Money and Output: Basc Facts and Flexble Prce Models Karel Mertens, Cornell Unversty Contents 1 Monetary Facts 3 1.1 Long run monetary facts............................. 4 1.2 Short run monetary facts............................ 8 2 Monetary Models 10 2.1 A Money-n-the-Utlty (MIU) Model...................... 10 2.1.1 A Basc MIU model........................... 10 2.1.2 An Extended MIU model wth Nonseparable Preferences and Elastc Labor Supply............................... 20 2.2 A Cash-n-Advance (CIA) Model........................ 24 2.3 A Shoppng Tme (ST) Model.......................... 29 1
References Char, V. V., Kehoe, P. J. and McGrattan, E. R. (2003), Stcky prce models of the busness cycle: Can the contract multpler solve the persstence problem?, Econometrca 68(5), 1151 1179. Cooley, T. and Hansen, G. (1989), The nflaton tax n a real busness cycle model, Amercan Economc Revew 79(4), 733 748. Fredman, M. and Schwartz, A. J. (1963), Monetary Hstory of the Unted States, 1867-1960, Prnceton Unversty Press. Kng, R. G., Plosser, C. I. and Rebelo, S. T. (1988), Producton, growth and busness cycles: I. the basc neoclasscal model, Journal of Monetary Economcs 21(2-3), 195 232. Kng, R. and Plosser, C. (1984), Money, credt, and prces n a real busness cycle, Amercan Economc Revew 74(3), 363 380. Kydland, F. E. and Prescott, E. C. (1990), Busness cycles: Real facts and a monetary myth, Quarterly Revew, Federal Reserve Bank of Mnneapols. McCandless, G. T. and Weber, W. (1995), Some monetary facts, Federal Reserve Bank of Mnneapols, Quarterly Revew. Stock, J. H. and Watson, M. W. (1999), Busness cycle fluctuatons n us macroeconomc tme seres, Handbook of Macroeconomcs 1. 2
Fgure 1: Defnton of Monetary Aggregates 1 Monetary Facts The prevous chapter presented models wth no role for money and no predctons for nomnal varables. In ths chapter we wll analyze models that ncorporate monetary factors that allow for the analyss of prce-level determnaton, nflaton and monetary polcy and have mplcatons for the behavor of nomnal varables n the short and long run. The broad queston that these models am to address s whether money matters. 1 Cooley and Hansen (1989) gve ths fundamental queston three dfferent nterpretatons: Do money and the form of the money supply rule affect the nature and ampltude of the busness cycle? How does antcpated nflaton affect the long-run values of macroeconomc varables? What are the welfare costs assocated wth dfferent money supply rules? Early busness cycle models that have addressed these questons have been heavly nspred by the monetarst tradton ntated by the emprcal and theoretcal work of Mlton Fredman and Anna Schwartz, whch attrbuted a great role to money for generatng cyclcal fluctuatons. In ths chapter we wll address these questons n varous flexble-prce dynamc stochastc general equlbrum (DSGE) models. 1 For the defntons of the most common monetary aggregates, see Fgure 5. 3
Here are some monetary facts, whch one should stress do not suggest any drecton of causaton whatsoever: 1.1 Long run monetary facts Growth rates of monetary aggregates and nflaton are extremely hghly and postvely correlated across countres and wthn countres n the long run, regardless of the defnton of money. In the long run, there s a one-to-one relatonshp between money growth and nflaton. Long-run average growth rates of monetary aggregates and real output are not correlated across most countres. However, ths fact s not entrely robust across subsamples of countres. McCandless and Weber (1995) fnd a postve relaton for OECD countres and a negatve relaton for Latn Amercan countres. McCandless and Weber (1995) fnd that nflaton rates are not correlated wth real output growth across countres. Other studes present some evdence for a slght negatve correlaton. 4
Fgure 2: (1995) Correlaton of money growth and nflaton, source: McCandless and Weber Table 1 Correlaton Coeffcents for Money Growth and Inflaton* Based on Data From 1960 to 1990 Coeffcent for Each Defnton of Money Sample M0 M1 M2 All 110 Countres.925.958.950 Subsamples 21 OECD Countres.894.940.958 14 Latn Amercan Countres.973.992.993 *Inflaton s defned as changes n a measure of consumer prces. Source of basc data: Internatonal Monetary Fund Table 2 Prevous Studes of the Relatonshp Between Money Growth and Inflaton Study Characterstcs Tme Seres Author Tme Data (and Year Publshed) Money Inflaton Countres Perod Frequency Fndng Vogel (1974) Currency + Consumer 16 Latn 1950 69 Annual Proportonate changes n nflaton Demand deposts prces Amercan rate wthn two years of changes n countres money growth Lucas (1980) M1 Consumer Unted States 1955 75 Annual Strong postve correlaton: prces Coeffcent closer to one the more flter stresses low frequences Dwyer and Hafer n.a. GDP 62 countres 1979 84 Fve-year Strong postve correlaton (1988) deflator averages Barro (1990) Hand-to-hand Consumer 83 countres 1950 87 Full-perod Strong postve assocaton currency prces averages Pakko (1994) Currency + Consumer 13 former 1992 and Four-quarter Postve relatonshp Bank deposts prces Sovet republcs 1993 averages Poole (1994) Broad money n.a. All countres n 1970 80 and Annual Strong postve correlaton World Bank tables 1980 91 averages Rolnck and Weber Varous Varous 9 countres Varous Long-perod Strong postve correlaton (1994) averages for fat money regmes n.a. = not avalable 5
