working R Timing and Real Indeterminacy in Monetary Models by Charles T. Carlstrom and Timothy S. Fuerst FEDERAL RESERVE BANK OF CLEVELAND

Transcription

1 working p a p e r 9 9 0R Timing and Real Indeerminacy in Moneary Models by Charles T. Carlsrom and Timohy S. Fuers FEDERAL RESERVE BANK OF CLEVELAND

2 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 sie on he World Wide Web:

3 Working Paper 990R Ocober 200* Timing and Real Indeerminacy in Moneary Models by Charles T. Carlsrom and Timohy S. Fuers An increasingly common approach o he heoreical analysis of moneary policy is o ensure ha a proposed policy does no inroduce real indeerminacy and hus sunspo flucuaions ino he model economy. Policy is ypically conduced in erms of direcives for he nominal ineres rae. This paper uses a discree-ime money-in-he-uiliy funcion model o demonsrae how seemingly minor modificaions in he rading environmen resul in dramaic differences in he policy resricions needed o ensure real deerminacy. These differences arise because of he differing pricing equaions for he nominal ineres rae. JEL Codes: D5, E42, E52 Key Words: general equilibrium, money and ineres raes, moneary policy Charles T. Carlsrom is a he Federal Reserve Bank of Cleveland. Timohy S. Fuers is a Bowling Green Sae Universiy. The auhors hank Jess Behabib, Ben Eden, Bob King, an anonymous referee, and paricipans in he 999 NBER Summer Insiue for helpful commens. Charles T. Carlsrom may be reached a ccarlsrom@clev.frb.org or (26) *Originally published Sepember 999 Revised Ocober 200

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5 . Inroducion. An increasingly common approach o he heoreical analysis of moneary policy is o ensure ha a proposed policy does no inroduce real indeerminacy and hus sunspo flucuaions ino he model economy. Policy is ypically conduced in erms of direcives for he nominal ineres rae. For example, a simple Taylor (993) ype rule posis ha he cenral bank conducs policy according o he following rule: R = A. (π + ) τ for τ 0, where R and π + denoe he (gross) nominal ineres and inflaion rae (beween and +). Tha is, he cenral bank varies he nominal rae in relaion o movemens in inflaion wih an elasiciy of τ. In his conex, an imporan policy quesion is wha resricions on τ are needed o ensure real deerminacy. A sandard resul in he lieraure is ha a necessary condiion for real deerminacy is τ > (he sufficien condiion is for τ o exceed one, bu o no be oo large). See for example, Bernanke and Woodford (997) and Clarida, Gali, Gerler (998). These analyses are all reduced-form sicky price models, where he underlying srucural model is a money-in-he-uiliy funcion (MIUF) model wih a zero cross-parial beween consumpion and real balances. 2 In sharp conras, Carlsrom and Fuers (999a) analyze a flexible price, cash-in-advance (CIA) model and demonsrae ha a necessary and sufficien condiion for deerminacy is τ <. 3 The moivaion for his paper is o help This corresponds o argeing he expeced inflaion rae, a policy consisen wih he pracice of cenral banks wih explici inflaion arges. In conras, Taylor s (993) original rule has he cenral bank responding o curren and pas inflaion raes. As we demonsrae his difference is criical. 2 A more complee analysis of hese condiions is provided by Benhabib, Schmi-Grohe, and Uribe (998). Bu as argued below, heir coninuous ime analysis ignores he cenral issue of his paper. 3 Carlsrom and Fuers (999a) consider ineres rae rules in which he nominal rae responds o he real rae wih an elasiciy of γ, and find ha a necessary and sufficien condiion for deerminacy is γ < ½. There is a one-o-one mapping beween a policy rule in erms of he real rae and a policy rule in erms of he inflaion rae, τ = γ/(γ-), so ha γ < ½ implies τ <.

