The Role of Money: Credible Asset or Numeraire? Masayuki Otaki (Institute of Social Science, University of Tokyo)

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1 DBJ Disussion Paper Series, No.04 The Role of Money: Credible Asse or Numeraire? Masayuki Oaki (Insiue of Soial Siene, Universiy of Tokyo) January 0 Disussion Papers are a series of preliminary maerials in heir draf form. No quoaions, reproduions or irulaions should be made wihou he wrien onsen of he auhors in order o proe he enaive haraers of hese papers. Any opinions, findings, onlusions or reommendaions expressed in hese papers are hose of he auhors and do no refle he views of he Insiue.

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3 The Role of Money: Credible Asse or Numeraire? Masayuki Oaki Insiue of Soial Siene, Universiy of Tokyo E mail: ohaki@iss.u okyo.a.p Absra I is well known ha money is neural if (i) people hold he exraneous belief ha i is only a numeraire and does no possess inrinsi value, and (ii) new money is ineed ino an eonomy as is own ineres in he OLG model under perfe informaion (Luas [] Theorem ). We find ha whenever (i) is no saisfied and money is raionally held o have subsane value, i beomes non neural even if we use he same model as Luas []. Inroduion This arile shows ha he mulipliiy of raional beliefs onerning he value of money deides wheher money is neural under perfe informaion sruures in he wo period OLG model. The resul is kep ina, even if he new money ineed ino he eonomy is sube o he model of Luas []. This resul onrass wih hose of Oani [] and Oaki [3], [4], [5]. If he nominal rae of ineres on money inreases (i.e., money supply inreases), people an onsisenly believe ha he purhasing power of money (he inverse of he nex period prie level) is reained. Then, he real ineres rae beomes higher, hereby mahing he supply, and he demand for money inreases. From assumpions onerning he uiliy funion, This siuaion also implies he reduion of urren onsumpion and leisure. Thus, he moneary expansion inreases urren oal oupu, and hene, money beomes non neural. We mus noe ha he aained equilibrium is saionary in he sense ha he values of real endogenous variables, suh as urren/fuure onsumpion and leisure, are enirely ime independen. This asserion holds, sine one he real ineres rae is raised beause of a hange of he nominal ineres rae (inremen of nominal money supply per apia), one may expe ha he hange in he inflaion rae will equal ha of he nominal ineres rae; he heighened real ineres rae is kep ina, and hus, he equilibrium beomes self enforing and saionary. The res of paper is organaized as follows. In Seion, I onsru he same model as Luas [], exep for he formaion of raional expeaions onerning he value of money, and proves he non neuraliy of money. A welfare eonomis impliaion is also analyzed. Seion 3 onains brief onluding remarks. The Model. The Sruure of he Model Oaki [5] defines suh a raional expeaion as money being redible.

4 We use essenially he same model as Luas [], exluding unerainy. In every period a uni individual is born and lives wo periods. Eah individual has an indenial uiliy funion U : U U(, n) V( ) () where and are he urren and fuure onsumpion level, and n denoes he hours worked per individual. Furhermore, U and V saisfy he following properies: U >0, U n <0, () U <0, Unn 0, U Un <0, Un Unn <0, (3) V ( ) V ( ) V ( )>0, a <0, V ( ) (4) limv ( )=, limv ( )=0. (5) 0. The Maximizaion Problem of Represenaive Individual Eah individual maximizes his/her lifeime uiliy U, sube o he following budge onsrain: m n, p m x, p n, (6) p, (7) px where x denoes he inremen of money supply per apia, and is he inverse of he real ineres rae. m is he money per apia arried over from he previous period. The Kuhn Tuker ondiions imply ha he opimal deision (,, n ) saisfies dn U(, n) =, (8) d Un(, n ) V ( ) = (9) U(, n ) n =. (0).3 Marke Equilibrium There are wo markes in he above model: he money marke and goods marke. By Walras' law, we an negle he equilibrium ondiion for he goods marke. The money marke equilibrium ondiion is m x= p. () Insead of he quaniy heorei equilibrium prie funion imposed by Luas [],