Fgure 3: Correlaton of money growth and real output growth, source: McCandless and Weber (1995) Table 3 Correlaton Coeffcents for Money Growth and Real Output Growth* Based on Data From 1960 to 1990 Coeffcent for Each Defnton of Money Sample M0 M1 M2 All 110 Countres.027.050.014 Subsamples 21 OECD Countres.707.511.518 14 Latn Amercan Countres.171.239.243 *Real output growth s calculated by subtractng changes n a measure of consumer prces from changes n nomnal gross domestc product. Source of basc data: Internatonal Monetary Fund Table 4 Prevous Studes of the Relatonshp Between Money Growth and Real Outpu t Growth Study Characterstcs Tme Seres Author Tme Data (and Year Publshed) Money Output Countres Perod Frequency Fndng Kormend and M1 Real GDP 47 countres 1950 77 Perod Negatve correlaton Megure (1985) averages Geweke (1986) M2, NNP, ndustral Unted States 1870 1978, Annual, Money superneutral M1 producton Postwar perod monthly Dwyer and Hafer n.a. Real GDP 62 countres 1979 84 Fve-year Slght negatve correlaton (1988) and GNP averages (not statstcally sgnfcant) Porer (1991) M1 Real GDP 47 countres 1873 Annual Money neutral n some countres, not n others n.a. = not avalable 6
Fgure 4: (1995) Correlaton of nflaton and real output growth, source: McCandless and Weber Table 5 Correlaton Coeffcents for Inflaton and Real Output Growth* Based on Data From 1960 to 1990 Coeffcent Wth Outler** Sample Included Excluded All 110 Countres.243.101 Subsamples 21 OECD Countres.390.390 14 Latn Amercan Countres.342 *Inflaton s defned as changes n a measure of consumer prces. Real ou tput growth s calculated by subtractng those nflaton rates from changes n nomnal gross domestc product. **The outler s Ncaragua. Source of basc data: Internatonal Monetary Fund Table 6 Prevous Studes of the Relatonshp Between Inflaton and Real Output G rowth Study Characterstcs Tme Seres Author Number of Tme Data (and Year Publshed) Inflaton Output Countres Perod Frequency Fndng Fscher (1983) n.a. n.a. 53 1961 73, Annual Negatve contemporaneous 1973 81 relatonshp; postve correlaton wth one lag Kormend and Consumer Real GDP 47 1950 77 Perod Negatve correlaton Megure (1985) prces averages Fscher (1991) GDP deflator GDP 73 1970 85 Annual Negatve relatonshp Altg and Bryan (1993) GDP deflator Per capta 54 and 73 1960 88 Annual Negatve correlaton GDP Ercsson, Irons, GDP deflator GDP 102 1960 89 Annual Weak negatve correlaton and Tryon (1993) Barro (1995) Consumer Per capta 78, 89, 1965 90 Fve- or Negatve correlaton prces real GDP and 84 ten-year averages n.a. = not avalable 7
1.2 Short run monetary facts M0 (= the monetary base,.e. currency n crculaton + total reserves held by banks), M1 and M2 are all pretty volatle and procyclcal. M0, M1 and M2 veloctes are all volatle and procyclcal. Note the velocty of money V s V = P Y/M where P Y s nomnal GDP and P s the prce level. M0, M1 and M2 lead real output (see Fredman and Schwartz (1963)). Stock and Watson (1999) fnd the log level of nomnal M2 s procyclcal wth a lead of two quarters, and the nomnal monetary base s weakly procyclcal and also leadng. In contrast, the growth rates of nomnal M2 and the nomnal monetary base are countercyclcal and laggng. Kydland and Prescott (1990) dsagree that monetary aggregates are leadng ndcators (monetary myth). Stock and Watson (1999) fnd that the cyclcal component of the prce level, measured for nstance by the CPI, s countercyclcal and leads the cycle by approxmately two quarters. The cyclcal components of nflaton rates nstead are strongly procyclcal and lag the busness cycle. The nomnal wage ndex exhbts a pattern qute smlar to the CPI prce level. Real wages have essentally no contemporaneous comovement wth the busness cycle. Growth rates of monetary aggregates and nflaton are not correlated n the short run, except n epsodes of hypernflaton, durng whch the relatonshp s one-to-one. Short term nomnal nterest rates are procyclcal. 8
Fgure 5: source: Kydland and Prescott (1990) 9
2 Monetary Models The fundamental problem n monetary models s how to model the demand for fat money. Seeng fat money as an asset, t may assume the role as a store of value. However, money s almost always domnated n return by other assets. A second potental motve for holdng money s as a unt of account, but any other good can serve that purpose. A thrd, more plausble reason for holdng money s that t facltates economc transactons as a medum of exchange, for nstance by avodng the double concdence of wants. Although there s a szeable lterature n whch money fulflls one of the above roles n very well-specfed way, many macroeconomc busness cycle models take a more ad-hoc approach. Ths chapter covers three wdely used models: the Money-n-Utlty (MIU) model, the Cash-n-Advance (CIA) Model and the Shoppng Tme (ST) Model 2.1 A Money-n-the-Utlty (MIU) Model 2.1.1 A Basc MIU model Households represented by The economy s populated by a representatve household wth preferences U = E 0 β t u(c t, m t ), 0 < β < 1 (1) t=0 where C t > 0 s commodty consumpton n perod t, m t = M t s the real value of money holdngs, M t > 0 denotes nomnal money balances and s the nomnal prce level. Assume that u c > 0 and u m > 0 and that u(c, m) s strctly concave n both arguments, twce contnuously dfferentable and satsfes lm m 0 u m (C, m) =. Note that (1) mples that, holdng constant the path of real consumpton C t for all t, the ndvdual s utlty s ncreased by an ncrease n real money holdngs. Even though the money holdngs are never used to purchase consumpton, they yeld utlty. The household supples ts untary endowment of tme nelastcally n the labor market. The fnal good can be ether consumed or used for nvestment,.e. t can be added to the captal stock K t, whch evolves accordng to K t+1 = I t + (1 δ)k t, 0 < δ < 1 (2) where I t denotes gross nvestment. 10
