working Money Growth Rules and Price Level Determinacy by Charles T. Carlstrom and Timothy S. Fuerst FEDERAL RESERVE BANK OF CLEVELAND

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1 working p a p e r Money Growh Rules and Price Level Deerminacy by Charles T. Carlsrom and Timohy S. Fuers FEDERAL RESERVE BANK OF CLEVELAND

2 Working Paper Money Growh Rules and Price Level Deerminacy by Charles T. Carlsrom and Timohy S. Fuers Charles T. Carlsrom is a he Federal Reserve Bank of Cleveland. Timohy S. Fuers is a Bowling Green Sae Universiy, Bowling Green, Ohio and is a visiing research associae a he Federal Reserve Bank of Cleveland. Working papers of he Federal Reserve Bank of Cleveland are preliminary maerials circulaed o simulae discussion and criical commen on research in progress. They may no have been subjec o he formal ediorial review accorded official Federal Reserve Bank of Cleveland publicaions. The views saed herein are hose of he auhors and are no necessarily hose of he Federal Reserve Bank of Cleveland or of he Board of Governors of he Federal Reserve Sysem Working papers are now available elecronically hrough he Cleveland Fed's home page on he World Wide Web: hp:// Augus 2000

3 Money Growh Rules and Price Level Deerminancy by Charles T. Carlsrom and Timohy S. Fuers This paper demonsraes ha in a plausibly calibraed moneary model wih explici producion, exogenous money growh rules ensure real deerminacy and hus avoid sunspo flucuaions. Alhough i is heoreically possible o consruc examples in which real indeerminacy does arise, hese examples rely on implausible money demand elasiciies or ignore he effec of producion on he model s dynamics.

4 Money Growh Rules and Price Level Deerminacy Charles T. Carlsrom a Timohy S. Fuers b a Federal Reserve Bank of Cleveland, Cleveland, Ohio, 44101, USA. b Deparmen of Economics, Bowling Green Sae Universiy, Bowling Green, Ohio, 43403; and Consulan, Federal Reserve Bank of Cleveland. June 21, 2000 ABSTRACT: This paper demonsraes ha in a plausibly calibraed moneary model wih explici producion, exogenous money growh rules ensure real deerminacy and hus avoid sunspo flucuaions. Alhough i is heoreically possible o consruc examples in which real indeerminacy does arise, hese examples rely on implausible money demand elasiciies or ignore he effec of producion on he model s dynamics. The views saed herein are hose of he auhors and no necessarily hose of he Federal Reserve Bank of Cleveland or hose of he Board of Governors of he Federal Reserve Sysem.

5 I. Inroducion. I has become convenional wisdom ha cenral banks should arge ineres raes and no moneary aggregaes. There are a leas wo reasons for his preference. Firs, following he classic argumens of Poole (1970), he apparen evidence of exogenous shocks o velociy leads o a preference for ineres rae argeing. Second, a more recen line of research suggess ha even in he absence of velociy shocks, money growh argeing may be problemaic because i is more prone o real indeerminacy. For example, Masuyama (1990) and Woodford (1994) show ha money growh argeing can allow exrinsic uncerainy ( sunspos ) o be inroduced ino an oherwise deerminae real economy. The purpose of his paper is o challenge he asserion ha real indeerminacy is likely wih money growh argeing. Alhough i is heoreically possible for an exogenous money growh policy o inroduce sunspo equilibria, his paper demonsraes ha in a reasonably calibraed moneary model wih explici producion, money growh rules produce real deerminacy. Tha is, money growh rules avoid he possibiliy of sunspo equilibria. We see he avoidance of sunspos as a necessary condiion for any good moneary policy rule. Hence, exogenous money growh rules saisfy his minimalis crierion. In conras, ineres rae rules do no generally saisfy his minimalis crierion. Ineres rae rules are prone o sunspo equilibria because money growh is endogenous under such a policy. For example, consider he exreme case of an ineres rae peg in which he money supply is passively varied o hi an ineres rae direcive. The wellknown nominal indeerminacy under such a rule means ha sunspo flucuaions in he 2

