Cochrane s Defence of the Fiscal Theory of the Price Level Clarification and Critique

Transcription

1 The Universiy of Adelaide School of Economics Research Paper No Cochrane s Defence of he Fiscal Theory of he Price Level Clarificaion and Criique Colin Rogers

2 Cochrane s defence of he fiscal heory of he price level clarificaion and criique Colin Rogers School of Economics Universiy of Adelaide Absrac Cochrane (2005) employs a well specified Walrasian general equilibrium model o defend he fiscal heory of he price level agains wo forms of criicism; ha: (1) i implies he violaion of Walras s Law and, (2) he compleely cashless, fricionless version of he model canno deermine he price level. Of he wo criicisms only he second is subsanive. This paper explains why no version of Cochrane s model offers a heory of he price level. The model can deermine he consumpion good value of an asse and a numeraire denominaed price level. A numeraire denominaed price level is of no heoreical significance. Key Words: Fricionless; Cashless; fiscal heory of price level; Walras s Law. JEL classificaion: E31; E42. Corresponding auhor: Colin Rogers, School of Economics, Universiy of Adelaide, Norh Terrace, Adelaide, Souh Ausralia, Tel: (08) fax: colin.rogers@adelaide.edu.au

3 Cochrane s defence of he fiscal heory of he price level clarificaion and criique Colin Rogers School of Economics Universiy of Adelaide Revised February 9, 2006 Inroducion The fiscal heory of he price level (FTPL) is he idea ha he governmen s ineremporal budge consrain can be employed o deermine he general price level and ha his relaionship applies even in a cashless or moneyless, fricionless world where he ineremporal budge consrain is reaed as a pricing kernel for he public deb. In a recen paper Cochrane (2005) presens a defence of he FTPL agains he heoreical criique by Buier (1998, 1999, 2002, 2004) and McCallum (2003, 2004). McCallum argues ha he FTPL is misleading and no applicable o real economies while Buier (1999, p. 1) represens he mos sriden saemen of he criicism when he saes: I is no common for an enire scholarly lieraure o be based on a fallacy, ha is on fauly reasoning; misleading or unsound argumen. The recenly revived fiscal heory of he price level is an example of a research programme ha is faally flawed, concepually and logically. Buier s criicism is based on wha he considers o be an economic misspecificaion he proposiion ha he governmen s ineremporal budge consrain need only be saisfied in equilibrium. In a Walrasian general equilibrium model ha is equivalen o he claim ha Walras s law applies only in equilibrium. Buier (1999, p. 1, 2002, p. 478) goes on o argue ha his economic misspecificaion is he cause of concepual anomalies such as he apparen abiliy o price money in a world wihou money. 2

4 In his paper I explain why he quesion of Walras s Law is largely a red herring ha has no direc implicaions for he FTPL. In a well-specified Walrasian general equilibrium model, as employed by Cochrane (2005), Walras s Law applies in all saes of he model and all budge consrains are always saisfied. Tha effecively dispenses wih Buier s argumen ha he FTPL implies ha he governmen budge consrain only holds in equilibrium. Despie his conclusion i does no absolve he FTPL of he charge ha i produces a rash of conradicions and anomalies. The reason is ha hose anomalies originae wih he fricionless model ha Cochrane uses o suppor he FTPL and do no derive from he alleged failure of he model o saisfy Walras s Law in disequilibrium as claimed by Buier (2002). The anomalies generaed by he FTPL are examples of heoreical and concepual hurdles ha resul from he applicaion of Walrasian perfec or efficien barer general equilibrium models o quesions of moneary and fiscal heory 1 - as for example in Woodford (1998, 2003). The basis of Cochrane s (2005) defence of he FTPL is an analogy beween he pricing of sock and he pricing of governmen bonds: he belief ha nominal deb is mahemaically idenical o sock, Cochrane (2005, p. 526). Cochrane (2005, p. 504, emphasis added) applies ha idea in he conex of a: perfecly sandard and well-specified Walrasian economic model, one in which he governmen has no special saus, and one in which all budge consrains are saisfied a boh equilibrium and disequilibrium price levels. I may or may no apply o a given ime and place bu i is a leas a heoreically coheren possibiliy. In his paper I argue here are serious logical and concepual hurdles raised by Cochrane s analysis ha undermine is abiliy o provide a heoreically coheren 1. As Lucas (1984, p. 32) explained, hese perfec barer or fricionless models appeal because i was hough ha he successful empirical applicaion of hese fricionless models in financial economics could be repeaed in moneary economics. However, Cochrane does no offer an empirical defence of he fricionless barer model bu aims only o presen a heoreically coheren defence of he FTPL. 3