The household s perod budget constrant s: C t + I t + B t + M t w t + r t K t + (1 + R t 1 ) B t 1 + M t 1 + T t or C t + I t + b t + m t w t + r t K t + 1 + R t 1 1 + π t b t 1 + m t 1 1 + π t + t t where b t = B t bond, R t s the nomnal nterest rate on the bond, t t > b denote real holdngs of a one perod uncontngent government ssued = Tt transfers by the government and π t s the rate of nflaton. 2 denote any real lump-sum Wthout loss of generalty, we abstract from modelng the market for frms shares (see prevous chapter). The assets avalable to the households are physcal captal, government bonds and money balances. The household s problem s to choose the real quanttes {C t, b t, m t, K t+1 } t=0 to maxmze (1) subject to the law of moton for captal, the budget constrants and takng as gven nflaton and nomnal nterest rates {π t, R t } t=0 as well as the real factor prces {w t, r t } t=0, the real transfers {t t } t=0 and the ntal captal stock K 0 and ntal real bond and money holdngs and nomnal nterest rate b 1, m 1, R 1. 3 Frms There s only one fnal good n the economy that s produced by frms accordng to a producton technology gven by Y t = A t (K d t ) 1 α (N d t ) α where K d t s the captal nput and N d t s labor nput, whch are rented n compettve markets at real prces w t and r t respectvely. Total factor productvty evolves accordng to the followng stochastc process, A t = Āeat a t = ρ a a t 1 + ϵ a t (3) where ϵ a t s a whte nose random varable wth standard devaton σ a ϵ and 0 < ρ a < 1 measures the shock persstence. The frm s problem n each perod t s to choose N t and 2 Note the constrant b t > b on bond holdngs, whch s a suffcent condton to exclude Ponz-scheme solutons to the households problem. For dscusson, see Wouter Denhaan s lecture notes. 3 We abstract from the labor lesure choce snce t s assumed that the household supples one unt of labor nelastcally. 11
K t to maxmze real profts Y t r t K d t w t N d t takng the factor prces as gven. Government The government s the monopoly suppler of money, whch t uses to fnance the lump-sum transfers and the net debt oblgatons to the household. The government budget constrant s therefore M s t M s t 1 + Bs t = T t + (1 + R t 1 ) Bs t 1 or m s t ms t 1 1 + π t + b s t 1 + R t 1 1 + π t b s t 1 = t t Accordng to ths government budget constrant, the transfers T t to the households and the nterest payments R t 1 Bt 1 s on outstandng government debt must be funded ether by addtonal borrowng Bs t Bs t 1 or by expandng the real money supply M t s M t 1 s ( prntng money, segnorage ). For now we wll assume for smplcty that B 1 = 0 and Bt s = 0 for all t, such that the government budget constrant reduces to t t = m s t ms t 1 1 + π t Assume that the growth rate of the money supply n devaton of the steady state growth rate, denoted by θ t = M t M t 1 µ 1, s exogenous and evolves accordng to θ t = ρ θ θ t 1 + ϵ θ t where µ > 0 s the average growth rate of the money supply, ϵ θ t s a whte nose random varable wth standard devaton σ θ ϵ and 0 < ρ θ < 1 measures the shock persstence. Equlbrum An equlbrum s defned as an nfnte sequence of allocatons of consumpton, captal, labor nputs and real bond and money holdngs and a system of prce sequences contanng the real factor prces, a nomnal nterest rate and nflaton such that for all t: The goods market clears: C t + I t = Y t The market for money clears: m s t = m t The bond market clears: b s t = b t = 0 12
The factor markets clear. Kt d = K t and N d = 1. The government satsfes ts budget constrant. and the households and frms solve ther respectve problems for every sequence of nnovatons to productvty and the money growth rate. Money Demand In equlbrum, the followng condtons must be satsfed n every perod t at an nteror soluton: [ u c (C t, m t ) + βe t uc (C t+1, m t+1 ) ( (1 α)a t+1 Kt+1 α + 1 δ)] = 0 (4a) [ u c (C t, m t ) + βe t u c (C t+1, m t+1 ) 1 + R ] t = 0 (4b) 1 + π t+1 ] 1 u m (C t, m t ) + βe t [u c (C t+1, m t+1 ) u c (C t, m t ) = 0 (4c) 1 + π t+1 together wth C t = A t Kt 1 α + (1 δ)k t K t+1 and the government budget constrant. Equaton (4a) s the famlar Euler equaton descrbng the optmal consumpton-nvestment choce, (4b) s the bond Euler equaton, (4c) s the money demand equaton. The transversalty condtons are lm T βt E t [u c (C T, m T )K T +1 ] = 0 lm T βt E t [u c (C T, m T )b T ] = 0 lm T βt E t [u c (C T, m T )m T ] = 0 Let s have a closer look at the money demand equaton: ] 1 u m (C t, m t ) = βe t [u c (C t+1, m t+1 ) + u c (C t, m t ) 1 + π t+1 [ ] u m (C t, m t ) u c (C t, m t ) = 1 βe uc (C t+1, m t+1 ) 1 t u c (C t, m t ) 1 + π t+1 u m (C t, m t ) u c (C t, m t ) = 1 1 by (4b) 1 + R t u m (C t, m t ) u c (C t, m t ) = R t (5) 1 + R t The term R t 1+R t consttutes the prce of money n the sense that t s the dollar opportunty cost of holdng an addtonal unt of money. Instead the household could purchase a bond and earn nterest tomorrow, the real present value of whch today s 13 R t 1+R t. Equaton (5)
