Money, Interest Rates, and Exchange Rates with Endogenously Segmented Markets

Transcription

1 Money, Ineres Raes, and Exchange Raes wih Endogenously Segmened Markes Fernando Alvarez Universiy of Chicago, Universidad Torcuao di Tella, and Naional Bureau of Economic Research Andrew Akeson Universiy of California, Los Angeles, Federal Reserve Bank of Minneapolis, and Naional Bureau of Economic Research Parick J. Kehoe Federal Reserve Bank of Minneapolis, Universiy of Minnesoa, and Naional Bureau of Economic Research We analyze he effecs of money injecions on ineres raes and exchange raes when agens mus pay a Baumol-Tobin-syle fixed cos o exchange bonds and money. Asse markes are endogenously segmened because his fixed cos leads agens o rade bonds and money infrequenly. When he governmen injecs money hrough an open marke operaion, only hose agens ha are currenly rading absorb hese injecions. Through heir impac on hese agens consumpion, hese money injecions affec real ineres raes and real exchange raes. The model generaes he observed negaive relaion beween expeced inflaion and real ineres raes as well as persisen liquidiy effecs in ineres raes and volaile and persisen exchange raes. We hank Rober E. Lucas, Jr., Dean Corbae, and Sanley Zin for heir commens and he Naional Science Foundaion for is suppor. The views expressed herein are hose of he auhors and no necessarily hose of he Federal Reserve Bank of Minneapolis or he Federal Reserve Sysem. [Journal of Poliical Economy, 2002, vol. 110, no. 1] 2002 by The Universiy of Chicago. All righs reserved /2002/ $

2 74 journal of poliical economy Several feaures of he observed relaionships beween money and boh ineres raes and exchange raes are difficul o accoun for in sandard moneary models. Moivaed by he work of Baumol (1952) and Tobin (1956), economiss have argued ha adding fricions ha lead o a segmened marke for rading money and ineres-bearing asses migh help improve hese models (see Grossman and Weiss [1983], Roemberg [1984, 1985], and Lucas [1990], among ohers). Here we build on his lieraure by developing a model wih endogenously segmened asse markes. Our model is boh simple and promising as a way o accoun for he daa. In our model, agens mus pay a fixed cos o ransfer money beween he asse marke and he goods marke. This fixed cos leads agens o rade bonds and money only infrequenly. In any given period, only a fracion of agens are acively rading; ha is, he asse marke is segmened. When he governmen injecs money hrough an open marke operaion, only he currenly acive agens are on he oher side of he ransacion, and only heir marginal uiliies deermine ineres raes and exchange raes. Money injecions are absorbed exclusively by hese acive agens: he injecions increase acive agens curren consumpion; hence, real ineres raes fall and he real exchange rae depreciaes. We refer o his effec of money injecions on real ineres raes and real exchange raes as he segmenaion effec. Our main conribuion here is o derive wih pencil and paper he implicaions of segmened asse markes for he relaionships of money, ineres raes, and exchange raes for sochasic processes for shocks moivaed by he daa. Our derivaion sheds ligh on how he complicaed relaionships beween money, ineres raes, and exchange raes are all driven by a simpler one, namely, he relaionship beween money injecions and he marginal uiliy of acive agens. We also show ha some predicions of a simple, quaniaive version of our model come close o maching feaures of he daa ha sandard models wihou segmenaion have no been able o produce. We begin wih wo feaures of ineres raes ha have been difficul o accoun for in sandard moneary models. Firs, expeced inflaion and real ineres raes generally move in opposie direcions. This has been documened by Barr and Campbell (1997) using indexed and nominal bonds (see also Pennacchi 1991; Campbell and Ammer 1993). Second, a leas since Friedman (1968), open marke operaions have been hough o have liquidiy effecs: money injecions lead iniially o a decline in shor-erm nominal ineres raes, a decline ha is hough o decay over ime, wih shor-erm raes evenually rising o normal levels or higher. Accordingly, money injecions are hough o seepen he yield curve, lowering long-erm raes less han shor-erm raes, or even o wis he yield curve by raising long-erm raes. The vecor au-

3 money 75 oregression (VAR) lieraure has been somewha successful in confirming hese paerns in he daa (see Cochrane 1994; Chrisiano, Eichenbaum, and Evans 1998). Our model wih segmened asse markes can produce boh of hese feaures whereas a sandard model canno. In a sandard model wihou marke segmenaion, persisen money injecions increase expeced inflaion bu have no effecs on real ineres raes, so he model induces no relaion beween hem. In addiion, hese injecions raise nominal ineres raes of all mauriies and flaen or even inver he yield curve. In our model, however, money injecions move expeced inflaion and real ineres raes in opposie direcions. These injecions hus generae he negaive correlaion beween expeced inflaion and real ineres raes ha is observed in he daa. Also, if asse markes are sufficienly segmened, money injecions in our model have liquidiy effecs: money injecions lower shor-erm nominal ineres raes and seepen or even wis he yield curve by lowering shor raes and raising long ones. We show ha wih moderae amouns of segmenaion, our model can produce dynamic responses similar o hose found in he VAR lieraure. Moreover, our model generaes persisen real effecs from marke segmenaion even from anicipaed shocks. Cochrane (1998) argues ha a reasonable inerpreaion of he VAR resuls may require models wih his propery. Afer our look a money and ineres raes, we urn o some prominen feaures of money and exchange raes. These feaures are differen for counries wih differen raes of inflaion. For low-inflaion counries, real and nominal exchange raes have similar variabiliy, hese raes are highly correlaed, and boh are persisen. For high-inflaion counries, real exchange raes are much less volaile han nominal exchange raes. A sandard model can produce none of hese feaures, bu our endogenously segmened model can produce hem all. In a sandard model, money injecions do no affec real exchange raes, and hey affec nominal exchange raes only hrough heir impac on inflaion. In our model, however, when inflaion is low, asse markes are segmened and money injecions have a subsanial impac on real exchange raes. Wih moderae amouns of segmenaion, herefore, real and nominal exchange raes have similar variabiliy, hey are highly correlaed, and boh are persisen, jus as in he daa. When inflaion is high, agens rade more frequenly, markes become less segmened, and money injecions have a smaller impac on real exchange raes. Hence, in our model as in he daa on high-inflaion counries, real exchange raes are significanly less volaile han nominal exchange raes. Our model wih segmened markes is a sandard cash-in-advance model wih he addiion of fixed coss for agens o exchange money