6 reconcile hese differing resuls. The resoluion does no come solely from he sicky price vs. flexible price assumpion. For example, if we dropped he sick price assumpion from he Clarida e al (998) environmen, hen any τ yields deerminacy. Surprisingly, he major difference comes from an even more fundamenal assumpion which money balances ener he uiliy funcion? The radiional MIUF assumpion and ha assumed by Clarida e al is ha end-of-period balances ener he funcional. Bu a direc exension of a ypical cash-in-advance economy suggess ha he money he household has lef afer leaving he bond marke and before enering he goods marke is more appropriae. Remarkably his disincion is criical for quesions of deerminacy. We analyze a MIUF endowmen economy under differing assumpions abou which money balances ener he uiliy funcion. 4 The firs iming convenion we analyze is a direc exension of ypical CIA iming. Tha is, he money available o saisfy consumpion needs is he money he household has lef afer leaving he bond marke bu before enering he goods marke. In conras, we also analyze cash-when-i m-done (CWID) iming where end-of-period money balances (ne of curren income and curren consumpion) ener he uiliy funcional. These differing assumpions lead o differen pricing equaions for he nominal ineres rae. In a model in which he cenral bank operaes moneary policy via he nominal ineres rae hese differences have imporan effecs on he condiions for deerminacy. We hen exend his analysis o a model wih producion. Surprisingly, a sandard 4 We uilize a MIUF model because of is generaliy. Feensra (986) demonsraes ha any ransacions cos (TC) economy can be wrien as a MIUF economy. Similarly, a shopping-ime (ST) model can be rewrien as a MIUF economy. Finally, cash-in-advance (CIA) models are exreme versions of MIUF and

7 producion model wih capial generaes he same condiions for deerminacy irrespecive of how money affecs uiliy. Tha is, he condiions for deerminacy are idenical o he endowmen economy model where consumpion and real cash-balances are separable. Thus for CIA iming he τ < rule of Carlsrom and Fuers (998) coninues o hold. Wih CWID iming, however, he model is deerminae for all τ. These resuls sugges ha as moneary heoriss we mus be more careful in wriing down he basics of our models. Relaedly, i suggess ha here are concerns wih coninuous-ime analyses ha simply sweep his fundamenal iming issue under he rug, i.e., in a coninuous ime model he ime inerval beween bond and goods marke ransacions collapses o zero. Since hese indeerminacy issues arise for any discree bu arbirarily small ime period his resoluion of he iming quesion is arificial. The paper proceeds as follows. Secion 2 lays ou he basic environmen. Secion 3 presens he deerminacy resuls for differen modeling assumpions. Secion 4 adds producion o he model. Secion 5 akes a brief look a a sicky-price model, and Secion 6 concludes. 2. A MIUF 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 key issue is wha measure of money appears in he uiliy funcion. We will urn o his TC economies. Thus TC, ST, and CIA models imply paricular funcional forms for he MIUF economy. 2

8 shorly. The household begins he period wih M cash balances and B - holdings of nominal bonds. Before proceeding o he goods marke, he household visis he financial marke where i carries ou bond rading and receives a cash ransfer of X from he moneary auhoriy. Hence, before enering goods rading, he household has cash balances given by M + X + B - R - B, where R - denoes he nominal ineres rae from - o. Afer engaging in goods rading, he household ends he period wih cash balances given by he ineremporal budge consrain. M + = M + X + B - R - - B - P c + P y, () We now urn o he cenral issue of he paper wha money balances aid in conemporaneous ransacions, i.e., wha is A? The exising lieraure conains wo prominen choices 5 : Model : CIA Timing A = M + X + B - R - B (2) Model 2: CWID Timing A = M + = M + X + B - R - - B - P c + P y (3) Model assumes ha wha maers for ime- ransacions is he money wih which one eners he ime- goods marke, i.e., cash held in advance of goods marke rading. This iming assumpion is always used in models wih sric CIA consrains (eg., Lucas (982), and Lucas and Sokey (987)), bu is ypically no used in MIUF 5 The working-paper version of his paper also considers a relaed CIA model in which ime- goods marke rading precedes ime- bond marke rading so ha A = M + X. As noed in Farmer s ex (993), i is quie easy o ge real indeerminacy in such a model. This iming seems arificial since agens would prefer 3

9 environmens. In conras, he radiional MIUF approach is o assume Model 2 iming, i.e., ha end-of-period balances, A = M +, ener ino he uiliy funcion. 6 I is very difficul o jusify CWID iming on heoreical grounds. Using end-of-period money implies ha money a he beginning of + reduces ransacions coss in period. Equivalenly, Model 2 implies ha wha maers for ransacions purposes is he money you leave he goods marke wih, ne of curren consumpion and curren income. Tha is, wha aids in curren ransacions is he money I leave he supermarke wih no he money I enered he marke wih. Wih CWID iming curren income is included as par of curren money balances. This violaes Clower s (967) dicum ha money buys goods, and goods buy money, bu goods do no buy goods. One can imagine rading environmens in which his violaion is possible. Bu i is very difficul o defend he subracion of curren consumpion from curren money balances. To see hese differences more sarkly, consider a Leonief MIUF specificaion so ha opimal behavior is given by he consrain P c = A. In he case of Model iming, he subsiuion of A yields he sandard CIA consrain, P c = M + X, where we have dropped he bond rading for simpliciy. Bu in he case of CWID iming his subsiuion yields P c = M + X + P y P c. This is a peculiar ransacions consrain. 7 o have he bond marke precede he goods marke. 6 In fac, Model 2 is ypically used in all moneary models (MIUF, TC, ST) excep CIA models. 7 This consrain is so peculiar ha i is naural o ask why he profession ever made Model 2 is choice for MIUF, TC, and ST models. One explanaion suggesed by Ben Eden is ha when Painkin (965) and ohers firs wroe down MIUF models, for mahemaical simpliciy he profession used a finie horizon model as a proxy for an infinie-horizon model. In order for money o be held in he final period, he 4