6 price level naurally arise. In environmens wih nominal rigidiies, hese nominal flucuaions induce real flucuaions and are welfare-reducing. 1 In conras, we show ha money growh rules ensure deerminacy in a general moneary environmen for all plausible calibraions. We uilize a generic money-in-heuiliy funcion (MIUF) model because of is generaliy as i encompasses rigid cash-inadvance (CIA) models, ransacions cos models (see Feensra (1986)), shopping ime models, and he cash-credi model pioneered by Lucas and Sokey (1983,1987). 2 These models differ in he micro deails of he rading arrangemens, bu since we calibrae he models o aggregae moneary daa (eg., he ineres elasiciy of money demand), hese micro differences are irrelevan. We resric he analysis o an infiniely-lived represenaive agen economy because his has become he workhorse in heoreical moneary policy analysis. 3 Since we are concerned wih issues of deerminacy wihou loss of generaliy we limi he discussion o a deerminisic model. As is well known, if he deerminisic dynamics are no unique, hen i is possible o consruc sunspo equilibria in he model economy. Below we will use he erms real indeerminacy and sunspo equilibria inerchangeably. Under he assumpion of an exogenous money growh rae, we consider wo ypes of real indeerminacy. Firs we analyze he possibiliy of self-fulfilling hyperinflaions in 1 I is possible o design more complex ineres rae operaing procedures ha avoid hese problems. For example, Carlsrom and Fuers (2000) argue ha if he cenral bank uses an ineres rae operaing procedure, hen he only way of ensuring real deerminacy in a sicky price model is for he cenral bank o respond aggressively o lagged inflaion. For a relaed analysis see Benhabib, Schmi-Grohe and Uribe (2000). 2 For he case of he cash-credi model see foonoe 7 of Lucas and Sokey (1983). 3 There is a vas lieraure on real indeerminacy in overlapping generaions models of money. See, for example, Azariadis (1981). 3

7 which he economy becomes de-moneized in he limi. We show ha hese can only arise if he limiing elasiciy of money demand is quie high. This is in conras o he classic conribuion of Obsfeld and Rogoff (1983) in which hyperinflaions arise for all money demand elasiciies. The difference arises because, following Carlsrom and Fuers (2000), we use cash-in-advance (CIA) iming in which he money ha faciliaes ransacions is he money he economic agen has in advance of enering he goods marke. In conras, Obsfeld and Rogoff assume ha money balances held a he end of he period faciliaes rading earlier in he period or wha we call cash-when-i m-done (CWID) iming. The remainder of he analysis focuses on he second form of real indeerminacy; he possibiliy of saionary muliple equilibria. A key innovaion is ha we add a sandard CRS producion echnology o he environmen. In his case, muliple saionary equilibria arise only wih implausibly low money demand ineres elasiciies. This conrass wih he high elasiciies needed for raional hyperinflaions. To undersand our resuls in he case of saionary equilibria, i is helpful o compare hem wih he work of Masuyama (1990) and Woodford (1994). Masuyama analyzes an endowmen MIUF model wih an exogenous money growh policy. 4 He demonsraes ha a necessary condiion for saionary sunspo equilibria is ha he crossparial of he uiliy funcion U cm be sufficienly negaive. Woodford (1994) analyzes a Lucas-Sokey (1983,1987) cash-credi economy. Surprisingly, Woodford s analysis is consisen wih he exisence of sunspo equilibria in 4 Anoher difference beween he curren paper and Masuyama (1990) is ha we uilize CIA iming, while he uses CWID iming. This difference has a small effec on he exisence of muliple saionary equilibria, and is foonoed when appropriae. See Carlsrom and Fuers (2000) for a complemenary analysis for ineres rae rules. 4