4 76 journal of poliical economy and bonds. In our model, he household begins each period wih some cash in he goods marke; he money injecion is hen realized, and he household hen splis ino a worker and a shopper. The worker sells he curren endowmen for cash, and he shopper decides eiher o buy goods wih jus he curren real balances or o pay he fixed cos o ransfer cash o or from he asse marke and hen buy goods. The household s endowmen and, hus, he household s cash holdings are random and idiosyncraic. The shopper follows a cuoff rule ha defines zones of aciviy and inaciviy for rading cash and ineres-bearing asses. In he zones of aciviy, shoppers wih high real balances pay a fixed cos o ransfer cash o he asse marke, whereas shoppers wih low real balances pay a fixed cos o obain cash from he asse marke. Shoppers wih inermediae real balances are in he zone of inaciviy. They do no pay a fixed cos; hey simply spend heir curren real balances. Over ime, households sochasically cycle hrough he zones of aciviy and inaciviy as heir idiosyncraic shocks vary. If he fixed cos is zero, all agens are acive, and he model reduces o a sandard one similar o ha of Lucas (1984). Ours is a fully sochasic model wih boh aggregae money shocks and idiosyncraic endowmen shocks. In i, agens rade a complee se of sae-coningen bonds in he asse marke, and hus markes are as complee as hey can be subjec o he rading fricion. Even wih he complee markes, however, he rading fricion leads agens wih differen idiosyncraic endowmen shocks o have poenially differen consumpion. Agens use hese bonds o insure away he effecs of heir idiosyncraic endowmen realizaions on heir porfolios of claims in he asse marke, and hence, in equilibrium, all agens have he same wealh. This feaure of he model vasly simplifies he analysis. When discussing exchange raes, we use a wo-counry version of our segmened markes economy. In his wo-currency, cash-in-advance model, shoppers mus use he local currency o purchase he local good. We absrac from rade in goods across counries in order o focus on he role of asse marke segmenaion. By doing so, we follow he spiri of Lucas (1978) in using marginal raes of subsiuion o price asses even hough here is no rade in equilibrium. There is a large lieraure in his general area. Our paper is clearly relaed o he work of Baumol (1952) and Tobin (1956). More recenly, Jovanovic (1982), Romer (1986), and Chaerjee and Corbae (1992) have developed general equilibrium versions of Baumol s and Tobin s models and have used heir versions o sudy how differen consan inflaion raes affec he seady sae. In conras o hese sudies, however, ours examines he dynamic responses of ineres raes and exchange raes o money growh shocks.

5 money 77 Grossman and Weiss (1983) and Roemberg (1984, 1985) sudy he dynamic responses of ineres raes and exchange raes in deerminisic models wih exogenous segmenaion. In addiion o his segmenaion, he Grossman-Weiss-Roemberg models exogenously limi asse rade o unconingen bonds. Because of ha marke incompleeness, hese models have besides he pure liquidiy effecs from he rading fricions complicaed wealh effecs ha effecively limi hese sudies o one-ime unanicipaed shocks in deerminisic environmens. Grossman (1987) exends his work o include proporional coss of rading money and asses and, hence, endogenous segmenaion, bu because of he marke incompleeness, his work is also limied o one-ime unanicipaed shocks in deerminisic environmens. We go beyond his lieraure by analyzing a fully sochasic model wih shocks moivaed by he processes in he daa. Such a sep is clearly required o develop he empirical implicaions of marke segmenaion. We ake his sep by drawing on a device of Lucas (1990) ha les us absrac from wealh effecs. Lucas organizes agens ino coaliions in which agens pool heir resources and choose consumpion subjec o a single budge consrain for he coaliion as a whole, subjec o resricions on he rading echnology. Given he rading echnology, hen, markes are complee. Thus money injecions have real effecs only because of he rading fricions and no because of addiional exogenous marke incompleeness. We follow Lucas and allow agens o rade a complee se of sae-coningen bonds in he asse marke in order o eliminae complicaed bu inessenial wealh effecs. We differ from Lucas in erms of boh he rading fricion used and he resuls obained. He assumes ha he coaliion mus divide is cash each period ino one porion o be used o purchase goods and anoher porion o be raded for bonds in he asse marke before he size of he curren open marke operaion is announced. Unforunaely, in ha model, only unexpeced money shocks have real effecs. Hence, he model canno produce he Barr and Campbell (1997) observaions on expeced inflaion and real ineres raes. Moreover, in ha model, liquidiy effecs las only one period. Fuers (1992) and Chrisiano and Eichenbaum (1995) exend Lucas s (1990) model o include producion, and hey ge similar resuls. Grilli and Roubini (1992) and Schlagenhauf and Wrase (1995a) exend his work o he open economy. They also find ha he response of real exchange raes o money injecions lass only one period. In relaed work, Alvarez and Akeson (1997) use coaliions o exend he work of Roemberg (1985), bu wih his fricion, markes can be highly segmened only if velociy is exremely low. In an exension of heir basic model, Chrisiano and Eichenbaum (1992) and Chari, Chrisiano, and Eichenbaum (1995) add quadraic