mplctly defnes the money demand and taken together wth the exogenous money supply process may remnd you of the LM curve descrbng money market equlbrum n undergraduate macro. Smlarly, you can thnk of (4a), (4b) together wth the resource constrant as determnng a dynamc verson of the IS curve descrbng goods market equlbrum. The Determnstc Steady State Consder for a moment a dfferent verson of the model n whch there are no stochastc shocks. In a determnstc steady state β ( (1 α)ā K α + 1 δ ) = 1 (6) The steady state level of captal, consumpton, nvestment and output n the nonstochastc model are determned by equaton (6), the law of moton for captal Ī = δk, Ȳ = Ā K 1 α and the resource constrant C + Ī = Ȳ. They are ndependent of any utlty parameter other than β and do not depend on the rate of nflaton or the money growth rate. Next, note that to ensure that a steady state monetary equlbrum exsts n whch m > 0 s constant, there must exst a postve value of m that solves u m ( C, m) = ( 1 β ) u c ( 1 + µ C, m) Dependng on the nstantaneous utlty functon, there mght be no soluton, a unque soluton or even multple solutons. Consder for nstance the smpler case where utlty s addtvely separable, such that u(c, m) = u 1 (C) + u 2 (m) and u 2 m( m) = ( 1 β ) u 1 1 + µ c( C) > 0 A unque soluton where m > 0 exsts gven that our earler assumptons mply that lm m 0 u 2 m(m) =, u 2 mm < 0 and provded there exsts some m such that u 2 m( m) < ( ) 1 β 1+µ u 1 c( C). In order to analyze the dynamcs of the nomnal prce level around the 14
determnstc steady state, consder that β u 1 1 + π c( C) t+1 = u 1 c( C) u 2 m(m t ) β u 1 1 + π c( C)M t+1 t+1 = ( u 1 c( C) u 2 m(m t ) ) M t+1 β u 1 1 + π c( C) M t+1 +1 = ( u 1 t+1 +1 P c( C) u 2 m(m t ) ) (1 + µ) M t t β 1 + µ u1 c( C)m t+1 = ( u 1 c( C) u 2 m(m t ) ) m t m t+1 = 1 + µ ( 1 u2 m(m t ) β u 1 c( C) ) m t Φ(m t ) (7) Prce level determnacy for a gven exogenous path of the money supply M t depends on the propertes of the functon Φ. Frst notce that we can rule out any prce paths solvng equaton (7) that lead to m s gong to nfnty for s because of the transversalty condton. Ths rules out solutons where the prce level does not grow at least at the same rate as the money supply, or n other words, nflaton s at least the money growth rate. Suppose lm Φ(m) < 0 (or equvalently lm u 2 m 0 m 0 mm > 0), then gven all our earler assumptons, there can only exst one soluton, snce real money balances cannot be negatve. In ths case, the prce level s determned, and as a jump varable, always adjusts such that m = m > 0. However, f lm u 2 m 0 mm = 0 such that lm Φ(m) = 0, there exst prce m 0 paths for whch money balances are postve for all s and where m s 0 as s 0. These solutons, where nflaton exceeds the rate of money growth, are characterzed by speculatve hypernflatons. They dffer from regular hypernflatons because they are not caused by hgh growth rates of the money supply. The steady state prce level and nflaton rates are not determned n those cases. Unfortunately, the condton lm u 2 m 0 mm > 0 necessary to rule out these hypernflatonary solutons mples that lm u 2 (m) =. Money must be m 0 such an mportant good that utlty goes to mnus nfnty when real balances drop to zero! Nevertheless we wll assume that ths condton s satsfed, unless otherwse mentoned. Consder the followng defntons: Neutralty: A model has the property of neutralty when a once and for all change n the level of the money supply changes the prce level proportonally such that real money holdngs are constant. Superneutralty: A model has the property of superneutralty when a change n the growth rate of the money supply only affects real money balances but leaves all other 15
real varables unchanged. The MIU model wthout stochastc shocks dsplays long run superneutralty because the money growth rate µ does not affect the steady state level of captal, output and consumpton. Note however that except for very specfc utlty functons, there s nonsuperneutralty durng the transton to the steady state. The MIU model wthout stochastc shocks also dsplays monetary neutralty both n the long and short run. Monetary neutralty s a general property of flexble prce monetary models. In a later chapter, we wll see models wth nomnal rgdtes (stcky prce models) n whch money s not neutral. Dynamcs n the stochastc model Consder agan the stochastc model, n whch there are random nnovatons to productvty and monetary growth. We wll consder the followng parametrzaton of the utlty functon u(c t, m t ) = C1 σ t 1 1 σ + ϕ m1 χ t 1 1 χ where σ > 0, χ > 0, ϕ > 0. The frst order necessary condtons are C σ t [ ( + βe t C σ t+1 (1 α)at+1 Kt+1 α + 1 δ)] = 0 (8a) [ ] Ct σ + βe t Ct+1 σ 1 + R t = 0 (8b) 1 + π t+1 ϕm χ t R t Ct σ 1 + R t = 0 (8c) A t Kt 1 α + (1 δ)k t K t+1 = C t (8d) where M t M t 1 = 1 + µ + θ t (8e) θ t = ρ θ θ t 1 + ϵ θ t (8f) log(a t ) = (1 ρ a ) log(ā) + ρa log A t 1 + ϵ a t (8g) 16