5 accoun of he price level in he fricionless, well-specified Walrasian general equilibrium sysem. In ha conex he concepual hurdles noed by Buier and McCallum are no merely semanic (e.g., wha is mean by he price level, cashless or fricionless ) bu raise serious doubs abou he logical and heoreical coherence of Cochrane s model of he FTPL. They also raise fundamenal quesions abou he use of he fricionless model as a basis for moneary and fiscal heory. The concepual hurdles originae wih, and are inheren in, he version of he Lucas ree model wih a cash-in-advance consrain employed by Cochrane, bu are highlighed by his exension of he model o a moneyless, fricionless sae. The model has all he feaures of a Walrasian general equilibrium sysem. Tha is, budge consrains are always saisfied and Walras s Law holds. However, i is anoher feaure of he model ha effecively undermines Cochrane s use of he model as he basis for he FTPL. Fia or inconverible money only exiss, as Sargen (1987, chaper 5) poins ou, because resricions are placed on household exchange paerns. Wihou hese resricions fia money has zero exchange value in such models. This is an example of a problem firs noiced by Hahn (1965, 1973a, 1973b, 1982) wih respec o Painkin s model bu i applies o Walrasian general equilibrium sysems in general (see Sargen 1987, chaper 3). Models wih his feaure are bes described as models of efficien or perfec barer (as opposed o real world barer) and when money is incorporaed ino he model, in he form of a cash-in-advance (CIA) consrain or in he uiliy funcion, i suppors an inferior se of equilibria as a resul of he resricions i places on exchanges. This feaure of Walrasian general equilibrium sysems puzzled moneary heoriss in he pas bu is now widely undersood 2. 2 Clower (1984, p. 267) was puzzled by he fac ha money appeared o be a welfare reducing fricion in his model: he choice alernaives confroning households were more resricive in a money han in a barer economy, which mean ha moneary exchange is less efficien ha barer exchange, 4

6 The relevance of all his o Cochrane s defence of he FTPL arises direcly from his applicaion of a well specified Walrasian general equilibrium sysem in boh is moneary and non-moneary and fricionless versions. For Cochrane (2005, p.503): [F]ricionless compeiive models are he benchmark, he foundaion upon which we add ineresing fricions. Ye moneary economics has so far crucially relied on a big fricion a he shor end of he yield curve in order o even sar alking abou a price level. Cochrane s obecive is o eliminae he big fricion a he shor end of he yield curve, inroduced by he quaniy equaion or a CIA consrain, o produce his fricionless benchmark model, and o employ ha model o deermine he price level. Bu by eliminaing he big fricion Cochrane creaes he concepual and logical anomalies noed by Buier (1999, 2002, 2004) and McCallum (2003, 2004). The concepual difficulies arise because by eliminaing he medium of exchange funcion of money Cochrane is forced o change he concep of money, and along wih i he concep of he price level, when moving beween he differen versions of he model. Tradiional quaniy heory inerpreaions of he model include fia or inconverible money and impose consrains on households o force hem o hold inrinsically worhless money. The price level in ha conex is he inverse of he purchasing power of money and is deermined by he quaniy equaion or defined in erms of prices denominaed in he medium of exchange. Moving o he fricionless version of he model eliminaes all he consrains ha give fia money posiive exchange value conrary o boh common sense and wo hundred years of convenional wisdom. Somehing obviously was wrong. Bu wha? Wha is wrong is ha money has been added o a model where i is no needed a model of perfec barer where a medium of exchange is redundan. Walrasian, Arrow-Debreu models based on a ime-0 aucion have no role for fia money as a medium of exchange, as Hahn (1982, p.1) noed. Hence imposing he medium of exchange funcion reduces money o a fricion. 5

7 and money is reinerpreed as an income-earning asse. The price level can no longer be deermined by he quaniy equaion and ha concep is abandoned, and replaced by he concep of a numeraire price level he price level defined in erms of he numeraire prices of all commodiies. I is here ha he concepual problems arise because numeraire prices are arbirary and have no economic significance. Consequenly any price level defined in erms of hem has a similar inerpreaion. This, in essence is he cause of he obecions o he FTPL by Buier and ohers he FTPL is no a heory a all bu an arbirary definiion of no economic significance. To make he case he remainder of he paper proceeds as follows. Secion II oulines he hree saes of he model idenified by Cochrane he moneary, cashless and compleely cashless saes. I sugges a clarificaion of Cochrane s erminology: ha he erms cashless and fricionless are of analyical significance only in he conex of he compleely cashless and fricionless model when all fricions, including he CIA consrain Cochrane s big fricion -have been eliminaed. From his perspecive here are only wo analyically disinc versions of Cochrane s model he version wih he CIA consrain (he moneary and cashless versions) and he compleely cashless and fricionless version where he CIA consrain has been removed. The cashless version is analyically similar o he moneary version because boh reain he big fricion on inra day rades. Secion III hen inerpres Cochrane s model. Firs, i applies Walras s Law o illusrae why he moneary and cashless versions canno differeniae beween he FTPL and he quaniy heory of he price level (QTPL). Second, and more imporanly, Painkin s (1965) disincion beween accouning prices and money prices is employed o show ha he compleely cashless and fricionless model 6