10 We now presen he differen bond-pricing and money demand and equaions ha arise from hese wo iming convenions. 8 Model : CIA Timing (A = M + X + B - R - B ). [U m ()+U c ()]/P = R β [U m (+)+U c (+)]/P + (4) U m ()/U c () = (R -) (5) Model 2: CWID Timing (A = M + X + B - R - - B - P c + P y ). U c ()/P = R βu c (+)/P + (6) U m ()/U c () = (R -)/R (7) Noe he fundamenal differences beween he Fisher equaions in Models and 2 (equaions (4) and (6)). In Model, if he household liquidaes a bond for cash wih he inenion of increasing curren consumpion his bond rade increases he uiliy from curren consumpion U c () and curren liquidiy U m (). In conras, in Model 2, a bond sale ha is used for consumpion purchases has no effec on curren liquidiy so ha only U c () eners he bond equaion. In effec, Model 2 assumes ha a marginal bond sale can direcly finance an incremen o consumpion. This seems o violae he essenial naure of a moneary economy. In Model 2, herefore, we ge he sandard Fisherian decomposiion ha resuls modeler had o assume ha he household had preferences over end-of-he-world money. Insead of simply acking on an addiional uiliy funcional, a more symmeric choice was o simply slide he enire sequencing backwards so ha ime T+ money was in he ime T uiliy funcional. 8 Equaion 4-5 (6-7) can be derived from subsiuing 2 (3) ino uiliy and maximizing subjec o () wih respec o c, B, and M +. The Lagrangian associaed wih () was hen subsiued ou. 5

11 from an enirely real model. Namely he real rae is given by he raio of he marginal uiliies of consumpion. Bu wih CIA iming, he fac ha his is a moneary economy is paramoun. The real rae of ineres now depends on he marginal uiliies of consumpion and real cash balances, i.e., he uiliy of $ is more han is abiliy o purchase consumpion bu also is liquidiy value. These differences in real rae deerminaion imply ha changes in real cash balances can direcly influence he real rae in Model, while hey do so only indirecly (via he cross parial U cm ) in Model 2. Noice ha in Model we can use he money demand relaionship (5) o rewrie he Fisher equaion (4) as U c ()/P = R + βu c (+)/P +. (8) Hence, one manifesaion of our CIA iming is ha one can use he radiional Fisher expression (equaion 6) bu wih he nominal rae scrolled forward one period. 9 This difference has imporan effecs on issues of equilibrium deerminacy. The remainder of he paper illusraes hese differences. Our focus will be on ineres rae rules Real Indeerminacy in an Endowmen Economy wih U cm = 0. Suppose ha he cenral bank conducs policy according o he following rule: R = A(π + ) τ, (9) where τ 0, A R π τ ss ss, and R ss and π ss are he seady-sae values. Tha is, he cenral bank varies he nominal rae in relaion o movemens in expeced inflaion wih an elasiciy of τ. Under such a policy we ge he sandard resul ha here is nohing o pin 9 In a model wih a sric CIA consrain he Fisher equaion is of he form given by (6). This arises because he implici Leonief ransacions echnology implies ha U m = 0 in equilibrium, ie., if m = c, addiional real balances have no effec on he abiliy o ransac, while if m < c, i is impossible o ransac. 0 These iming issues also have a dramaic effec on he condiions for real indeerminacy when cenral 6