The loglnearzed dynamcs around the determnstc steady state are descrbed by σĉ t = σe t ĉ t+1 + (1 β(1 δ)) E t (a t+1 αˆk t+1 ) σĉ t = σe t ĉ t+1 + ˆR t E tˆπ t+1 ( ) β χ ˆm t = σĉ t + ˆR t 1 + µ β s c ĉ t + s δ ˆk t+1 = a t + ( (1 α) + s 1 δ δ ˆm t = ˆm t 1 ˆπ t + θ t θ t = ρ θ θ t 1 + ϵ θ t a t = ρ a a t 1 + ϵ a t ) ˆk t (9a) (9b) (9c) (9d) (9e) (9f) (9g) Watch out: for the nomnal nterest and nflaton rate, we devate from our standard defnton of the hat varables and defne ˆR t = (R t R)/(1 + R) and ˆπ t = (π t π)/(1 + π). Calbraton The followng parameters appear n the equatons characterzng the model dynamcs n the neghborhood of the determnstc steady state: α, β, δ, σ, Ā, ρ a, σ a, ϕ, χ, µ, ρ θ, σ θ. Some of these parameters are famlar from the RBC model of the last chapter, and hence we wll adopt the same values as before: α = 0.58, β = 0.988, δ = 0.025, σ = 1 ρ a = 0.9 and σ a = 0.01. Ā s chosen such that the steady state level of output s unty. However, we need to choose values for the new parameters ϕ, χ, µ, ρ θ and σ θ that characterze money demand and the exogenous process for the money growth rate. Usng M1 as ther money measure, Cooley and Hansen (1989) estmate the followng process for money growth usng data wth quarterly frequency: log(m t ) = 0.00798 + 0.481 log(m t 1 ) and σ θ = 0.009, whch mples that ρ θ = 0.481 and 1 + µ = 1 + 0.00798 = 1.015,.e. a 1 ρ θ quarterly rate of money growth of 1.5%. An mportant parameter for the model dynamcs s χ. Note that we can wrte the loglnearzed verson of money demand as ˆm t = σ 1 ( ) β ˆR t χĉt χ 1 + µ β ( ) The nterest rate elastcty of money demand s therefore 1 β χ 1+µ β. The emprcal money demand lterature however usually focuses on the nterest rate sem-elastcty of 17
Fgure 6: Response to a +1% technology shock Percent Devatons 3 2 1 0 k c y a 1 0 5 10 15 20 25 Quarters Percent Devatons 0.2 0.1 0 0.1 r π m θ 0.2 0 5 10 15 20 25 Quarters money demand log(m t ) (1 + R t ) = 1 ( 4χ β 1 + µ β ) β 1 + µ Ths expresson takes nto account that the tme perod of the model s a quarter, whereas the elastcty s measured wth respect to the annualzed nterest rate. Many studes estmate dfferent absolute values for the nterest rate sem-elastcty of money demand, usually rangng from < 1 (χ > 10) up to 8 (χ 1). We wll take an ntermedate value of χ = 1/0.39, whch s the estmate of Char, Kehoe and McGrattan (2003). Fnally, ϕ s chosen n order to match velocty v = ȳ m (.e. the average rato of nomnal GDo M1) of about 1/0.16 from the steady state money demand relatonshp ϕ = ( ) m χ 1 + µ β C σ ȳ 1 + µ ȳχ The model s solved n MIUmodel.m usng the QZ decomposton. Fgure 6 dsplays the mpulse responses of the key macroeconomc varables to a 1% postve nnovaton n technology, and Fgure 7 shows the mpulse responses to a 1% postve nnovaton n money growth. The frst noteable feature of the model s the classcal dchotomy. The real allocatons are ndependent of monetary shocks (short-run superneutralty). Notce that equatons 18
Fgure 7: Response to a +1% money growth shock Percent Devatons 15 x 10 3 10 5 0 k c y a 5 0 1 2 3 4 5 6 7 8 9 10 Quarters Percent Devatons 2 1.5 1 0.5 0 0.5 r π m v θ 1 0 1 2 3 4 5 6 7 8 9 10 Quarters (9a), (9d) and (9f) descrbe the dynamcs of consumpton and captal ndependently of the other equatons. After a technology shock, the response of nflaton, real money holdngs, nomnal nterest rates and money velocty are consstent wth the data. Besdes the lack of real effects after a monetary expanson, there are two other dmensons n whch the model does poorly. Frst, there s no domnatng lqudty effect after a monetary expanson, whch means that because of hgher expected nflaton, an ncrease n money supply growth leads to hgher nstead of lower nomnal nterest rates. Second, nflaton s not persstent gven our (realstc) choce for ρ θ. These two shortcomngs together wth the lack of real effects are nconsstent wth most of the emprcal lterature on monetary shocks that wll be dscussed n the next chapter. A fnal unattractve feature s that, when we ntroduce technologcal progress, the model s nconsstent wth balanced growth unless χ = σ. To see why, note that n the money demand equaton (8c), growth n consumpton cannot be reconcled wth growth n real money balances n a way that leads to a statonary velocty of money. 19
2.1.2 An Extended MIU model wth Nonseparable Preferences and Elastc Labor Supply Ths secton addresses some of the weaknesses of the basc MIU model by ntroducng an elastc labor supply and by assumng non-separable preferences of the form u(c t, m t, 1 N t ) = ( ( C 1 χ t ) + ϕm 1 χ 1 ) 1 σ 1 χ t (1 N t ) η 1 σ Because consumpton and real money holdngs are no longer separable, changes n m t wll n general affect the margnal utlty of consumpton and lesure. These preferences are also appealng because they are consstent wth balanced growth. The frst order necessary condtons are now ( C 1 χ t ( λ t + C 1 χ t [ ( λ t + βe t λ t+1 ) χ σ + ϕm 1 χ 1 χ t (1 α)a t+1 ( Kt+1 N t+1 (1 N t ) (1 σ)η C χ t = 0 ) α + 1 δ)] = 0 ) 1 σ ( ) 1 α + ϕm 1 χ 1 χ t η(1 N t ) (1 σ)η 1 Kt + λ t αa t = 0 N t [ ] 1 + R t λ t + βe t ϕ ( mt C t λ t+1 = 0 1 + π t+1 ) χ R t = 0 1 + R t A t K 1 α t N α t + (1 δ)k t K t+1 = C t Determnstc Steady State Note that now the determnstc steady state levels of captal, output, etc. are no longer ndependent of the money growth rate. The reason s that labor supply s now affected by real money balances. The optmalty condton for the labor-lesure can be wrtten as η C1 χ t + ϕm 1 χ t (1 N t )C χ t = w t where w t s the real wage. Unless χ = 1, the steady state level of real balances, whch n turn depends on the average money growth rate µ through the money demand equaton, changes the steady-state labor supply. Hgher µ lowers the real demand for money, whch as long as χ > 1 rases the margnal utlty of lesure relatve to the margnal utlty of consumpton. Therefore hgher µ and hgher steady state nflaton lowers labor nput and output n the determnstc steady state. Money s no longer superneutral. 20