6 78 journal of poliical economy coss of adjusing he porfolio beween periods o he infinie adjusmen coss wihin he period. They show ha his seup can generae persisen liquidiy effecs. Evans and Marshall (1998) use ha exended model o analyze he responses of ineres raes of differen mauriies o money shocks. Dosey and Ireland (1995) and Schlagenhauf and Wrase (1995b) criicize he lack of symmery in such a model beween he adjusmen coss wihin a period and across periods. Dosey and Ireland show ha when a model has quadraic coss of adjusmen boh wihin and across periods, he liquidiy effecs are small. In conras o he rading fricions in he lieraure iniiaed by Lucas (1990), our rading fricions are close o hose of he Baumol-Tobin models. These fricions can generae Barr and Campbell s (1997) observaions and persisen liquidiy effecs even hough coss are symmeric. Moreover, in our sudy, all he resuls can be derived wih paper and pencil, so ha he essenial driving forces in he model are easily seen. I. A One-Counry Economy Firs we skech he basic ouline of our model economy, and hen we fill in he deails. A. The Ouline We begin wih a one-counry, cash-in-advance economy wih an infinie number of ime periods p 0, 1, 2,, a governmen, and a coninuum of households of measure 1. Trade in his economy occurs in wo separae locaions: an asse marke and a goods marke. In he asse marke, households rade cash and bonds ha promise delivery of cash in he asse marke in he nex period, and he governmen inroduces cash ino he asse marke via open marke operaions. In he goods marke, households use cash o buy goods subjec o a cash-in-advance consrain, and households sell heir endowmens of goods for cash. Households face a real fixed cos of g for each ransfer of cash beween he asse marke and he goods marke. Excep for his fixed cos, he model is a sandard cash-in-advance model like ha of Lucas (1984). This economy has wo sources of uncerainy: idiosyncraic shocks o households endowmens and shocks o money growh. The iming wihin each period 1 is illusraed in figure 1. We emphasize he physical separaion beween markes by placing he asse marke in he op half of he figure and he goods marke in he boom half. Households ener he period wih he cash P 1 y 1 hey obained from selling heir endowmens a 1, where P 1 is he price level and y 1 is heir idiosyncraic random endowmen a 1. The governmen conducs

7 Fig. 1. Timing in he wo markes

8 80 journal of poliical economy an open marke operaion in he asse marke, which deermines he realizaion of money growh m and he curren price level P. Each household hen splis ino a worker and a shopper. The worker sells he household endowmen y for cash Py and rejoins he shopper a he end of he period. The shopper akes he household s cash P 1 y 1 wih real value m p P 1 y 1/P and shops for goods. The shopper can choose o pay he fixed cos g o ransfer cash Px wih real value x o or from he asse marke. This fixed cos is paid in cash obained in he asse marke. If he shopper pays he fixed cos, hen he cash-inadvance consrain is c p m x, where c is consumpion; oherwise, his consrain is c p m. Here and elsewhere, we assume ha he shopper s cash-in-advance consrain binds. Thus consumers choose no o carry cash in he goods marke from one period o he nex. Insead, hey save by holding ineres-bearing securiies in he asse marke. In Appendix A, we provide sufficien condiions for his assumpion o hold. Each household also eners he period wih bonds ha are claims o cash in he asse marke wih payoffs coningen on boh he household s idiosyncraic endowmen y 1 and he rae of money growh m in he curren period. This cash eiher can be reinvesed in he asse marke or, if he fixed cos is paid, can be ransferred o he goods marke. Likewise, if he fixed cos is paid, hen cash from he goods marke can be ransferred o he asse marke and used o buy new bonds. In figure 1, he asse marke consrain is B p qb P(x g) if he fixed cos is paid and B p qb oherwise, where B denoes he curren realizaion of he sae-coningen bonds and qb he household s purchases of new bonds. A he beginning of he nex period, period 1, his household sars wih cash Py in he goods marke and coningen bonds B in he asse marke. In equilibrium, some households choose o pay he fixed cos o ransfer cash beween he goods and asse markes and ohers do no. We refer o households ha pay he fixed cos as acive and households ha do no as inacive. Households wih eiher sufficienly low real balances or sufficienly high real balances are acive. Households wih low real balances ransfer cash from he asse marke o he goods marke, whereas hose wih high real balances ransfer cash in he opposie direcion. Households wih inermediae levels of real balances are in a zone of inaciviy and simply consume heir curren real balances. In his economy, bonds are a complee se of coningen claims o cash in he asse marke. These complee coningen claims, however, pay off in he asse marke. Accordingly, households do no choose idenical consumpion because hey mus pay a fixed cos o ransfer cash beween he goods and asse markes.