8 deermines only he real relaive prices of commodiies and he accouning or numeraire prices are enirely arbirary. Defining he price level in erms of numeraire prices is possible bu he price level defined in his way is no deermined by he model- i is as arbirary as he numeraire prices of goods and has no analyical significance. This explains Buier s characerisaion of he FTPL as an aemp o deermine he price of a non-exisen eniy like phlogison. Finally, Secion III briefly oulines how he concep of money changes- from fia money o asse money- as he analysis moves beween he moneary and compleely cashless, fricionless versions of he model. Secion IV concludes ha he obecions o he FTPL raised by Buier and McCallum are subsanive, and no merely semanic, when ha heory is presened in he conex of a fricionless Walrasian general equilibrium model. Exponens of he FTPL need o look elsewhere for heoreical suppor. II Cochrane s moneary, cashless, and compleely cashless worlds Cochrane begins by sressing he analogy beween he pricing of equiy (sock) and he pricing of governmen bonds. The wo relaionships are lised as expressions (1) and (2) 3. Nominal value of shares = Expeced presen value of dividends. (1) Price level Nominal Governmen deb = Expeced presen value of primary surplus. (2) Price level Equaion (2) is an applicaion of he idea ha he real value of sock, as he expeced presen value of he dividend sream, can be applied direcly o deermine he real 3 Equaion numbers follow Cochrane (2005) for ease of comparison. I have amended expression (1) o include he nominal value of shares in he numeraor insead of he number of shares as in Cochrane. Boh expressions hen have real (purchasing power) value on he LHS. 7

9 value of governmen deb as he expeced presen value of he primary surplus. This is he basis of he idea ha nominal deb is mahemaically idenical o equiy. Tha may or may no be so. Bu even if we accep he mahemaical equivalence of expressions (1) and (2) here are several concepual hurdles ha need o be overcome before we can accep Cochrane s FTPL. These hurdles arise because Cochrane seeks o move seamlessly beween various saes of he model wihou realising ha he conceps of money and he price level change across differen saes of he model. The concepual problems arise wih he use of he compleely cashless and fricionless model as he benchmark o which ineresing fricions, like money, can be added. Cochrane (2005, p. 503) explains his sraegy as follows. More imporanly, a fricionless model can provide a useful benchmark for more complex and realisic analyses wih fricions. Throughou economics, fricionless compeiive models are he benchmark, he foundaion upon which we add ineresing fricions. Ye moneary economics has so far crucially relied on a big fricion a he shor end of he yield curve in order o even sar alking abou he price level. To see how he concepual hurdles arise we begin wih he moneary version of he well-specified Walrasian general equilibrium sysem. Cochrane s moneary model Cochrane (2005) employs a Lucas ree model wih a cash-in-advance consrain as presened by Sargen (1987, chaper 5). Sargen (1987, Table 5.1, p. 158) provides a deailed ouline of he rading paern and he resricions on households needed o usify a role for inrinsically worhless fia money in he model. Wihou hese resricions fia money would have zero exchange value (Sargen 1987, chapers 3, 5). The model has hree componens or Walrasian markes (o use radiional erminology) - goods, asses (bonds/sock) and money - in an ineremporal seing 8

10 wih rading days exending ou o an infinie horizon. Three ypes of securiies are raded in Sargen s model, shares in rees, governmen issued currency and one period sae coningen claims o currency. Bu only wo securiies, currency and nominal governmen bonds appear in budge consrains as Cochrane noes ha shares in rees will be in zero ne supply and consequenly have no effec on he equilibrium soluion. They are, however, imporan o his applicaion of he analogy beween sock and bonds and he mahemaical ideniy in pricing he wo asses. There are many idenical households and a governmen. Each household owns a ree ha produces frui (dividends) bu households are precluded from consuming heir own frui 4. Idenical households maximise a sandard uiliy funcion, 0 =0 max E β u( c ), have shopper and worker characerisics, and ener period wih money balances M 1 and one period nominal discoun bonds wih face value B 1. The governmen chooses a sae-coningen sequence of one-period nominal deb, s s money and primary surpluses, B, M, s } and each are random variables. The model { sars a = 0 so B 1 and M 1 are fixed (hey may boh be se o zero). In he asse marke he household sells bonds for currency, pays lump sum axes, p s, buys new bonds, B and leaves wih money, d M. In he goods marke he household receives an endowmen, e of he consumpion good (frui from he ree) and canno consume is own endowmen so mus rade wih oher idenical households. The household has worker and shopper characerisics. The shopper uses he money from he asse marke, M o buy goods, c subec o a cash-in-advance consrain, d 4 This eliminaes he auarky soluion. 9

