Money in a Real Business Cycle Model Graduae Macro II, Spring 200 The Universiy of Nore Dame Professor Sims This documen describes how o include money ino an oherwise sandard real business cycle model. I is no easy o ge agens o hold money in equilibrium. The inuiion is fairly sraighforward money earns no ineres and provides no uiliy, so here is no direc (uiliy) or indirec (savings) moivaion for holding i. There is a large lieraure exploring micro foundaions of money, wherein he fricions giving rise o money are explicily modeled (e.g. Kiyoaki and Wrigh (989)), ypically in a search heoreic framework. I goes wihou saying ha i is exremely di cul o incorporae such a framework ino a sandard DSGE model. As such, we ll be aking a shorcu (his is quie convenional in he lieraure). There are wo basic shorcus for geing agens o hold money cash in advance and money in he uiliy funcion. Boh ge a he medium of exchange funcion of money in he cash in advance framework, money is necessary o conduc exchange, while in he money in he uiliy funcion, having more money increases uiliy, presumably because i makes exchange easier. An example of money in he uiliy funcion is Sidrauski (967). Examples of cash in advance economies are Lucas (987) and Cooley and Hansen (989). The Budge Consrain Before going any furher we need o alk rs abou how money eners he households budge consrain. I will be seing his problem up as a decenralized equilibrium, so I will imagine a world in which households hold nominal bonds issued by rms o ge capial, B, and/or receive pro disribuions from rms. Households have hree sources of nominal income in period : nominal wage income, previous period bond holding plus nominal ineres, and previous period money holdings. Wih his nominal income households can eiher purchase more goods, more bonds, or hold more money. The consrain is: c + B + M = W n + + ( + i )B + M Take noe of he daing convenion here: B and M denoe bond and money holdings held beween period and period, wih i he nominal ineres rae on bond holdings. We wan o wrie he budge consrain in real erms. Divide everyhing by : c + B + M De ne b = B as real bond holdings, m = M he real wage. The consrain is hen: = W n + + ( + i ) B + M as real money balances, and w = W as c + b + m = w n + + ( + i ) B + M We need o play around o ge he righ hand side in appropriae erms. De ne + =. Using his, we have:
c + b + m = w n + + ( + i ) B + M c + b + m = w n + + ( + i ) b + + m + The Cash in Advance Consrain The cash in advance consrain is fairly simple. I says ha he household mus have enough cash held over from he previous period o nance oday s nominal consumpion expendiures. Formally: Wriing his ou in real erms yields: M c M c M c m ( + )c The Money Growh Rule We assume ha he quaniy of money is se exogenously by some hird pary (e.g. he Fed). Higher in aion is essenially a ax on he holders of money. For simpliciy, we assume ha he hird pary jus squanders any revenue i generaes by prining money. We ll come back o his issue when sudying scal policy. I is clear from he cash in advance consrain ha seady sae in aion will be equal o he seady sae growh rae of money. Since we wan o allow for posiive seady sae in aion, I m going o wrie he money rule as an AR() in he growh rae (i.e. log rs di erence) wih a posiive mean. Formally: ln M ln M = ( m ) + m (ln M ln M 2 ) + " m; Because we wan o wrie everyhing in erms of real balances (which removes he rend), le s play around wih his by adding and subracing logs of he price level a various lags: ln M ln + ln ln ln M + ln = ( m ) + ::: ::: + m (ln M ln + ln ln 2 ln M 2 + ln 2 ) + " m; Nohing ha ln ln = (as a rs order approximaion), we have: m m + = ( m ) + m + m (m m 2 ) + " m; 2
