory, 00, [3] Clower, R. (1967), A reconsderaton of the mcrofoundatons of monetary theory, Western Economc Journal, 6, 1-8. [4] P. Dubey and J.D. Geanakoplos (2005), Insde-outsde money, gans to trade and IS-LM, Economc Theory, 00, R, [5] Fredman, M. (1969), The optmum quantty of money, n M. Fredman (ed.), The Optmum Quantty of Money and Other Essays, Aldne, [6] Gale, D. (1973), Pure exchange equlbrum of dynamc economc models, Journal of Economc Theory, 5, [7] Malnvaud, E. (1953), Captal accumulaton and effcent allocaton of resources, Econometrca, 21, [8] Samuelson, Paul A. (1958), An exact consumpton-loan model of nterest wth or wthout the contrvance of money, Journal of Poltcal Economy, 66, [9] Wess, L. (1980), The effects of money supply on economc welfare n the steady-state, Econometrca, 48, [10] Wlson, C. (1981), Equlbrum n dynamc models wth an nfnty of agents, Journal of Economc Theory, 24, [11] M. Woodford, Monetary polcy and prce level determnacy n a cash-nadvance economy, Economc Theory, 4, ,