11 where d pc M v (5) v = velociy in a quaniy equaion and p is he price level. Cochrane iniially ses v = 1. The worker sells he endowmen e for money and ges cash p e in reurn. In he moneary sae of he model, he worker and shopper go home and ea c. Also hey mus hold, overnigh, any money balances no convered ino consumpion goods by he shopper, he balance would be; M p c, plus he money d balances earned by he worker, and equal o p e. Hence we have: M d = M p e c ) (6) ( To derive he household s budge consrains Cochrane noes ha hey can rade arbirary coningen claims (shares in rees) in he asse marke and so he price of a one period sae coningen share a ime is given by; 1 Q = pe ( m, 1 ) (7) 1 p 1 The expression m, is a real sochasic discoun facor or pricing kernel. As noed above, since all households are idenical, claims provided by households (shares in rees) are in zero ne supply, so do no appear in he household budge consrains and do no influence equilibrium prices or allocaions. Only governmen bonds appear in he household s period-by-period budge consrain; B 1 M 1 p ( e c ) = Q B M p s (9) Prohibiing he households from issuing money (o preven arbirage agains ineresbearing bonds) and applying a ransversaliy condiion produces wha Cochrane calls he presen value budge consrain: 10

12 11 = = 0 1, 1 ) ( e c s m E p B (11) In equilibrium he goods, money and asse markes are described by he following hree equaions in he case of he moneary version of he model. m e u e u =, ) ( ' ) ( ' β (12) e p c p v M = = (14) = = 0 1, 1 ( p M M s m E p B (15) An equivalen version of (15) is wrien as; = ) 1 (. 1 1 f f p M r r s m E p M B (16) where f r is he one period nominal rae of ineres such ha f r Q 1 1. Cochrane inroduces one change o Sargen s resricions o relieve wha he calls a fricion ha arises from he need for households o hold money balances overnigh, hereby foregoing poenial ineres income. To ha end he securiies marke reopens a he end of he day o allow households o conver any unused money balances ino governmen bonds. Cochrane describes his model as cashless and fricionless. Bu as I will explain below, boh of hese descripions are misleading. Cochrane (2005, p. 513) laer inroduces he concep of a compleely cashless model and i is ha model ha is also compleely fricionless and he source of he conroversy over he FTPL. Cochrane s cashless model. In he cashless version of he model Cochrane amends Sargen s specificaion so ha households can reurn o he securiies marke a he end of he day and rade any

13 unwaned cash (money) for more governmen bonds. In ha case expression (6) does no hold in he cashless version of he model. Cochrane (2005, p. 508) saes ha he absence of (6) is he only difference beween he wo versions of he model. In Cochrane s cashless model equilibrium in he goods, money and asses markes are described by he following equaions: u' ( e ) β = m,. (17) u' ( e ) M = 0 (18) B M 1 1 = E ( m, s ) p = 0 (19) The behaviour of households in he goods marke remains unchanged. The key difference is ha in he cashless model households have convered all heir money ino governmen bonds overnigh. Hence equaion (18) and he saemen ha money demand is zero from equaion (6). The idea here is ha if ineres raes are posiive, r f > 0, hen households hold no money overnigh hey buy governmen bonds when he financial markes open a he end of he day. In his sae of he model demand for overnigh money would only be posiive if nominal ineres raes were zero. However, households mus sill use money o buy goods he nex day subec o he CIA consrain on inraday rade. They sar he day by selling bonds for currency o he governmen, using currency o pay axes and hen using he res of he money so obained o buy goods and so on. Consequenly, I would argue ha i is no helpful o describe his model as cashless. I differs from he moneary version only in he 12

14 disribuion of money balances overnigh. From equaion (19) i is apparen ha he governmen holds he money overnigh. The same argumen explains why he cashless model is no fricionless - all he resricions imposed by Sargen (1987), in he form of he inraday CIA consrain, sill remain. So Clower s observaion ha money is a welfare reducing consrain applies o he cashless version of he model. Consequenly, alhough Cochrane has removed a lile fricion in he cashless version of he model i is also no helpful o describe his version of he model as fricionless. The big fricion generaed by he CIA consrain sill applies o inraday rade so he model is sill a moneary Walrasian general equilibrium model subec o he consrains imposed on households (he CIA consrain) o generae a non-zero exchange value for fia money. To eliminae all fricions ( mrs mr, and Sargen s resricions on exchanges) Cochrane akes he nex sep and removes he CIA consrain, he big fricion, o produce wha he calls he compleely cashless model. This model is a moneyless model and is ruly fricionless, as all he consrains on households o generae a nonzero exchange value for money have been removed. Cochrane s compleely cashless and fricionless model Cochrane (2005, p. 513) offers wha he calls he compleely cashless and herefore, fricionless inerpreaion of his model when he saes: The cash in advance consrain plays no essenial role in he equilibrium. For any equilibrium of he fricionless [i.e. cashless ] model saed so far, he same equilibrium holds if we eliminae he cash consrain and eliminae inraday cash. In his case money is eliminaed alogeher, money demand disappears and along wih i any M in equaion (19). The model is reduced o he wo equaions: 13