12 down he iniial π P / P -. This is a pure nominal or price level indeerminacy which in his flexible price economy has no effec on real behavior. Insead our focus is on real indeerminacy is he pah for expeced inflaion, he nominal ineres rae, and hus real cash balances pinned down by he ineres rae policy? If no, hen we will conclude ha, 2 he economy suffers from real indeerminacy and hus sunspo flucuaions. How could real indeerminacy arise? Consider he following scenario. Suppose ha here is an increase in expeced inflaion of %. The cenral bank responds by increasing he nominal rae by τ% hus producing a decline in real cash balances (because of he higher nominal rae) and a change in he real ineres rae of (τ )%. This expeced inflaion change, herefore, can only be self-fulfilling if he movemen in real cash balances produces a movemen in he real rae of (τ-)%. Since real ineres rae deerminaion is differen in he wo models, wheher or no his is possible will depend crucially upon he iming assumpion. CWID Timing: Suppose ha uiliy is separable and we normalize U c () in his endowmen economy o uniy. In Model 2, his implies ha he real rae of ineres is consan so ha real indeerminacy is no possible (excep for τ = ). To be specific, subsiuing he ineres rae rule ino he Fisher equaion for CWID iming implies he following: β Aπ π τ = + +. (0) banks uilize money growh rules. Wih an endowmen economy and separable preferences his indeerminacy has no effec on consumpion bu does affec uiliy (hrough real cash balances), so we classify his as real indeerminacy. 2 As is well known, in some moneary models here are also he possibiliy of equilibria wih self-fulfilling hyperinflaions in which real balances go o zero in he limi. We ignore hese non-saionary equilibria as hey are easily eliminaed if he cenral bank sands ready o redeem fia money a an arbirarily small bu nonzero real value. Our focus is on saionary sunspo equilibria. 7

13 Noice ha while π is free (nominal indeerminacy) fuure inflaion and nominal ineres raes are nailed down (when τ ). (Since βa = π τ ss, π +j = π ss for all j.) Hence, wih Model 2 and separable preferences here is never real indeerminacy excep for he borderline case (τ = ). 3 The key is ha oday s real ineres raes are no affeced by an increase in expeced inflaion. The exreme borderline case does produce indeerminacy since nominal ineres raes and expeced inflaion move one-for-one so ha real ineres raes in his case are no affeced by changes in expeced inflaion. CIA Timing: The sory is quie differen in Model. Even wih separable uiliy (so ha U c () is consan) he real rae is sill variable since from (4) he marginal uiliy of money (U m ()) affecs he real rae of ineres. This feedback o he real rae of ineres creaes he possibiliy of sunspos. Turning o he deails, subsiuing he assumed moneary reacion funcion (9) ino (8) for CIA iming yields he following: β Aπ π τ + 2 = +. () Once again we always have nominal indeerminacy since π is free. For real deerminacy, we need π + and hence R o be pinned down. This arises if and only if he above mapping is explosive or if τ <. In conras, τ > generaes real indeerminacy since all choices of π + converge o π ss. The inuiion is as follows: Suppose here is a sunspo-driven increase in expeced inflaion of %. Given he policy rule (9) his increases he nominal rae by τ% and he real rae of ineres by (τ )%. The increase in he nominal rae leads o a decline in real 3 Real indeerminacy could arise if R depended on π +k for k 2. 8

14 cash balances, which in urn increases he real rae (since U is concave in m). When τ > he iniial sunspo-driven increase in expeced inflaion is equivalen o an increase in he real rae so ha he circle is complee and here is real indeerminacy. The above analysis made a very special, and implausible, assumpion. Namely ha uiliy is separable beween money and consumpion. The nex secion generalizes he above analysis by adding producion o he above endowmen economy. Remarkably his general model generaes he exac same condiions for indeerminacy as derived above Real Indeerminacy in a Producion Economy wih U cm 0. Carlsrom and Fuers (999a) examine a sandard real business cycle model wih producion in which money is added wih a CIA consrain. They demonsrae ha indeerminacy arises if and only if τ >, he same condiion derived above for CIA iming in an endowmen economy and separable preferences. 5 This secion illusraes ha he naure of his resul does no depend on a sric CIA consrain bu merely on CIA-iming. The key is ha wih some fairly sandard assumpions a model wih producion will generae he same condiions for indeerminacy as a model where U cm = 0. This equivalence will be rue for boh he CWID- and CIA-iming MIUF models. Assume ha preferences are separable and linear in labor (L) and given by U ( c, m, L) V ( c, m) AL, and ha producion akes he sandard Cobb-Douglas form: y = K L wih a consan depreciaion rae of δ. 4 The working paper version of his paper derives he condiions for indeerminacy in he endowmen economy when uiliy is no separable beween real money and consumpion. 5 This resul is exacly rue for linear leisure. For more general preferences he numerical differences are rivial. 9