9 money 81 B. The Deails Now we flesh ou he ouline of his economy. Each household s endowmen y is independen and idenically disribued (i.i.d.) across households and across ime wih disribuion F, which has densiy f. Le Y p yf(y)dy be he consan aggregae endow men. Le y p (y 0,,y ) denoe a ypical hisory of individual shocks o endowmens up hrough period and f(y ) p f(y 0)f(y 1) f(y ) he probabiliy densiy over such hisories. Le M denoe he aggregae sock of money in period and m p M /M 1 he growh rae of ha money supply. Le m p (m 1,,m ) denoe he hisory of money growh shocks up hrough period and g(m ) he probabiliy densiy over such hisories. To make all households idenical in period 0, we need o choose he iniial condiions carefully. In period 0, households have B unis of governmen deb, which is a claim on B dollars in he asse marke in period 0. In his period, households rade only in bonds, no in goods. In period 1, households also have y 0/m 1real balances in he goods marke, where y 0 also has disribuion F and m 1 is he money growh shock a he beginning of period 1. The governmen issues one-period bonds wih payoffs coningen on he aggregae sae m. In period, given sae m, he governmen pays off ousanding bonds B(m ) in cash and issues claims o cash in he nex asse marke of he form B(m, m 1) a prices q(m, m 1). The gov- ernmen budge consrain in period 1, given sae m,is 1 B(m ) p M(m ) M(m ) q(m, m 1)B(m, m 1)dm 1. (1) m 1 In period 0, his consrain is B p m1 q(m 1)B(m 1)dm 1. In he asse marke in each period and sae, households rade a complee se of one-period bonds ha have payoffs nex period ha are coningen on boh he aggregae even m 1 and he household s en- dowmen realizaion y. A household in period wih aggregae sae m 1 1 and individual shock hisory y purchases B(m, m 1, y, y ) claims o cash ha pay off in he nex period coningen on he aggregae shock m 1 and he household s endowmen shock y. We le q(m, m 1, y ) be he price of such a bond ha pays $1.00 in he asse marke in period 1 coningen on he relevan evens. Because individual endowmens are i.i.d., we assume ha hese bond prices do no depend on he individual shock hisory y 1. Insead of leing each household rade in all possible claims coningen on oher households endowmens, we suppose ha each household rades only in claims coningen on he household s own endowmen wih a financial inermediary. This inermediary buys governmen

10 82 journal of poliical economy bonds and rades in he household-specific coningen claims. The laer approach is much less cumbersome han he former and yields he same oucomes. Specifically, he inermediary buys governmen bonds 1 1 B(m ) and sells household-specific claims of he form B(m, y ) o all he households in order o maximize profis for each aggregae sae 1 m : q(m, y )B(m, y, y )f(y )dy q(m )B(m, m 1) y 1 1 subjec o he consrain B(m ) p y B(m, y )f(y )dy. Arbirage im- 1 1 plies ha q(m, y ) p q(m )f(y ). Consider now he problem of an individual household. Le P(m ) denoe he price level in he goods marke in period. In ha marke, 1 in each period 1, a household sars wih real balances m(m, y ). I hen chooses ransfers of real balances beween he goods marke 1 1 and he asse marke x(m, y ), an indicaor variable z(m, y ) equal o zero if hese ransfers are zero and one if hey are no, and consumpion c(m, y ) subjec o he cash-in-advance 1 consrain: c(m, y ) p m(m, y ) x(m, y )z(m, y ), (2) 1 where in (2), when p 1, he erm m(m, y ) is given by y 0/m 1. New 1 money balances in period 1 are given by m (m, y ) p 1 P(m )y /P(m ). In he asse marke, each period a household sars wih coningen 1 claims B(m, y ) o cash delivered in he asse marke. The household purchases new bonds and makes cash ransfers o or from he goods marke subjec o he sequence of budge consrains for 1: 1 1 B(m, y ) p q(m, m 1)B(m, m 1, y, y )f(y )dm 1dy m 1 y 1 1 P(m )[x(m, y ) g]z(m, y ). (3) In period p 0, his asse marke consrain is 1 B p q(m )B(m, y )f(y )dy dm. m1 y Assume ha boh consumpion and real bond holdings B(m, 1 y )/P(m ) are uniformly bounded by some large consans. The problem for a consumer is o maximize uiliy p0 b U(c(m, y ))g(m )f(y )dm dy (4)

11 money 83 subjec o he consrains (2) and (3). The economy has a firm ha ransfers cash beween he asse marke and he goods marke. Since each ransfer of cash consumes g unis of goods, he oal resource cos of carrying ou all ransfers a is g z(m, y )f(y )dy. The firm purchases hese goods in he goods marke wih cash obained from consumers. The resource consrain is given by [c(m, y ) gz(m, y )]f(y )dy p Y (5) for all, m, and he money marke clearing condiion is given by M(m ) P(m ) p {m(m, y ) [x(m, y ) g]z(m, y )}f(y )dy (6) for all, m. An equilibrium is defined in he obvious way. II. Characerizing Equilibrium Here we solve for he equilibrium consumpion and real balances of acive and inacive households. In Secion III, we characerize he link beween he consumpion of acive households and asse prices. Again, hroughou we assume ha he cash-in-advance consrain always binds and he households hold only ineres-bearing securiies in he asse marke. Under his assumpion, a household s decision o pay he fixed cos o rade in period affecs only is curren consumpion and bond holdings and no he real balances i holds in he goods marke in laer periods. Inacive households simply consume he real balances hey currenly hold in he goods marke. More ineresing is he consumpion of acive households. Since he economy has a complee se of sae-coningen bonds, once a household pays he fixed cos o ransfer cash beween markes, i equaes is ineremporal marginal rae of subsiuion o ha of oher acive households. Since all households are idenical ex ane, all acive households have a common consumpion level c A(m ) ha de- pends only on he aggregae money shock m and no on heir idiosyncraic endowmens. We firs consruc he zones of aciviy and inaciviy for an arbirary consumpion level c A, and hen we use he resource consrain o deermine he equilibrium level of c A. Define he funcion h(m; c A) p [U(c A) U(m)] U (c A)(c A g m). (7) This funcion measures he ne gain o a household from swiching from being an inacive household wih consumpion m o an acive