8 a model in which he isomorphic MIUF has a posiive cross parial, U cm > 0. Below we will show ha his discrepancy beween Masuyama (1990) and Woodford (1994) arises because he former uses CWID iming, while he laer uses CIA iming. More imporanly, he paper demonsraes ha hese sunspo equilibria wih U cm 0 arise only wih implausibly low ineres elasiciies. As for he sunspos when U cm < 0, a second conribuion of he paper is o demonsrae ha when a sandard CRS producion echnology is added o he model his possibiliy disappears. The paper proceeds as follows. The nex secion considers an endowmen economy, and develops condiions for indeerminacy. Secion hree exends he analysis o an environmen wih producion and demonsraes ha once we resric he analysis o he plausible parameer space ha he sunspos of Masuyama (1990) and Woodford (1994) disappear, and money growh rules ensure real deerminacy. Secion four concludes. II. A MIUF Endowmen Economy. The economy consiss of numerous infiniely-lived households wih preferences given by = 0 β U(c, A /P ), where c and A /P denoe consumpion and real money balances, respecively. The household begins he period wih M cash balances and B -1 holdings of nominal bonds. Before proceeding o he goods marke, he household visis he financial marke s where i carries ou bond rading and receives a cash ransfer of M ( G 1) from he moneary auhoriy where s M denoes he per capia money supply and G is he gross 5

9 money growh rae. Hence, before enering goods rading, he household has cash balances given by A M + M s ( G 1) + B R B, 1 1 where R -1 denoes he gross nominal ineres rae from -1 o. Noice ha following Carlsrom and Fuers (2000) we uilize CIA iming. Tha is, he money balances ha aid in ransacions are he money balances ha he household has upon enering goods marke rading. In conras, Masuyama (1990) uilizes end-of-period money balances, wha Carlsrom and Fuers (2000) call CWID iming. We will commen on hese differences below. Afer engaging in goods rading, he household ends he period wih cash balances given by he ineremporal budge consrain. M B Pc + P y s + 1 = M + M G 1) + B 1R 1 (, where y = y denoes real household endowmen income. We will endogenize producion in he nex secion. The firs order condiions o he household s problem include he following: [U m ()+U c ()]/P = R β [U m (+1)+U c (+1)]/P +1 (1) U m ()/U c () = (R -1). (2) 6

10 Equaion (1) is he Fisherian ineres rae deerminaion in which he nominal rae varies wih expeced inflaion and he real rae of ineres on bonds. Equaion (2) is he model s money demand funcion. 5 Money demand elasiciy η -dlnm/dlni is given by U m η = m[ U iu cm ] > 0, mm where i = R 1 is he ne nominal rae. Suppose ha he cenral bank expands he money supply a a consan (gross) growh rae of M +1 /M = G > β. Since he nominal ineres rae is endogenous, one can combine (1)-(2) o yield he following difference equaion in real balances m M /P. G mu c ( m ) = m + 1 [ U c ( m+ 1) + U m ( m + 1)] β. (3) An equilibrium consiss of a non-negaive m sequence ha saisfies (3) and he sandard ransversaliy condiion. Expressing m +1 as an implici funcion of m, m +1 = g(m ), we noe firs ha g is non-negaive. Under he assumpion ha money demand slopes down (η > 0), here is a unique posiive seady-sae soluion o (3) given by he fixed poin g(m ss ) = m ss. One equilibrium is of course m = m ss for all. The key issue is wheher here are oher equilibria. There are hree possibiliies. Firs, hyperdeflaions in which m explodes and goes o infiniy in he limi. These pahs are ypically no equilibria as hey violae he household s ransversaliy condiion (see Obsfeld and Rogoff (1986)). 6 5 In he case of cash-when-i m done iming, he corresponding equaions are U c ()/P = R βu c (+1)/P +1, and U m ()/U c () = (R -1)/R. 6 A necessary condiion for ruling ou hese equilibria is ha G > 1, ie., ha money growh is posiive. 7

11 Second, hyperinflaions where m goes o zero (see Obsfeld and Rogoff (1983)). We will discuss hese in he nex subsecion. Third, and finally, saionary muliple equilibria in which for all saring values of m, he pah converges o m ss in he limi. The bulk of our analysis will revolve around hese equilibria. Self-fulfilling hyperinflaions: Self-fulfilling hyperinflaions (pahs wih m converging o zero) are possible if and only if lim g( m) = 0, ha is if here exiss a non-moneary seady-sae. We adop he 0 m mild assumpion ha sup ( y, m) is posiive and finie. Given his assumpion boh U c m 0 sides of equaion (3) go o zero (as m goes o zero) if and only if lim mu m ( y, m) = 0. 0 m (4) From he money demand relaionship (2) and he assumpion ha sup ( y, m) is finie and posiive, (4) is equivalen o lim mi( m) = 0, 0 m U c m 0 where i(m) denoes he invered money demand curve. This condiion has an elasiciy inerpreaion: in he limi, money demand ineres elasiciy mus exceed uniy so ha he decline in real balances can occur wihou oo large a movemen in he nominal rae. Obsfeld and Rogoff s (1983) analysis of hyperinflaions is quie differen. In paricular, money demand ineres elasiciy has no role in heir analysis. The essenial 8