15 The addiional Euler equaions for labor choice (2) and capial accumulaion (3) are familiar: U L ( ) U ( ) c = f ( ) (2) L U c ( ) = βu c ( + )[ f K ( + ) + ( δ )]. (3) c. (4) = K L + ( δ) K K + These Euler equaions are common across he wo models because consumpion and oupu ener symmerically in boh models. 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 he nominal ineres rae (and hence real balances) ypically disors he economy s behavior relaive o an oherwise sandard real business cycle (RBC) model. We now sae our principle resul: Proposiion : Assume ha preferences are separable and linear in leisure, U(c,m,-L) = V(c,m) AL, and ha he producion echnology is CRS and Cobb-Douglas. Then wih CIA iming a necessary and sufficien condiion for deerminacy is τ < ; wih CWID iming, he equilibrium is deerminae for all values of τ. Proof: See he Appendix. Alhough he proof of he proposiion explois he lineariy in labor preferences, his assumpion is heoreically convenien bu compuaionally irrelevan. For example, if insead of linear leisure here was a consan labor supply elasiciy of 0. hen wih plausible calibraions he bounds for deerminacy are largely unchanged in boh models. The model wih CIA iming is deerminae if and only if τ <.004, while a search for a τ 0

16 ha produces indeerminacy in he model wih CWID iming proved fuile. The assumpion ha leisure is separable in uiliy also proved o have no quaniaive imporance. I is imporan o noe ha hese condiions for deerminacy are idenical o an endowmen economy wih U cm = 0. The addiional resricions on U c implied by endogenous labor choice and capial accumulaion cause he model o behave as if U cm = 0. For example, consider a model wihou capial and wih consan reurns o labor ( = 0). In his case linear leisure and (2) implies ha U c is consan! Wih capial and CRS Cobb-Douglas echnology his basic logic carries hrough. The key is ha (given he above assumpions) we can rewrie (2) and hen subsiue (2) ino (3) o obain. x U c ( ) =. (5) ( ) x = = β x+ + β ( δ ) x+, where x. (6) K L The equilibrium marginal uiliy of consumpion is no direcly affeced by real money because i is enirely deermined by he capial-labor raio. This implies ha he marginal produc of capial is unaffeced by changes in real money balances. Lineariy implies ha if U cm >0 an increase in real money balances increases he marginal uiliy of consumpion which elicis more producion unil he marginal uiliy of consumpion reurns o where i was before he increase in real money balances. 5. Timing and Sicky Prices. This paper s analysis has been conduced in he conex of flexible price models.

17 However, hese iming concerns can arise in sicky price models as well. One example will illusrae his poin. Clarida, Gali, and Gerler (998) analyze a sicky-price moneary model and conclude ha a necessary condiion for real deerminacy is τ > he exac opposie condiion for deerminacy in a flexible price economy wih CIA iming! Assuming consan poenial oupu, heir log-linearized sysem of equaions is ~ y ~ σ ~ ~ (7) = [ R π + ] + y+ ~ π ~ ~ = λy + βπ + (8) ~ R τ ~ (9) = π + where ildes denoe log-deviaions from seady-sae, λ is he slope of he Phillips curve, and σ is he ineremporal elasiiciy of subsiuion. Equaion (7) is he Fisher equaion wih CWID iming and U cm = 0 (wih no invesmen consumpion equals oupu); equaion (8) is he Phillips curve, and equaion (9) is he forward-looking ineres rae rule. Unlike he flexible-price model, he Phillips curve implies ha for here o be real deerminacy π mus be pinned down. Sraighforward calculaions imply ha here is real deerminacy if and only if 2( β + ) + σλ < τ <. σλ For reasonable calibraions, he upper bound is quie high, abou 4, so ha he basic conclusion is ha a τ greaer han uniy will achieve deerminacy. This is he Clarida, Gali, and Gerler (998) resul. Wih CIA iming, however, equaion (8) implies ha we mus replace (7) wih 2