12 84 journal of poliical economy household wih consumpion c A. Noe ha his measure of he ne gain is simple and saic, wih only curren variables; i is no dynamic. This simpliciy sems from our assumpion ha he cash-in-advance consrain binds, so ha a household s decision o pay he fixed cos in period does no affec is real balances and consumpion in fuure periods. The firs wo erms on he righ side of (7) measure he direc uiliy gain wihin he curren period from he household s swich from inaciviy o aciviy, and he hird erm measures he uiliy cos of he required ransfer of real balances from he asse marke. Wih c A fixed, i is opimal for a household wih real balances m o rade cash and bonds and consume c A if h is posiive and no o rade and insead consume m if h is negaive. Noe ha h is sricly convex in he argumen m; i aains is minimum a m p c A and is negaive a his minimum if g 1 0. Thus h ypically crosses zero wice. Define low and high cuoffs for rade, y L(c A, m) and y H(c A, m), as he soluions o ( ) y h ; ca p 0 (8) m when boh of hese soluions exis. If (7) is negaive for all m! c A, hen se y L(c A, m) p 0; if i is negaive for all m 1 c A, hen se y H(c A, m) p. This cuoff rule is illusraed in figure 2. Noe ha as he fixed cos g goes o zero, y L(c A, m)/m and y H(c A, m)/m converge o c A, so ha all house- holds become acive. Given his form for he zones of aciviy and inaciviy, we use he resource consrain o deermine he equilibrium values of acive households consumpion and corresponding cuoffs. Togeher, he cash-inadvance consrain and consrains (5) and (6) imply ha he price level is P(m ) p M(m )/Y, he inflaion rae is p p m, real money holdings 1 are m(m, y ) p y 1/m, and he consumpion of inacive households 1 is c(m, y ) p y 1/m. Subsiuing he inacive household s consump- ion ino he resource consrain (5) and using he cuoff rule defined in (8) gives y 1 H A L H m yl (c g)[f(y ) 1 F(y )] yf(y)dy p Y, (9) where we have suppressed he explici dependence of c A, y H, and y L on m. Clearly, hese cuoff poins and consumpion levels of acive households depend only on m, whereas he consumpion level of inacive households depends only on (m, y 1 ). If we fix m 1 and use (8) o solve for y L and y H as funcions of c A, we see ha he lef side of (9) is coninuous and sricly monoonic in c A. Thus any soluion o he equaions for he equilibrium values of

13 money 85 Fig. 2. Cuoff rule defining zones of aciviy and inaciviy acive households consumpion and cuoffs is unique. These argumens give he following proposiion. (For deails, see App. A.) Proposiion 1. The equilibrium consumpion of households is given by 1 y 1 if y (y (m ), y (m )) 1 L H m c(m, y ) p {c (m ) oherwise, A where he funcions y L(m), y H(m), and c A(m) are he soluions o (8) and (9). In our analysis of asse prices, we can use he sequence of budge consrains (3) o subsiue ou for he household s bond holdings and replace hese consrains wih a single period 0 consrain on household ransfers of cash beween he goods and asse markes. As we show in Appendix A, period 0 nominal asse prices are deermined by he firsorder condiion for acive households: b U (c A(m ))g(m ) p lq(m )P(m ), (10) where l is he Lagrange muliplier on households period 0 budge

14 86 journal of poliical economy consrain and Q(m ) is he price in dollars in he asse marke in period 0 for a dollar delivered in he asse marke in period in sae m. Since all households are idenical in period 0, he mulipliers in he Lagrangian are he same for all of hem. In wha follows, we suppress reference o he sae m and wrie he n price of an n-period bond ha coss q dollars in period and pays $1.00 in all saes in period n as U (c A n) P n n q p b E. (11) U (c ) P A n There is a key difference beween his formula and he one ha arises in he sandard cash-in-advance model. In he sandard model, he relevan marginal uiliy for asse pricing is ha of he represenaive household, and he corresponding consumpion is aggregae consumpion. Here, he relevan marginal uiliy for asse pricing in period is ha of he acive households in period and ha expeced for hem in period n. These marginal uiliies in periods and n are no hose of any single household, bu raher hose of whichever households happen o be acive in hose periods. This disincion is criical for he resuls ha follow. III. Asse Prices Now we develop he economy s links beween money injecions and asse prices. The link inroduced wih marke segmenaion is how an acive household s consumpion responds o a money injecion. We sar wih his link and hen develop formulas for asse prices. A. Money Injecions and Consumpion We develop sufficien condiions for a money injecion o raise he consumpion of acive households. We begin wih a discree example and follow wih a coninuous example. Consider firs a simple example in which y akes on hree values, y 0! y 1! y 2, wih probabiliies f 0, f 1, and f 2, respecively. We conjecure an equilibrium in which, when money growh is m, households wih he cenral value of he endowmen y 1 choose no o rade and hose wih low and high endowmens y 0 and y 2 do choose o rade. Under his conjecure, for money growh shocks m close o m, we know from he resource consrain ha each acive household consumes an equal share