The loglnearzed dynamcs around the determn- Dynamcs n the stochastc model stc steady state are now descrbed by N ˆλ t = (1 σ)η 1 N ˆn t + ( χ + ˆλ t = E tˆλt+1 + (1 β(1 δ)) E t (a t+1 αˆk t+1 + αˆn t+1 ) ] 1 χ N a t (1 α)ˆn t + (1 α)ˆk t = 1 N ˆn (1 χ) C t + [χ + ĉ C 1 χ + ϕ m 1 χ t + ˆλ t = E tˆλt+1 + ˆR t E tˆπ t+1 ( ) β χĉ t χ ˆm t = ˆR t 1 + µ β s c ĉ t + s ( ) δ ˆk 1 δ t+1 = a t + (1 α) + s ˆk t + αˆn t δ ˆm t = ˆm t 1 ˆπ t + θ t θ t = ρ θ θ t 1 + ϵ θ t a t = ρ a a t 1 + ϵ a t ) ( ) 1 χ (χ σ) C (χ σ)ϕ m 1 χ ĉ C 1 χ + ϕ m 1 χ t + ˆm C 1 χ + ϕ m 1 χ t [ ] (1 χ)ϕ m 1 χ ˆm C 1 χ + ϕ m 1 χ t Calbraton We wll adopt the same values as before: α = 0.58, β = 0.988, δ = 0.025, σ = 1, ρ a = 0.9, σ a = 0.01, ρ θ = 0.481, σ θ = 0.009 and 1 + µ = 1.015. Ā s chosen such that the steady state level of output s unty. We chose η such that N = 0.20. As before, ϕ s chosen n order to match the average M1 velocty of about 1/0.16 and χ = 1/0.39. The model s solved n MIUmodel2.m usng the QZ decomposton. Fgure 8 dsplays the mpulse responses of the key macroeconomc varables to a 1% postve nnovaton n technology, and Fgure 9 shows the mpulse responses to a 1% postve nnovaton n money growth. The mpulse responses to a technology shock are very smlar to those of the standard RBC model. Ths s despte the fact that there s no longer a classcal dchotomy. Snce real money balances affect the margnal utltes of consumpton and lesure, they also affect the nvestment and labor decson. That money s no longer superneutral n the short run wth nonseparable preferences s also clear from the fact that the real varables respond to the monetary shock. However, the real effects of a monetary expanson are very weak quanttatvely. A monetary expanson reduces output and the response of nvestment has the opposte sgn of the output response. As before, there s no domnant lqudty effect and nflaton s not persstent after a monetary nnovaton. You can verfy that whether consumpton, output and hours decrease or ncrease after a postve money growth shock s determned by the sze of χ. If χ > 1 and the nterest rate elastcty s relatvely low, then a decrease n m t after a monetary expanson ncreases the 21
Fgure 8: Response to a +1% technology shock Percent Devatons 5 4 3 2 1 0 k c y n a 1 0 5 10 15 20 25 Quarters Percent Devatons 0.6 0.4 0.2 0 0.2 r π m θ 0.4 0 5 10 15 20 25 Quarters Percent Devatons 0.02 0.015 0.01 0.005 0 0.005 0.01 Fgure 9: Response to a +1% money growth shock k c y n a 0.015 0 1 2 3 4 5 6 7 8 9 10 Quarters Percent Devatons 2 1.5 1 0.5 0 0.5 r π m v θ 1 0 1 2 3 4 5 6 7 8 9 10 Quarters 22
margnal utlty of lesure relatve to consumpton, whch lowers labor supply and output. If χ < 1 and the nterest rate elastcty s relatvely hgh, then a decrease n m t after a monetary expanson lowers the margnal utlty of lesure relatve to consumpton, whch ncreases labor supply and output. EXERCISE: Consder the envronment above, but suppose now that the household s utlty functon s u(c t, m t, 1 N t ) = ( C 1 χ t ) + ϕm 1 χ 1 σ 1 χ t 1 σ + θ l(1 N t ) 1 η 1 η 1. Wrte down the equlbrum condtons. 2. Analyze the determnstc steady state and explan whether/why there s long-run nonsuperneutralty. Are these preferences consstent wth balanced growth? 3. Loglnearze the equlbrum condtons and explan whether/why there s short-run nonsuperneutralty. 4. Calbrate the model assumng that σ = 5, χ = 1/0.39 and η = 5. Analyze the dynamcs after an nnovaton n money growth. 5. Verfy that whether consumpton, output and hours decrease after a postve money growth shock s determned by the sgn of u cm and explan. 23
2.2 A Cash-n-Advance (CIA) Model Ths secton ncorporates an alternatve way of modelng money by assumng a cash-nadvance (CIA) constrant. One example of ths approach to modelng money s Cooley and Hansen (1989). Envronment The economy s populated by a representatve household wth preferences represented by U = E 0 β t u(c t, L t ), 0 < β < 1 (12) t=0 where C t > 0 s consumpton n perod t and L t denotes tme devoted to lesure. As n Kng et al. (1988), assume that the household has nstantaneous utlty: u(c t, L t ) = { (Ct v(l t )) 1 σ 1 σ f σ 1 log(c t ) + v(l t ) f σ = 1 (13) and that the tme constrant s L t = 1 N t. The representatve household enters every perod t wth nomnal money balances M t 1. In addton, these balances are augmented by a lump-sum money transfer by the government T t. The key assumpton n the CIA model s that households are requred to acqure money balances to purchase goods ntended for consumpton. The household s consumpton choce must satsfy the constrant: C t M t 1 + T t or equvalently The households perod t budget constrant s C t m t 1 1 + π t + t t (14) or equvalently C t + I t + B t + M t w t N t + r t K t + (1 + R t 1 ) B t 1 + M t 1 + T t C t + I t + b t + m t w t N t + r t K t + 1 + R t 1 1 + π t b t 1 + m t 1 1 + π t + t t and as before, the law of moton for the captal stock s K t+1 = I t + (1 δ)k t, 0 < δ < 1 24