15 u' ( e ) β = m,. (17) u' ( e ) B p 1 = E ( m, s ) = 0 (19 ) In equaion (19 ) he absence of M 1 indicaes ha he model is compleely cashless and hence fricionless in he sense of boh Cochrane and Sargen/Clower. I is on he basis of he compleely cashless and fricionless version of he model ha Cochrane makes his sronges claims abou he FTPL and i is in his conex ha large concepual hurdles emerge. III Inerpreing Cochrane s model. The firs poin o noe abou he moneary and cashless versions of Cochrane s model (which I rea as analyically equivalen for he reasons given above) is ha Walras s Law is an ineviable propery of a well-specified Walrasian general equilibrium model so applies o Cochrane s analysis. Hence, we can follow Painkin (1965, chapers 3, 10, 11) and use he hree equaions (12), (14) and (15) o derive marke-clearing loci for Cochrane s moneary version of he model in (r, p) space (and he cashless version for reasons explained above). The marke clearing condiions provided by Cochrane (2005, p. 509) are simply: c = e goods marke (12 ) s M = M money marke (14 ) s B = B bonds marke (15 ) For each dae and sae of naure, expressions (12 ) o (15 ) can be used o conduc wha Painkin called marke experimens o derive he marke clearing loci for each of he goods, money and bonds markes in (r, p) space. By Walras s Law only wo of 14

16 hese hree marke-clearing loci are independen so only wo are needed o provide he equilibrium soluion (r, p). This analysis is familiar from Painkin s Money Ineres and Prices, chapers 10 and 11. See, in paricular, Figure 11-2, Painkin (1965, p. 259). Which of he marke-clearing loci is dropped is enirely arbirary. Thus, in Chaper 10 Painkin drops he bond marke and, before proceeding o drop he money marke in chaper 11, he noes, Painkin (1965, p. 253): We shall now show ha he conclusions reached in he preceding chapers by an analysis of he commodiy and bond markes can also be reached in a somewha more familiar way by an analysis of he money marke. This equivalence is, of course, a simple implicaion of Walras s Law Thus in a well-specified Walrasian model wih fia money, where money is incorporaed eiher in he uiliy funcion or as a cash-in-advance consrain 5, here is no basis in he model for disinguishing beween a quaniy heory of he price level or a bond (fiscal) heory of he price level or, for ha maer, he commodiy heory of he price level. If we drop he goods marke and leave he money and bonds markes wha do we call he heory of he price level in ha case? Noice, ha by Walras s Law he money marke locus for he cashless version of he model could be dropped irrespecive of who is holding he fia money balances so, a leas from he perspecive of Walras s Law, he laer refinemen adds nohing of analyical significance. The siuaion changes significanly when we consider he compleely cashless, fricionless version of he model. In his version of Cochrane s model, fia money (cash) has been eliminaed alogeher, and money is redefined as an income earning asse; hence he ile of Cochrane s paper money as sock. This avoids Clower s 5 As Feensra (1986) explained, under cerain condiions he cash-in-advance consrain is simply a special case of he money in he uiliy funcion approach. See Painkin (1989, p, xxxiii). 15

17 problem and he fac ha fia money has zero exchange value in Walrasian general equilibrium models (Hahn, 1965, Sargen 1987, p. 136) bu raises concepual hurdles of he sor noed by Buier and McCallum. Before we consider hose, noe ha in is compleely cashless, fricionless sae, Cochrane s model sill saisfies Walras s Law. Bu as he model has been reduced o only wo marke-clearing loci, hose ha can be derived from expression (12 ) and (15 ) backed by expressions (17) and (19 ), he relevance of Walras s Law is moo boh marke clearing loci are needed o solve for he equilibrium (r, p) soluion. The variables r and p do, however, require careful re-inerpreaion (and re-definiion) in he compleely fricionless model. The quaniy of money no longer exiss so he quaniy heory canno be proposed as a heory of he price level. Wha hen deermines he price level in his version of he model? The simple answer is ha he price level, as defined in he moneary and cashless versions of he model, no longer exiss. I is replaced by somehing else. Hence he obecion by McCallum (2003, p.634) ha he analysis is misleading and Buier s (2002) obecion ha he fiscal heory of he price level is an inellecual bridge oo far. They obec o he reinerpreaion of he concep of he price level offered by Cochrane. Specifically, Buier (2004, p. 31, emphasis added) argues ha alhough Cochrane s model can deermine a relaive price of wo commodiies expressed in erms of he numeraire (e.g., phlogison) i is obviously no possible for he model o deermine he price of phlogison (he numeraire): Any wo commodiies priced in phlogison (or any imaginary and non-exisen numeraire) will have a well-deermined relaive price. Deermining he price of phlogison (he numeraire) when phlogison does no exis excep as a word, is an inellecual bridge oo far. 16

18 To clarify he maer, recall Painkin s (1965, pp ) disincion beween ypes of money and prices in he Walrasian general equilibrium model. He disinguished beween wo ypes of money in his model. The absrac uni of accoun for purposes of record keeping bu which had no physical exisence -we could follow Buier and call i phlogison- and fia money ha acs as he physical medium of exchange. Corresponding o he wo ypes of money were wo ypes of prices and Painkin labelled hem accouning prices and money or absolue prices respecively. In addiion Painkin noed ha here were real or relaive prices ha represen he prices of commodiies in erms of one anoher. Finally, Painkin (1965, p. 403) defined he n 1 price level in his model as p = w p where he p are he prices of goods in = 1 erms of he uni of accoun, he paper money. w are known weighs and he n h good is reaed as Applying hese definiions o Cochrane s model reveals he concepual hurdles o which Buier and ohers obec. The fricionless sae of Cochrane s model is one in which fia money has been eliminaed so we are back in Sargen s (1987, secion 3.3) saemen of a Lucas model of asse prices. Consequenly, all asses, shares in rees and governmen bonds mus be paid for wih frui even if heir prices are quoed in $s or unis of phlogison. In erms of he absrac uni of accoun, phlogison, he prices of he frui and he bond and sock can be wrien as p, p, p ). If bonds have aken over he role of money, ( c b s as Cochrane (2005, p. 506) suggess when he claims ha he compleely cashless 17