15 money 87 of he acive households aggregae endowmen plus he inflaion ax levied on inacive households minus he fixed cos, or from (9), ) y0f0 y2f2 1 y1f1 c A (m) p ( 1 g. (12) f f m f f The corresponding cuoffs y L(c A(m), m) and y H(c A(m), m) are found from (8). This conjecure is valid as long as y 0! y L(c A(m), m)! y 1! y H(c A(m), m)! y 2. Clearly, an increase in he money growh rae m raises he inflaion ax levied on each inacive household s real balances. In equilibrium, asse prices adjus o redisribue hese inflaion ax revenues o acive households. In his example, he number of acive households does no vary wih he money injecion, so he consumpion of each acive household increases. Specifically, d log c A (y1f 1)/m p, (13) d log m c A(f 0 f 2) which is he raio of he oal consumpion of inacive households o ha of acive households. Now consider an example in which y has a coninuous densiy. Here, as before, an increase in he money injecion reduces beginning of period real balances for every household. Bu now wih a coninuous densiy of hese real balances across households, some households swich zones. Because inflaion has reduced real balances, he iniially inacive households near he lower cuoff y L find i opimal o pay he fixed cos and become acive, whereas he iniially acive households near he upper cuoff y H find i opimal no o pay he fixed cos and become inacive. Boh of hese swiches end o reduce he level of acive households consumpion. Inuiively, acive households as a group pool heir real balances and have equal consumpion. Inacive households ha become acive bring lower han average real balances o his group, whereas acive households ha become inacive ake away higher han average real balances. As long as he fracion of households near hese cuoffs is no oo large, he consumpion of acive households increases wih a money injecion.

16 88 journal of poliical economy More formally, differeniaing (8) and (9) gives y { L [F(y L) 1 F(y H)] mf(y L) ( ca g ) hl m yh dca mf(y H) ca g h H (14) m dm where y H A y L ( ) } ( ) ( ) 1 y y L y L y H y H p f(y)dy c g f(y L) ca g f(y H), m m m m m m [ ] ca g (y i/m) hi p U (c A ). U (c ) U (y /m) From (7) and (8) we know ha y L/m! c A! (y H/m) g. Thus h H and h L are posiive, and so is he erm in braces on he lef side of (14). On he righ side of (14), he firs erm is posiive and he las wo erms are negaive, so wihou furher resricions, he sign of he righ side is ambiguous. The firs erm measures he effec of he inflaion ax on he consumpion of inacive households when he zone of inaciviy is held fixed. The las wo erms measure he change in he consumpion of inacive households ha resuls from a change in he zone of inaciviy. The fracion f(y L ) of households a he lower edge of he zone wih real balances y L/m become acive, and he fracion f(y H) of house- holds a he upper edge of he zone wih real balances y H/m become inacive. As long as he fracion of households a hese edges is no oo large, he consumpion of acive households increases. In Appendix B, we give an example in which dc A/d log m is posiive and y has a lognormal disribuion. Examples can, of course, also be consruced in which he fracion of households a he edges of he zone is large and an increase in money growh decreases he consumpion of acive households. Here, hough, we focus on wha we consider he sandard case when he opposie holds. A i B. Money Injecions and Asse Prices We now urn o he link beween money injecions and asse prices. In order o ge analyical resuls, we make several assumpions. Le he log of money growh in period be normally disribued and have consan condiional variance over ime. Le m be defined by log m p E log m, 1 j where E is he uncondiional expecaion. Le U(c) p c /(1 j), where he risk aversion parameer j 1 0. Le c A denoe he consumpion of acive households when money growh is equal o m. To a firs-order

17 money 89 approximaion, he log of an acive household s marginal uiliy is given by where log U (c ) p log U (c ) f(log m log m), A A d log c A f p j (15) d log m evaluaed a m p m. The parameer f is he elasiciy of an acive house- hold s marginal uiliy wih respec o a money injecion. Given hese assumpions, we shall analyze he relaion beween money and ineres raes. IV. Ineres Rae Dynamics Now we illusrae he dynamics of money injecions, expeced inflaion, and ineres raes. We firs show ha he model can produce he negaive relaion beween expeced inflaion and real ineres raes noed by Barr and Campbell (1997). We hen give condiions under which he effec of money injecions on real ineres raes dominaes heir effec on expeced inflaion, so ha money injecions have liquidiy effecs. We work ou he model s implicaions for he dynamics of he ineres rae erm srucure for wo common processes for money growh and inflaion: an auoregressive process and a long-memory process. We begin wih he auoregressive process because i is simple and generaes he well-known Vasicek (1977) model for he dynamics of he erm srucure. Moreover, according o Chrisiano e al. (1998), firs-order auoregressive processes do a good job of approximaing he responses of money growh and ineres raes o a money shock. Using a differen VAR, however, Cochrane (1994) has found a more proraced response for money growh o a money shock. Moivaed by his finding, we sudy a process for money growh wih impulse responses ha decay more slowly han hose of a firs-order auoregressive process. For simpliciy, we consider a long-memory process. We show ha wih such a process, a money injecion leads o a fall in he shor-erm nominal rae followed by a rise. We show ha he shock also wiss he yield curve: on impac, shor raes fall and long raes rise. A leas since Friedman (1968), economiss have argued ha money injecions have hese effecs on ineres raes. Moreover, Cochrane has found such a response for ineres raes in his VAR. Throughou he following analysis, money injecions have wo effecs on nominal ineres raes: an expeced inflaion effec and a segmenaion effec, as can be seen from he Fisher equaion: ˆı p rˆ E p ˆ, where i 1