12 difference is ha Obsfeld and Rogoff use CWID iming (uiliy depends on end-ofperiod money, A M +1 ) so ha he counerpar o (3) is G m[ U c ( m ) U m ( m )] = m+1 U c ( m + 1) β. (5) Le m +1 = h(m ) denoe his mapping. Obsfeld and Rogoff assume ha U cm = 0, and ha here exiss an mˆ > 0 such ha h (mˆ ) = 0. This laer assumpion arises from he reasonable asserion ha as m decreases U m evenually exceeds he consan U c. 7 In his case h becomes negaive for small 0 < m < mˆ, and here are hyperinflaionary equilibria if and only if h(0) = 0, or lim mu m ( y, m) = 0. 0 m (6) If condiion (6) holds, hen here are a counable infiniy of equilibria ha have as a penulimae poin mˆ. Aferwards he economy jumps disconinuously o a compleely demoneized economy where m = 0. All of hese equilibria are found by backing up he ransiion pah from mˆ a counable number of periods. Alhough condiion (6) is mahemaically he same as condiion (4), he economics are quie differen. Condiion (6) does no have an elasiciy inerpreaion. This is because he economy has become demoneized in he previous period when m T = m ~ and nominal raes are infinie. Afer his money demand ceases o hold as boh ineres raes and prices are infinie. Why hen is money held in his penulimae period (T) if nominal raes are infinie and he price level omorrow (T+1) is infinie? Equivalenly, 7 If U m never exceeds he consan U c, hen h(m) is always nonnegaive, and here are a coninuum of equilibria in which real balances go o zero only in he infinie limi. Since i(m) = [U m /(U c -U m )], hese equilibria have he propery ha nominal raes are ypically finie even wih zero real balances. In he hyperinflaionary equilibria considered by Obsfeld and Rogoff (1983), real balances jump o zero in finie ime, and in he penulimae period nominal raes are infinie. 9

13 why are money balances held a he end of his penulimae period even hough hey can never be used for ransacions? Because under he peculiar assumpion of CWID iming households receive uiliy from end-of-period money, ha is, ransacions in ime- are faciliaed wih he nominal money balances he household has a he beginning of ime +1. Condiion (6) is hus a resricion devoid of economic conen. The usual argumen ha hyperinflaionary pahs are always possible in infinie horizon moneary models resuls from a very peculiar iming assumpion. In conras, in he case of CIA iming a coninuum of hyperinflaionary pahs are possible if and only if he limiing ineres elasiciy is quie large, in excess of uniy. 8 Muliple Saionary Equilibria: Our primary focus for he remainder of he paper is on he hird equilibrium possibiliy: muliple saionary equilibria. We find hese more compelling han he hyperinflaionary equilibria because, as noed by Obsfeld and Rogoff (1983), even if he ineres rae elasiciy of money demand exceeds uniy, hese hyperinflaions can sill be ruled ou under he mild assumpion ha he governmen guaranees a minimal real redempion value for money. In conras, he exisence of saionary sunspo equilibria is much more roubling. Since all of hese pahs converge o he moneary seady-sae, simple limiing argumens canno rule hem ou. Reurning o condiion (3) and he implici g-mapping, i is sraighforward o calculae he slope of g a m ss : 8 Typical esimaes of money demand elasiciy are far below uniy. However, in he case of hyperinflaions he evidence is less clear. Cagan (1956) esimaes a semi-elasiciy of abou 4.5, implying an elasiciy of 4.5 imes he nominal rae. During he high inflaion periods his elasiciy will exceed uniy. 10