19 economy can work us as well as a moneary economy if governmen bonds ac as he medium of exchange, he respecive relaive prices are simply p / p,1, p / p ) ( c b s b where he p ; = s, b, c are he accouning prices. Painkin s definiion of he price level in erms of any numeraire prices hen reduces o p = = w c p. ( b). So in a simple echnical sense he price level in Cochrane s model can be defined in erms of numeraire prices, even unis of phlogison. Why can his be inerpreed as a proxy for he price level as defined in he moneary version of he model? The answer is ha he price level defined in erms of accouning, or numeraire, prices is arbirary or indeerminae. I has no heoreical significance. As Painkin (1965, p. 16) explains: The accouning price of a given good is disincive in having no operaional significance for he marke.. he saemen ha he accouning price of X is 4 in he absence of addiional informaion on he accouning price of a leas one oher good gives us no idea wha we mus do o acquire a uni of good X. A similar argumen applies o he definiion of he price level in he compleely cashless, fricionless version of Cochrane s model. The arbirary naure of he numeraire or accouning prices, hey can ake on any absolue values so long as hey reflec he relaive commodiy exchange raios generaed by he model, is ransferred direcly o he definiion of he price level, i is enirely arbirary and has no analyical or heoreical significance. In his sense Cochrane fails in his aemp o demonsrae ha he FTPL is a heoreically coheren possibiliy in he conex of a well-specified Walrasian general equilibrium model. In he compleely cashless and fricionless model he FTPL is reduced o an arbirary definiion of he price level in erms of accouning prices. This is one elemen of he subsance o he obecion by Buier and McCallum 6. s 6 Buier (1998, p. 29) makes his quie clear when he argues: In he demoneised economy, money has become solely he numeraire. We know from general equilibrium heory ha he numeraire need no be a good or bundle of goods in he commodiy space. Indeed he numeraire could be somehing enirely ficiious, such as phlogison. Only relaive prices maer and are deerminae. 18

20 Wha he model does deermine is he relaive price of he consumpion commodiy, frui, in erms of bonds p / p ) or he relaive price of shares in erms of bonds ( c b ( p s / pb ). And, alhough hese prices are expressed in erms of unis of phlogison, hey represen quaniies of frui so he model is essenially one of barer (acually perfec barer). Cochrane s compleely cashless and fricionless model is obviously a model of asse prices. I deermines he price of an asse in erms of he consumpion good. Redefining he asse as money hen enables Cochrane o define he price level as he relaive price of consumpion goods and asse-money. To see his, consider Cochrane s formal derivaion of wha he calls he price level in his fricionless model. Cochrane (2005, p. 511) draws ou hese feaures of his model when he observes ha, wih a uiliy funcion of he form, u ( c) = c 1 γ and e e =, B s s = B, M = 0 and s = s, all posiive and consan over ime, equaions (17) and (19 ) simplify o: m = β (17 ),1 B p = p = ( 1 β ) (19 ) s The discoun facor is consan, nominal ineres raes are said o be posiive and he price level is said o be posiive and consan. So if rue, his would amoun o he deerminaion of he price level in he perfecly cashless and fricionless model. However, a momen s reflecion will reveal ha his argumen fails a several poins. Firs, he variable p in equaion (19 ) obviously does no correspond o he price level, p as defined in equaion (14). Equaion (19 ) is simply a special case of he 19

21 asse pricing formula of he ype discussed by Sargen (1987, p. 96) so a new variable, q -he consumpion good value of a share, should be inroduced here. Tha is: u'( d ) q = E β d. (20 ) = 1 u'( d ) Where q is defined as he real asse price (measured in erms of he consumpion good, frui), no he price level, p from equaion (14), and d are he dividends, he frui from he ree, paid o holders of iles o hese asses, all measured in erms of he consumpion good. For he simple uiliy funcion u( c) = ln c Sargen s asse pricing formula becomes: β q = E β d or q = d (21 ) 1 β = 1 Expression (19 ) is simply anoher special case of his form of soluion represened by (21 ). The Lucas ree model wihou a CIA consrain deermines he real price (measured in erms of he consumpion good) of he securiy backed by he ree. Thus Cochrane s compleely cashless and fricionless model deermines he price of he socks (shares in rees), q, as measured in erms of consumpion goods a period per share held during period and where he dividends are also paid in period consumpion good. Consequenly, he LHS variable in equaion (19 ) should be q - he consumpion good value of shares in he rees and is clearly no wha is radiionally undersood by he price level 7. 7 Hoover (1988) poined o a similar concepual hurdle in Fama s (1980) aemp o apply Walrasian general equilibrium heory o moneary economics. 20