5 le us assume ha money is redible in he sense of Oaki [5] ha is, he raional expeaion onerning he urren purhasing power of money is no pururbed by an inrese of x : p dp =0. () dx The general equilibrium of markes is aained by five equaions: (8), (9), (0), () and (). Endogenous variables are (,, p, p, n ). The parial equilibrium of labor n and he younger generaion's onsumpion is illusraed by Figure. The downward sloping urve AA is he lous of Equaion (8), whih is easily derived from Assumpion (3). The upward sloping urve BB is he lous of Equaion (0), whih is ombined wih Equaion (9). The proedure is as follows: Subsiuing Equaion (9) ino (0), we obain V n =. U In differeniaing boh sides of he above equaion, dn d d[ V ] Ud Undn = n V U dn [ U Un ] d d[ V ] d d V U dn [ U Un] d d[ V ] =[ ] n U V holds. Hene, Curve BB is upward sloping for any fixed. When he money marke equilibraes, he equilibrium onsumpion of he younger generaion and oupu is deermined a he inerseion of Curves AA and BB (Poin E 0 ). Whenever money is redible, i is faile o depi he propery of money marke equilibrium. From Equaions () and (), we obain d dx =. (4) x Sine V is an inreasing funion of from Assumpion (4), Equaions (3) and (4) imply ha Curve BB shifs oward he souh eas like, Curve B B by an inrease of x. Thus, he eonomy moves from Poin E 0 o E. Aordingly, as long as money is redible, a moneary expansion inreases he oupu n and fuure onsumpion, and dereases he younger generaion's onsumpion. To sum up: (3) 3

6 Theorem. If money is a redible asse, i beomes non neural o he real eonomy. An aelaraion of moneary growh heighens he real ineres of money, and hene simulaes fuure onsumpion and oupu/labor supply, and eonomizes urren onsumpion. Nex, we shall show ha he equilibrium depied above is a saionary raional expeaion equilibrium. Suppose ha he eonomy is loaed a Poin E by an inrease of x, and individuals believe ha he higher equilibrium real ineres rae / prevails hereafer. Then by he definiion of (Equaion (7)), d dx dp dp = [ ]=0, x p p (5) holds. Sine m x= p m = m x, dx dp dp d d =[ ] [ ] x p p (6) also holds. Combining Equaion (6) wih (5), we finally obain d d =.. (7) Thus, he equilibrium onsumpion of an old individual is ime independn. I is lear from Equaions (8) and (3) ha he res of he wo endogenous variables (, n ) are also ime independen. Consequenly, he equilibrium illusraed by Poin E is saionary in he sense ha every equilibrium value of endgenous variables is ime independen. One an hus affirm Theorem. The raional expeaion equilibrium defined by Equaions (8), (9), (0), (), and (5) is saionary (i.e., ime independen). Hene he heighened real ineres rae aused by an inrease of he nominal ineres on money x permanenly affes he real variables..4 A Welfare Impliaion of he Model By Theorem, a moneary expansion (an inrease in x ) simulaes he equilibrium real GDP n hrough he rise of he real rae of ineres. Here, we onsider is welfare eonomis impliaion. Le he Lagrangean of individual deision L ha is evaluaed a he equilibrium value. Then, applying he envelop heorem, we obain du dl L = = = ( x ) ( x)<0, d d where is he Lagrangean muliplier. Aordingly, a moneary expansion improves eonomi welfare, sine i makes fuure goods heaper. 4

7 3 Conluding Remarks This paper shows ha money is non neural as long as i is redible even if we obey he money supply rule proposed by Luas []. A moneary expansion (an aelaraion of he money growh rae) surely highens he real rae of ineres of money whenever people believe ha money is redible. The effe of ineremporal subsiuion leads hem o work more o prepare for more fuure onsumpion, and hus, he aggregae produs inreases. I also implies ha eonomi welfare is improved by a moneary expansion. Referenes [] R. E. Luas, Jr., ``Expeaions and he Neuraliy of Money,'' Journal of Eonomi Theory, Vol. 4, No., 97, pp [] K. Oani, ``Raional Expeaions and Non Nueraliy of Money, Welwirshaflihes, Vol., 985, pp ' [3] M. Oaki, ``The Dynamially Exended Keynesian Cross and he Welfare Improving Fisal Poliy,'' Eonomis Leers, Vol. 96, No., 007, pp.3 9. [4] M. Oaki, ``A Welfare Eonomi Foundaion for he Full Employmen Poliy,'' Eonomis Leers, Vol.0, No., 009, pp. 3. [5] M. Oaki, ``A Pure Theory of Aggregae Prie Deerminaion,'' Theoreial Eonomis Leers, Vol., 0, pp. 8. 5

8 AA BB E 0 E B B O n Figure 6