I is sraighforward o verify ha his will hold in he seady sae wih he growh rae of real balances being equal o zero. I can de ne a new variable o be dm = m m so ha I can make here be only one lag in he sysem. The Res of he Model The res of he model is our sandard real business cycle model, plus he addiional consrains. I begin wih he household problem. The household chooses consumpion, labor, real bond holdings, and real money balances o saisfy: max c ;n ;b ;m E 0 X =0 c + ( n ) s..! c + b + m w n + + ( + i ) b + m + + ( + )c m I solve he model using a Lagrangian: L = E 0 X =0 ( c + ( n) + w n + + ( + i ) b ::: + (m ( + )c ) + + m + c b m + ::: ) The rs order condiions are: = 0, c = + @c ( + ) = 0, ( n ) = w @n + i = 0, = + @b + + + = 0, = + @m + + + Before proceeding, ake a sep back o noe ha hese FOCs make sense. Firs, noe ha he Fisher relaionship is: + r = +i + +, so he Euler equaion looks idenical o wha we ve had before. Second, suppose ha he cash in advance consrain weren here, which would mean ha = 0 a all imes. Then he rs FOC would look normal, and he hird and fourh would be equivalen if and only if i = 0. If he nominal rae were posiive, we d be in a corner soluion (i.e. no one would ever hold money), and so he nal FOC wouldn be necessary for a soluion. This makes inuiive sense he opporuniy cos of holding money is he nominal ineres rae. If here is no reason o hold money (i.e. no cash in advance consrain), hen no one would ever hold money a a posiive ineres rae. 3
Now le s go o he rm problem. I assume ha he rm owns is capial sock and chooses employmen and invesmen each period o maximize he presen discouned value of (real) pro s, wih a Cobb-Douglas producion funcion, and subjec o he capial accumulaion equaion. The value of he rm can be wrien as curren pro s plus he discouned value of fuure pro s: V = a k n w n I + X j= k= The capial accumulaion equaion is sandard: jy ( + r +k ) a +j k+j n +j w +j n +j I +j k = I + ( )k I can solve he problem using a Lagrangian: L = a k n w n I + X ::: + q (I + ( )k k ) The FOC are: j= k= jy ( + r +k ) a +j k+j n +j w +j n +j I +j + ::: + = 0, w = ( )a k @n n = 0, q = @I = 0, q + a + k n+ + q + ( ) @k + r ) + r = a + k n+ + ( ) These are all (or should be) familiar condiions by now. I assume ha echnology follows a saionary AR() process (i.e. I absrac from rend growh, which doesn make much of a di erence anyway): ln a = ln a + " a; Because he uncondiional mean of he log of echnology is zero, he seady sae of he level of echnology is one. Aggregae marke-clearing requires: y = c + I The full model can be characerized by he following foureen equaions: c = + ( + ) () 4
( n ) = w (2) + i = + + + (3) + = + + + + (4) w = ( )a k n (5) + r = a + k n + + ( ) (6) k + = I + ( )k (7) + r = + i + + (8) y = a k n (9) y = c + I (0) dm + = ( m ) + m + m dm + " m; () dm = ln m ln m (2) ln a = ln a + " a; (3) m = ( + )c (4) Noe ha, as speci ed, I do no explicily include he price level or he nominal money supply in he rs order condiions. This is a maer of convenience, as Dynare will no solve he model wih non-saionary variables, and boh of hese variables will be non-saionary. If I desire o reconsruc hese variables (as I will in he impulse responses below), I can consruc hese series afer running Dynare by de ning hem as follows: ln M = ln m + ln (5) I calibrae he parameers as follows: ln = + ln (6) 5
= 0:98 = 0:33 = 0:03 = 0:95 = = 3:5 = = 0:0 m = 0:75 This means ha we re in he familiar log-log case for preferences and ha seady sae annual in aion is abou 4 percen. In addiion, I se he sandard deviaion of he echnology shock o 0.007 and he sandard deviaion of he money growh shock o 0.002. I solve he model using a rs order log-linear approximaion in Dynare. The impulse responses o he wo shocks are below. As noed above, I compue he responses of he price level and he nominal money supply ouside of he Dynare.mod le. I is ineresing o noe ha he responses of he real variables of he model are idenical in he model here as hey would be in an RBC model wihou money explicily included. This is comforing absracing from money before doesn really change our conclusions 6
(a leas in his framework). We see ha he echnology shock lowers he price level (i.e. in aion jumps down immediaely) and leads o an increase in real balances. Nominal money, by consrucion, does no respond. Below are he impulse responses o a money growh shock: Here we see somehing very ineresing and no very inuiive. Money is no neural here i has real e ecs, albei hese real e ecs are quie small. Furher, we observe ha a emporary increase in he growh rae of money (i.e. a permanen change in he level of he nominal money supply) acually lowers oupu, hours, and consumpion, which is no paricularly inuiive. We ll come back o his in a momen. Somehing else worh playing around wih is o change he seady sae in aion rae,. Below is a able showing he seady sae level of oupu for di eren in aion raes: y -0.0 0.5443 0.00 0.5400 0.0 0.5359 0.02 0.537 0.05 0.596 We see ha seady sae oupu is decreasing in seady sae in aion. We mus impose ha + > for a soluion o exis, so i is no possible o have very large seady sae de aion. This able (ha long run oupu is decreasing in long run in aion) acually 7