where I t denotes gross nvestment. The household s problem s to choose the real quanttes {C t, b t, m t, N t, K t+1 } t=0 to maxmze (12) subject to b t > b, the law of moton for captal, the CIA constrants and the budget constrants and takng as gven nflaton and nomnal nterest rates {π t, R t } t=0 as well as the real factor prces {w t, r t } t=0, the lumps sum transfers and the ntal captal stock K 0 and real bond and money holdngs and nomnal nterest rate b 1, m 1, R 1. The behavor of the frms and the government as well as the defnton of equlbrum are dentcal to the MIU model. Money Demand In equlbrum, the followng condtons must be satsfed n every perod t at an nteror soluton,.e. under the assumpton that the CIA constrant s always bndng: λ t + βe t [ λ t u c (C t, 1 N t ) + Ξ t = 0 (15a) ( ( ) α Kt+1 λ t+1 (1 α)a t+1 + 1 δ)] = 0 (15b) N t+1 ( ) 1 α Kt u l (C t, 1 N t ) + λ t αa t = 0 (15c) N t [ ] 1 + R t λ t + βe t λ t+1 = 0 (15d) 1 + π t+1 ] 1 λ t + βe t [u c (C t+1, 1 N t+1 ) = 0 (15e) 1 + π t+1 C t = m t (15f) together wth C t = A t K 1 α t N α t + (1 δ)k t K t+1. Equaton (15a) lnks the margnal utlty of wealth λ t to the margnal utlty of consumpton. Note that both are no longer dentcal: the margnal utlty of consumpton exceeds λ t by Ξ t > 0, the value of lqudty servces. Ξ t s the multpler assocated wth the CIA constrant. Equaton (15b) s the famlar Euler equaton descrbng the optmal consumpton-nvestment choce, (15c) descrbes the optmal labor lesure trade-off, (15d) s the bond Euler equaton, (15e) s the money demand equaton, and (15f) s the CIA constrant. Note that (15a), (15d) and (15e) mply that Ξ t+1 = 0 whenever R t = 0. A requrement for an nteror soluton n whch the CIA constrant bnds s that the nomnal nterest rate s strctly postve! The money demand equaton reduces to βe t [ Ξt+1 1 + π t+1 ] = R t 1 + R t λ t whch states that the margnal beneft of holdng money (.e. the expected lqudty servces 25
provded tomorrow) equals the opportunty cost The Determnstc Steady State R t 1+R t. Frst note that unlke n the MIU model, the CIA constrant rules out speculatve hypernflatons and π = µ. Second, the steady state values of captal, consumpton, nvestment, labor nput and output n the determnstc CIA model wth elastc labor supply are dfferent from those n the correspondng RBC model, and wll depend on the money growth/nflaton rate n the steady state. As n the extended MIU model, the determnstc CIA model wth elastc labor supply dsplays long-run nonsuperneutralty. The key reason s that a bndng CIA constrant and therefore Ξ > 0 dstorts the labor supply decson: u l ( C, 1 N) < u c ( C, 1 N) w (16) snce λ = β 1+µ u c( C, 1 N) < u c ( C, 1 N). The hgher the money growth rate µ, the lower the margnal value of the real wage, and therefore the lower the labor supply. Had we assumed an nelastc labor supply, the CIA model would be superneutral as n the basc MIU model. Fnally, note that the requrement of a postve nomnal nterest rate requres 1 + R = 1 + µ β > 1 The loglnearzed dynamcs around the determn- Dynamcs n the Stochastc Model stc steady state are descrbed by a t (1 α)ˆn t + (1 α)ˆk t + ˆλ t = ξ ll N 1 N ˆn t + ξ lc ĉ t ˆλ t = E tˆλt+1 + (1 β(1 δ)) E t (a t+1 αˆk t+1 + αˆn t+1 ) ˆλ t = E tˆλt+1 + ˆR t E tˆπ t+1 ˆλ t = N σe t ĉ t+1 ξ cl 1 N E tˆn t+1 E tˆπ t+1 ĉ t = ˆm t s c ĉ t + s δ ˆk t+1 = a t + ˆm t = ˆm t 1 ˆπ t + θ t θ t = ρ θ θ t 1 + ϵ θ t a t = ρ a a t 1 + ϵ a t ( ) 1 δ (1 α) + s ˆk t δ Calbraton We wll adopt the same values as before: α = 0.58, β = 0.988, δ = 0.025, σ = 1, ρ a = 0.9, σ a = 0.01, ρ θ = 0.481, σ θ = 0.009 and 1 + µ = 1.015. Ā s chosen such that 26
Fgure 10: Response to a +1% technology shock Percent Devatons 5 4 3 2 1 0 k c y n a 1 0 5 10 15 20 25 Quarters Percent Devatons 0.6 0.4 0.2 0 0.2 r π m θ 0.4 0 5 10 15 20 25 Quarters the steady state level of output s unty. We set v(l) = θ l log(l) and chose θ l such that N = 0.20. Our assumptons mply that ξ cl = ξ lc = 0 and ξ ll = 1. The model s solved n CIAmodel.m usng the QZ decomposton. Fgure 10 dsplays the mpulse responses of the key macroeconomc varables to a 1% postve nnovaton n technology, and Fgure 11 shows the mpulse responses to a 1% postve nnovaton n money growth. Monetary shocks have a szeable but opposte effect on consumpton and nvestment, but only weak effects on output or hours. However, Cooley and Hansen (1989) show that the mplcatons for the busness cycle moments are only mnor, see Fgure 12. 4 Note that, unlke n the MIU model, the decrease n m t after a monetary expanson always ncreases the margnal utlty of consumpton, ncreases the demand for lesure and therefore lowers the labor supply and output. As n the MIU model, there s no domnant lqudty effect and nflaton s not very persstent after a monetary nnovaton. 4 The results of Cooley and Hansen (1989) are not drectly applcable to the model n the notes because of dfferences n calbraton, most mportantly of the value of the labor supply elastcty. 27
Percent Devatons Fgure 11: Response to a +1% money growth shock 1.5 1 0.5 0 0.5 k c y n a 1 0 1 2 3 4 5 6 7 8 9 10 Quarters Percent Devatons 2 1.5 1 0.5 0 0.5 r π m v θ 1 0 1 2 3 4 5 6 7 8 9 10 Quarters Fgure 12: Source: Cooley and Hansen (1989) 28