22 Second, he mahemaical ideniy beween sock and bonds occurs because bonds and sock are effecively perfec subsiues (rade one for one) in he sense ha arbirage mainains equaliy beween heir prices. Hence he relaionship beween expressions (1) and (2) in he fricionless model is no by analogy bu an ineviable propery of he model. Yields on governmen bonds, in erms of frui, mus be he same as on shares in rees. Pricing shares in erms of frui simulaneously prices governmen bonds in erms of frui. We could us as easily use he model o deermine he sock heory of he price level raher han he FTPL. Inerpreing sock (an income earning asse) as money is hen he basis for Cochrane s claim ha as he model deermines he consumpion good value of asse-money i is a heory of he price level. This logic fails because he price level and he consumpion good value of an asse are concepually and analyically wo quie differen hings. Third, equaion (19 ) conains B which was iniially defined as he face value of a nominal governmen bond ha was ile o a flow of ineres paid in money. Bu in he compleely cashless, fricionless model here is no fia money in which B can be paid. Buier (1999) makes his poin clear. Consequenly here are no nominal ineres raes; how much money mus be given up oday in exchange for a uni of money omorrow. Wha ineres raes here are, are commodiy raes, how much frui mus be given up oday in exchange for a uni of frui omorrow. Expressing his ineremporal exchange raio in erms of numeraire prices, e.g., phlogison, does no conver i ino a moneary or nominal rae. Finally, he fricionless model has no role for money as a medium of exchange - be i fia money or asse money. This follows from he ime-0 form of he aucion ha 21

23 underlies he fricionless model and i is his form of aucion ha effecively renders he model fricionless (see Lungqvis and Sargen (2004) for a discussion of he properies of his form of aucion.) As a model of efficien or perfec barer here is simply no need for any ransacions echnology he model does no specify any and none is required (Rogers 2005). Cochrane s suggesion ha asse-money can become he medium of exchange is herefore inconsisen wih his propery of he compleely cashless and fricionless model. Asses canno be imbued wih he medium of exchange funcion of money in his version of he model wihou re-inroducing fricions. Consequenly, Buier, McCallum and ohers are righ o reec Cochrane s concep of he price level, no on semanic grounds, bu on concepual and heoreical grounds. The concep of money changes as we move hrough he various versions of Cochrane s model as does he concep of he price level. The conceps are no comparable across versions of he model. Furhermore, he numeraire prices of asses and frui are enirely arbirary as is any price level defined in erms of hem. Cochrane s compleely cashless and fricionless model deermines he consumpion good value of asses and relaive prices of asses and he consumpion good all measured in unis of frui. I does no deermine he price level as defined in moneary Walrasian general equilibrium models or as measured by he GDP deflaor or CPI. IV Concluding remarks In his paper I have argued ha neiher he cashless nor he compleely cashless and fricionless versions of he model presened by Cochrane can provide a heoreically coheren framework for he FTPL. 22

24 Cochrane s formal cashless model is no compleely cashless or fricionless. I is a moneary model of he Painkin ype in which money is a welfare reducing fricion. The formal cashless model, equaions (17) o (19), does no become compleely cashless or fricionless when money or cash is convered o bonds overnigh. I is no compleely cashless because ownership of cash has simply been ransferred o he governmen and he CIA consrain (a big consrain) sill applies o inraday rade. I is no fricionless because he CIA consrain imposes he medium of exchange funcion of money as a welfare reducing fricion on inraday rade. In ha respec Cochrane s formal cashless version of he model has all he properies of Painkin s model in Money, Ineres and Prices. Cochrane s model in boh is cashless and moneary versions is a moneary model of sors bu one in which money has been imposed when i is no needed. Hence in a well-specified Walrasian general equilibrium model fia money is a big fricion. The model has he propery ha he price level, as he inverse of he purchasing power of he money sock, can be deermined bu here is no basis in he model o claim precedence for he bond marke over he money marke, or indeed he goods marke. The FTPL canno ake precedence over he QTPL in Cochrane s cashless model equaion (19) has no precedence over equaion (14). Cochrane s informal discussion of he model relies more subsanially on he compleely cashless and fricionless version. Bu ha version canno provide a foundaion for any heory of he price level, le alone he FTPL. Conrary o Cochrane s claim, he FTPL is no a coheren heoreical possibiliy in a compleely cashless, fricionless and well-specified Walrasian equilibrium sysem. The 23