18 90 journal of poliical economy is he nominal ineres rae and r is he real ineres rae. (Here and elsewhere, a care over a characer denoes a log deviaion.) Using a log-linear approximaion o (11), we can express he expeced inflaion effec as E pˆ p E m ˆ. (16) 1 1 This holds because in he model, boh oupu and velociy are consan, so expeced inflaion is simply expeced money growh. Similarly, we can express he segmenaion effec as r p U (c ) EU(c ) p f(e m m ), (17) ˆ A A+1 ˆ 1 ˆ where fmˆand fe mˆ 1 are he effecs of he money injecion on he acive households marginal uiliy in periods and 1. In he sandard model, g p 0, so f p 0 and real ineres raes are consan. In our model, g 1 0, so f 1 0; hus a money growh shock ha increases m also increases he consumpion of acive households in and drives down heir marginal uiliy in. If he money growh shock raises expeced money growh in 1 as well, hen i raises expeced consumpion and lowers expeced marginal uiliy for acive households in 1. As long as he money growh process is mean revering, so ha Emˆ 1 mˆ is decreasing in m, ˆ an increase in money growh drives down real ineres raes. Wih such processes, he model reproduces he negaive relaion beween expeced inflaion and real raes found by Barr and Campbell (1997), since a money injecion drives expeced inflaion up and real raes down. Our model produces liquidiy effecs when he segmenaion effec (17) dominaes he expeced inflaion effec (16). The overall magniude of he segmenaion effec depends on wo parameers: he elasiciy of he marginal uiliy of acive households wih respec o money growh f and he persisence of a money growh shock as measured by Emˆ 1 m ˆ. The segmenaion effec increases he higher f is, ha is, he more responsive an acive household s marginal uiliy is o a money injecion. This effec is smaller he greaer he persisence of money growh. If money growh is emporary, hen a given money injecion will lead o a emporary increase in acive households consumpion and, hence, o a relaively large drop in he real ineres rae. As he shock o money growh becomes more persisen, a given money injecion leads o a more permanen increase in acive households consumpion and, hence, o a smaller drop in he real ineres rae. We urn now o an analysis of he wo common processes for money growh and inflaion.

19 money 91 A. Example 1: Auoregressive Process Assume ha money injecions saisfy mˆ 1 p rmˆ e 1, where r is he persisence of he money shock and e 1 is a normal, i.i.d. innovaion 2 wih mean zero and variance j e. The expeced inflaion effec is given by E pˆ p rm ˆ, and he segmenaion effec is given by rˆ 1 p f(r 1)mˆ so ha ˆ i p [f(r 1) r]m ˆ. As long as money growh is mean revering, so ha r! 1, expeced inflaion and real raes move in he opposie direcion. Noice ha if r f 1, (18) 1 r hen he segmenaion effec dominaes he expeced inflaion effec, and a money injecion leads o a fall in nominal ineres raes on impac. Consider nex he dynamics of he shor-erm ineres rae. Since k k k E ˆ ˆ and ˆ ˆ we have ha ˆ ˆ p k 1 p r Ep 1 Er k p r r, Ei k p r i. Thus real and nominal ineres raes have he same persisence as money shocks. If (18) holds, hen a money injecion leads nominal raes o iniially fall and decay back o zero a rae r. Clearly, hese liquidiy effecs are persisen whenever money shocks are persisen. Consider he effecs on he yield curve. In our model, he dynamics of he erm srucure saisfies he expecaions hypohesis wih a consan risk premium: movemens in long-erm raes are an average of movemens in expeced fuure shor-erm raes. In fac, his is rue for any log-linear model wih consan condiional variances. When (18) holds, so ha he segmenaion effec dominaes he expeced inflaion effec, a money injecion lowers he shorer yields by more han he longer yields and hus seepens he yield curve. Each yield follows an auoregressive process and reurns o is mean value a rae r. For his example, hen, our general equilibrium model generaes he dynamics of he erm srucure summarized by he Vasicek (1977) model. (Of course, here is subsanial evidence ha he expecaions hypohesis is a poor approximaion of he dynamics of he erm srucure; see Campbell, Lo, and MacKinlay [1997]. Addressing ha problem, however, is beyond our scope here.) Consider he magniude of f required for liquidiy effecs for his auoregressive example. Chrisiano e al. (1998) argue ha he impulse response for M2 growh following a money shock is well approximaed by an auoregressive process wih r p.5. Wih his persisence, (18) implies ha he model produces liquidiy effecs for f 1. Geing a handle on he level of segmenaion in he daa is harder. To ge a rough