seems inuiive, especially in comparison o he resul abou ha increase in he money supply reduce oupu in he shor run. The inuiion for hese resuls is as follows. In aion is essenially a ax on he holders of money. The more in aion here is, he more you penalize people who hold money, and hereby he less money people would like o hold, oher hings being equal (of course in equilibrium hey have o hold whaever money he cenral bank prins). Because money is necessary o consume (i.e he cash in advance consrain), here is a non-neuraliy here. When in aion is higher, people wan o hold less money. Since in equilibrium hey can hold less money han he cenral bank prins, hey end up subsiuing away from hings which require money (consumpion) and ino hings ha don (leisure). This ends up reducing oupu in he long run. The emporary increase in he money supply works he same way, albei only for a while. People ry o ge away from money and ino leisure, so consumpion and employmen boh go down immediaely when he growh rae of money jumps up. For hese parameers, here is a emporary increase in invesmen following he increase in money (i.e. he reducion in consumpion dominaes he reducion in hours in erms of wha happens o invesmen). The Dynare code I used o produce hese gures is called money_cia.mod. Money in he Uiliy Funcion The oher popular way of geing people o hold money in equilibrium is o explicily pu he quaniy of real balances ino he ow uiliy funcion. This may seem somewha ad-hoc, bu i ges a he idea ha he exisence of money makes exchange easier, hereby increasing uiliy. Mos of he model is idenical o above, bu preferences are di eren. The household faces he following problem:! X max E 0 c c ;n ;b ;m + ( n ) + m =0 s.. c + b + m w n + + ( + i ) b + + m + Unlike he se up above, here is no cash in advance consrain here. for he problem can be wrien: The Lagrangian L = E 0 X =0 8 < : c + m + ( n) + ::: ::: + w n + + ( + i ) b + + m + c b m 9 = ; The rs order condiions are: 8
= 0, c = @c = 0, ( n ) = w @n + i = 0, = + @b + + = 0, m + + + = 0 @m + + ) m = + + + Given he Fisher relaionship, he rs hree rs order condiions are idenical o wha would obain in a real model wihou menion of money a all. We can simplify he rs order condiion for real balances by noing ha + + + = +i from he Euler equaion. Le s plug his in and simplify: m = + i m = m = i m = + i + i ) m = c i + i + i We see ha he demand for real balances is increasing in consumpion and decreasing in he nominal ineres rae, which makes sense as he nominal ineres rae is he opporuniy cos of holding money. If we re in he familiar log-log case in which = =, hen his reduces even furher o: + i i m = c This begins o look a lo like he quaniy equaion (i.e. replaced by consumpion. We can rewrie i: i MV = P Y ), wih oupu M v = c v = i ( + i ) 9
This yields he inuiive resul ha velociy is increasing in he nominal ineres rae (i.e. he more cosly i is o hold money, he more imes you d like o spend each individual uni of money). As noed above, he res of he model is idenical o above, absen he cash in advance consrain. Hence he model has one fewer variable, as here is no Lagrange muliplier on he cash in advance consrain. I calibrae he parameers he same as above, now wih = and =. The impulse responses o boh a echnology shock and a money growh shock are shown below: The money growh shock responses are: 0
The main di erence here is in he responses o he money growh shock. In his model, money is compleely neural wih respec o real variables (boh in he shor run and in he long run, as here is no e ec of more or less in aion on he seady sae values of he real variables). The responses of he real variables o he echnology shock are also idenical o wha would obain in a real model. The reason money is neural here is ha here is no fricion ha gives rise o money maering in he sense of a ecing he values of real variables, hough i does maer for uiliy purposes.