14 g ( m ss i /(1 + i) ) = 1 η(1 + mu cm / U c ) 1, (7) where η = -dlnm/dlni > 0 is he ineres elasiciy of money demand. Recall ha U m = m[ U iu cm ] η. (8) mm For wha follows, i is helpful o noe ha he values of η and U cm are logically disinc. Alhough economic heory and empirical evidence implies ha η is posiive, Feensra (1986) demonsraes ha here is no heoreical resricion on he sign of U cm. 9 For any given value of U cm, here exiss a value of U mm < 0 ha maps ino any esimaed η. Holding U cm fixed, η varies inversely wih he absolue value of U mm. A necessary and sufficien condiion for local real indeerminacy is ha g (m ss ) is wihin he uni circle. The expression for g (m ss ) hus implies ha here exis saionary sunspo equilibria in his endowmen model only if (1+mU cm /U c ) < 0 or if η is sufficienly small. Wha is he inuiion for hese muliple saionary equilibria? Suppose ha real balances begin below seady sae, m < m ss. Since he pah is saionary, i mus be he case ha m < m +1, ie., he pah is moving back o he seady-sae. For m < m ss, i mus be he case ha he nominal rae a ime- is above seady-sae. Bu given a consan money growh rule i also mus be he case ha inflaion is below seady-sae. Therefore he real rae of ineres mus be sufficienly above seady-sae. Hence, saionary sunspo 9 For example, using Feensra s (1986) ransacions cos model, c denoes oal consumpion expendiures, including ransacions coss. These expendiures are urned ino acual consumpion (ac) wih he assisance of real cash balances, ac = φ(c,m). Uiliy is hus given by U(ac) = U(φ(c,m)). Since U is concave, assuming ha he cross parial of φ is posiive does no guaranee ha he cross parial of U is posiive. 11

15 equilibria are possible only if lower real balances (m < m ss ) lead o increases in he real rae of ineres. From (1), he effec of real balances on he real rae of ineres depends upon he sign of (U mm + U cm ). If his erm is sufficienly negaive, hen here are sunspo equilibria. There are hus wo cases, one corresponding o U cm being sufficienly negaive and he oher o U mm being sufficienly negaive. Firs, if U cm is sufficienly negaive so ha (1+mU cm /U c ) < 0, hen we have non-oscillaory sunspo equilibria (0 < g (m ss ) < 1) for all values of η. These are akin o hose discussed in Masuyama (1990). 10 Second, even if (1+mU cm /U c ) >0, here are sunspo equilibria if U mm is sufficienly negaive. Recalling ha η varies inversely wih U mm, his corresponds o an η ha is sufficienly small. These equilibria are oscillaory because wih an exremely small money demand elasiciy a given movemen in real balances requires an exremely large movemen in he nominal rae and hence he real rae. Such a large real rae movemen requires real balances beween and +1 o be sufficienly differen (m < m ss < m +1 ), ha is for real balances o be oscillaory. In paricular, if η saisfies η i 1 < 2(1 + i) 1+ mu cm / U, (9) c hen -1 < g (m ss ) < 0 and we have oscillaory sunspo equilibria. As η increases and g (m ss ) falls below -1, he Hopf-Bifurcaion heorem implies ha he wo-period cycles associaed wih an Eigenvalue of 1 evenually increase unil he cycles become infinie, 10 Masuyama (1990) assumes CWID iming, which implies i /(1 + i) g ( m ) = 1 +. Masuyama hus concludes ha (1+mU ss cm /U c ) < 0 is necessary for η(1 + mu cm / U c ) real indeerminacy bu no sufficien. In he case of CIA iming, his negaiviy condiion is sufficien, bu 12