2.3 A Shoppng Tme (ST) Model Shoppng tme models assume that the purchase of goods requres the nput of transacton servces whch n turn are provded by money and tme. One example of ths approach to modelng money s Kng and Plosser (1984). Envronment As before, the economy s populated by a representatve household wth preferences represented by U = E 0 β t u(c t, L t ), 0 < β < 1 t=0 where C t > 0 s consumpton n perod t and L t denotes tme devoted to lesure. The nstantaneous utlty functon s agan gven by (13). The key assumpton n the shoppng tme model s that the household s tme constrant s L t + N t + S t = 1 The unt tme endowment conssts of tme devoted to lesure L t, work N t and shoppng tme S t. The amount of shoppng tme S t needed to purchase a partcular level of consumpton C t s assumed to be negatvely related to the household s holdngs of real money balances m t. 5 The shoppng or transacton technology s gven by S t = Φ (C t, m t ) (18) where Φ, Φ c, Φ cc, Φ mm 0 and Φ m, Φ cm 0. The households perod t budget constrant s C t + I t + b t + m t w t N t + r t K t + 1 + R t 1 1 + π t b t 1 + m t 1 1 + π t + t t and the law of moton for the captal stock s K t+1 = I t + (1 δ)k t, 0 < δ < 1 where I t denotes gross nvestment. The household s problem s to choose the real quanttes {C t, b t, m t, N t, S t, K t+1 } t=0 to maxmze (2.3) subject to b t > b, the law of moton for captal, the CIA constrants and the budget constrants and takng as gven nflaton and 5 Note that we assume that only the purchase of consumpton goods requres the nput of transacton servces. It s straghtforward to mpose the same requrement to the purchase of goods for nvestment. 29
nomnal nterest rates {π t, R t } t=0 as well as the real factor prces {w t, r t } t=0, the lump sum transfers and the ntal captal stock K 0 and real bond and money holdngs and nomnal nterest rate b 1, m 1, R 1. The behavor of the frms and the government as well as the defnton of equlbrum are dentcal to the MIU and CIA model. Money Demand The frst order necessary condtons are now λ t u c (C t, L t ) + u l (C t, L t )Φ c (C t, m t ) = 0 (19a) [ ( ( ) α Kt+1 λ t + βe t λ t+1 (1 α)a t+1 + 1 δ)] = 0 (19b) N t+1 ( ) 1 α Kt u l (C t, L t ) + λ t αa t = 0 (19c) N t [ ] 1 + R t λ t + βe t λ t+1 = 0 (19d) 1 + π t+1 [ ] 1 λ t u l (C t, L t )Φ m (C t, m t ) + βe t λ t+1 = 0 (19e) 1 + π t+1 L t + N t + Φ (C t, m t ) = 1 (19f) together wth C t = A t K 1 α t N α t + (1 δ)k t K t+1. Equaton (19a) lnks the margnal utlty of wealth λ t to the margnal utlty of consumpton. Agan, both are not dentcal: the margnal utlty of consumpton exceeds λ t by u l (C t, L t )Φ c (C t, m t ) > 0, the value of transacton servces. Equaton (19b) s the famlar Euler equaton descrbng the optmal consumpton-nvestment choce, (19c) descrbes the optmal labor lesure trade-off, (19d) s the bond Euler equaton, (19e) s the money demand equaton, and (19f) s the tme constrant. The money demand equaton reduces to u l(c t, L t )Φ m (C t, m t ) λ t = R t 1 + R t Usng the frst-order condton for the labor lesure choce, we can wrte money demand as w t Φ m (C t, m t ) = R t 1 + R t whch states that the margnal beneft of holdng money (.e. the tme saved valued at the real wage) equals the opportunty cost The Determnstc Steady State R t 1+R t. We can defne a functon ũ(c t, m t, L t ) = u(c t, 1 N t Φ(C t, m t )) that gves utlty as a functon of consumpton, lesure and real money 30
holdngs. Thus the shoppng tme model can motvate the presence of m t n the utlty functon, such that the analyss from the MIU model apples. The loglnearzed dynamcs around the determn- Dynamcs n the Stochastc Model stc steady state are now descrbed by ˆλ t (1 b)ω cm ˆm t = ( bσ + (1 b) (ξ lc + ω cc )) ĉ t + (bξ cl + (1 b)ξ ll ) ˆl t a t (1 α)ˆn t + (1 α)ˆk t + ˆλ t = ξ llˆlt + ξ lc ĉ t ˆλ t = E tˆλt+1 + (1 β(1 δ)) E t (a t+1 αˆk t+1 + αˆn t+1 ) ˆλ t = E tˆλt+1 + ˆR t E tˆπ t+1 a t (1 α)ˆn t + (1 α)ˆk t + ω mc ĉ t + ω mm ˆm t = β 1 + µ β ˆR t ˆlt + N L ˆn t = Φ( C, m) L s c ĉ t + s δ ˆk t+1 = a t + ˆm t = ˆm t 1 ˆπ t + θ t θ t = ρ θ θ t 1 + ϵ θ t a t = ρ a a t 1 + ϵ a t (ϕ c ĉ t + ϕ m ˆm t ) ( 1 δ (1 α) + s δ uc( C, L) where b =, ω λ ab s the elastcty of Φ a wth respect to b and ϕ a s the elastcty of Φ wth respect to a. ) ˆk t Calbraton We wll assume the followng functonal form for the transacton technology C t Φ(C t, m t ) = τ, τ > 0 1 + m t Ths functonal form mples that ω cc = 0, ω cm = m 1+ m, ω mc = 1, ω mm = 2 m 1+ m, ϕ c = 1 and ϕ m = m 1+ m. For the parameters, we take the same values as before: α = 0.58, β = 0.988, δ = 0.025, σ = 1, ρ a = 0.9, σ a = 0.01, ρ θ = 0.481, σ θ = 0.009 and 1 + µ = 1.015. Ā s chosen such that the steady state level of output s unty. We set v(l) = θ l log(l) and chose θ l such that N = 0.20. τ s chosen to match the M1 money velocty of about 1/0.16. Our assumptons on the preferences mply that ξ cl = ξ lc = 0 and ξ ll = 1. The model s solved n STmodel.m usng the QZ decomposton. Fgure 13 dsplays the mpulse responses of the key macroeconomc varables to a 1% postve nnovaton n technology, and Fgure 14 shows the mpulse responses to a 1% postve nnovaton n money growth. The results are very smlar to the earler extended MIU model. 31
Fgure 13: Response to a +1% technology shock Percent Devatons 5 4 3 2 1 0 k c y n a 1 0 5 10 15 20 25 Quarters Percent Devatons 0.5 0 0.5 1 r π m θ 1.5 0 5 10 15 20 25 Quarters Fgure 14: Response to a +1% money growth shock Percent Devatons 10 x 10 3 5 0 k c y n a 5 0 1 2 3 4 5 6 7 8 9 10 Quarters Percent Devatons 2 1.5 1 0.5 0 0.5 r π m v θ 1 0 1 2 3 4 5 6 7 8 9 10 Quarters 32