25 compleely cashless, fricionless version of a well-specified Walrasian equilibrium sysem, one ha compleely eliminaes he big fricion a he shor end of he yield curve, effecively rules ou any role for he concep of a price level employed in he moneary version of he model. A numeraire price level can be defined in erms of a se of numeraire prices for asses and commodiies bu ha price level is iself arbirary because i is defined in erms of he arbirary numeraire, as Painkin explained. Thus no version of Cochrane s well specified Walrasian general equilibrium model, moneary, cashless or compleely cashless and fricionless, offers a coheren heoreical foundaion for he FTPL. Cochrane s sock-bond analogy also raises concepual hurdles when subec o scruiny. Removing he big fricion represened by he CIA consrain reduces Cochrane s model o one equivalen o he Lucas ree model wihou a CIA consrain. Tha is essenially a model of asse pricing and no a model of he price level. The model can deermine he consumpion good value of shares in rees and by arbirage he consumpion good value of governmen bonds. Inerpreing he consumpion good value of a governmen bond as he price level requires ha we (a) inerpre he price level as a relaive price and (b) inerpre he bond as money so we have he consumpion good price of money as he concep of he price level. In addiion, he fricionless version of a well-specified Walrasian general equilibrium model has no role for asse money, any more han i does for fia money, as a medium of exchange. In his paper I have also poined ou he source of he concepual hurdles generaed by Cochrane s use of he compleely cashless and fricionless model. The laer is an inheren propery of he ime-0 aucion ha underpins he Walrasian general 24

26 equilibrium sysem employed by Cochrane- such a model is bes inerpreed as a model of perfec barer. Aemps o employ he Walrasian sysem as a basis for moneary heory have hrown up similar concepual hurdles in he pas, as wih he MIU of Painkin and CIA consrain of Clower. Failure o recognise hese properies of he various saes of his model leads o he concepual hurdles acknowledged by Cochrane. Thus, despie he disracion of he debae abou Walras s Law here are grounds for concluding ha he applicaion of he Walrasian general equilibrium model as a basis for he FTPL is logically and concepually flawed as Buier (1999, p.1) claimed- and no relevan o acual economies, as McCallum (2003, p. 634) suggesed. 25

27 References Buier, W. H., The young person s guide o neuraliy, price level indeerminacy, ineres rae pegs, and fiscal heories of he price level, Naional Bureau of Economic Research, Working Paper Buier, W. H., The Fallacy of he Fiscal Theory of he Price Level, Naional Bureau of Economic Research, Working Paper A shorer version appeared as Buier, Buier, W. H., The Fiscal Theory of he Price Level: A Criique, Economic Journal, 112, July, Buier, W. H., New Developmens in Moneary Economics: wo ghoss, wo eccenriciies, a fallacy, a mirage and a myhos, Royal Economic Sociey Annual Conference, Swansea. Clower, R. W., Money and Markes in Donald A. Walker, (Ed.), Money and Markes: Essays by Rober W. Clower, Cambridge UK, Cambridge Universiy Press. Cochrane, J. H., Money as Sock, Journal of Moneary Economics, 52: Fama, E., Banking in he heory of finance, Journal of Moneary Economics, 6,

28 Hahn, F. H., On some problems of proving he exisence of an equilibrium in a moneary economy, in F. H. Hahn and F. P. R. Brechling, (Eds.), The Theory of Ineres Raes, London: Macmillam, Hahn, F. H., 1973a. On Transacions Coss, Inessenial Sequence Economies and Money, Review of Economic Sudies, 40, Hahn, F. H., 1973b. On he foundaions of moneary heory, in Michael. Parkin (Ed.), Essays in Modern Economics, Longman UK, London, Hahn, F. H., Money and Inflaion, Basil Blackwell, Oxford. Hoover, K. D., The New Classical Macroeconomics: A scepical inquiry, Basil Blackwell, Oxford. Lungqvis, L, and Sargen T. J., Recursive macroeconomic heory, 2 nd ediion, M.I.T. Press, Cambridge MA. Lucas, R. E., 1984, Money in a heory of Finance, Carnegie-Rocheser Conference Series on Public Policy, 21, McCallum, B. T., Bank deregulaion, accouning sysems of exchange, and he uni of accoun: a criical review, in Karl Brunner and Allan Melzer (Eds.) The New Moneary Economics, Fiscal Issues and Unemploymen, Carnegie-Rocheser Conference Series on Public Policy, 23, Amserdam: Norh Holland,

29 McCallum, B. T., Is he fiscal heory of he price level learnable?, Scoish Journal of Poliical Economy, 50, 5, McCallum, B. T., Moneary Policy in Economies wih Lile or No Money, Pacific Economic Review, 9, Painkin, D., Money Ineres and Prices: An Inegraion of Moneary and Value Theory, (second ediion), New York: Harper and Row. Second ediion, abridged (1989), MIT Press, Cambridge, Massachuses. Rogers, C., Doing wihou money: a criical assessmen of Woodford s analysis, Cambridge Journal of Economics, 8 Augus advanced access. Sargen, T. J., Dynamic Macroeconomic Theory, Harvard Universiy Press, Cambridge, Massachuses. Woodford, M., Doing wihou money: conrolling inflaion in a pos-moneary World, Review of Economic Dynamics, 1, Woodford, M., Ineres and Prices: Foundaions of a heory of moneary policy, Princeon Universiy Press, Princeon. 28