16 ha is chaoic dynamics emerge. 11 In hese cases sunspo equilibria are also possible alhough he economy is no locally sable. If η becomes large enough his possibiliy disappears. For example, given (1+mU cm /U c ) >0, a sufficien condiion for deerminacy is i 1 η >, (10) 1+ i 1+ mu cm / U c so ha g (m ss ) > 1. For he remainder of he paper our focus will be on local analysis, bu he reader may noe ha here is a small range beween condiions (9) and (10) in which he local deerminacy condiions are no sufficien for global deerminacy. In summary, here exis saionary sunspo equilibria if and only if (i) U cm is sufficienly negaive, or (ii) U mm is sufficienly negaive. In he nex secion we will demonsrae ha he former equilibria disappear in a model wih explici producion as he opimizaion condiions for producion consrain he behavior of U c. As for he laer, hese equilibria arise only under implausibly low ineres elasiciies. The following example will provide a precise bound. An Example: Suppose preferences are given by 1 1 σ 1 σ 1 ρ 1 ρ [ c + Am ] ρ = 1 U. In his case here is a uni consumpion elasiciy, and η = 1/ρ is he ineres elasiciy. The sign of U cm is given by he sign of ρ-σ. As ρ goes o infiniy he uiliy funcion no necessary. 11 See Fukuda (1993), Masuyama (1991), and Michener and Ravikumar (1998). 13

17 becomes Leonief and he model collapses o a rigid cash-in-advance consrain. Sraighforward calculaions imply: g ( m ss ) = 1 R iρ( v + i) ( v + i(1 + ρ σ )) 1 where v = c/m denoes seady-sae velociy. Henceforh we will ypically assume v > 1. (However, as ρ goes o infiniy, v will converge o one.) There are wo cases: (1) If v + i( 1+ ρ σ ) < 0, hen 0 g '( m ) < 1, and we have non-oscillaory sunspo < ss equilibria. Since velociy is large relaive o he nominal rae, we need σ quie large for hese sunspos o arise, ie., U cm mus be sufficienly negaive. (2) If v + i( 1+ ρ σ ) > 0, hen we can have oscillaory sunspo equilibria ( 1 < g '( m ) < 0 ) if and only if v-1-r > 0 and if ρ is sufficienly large (η is ss sufficienly small): [ + i(1 )] [ v 1 R] 2R v σ ρ >. i This region is quie small. For σ =10, v = 3, R = 1.02, his region is ρ > 293, or η <.003. This is an implausibly low ineres elasiciy. A sufficien condiion for global deerminacy (and hence o rule ou chaoic equilibria) is ρ < 147, or η >.006 (see (10) vs. (9)). Noice ha for a rigid cash-in-advance consrain, ρ goes o infiniy, bu v goes o one, so ha he requiremen v > 1+R is no saisfied, and here is deerminacy. Similarly, as we shrink he ime period beween visis o he bank, hen v declines unil he condiion v > 1+R is no saisfied so ha he oscillaory sunspos disappear. III. A Producion Economy. 14

18 In his secion we will add a sandard producion echnology o he analysis. Assume ha preferences are separable and linear in labor (L) and given by 1+ γ L U ( c, m,1 L) V ( c, m) B, 1+ γ and ha producion akes he sandard Cobb-Douglas form: α 1 α y = K L wih a consan depreciaion rae of δ. We will consider more general preferences over labor below. are familiar: The addiional Euler equaions for labor choice (12) and capial accumulaion (13) U L ( ) U ( ) c = f ( ) (12) L U c ( ) = βu c ( + 1)[ f K ( + 1) + (1 δ )]. (13) c. (14) α 1 α = K L + ( 1 δ) K K + 1 Real money balances indirecly ener boh of hese marginal condiions via he cross parials (U cm ) of he uiliy funcion. As a resul he behavior of real balances ypically alers he economy s behavior relaive o an oherwise sandard real business cycle (RBC) model. For presen purposes, a criical issue is ha (12)-(13) place resricions on he behavior of U c. This is paricularly clear in he case of linear leisure (γ=0). Le x = (L /K ) denoe he labor-capial raio. Exploiing he lineariy in leisure preferences, we can use (12) o rewrie (13) as: x. α α = αβx+ 1 + β( 1 δ) x+ 1 15

19 Since (12) implies ha U c depends only on x, hen real balances, m, depend only on c and x so ha we can rewrie (14) as 1 α K+ = K x + (1 δ ) K c( x, m ) 1 Collecing hese resuls, we can express (3) and (13)-(14) as he following linearized funcions: x = q ( x ) m = q ( x, m ) K + 1 = q ( x, m, K ). I is immediaely obvious ha we have a block-recursive sysem, wih eigenvalues given by he diagonal elemens. For deerminacy we need wo explosive roos. The firs and hird eigenvalues are given by α e = β (1 α)(1 δ ) <, 1 β (1 α)(1 δ ) e 3 = > 1, αβ so ha we have deerminacy if and only if 2 q m is ouside he uni circle. Equivalenly, we need o evaluae he slope of he g-funcion given by (3), holding x, ha is U c (), consan. This resricion imposes a relaionship beween c and m, c = c(m), wih dc/dm = -Ucm/Ucc. Imposing his resricion on he analysis of he previous secion we have: 1 g ( m ) =. ss m 2 1+ [ U ] mmu cc U mc (1 + i) U cc 16