18 ~ y ~ σ ~ ~. (20) = [ R+ π + ] + y+ To analyze he condiions for deerminacy subsiue (8)-(9) ino (20) o eliminae oupu and he nominal rae: ( = σλτ β ) π ( + σλ + β ) π + π 0. (2) For real deerminacy we need boh roos of his equaion o lie ouside he uni circle. 6 A sufficien (bu no a necessary) condiion for all forward-looking rules o be indeerminae is for σλ+β. Esimaes of λ are exremely difficul o come by. According o Clarida, Gali, Gerler, esimaes for λ range from 0.05 o.22. Assuming ha β=0.99 hen even if we ake he lowes plausible esimaes for λ he model will always be indeerminae unless σ < 0.2. Recall ha he essenial difference beween CWID-iming and CIA-iming is ha he nominal ineres rae in he laer is scrolled forward one period. This implies ha he condiions for deerminacy wih CIA iming and a curren-looking ineres rae rule are idenical o hose for CWID iming and a forward-looking ineres rae rule. Thus if we used CIA iming in Kerr and King s (996) model, here would be deerminacy for τ > (unless τ is oo large) since heir policy rule depends on curren inflaion. Carlsrom and Fuers (999b), however, demonsrae ha if we add invesmen o his model hen an aggressive backward-looking Taylor rule is necessary for real deerminacy. 6. Conclusion. Hippocraes advised he docor o do no harm. This minimal advice is equally imporan o he cenral banker. In paricular, a necessary condiion for good moneary 6 Because prices are sicky boh π and π + mus be pinned down here is no pure nominal indeerminacy in a sicky price model. 3

19 policy is ha i does no inroduce sunspo flucuaions ino he real economy. This paper has demonsraed ha he class of policies ha are good in his regard depends on basic assumpions abou he modeling environmen. Hence, a cenral conclusion of his analysis is ha we need o hink much more carefully abou basic modeling assumpions when wriing down moneary models. A lo depends on apparenly rivial assumpions. These iming issues are of more han academic ineres. Cenral banks ha have adoped explici inflaion argeing all use inflaion forecass as an inegral par of heir decision-making. This corresponds o he forward-looking Taylor rule analyzed here. In a model wih sicky prices and CWID iming, here is no problem as long as τ >. In conras, wih CIA iming, such a policy is poenially disasrous since i inroduces real indeerminacy and sunspo flucuaions ino he real economy. This suggess ha cenral banks should use eiher curren or backward-looking Taylor rules. 4

20 Appendix Proposiion : Assume ha preferences are separable and linear in leisure, U(c,m,-L) = V(c,m) AL, and ha he producion echnology is CRS and Cobb-Douglas. Then wih CIA iming a necessary and sufficien condiion for deerminacy is τ < ; wih CWID iming, he equilibrium is deerminae for all values of τ. Proof: The proof for boh models begins wih he following condiions (5-6) from he ex: x U = L c ( ) =, where x (A) ( ) K x = βx + + β( δ) x+ (A2) Noice ha (A) implies ha U c depends only on x, so ha real balances, m, depend only on c and x. CIA iming: Defining z = U ( ) U ( ), and using he fac ha m depends only on c and c + x, he budge consrain can be wrien as K + = K x + ( δ) K c( x, z ) (A3) Subsiuing he money demand equaion (5) ino (A) m z R = x, where x ( ) = L K (A4) Subsiuing he moneary policy rule (9) ino he Fisher equaion (4) yields R = R ss z z + τ τ Subsiuing his ino (A4) gives 5

21 τ τ x τ R ss z z+ =. (A5) Equaions (A2), (A3) and (A5) can be wrien as x + = F(x ) K + = G(x,z,K ) z + = H(x,z ) The hree Eigenvalues of he characerisic marix are = = β( )( δ) < β( )( δ) e F x, e 2 = G K = >, β e 3 = H z =. τ F x and G k are he wo Eigenvalues for he canonical real business cycle model. Since here is only one predeermined variable, for he economy o be deerminae wo Eigenvalues need o lie ouside he uni circle. Therefore he economy is deerminae if and only if τ <. CWID iming: Assume τ. Subsiuing he labor equaion (A) and he moneary policy rule (9) ino he Fisher equaion (6) gives τ τ x R =. (A6) x+ Define z = U ( ) U ( ). From he money demand equaion (7) his yields c m U c z =. (A7) R (A7) herefore becomes z = x ( ) τ x + ( ) R ss τ τ 6

22 Subsiuing his ino (A3) gives x + = F(x ) K + = G(x,x +,K ) The Eigenvalues are = = β( )( δ) < β( )( δ) e F x, e2 = G K = >. β Since here is one predeermined variable he sysem is always deerminae. If τ =, he nominal rae drops ou of he Fisher equaion so ha he counerpar o (A7) is x + = x. Bu now R and hus z (from (A6)) are enirely free, so ha we have real indeerminacy. QED 7

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