20 2 Concaviy implies [ U U ] U mm cc mc > 0 and cc U < 0. Hence, he only possible equilibria are oscillaory. The non-oscillaory equilibria have disappeared because he implici condiions on U c make i impossible for (1+mU cm /U c ) < This suggess ha we have sunspos only if η is sufficienly small. This is he case, as an example will demonsrae: An Example: Suppose preferences are given by U 1 1 σ 1 σ 1 ρ 1 ρ [ c + Am ] 1 ρ BL =. Recall ha he consumpion elasiciy is uniy and ha η = 1/ρ is he ineres elasiciy. We hen have: 1 iρσ ( i + v) g ( mss ) = 1 ( ). R σv + iρ Noice ha he only ype of real indeerminacy possible is of he oscillaory ype. As a special case le ρ go o infiniy, ha is, η goes o zero so ha he uiliy funcion becomes Leonief. Wih v=1 he model collapses o a rigid CIA consrain. In his case 1 g ( mss ) = so ha he model is indeerminae if and only if σ > 2. This is 1 σ exacly he resul derived by Farmer (1999) in his ex and exended by Carlsrom and Fuers (1999) o a model wih capial. The resul is also implici in Woodford (1994) in he Lucas-Sokey model. These resuls suggesed ha money growh rules were likely 12 In he case of CWID iming we have (1 + i) m 2 g ( m ) = 1 [ U U U ] ss mm cc mc U cc which always exceeds one so ha we never have real indeerminacy. 17

21 prone o sunspos. Bu his resul does no hold up under more reasonable calibraions of money demand elasiciy. In general we have local indeerminacy if and only if σ(i+v) > 2R and if ρ is sufficienly large: 2Rσv ρ >. i σ [ ( i + v) 2R] Noice ha as in he case of a sric CIA consrain indeerminacy becomes more likely he larger is σ. Accordingly we choose σ =10 he upper end of plausible esimaes o demonsrae he implausibiliy of sunspos. Calibraing o quarerly daa we choose R=1.02 (8% annualized). Given hese choices he larger is v he easier i is o ge sunspos. Therefore we inerpre money o mean he moneary base so ha quarerly velociy is 3. Given hese choices he indeerminacy region is ρ > 108, so ha here are sunspos only if η <.009! This is an implausibly low money demand elasiciy. 13 As before a sufficien condiion for global deerminacy is ρ < 54 or η >.018. Noice ha we calibraed according o quarerly daa. Mos would conend ha if money is being held o faciliae ransacions ha he model should be calibraed o an even higher frequency. Calibraing o a higher frequency, however, makes indeerminacy even less likely since v and i decline so ha he condiion σ(i+v) > 2R no longer is saisfied. Moving away from linear leisure (γ > 0) has no quaniaive effec on our resuls. Remarkably, even wih an exremely small labor supply elasiciy (eg., γ = 100) he cuoff for local indeerminacy is unchanged o he firs decimal poin. We conclude ha 18

22 for all reasonable calibraions here is real deerminacy under an exogenous money growh process. IV. Conclusion. One of he firs papers o inegrae money ino a real business cycle model is Cooley and Hansen (1989). Tha paper assumed log preferences over consumpion, linear preferences over leisure, imposed a sric cash-in-advance consrain, and assumed an exogenous money growh rae. The model as wrien is deerminae. However, if he risk aversion coefficien is greaer han 2 (an enirely reasonable assumpion), hen he real economy is indeerminae. This has led Woodford (1994) and ohers o argue for he inheren insabiliy of money growh rules. A surprising conribuion of his paper is ha even hough esimaed money demand elasiciies are fairly small, he absolue zero elasiciy inheren in he cash-inadvance consrain is criical for he exisence of saionary sunspos in he Cooley- Hansen model. For all plausible money demand elasiciies and risk aversion coefficiens wihin he reasonable range, he Cooley-Hansen model is deerminae. For self-fulfilling hyperinflaionary equilibria o be raional, however, we show ha he ineres elasiciy of money demand mus be quie large exceeding uniy. This conrass wih he resuls of Obsfeld and Rogoff who use CWID iming. The paper s analysis was conduced wihin he conex of a generic aggregaive MIUF model, an environmen ha is incredibly general, encompassing ransacions cos models, shopping ime models, rigid cash-in-advance models, and cash-credi models. 13 This is low even for shor-run elasiciies. The relevan elasiciy for sabiliy analysis, however, is he 19

23 Hence, i is hard o imagine a plausibly calibraed moneary environmen in which money growh rules are prone o saionary sunspos. In conras, saionary sunspos are endemic under mos ineres rae operaing procedures. This resul provides some heoreical suppor for hose who favor money growh argeing. long-run elasiciy which is subsanially greaer. 20

24 References Azariadis, Cosas, Self-fulfilling prophecies, Journal of Economic Theory (25), 1981, Benhabib, Jess, Sephanie Schmi-Grohe, and Marin Uribe, Moneary Policy and Muliple Equlibria, 2000 working paper. Cagan, Phillip, The Moneary Dynamics of Hyperinflaion, in M. Friedman (ed.), Sudies in he Quaniy Theory of Money, 1956: Universiy of Chicago Press. Carlsrom, Charles T, and Timohy S. Fuers, Timing and Real Indeerminacy in Moneary Models, March 2000, forhcoming, Journal of Moneary Economics. Carlsrom, Charles T, and Timohy S. Fuers, Forward vs. Backward Looking Taylor Rules, Federal Reserve Bank of Cleveland working paper, Carlsrom, Charles T, and Timohy S. Fuers, Real Indeerminacy in Moneary Models wih Nominal Ineres Rae Disorions, Federal Reserve Bank of Cleveland working paper, Cooley, Thomas F., and Gary D. Hansen, The Inflaion Tax in a Real Business Cycle Model, American Economic Review 79(4), Sepember 1989, Press. Farmer, Roger E.A., Macroeconomics of Self-Fulfilling Prophecies, 1999: MIT Fukuda, Shin-ichi, The emergence of equilibrium cycles in a moneary economy wih a separable uiliy funcion, Journal of Moneary Economics 32(2), November 1993, Lucas, Rober E., Jr., and Nancy L. Sokey, Opimal fiscal and moneary policy in an economy wihou capial, Journal of Moneary Economics 12(1), July 1983, Lucas, Rober E., Jr., and Nancy L. Sokey, Money and Ineres in a Cash-in- Advance Economy, Economerica 55(3), May 1987, Masuyama, Kiminori, Sunspo equilibria (raional bubbles) in a model of Money-in-he-uiliy funcion, Journal of Moneary Economics 25 (1), January 1990, Masuyama, Kiminori, Endogenous Price Flucuaions in an Opimizing Model of a Moneary Economy, Economerica 59(6), November 1991, Michener, Ronald, and B. Ravikumar, Chaoic dynamics in a cash-in-advance economy, Journal of Economic Dynamics and Conrol 22, 1998,

25 Obsfeld, M. and K. Rogoff, Speculaive Hyperinflaions in Maximizing Models: Can we Rule hem ou? Journal of Poliical Economy 91, 1983, Obsfeld, M. and K. Rogoff, Ruling ou Divergen Speculaive Bubbles, Journal of Moneary Economics (17), 1986, Woodford, M., Moneary Policy and Price Level Deerminacy in a Cash-in- Advance Economy, Economic Theory 4, 1994,

26 Federal Reserve Bank of Cleveland Research Deparmen P.O. Box 6387 Cleveland, OH BULK RATE U.S. Posage Paid Cleveland, OH Permi No. 385 Address Correcion Requesed: Please send correced mailing label o he Federal Reserve Bank of Cleveland, Research Deparmen, P.O. Box 6387, Cleveland, OH 44101