20 92 journal of poliical economy feel for wha differen levels of f enail, noe ha combining he formula from our discree example (13) wih equaion (15) gives oal consumpion of inacive households f p j. (19) oal consumpion of acive households Consider f p 2. In order o inerpre his value, we need o ake a sand on he risk aversion parameer j. The lieraure uses a wide range of esimaes for j. The business cycle lieraure commonly uses j p 2, bu esimaes easily range as high as j p 8. Wih j p 2, (19) implies ha we need half of he households o be no acively rading money for ineres-bearing asses in any given period in order o generae f p 2. Wih j p 8, we need only one-fifh of he households o be inacive in order o ge f p 2. We illusrae he model s predicions in figure 3. In figure 3a, we graph he impulse responses o a money shock of money growh and (annualized) shor-erm nominal ineres raes wih f p 2. The re- sponses are similar o hose found by Chrisiano e al. using M2. In figure 3b, we graph he yield curves a hree differen imes: a he ime of he shock s impac, one quarer afer he shock, and hree quarers afer he shock. These responses show he yield curve seepening on impac and hen revering slowly o is normal posiion. Since ineres raes in he model saisfy he expecaions hypohesis, he impulse response plo for he shor-erm rae compleely deermines he dynamics of yields of long mauriies. (Acually, he impulse response of Ei ˆ k is he response of he one-period forward rae of mauriy k in period, and he yields are jus averages of he forward raes.) So he wo plos in figure 3 are jus wo ways o summarize he same informaion. So far, we have worked ou relaions beween money injecions and ineres raes for a simple firs-order auoregressive process for money growh. In ha example, a money injecion eiher lowers ineres raes a all mauriies or raises hem a all mauriies. This paern is no a general feaure of our model, bu raher resuls from he special naure of a firs-order auoregressive process. To illusrae he implicaions of our model more generally, we develop hese relaions when money growh has a general moving average represenaion ˆm p jp0 ve j j, where he shocks e j are independen and 2 N(0, j e ). In his case, equaions (16) and (17), characerizing he impac of money injecions on expeced inflaion and he real ineres rae, become ˆ and ˆ jp1 jp0 Accordingly, E p p ve r p f (v v)e. 1 j 1 j j 1 j j

21 Fig. 3. How our model responds o a money shock: Paerns implied by an auoregressive process. a, Impulse responses of shor-erm nominal ineres raes and money growh. b, Ineres rae yield curve on shock s impac and one and hree quarers laer.

22 94 journal of poliical economy he impulse responses of expeced inflaion and he real ineres rae following a moneary shock e in period are given by ˆ E p p v e, k 1 k 1 ˆ Er p f(v v )e. k k 1 k In general, hen, he srengh of he expeced inflaion effec following a moneary shock depends on he level of hese moving average coefficiens v k, and he srengh of he segmenaion effec depends on he difference (vk 1 v k) of hese moving average coefficiens. Thus a money injecion can cause ineres raes o fall a some horizons and rise a oher horizons. In paricular, a posiive money injecion e lowers he expeced nominal ineres rae a k if v f(v v )! 0. (20) k 1 k 1 k When money injecions are a firs-order auoregressive process, v k p r k and (20) reduces o (18). In his case, money injecions eiher lower ineres raes a all horizons k or raise hem a all horizons. Inuiively, his happens because when he moving average coefficiens decay geomerically, he relaive srenghs of he segmenaion effec and he expeced inflaion effec are he same a all horizons. A leas since Friedman (1968), economiss have argued ha money injecions lead o an iniial decline in shor-erm ineres raes followed by a rise. If money injecions are a moving average process in which he coefficiens v k decline rapidly a firs and more slowly laer, hen (20) implies ha he segmenaion effec is relaively sronger a shorer horizons and relaively weaker a longer horizons. Thus a money injecion wih such moving average coefficiens can lead o an iniial decline in nominal ineres raes followed by a subsequen rise. We nex provide a simple parameric example illusraing his poin. B. Example 2: Long-Memory Process Consider he moving average process ˆm p jp0 ve j j, where v j are he moving average coefficiens and e is a whie-noise process. The longmemory process is a moving average process in which he coefficiens saisfy he recursion ( ) 1 d vj p 1 v j 1 1 for j 1 and! d!, and he e j are independen and disribued N(0, j e ). The parameer d conrols he rae of decay of he moving average coefficiens. These coefficiens decay a a rae (1 d)/j! 1. For j 1

23 money 95 large j, his rae approaches zero, which is he source of he long memory. Using (16) and (17), we can show ha he shor-erm nominal ineres rae ˆi p jp1 ae, where j 1 j [ ( )] 1 d 1 d aj p f 1 v j 1. j j Here, in he brackes, he firs erm is he segmenaion effec and he second is he expeced inflaion effec. Since he coefficiens v j are all posiive, for large enough j he expeced inflaion effec mus dominae he segmenaion effec, and a j mus be posiive. If f 1 d/(1 d), hen, for j p 1, he segmenaion effec ouweighs he expeced inflaion ef- fec; so for small j, a j is negaive. If we ignore inegers, we see ha a j goes from negaive o posiive a j p (1 f)(1 d). Noice ha he more segmened he marke, he longer he period in which he segmenaion effec ouweighs he expeced inflaion effec. We illusrae he paern implied by he long-memory process wih 1 d p and f p 2 in figure 4. In figure 4a, we see ha he nominal rae 4 drops on he money shock s impac and hen rises in he hird quarer afer he shock. Ineresingly, his paern is similar o ha esimaed by Cochrane (1998), which he argues is represenaive of resuls in he VAR lieraure. In figure 4b, we plo he yield curves on impac, one quarer afer he shock, and hree quarers afer he shock. In his figure, we see ha on impac, he money growh shock wiss he yield curve, lowering shor yields and raising long ones. Afer several quarers, shor yields rise and all yields slowly move back o heir average values. The differen responses of nominal ineres raes o a money injecion, shown in figures 3 and 4, sem from he differen paerns of he moving average coefficiens implied by he wo processes for money growh. In figure 5 we plo hese moving average coefficiens. As we have discussed, wih he auoregressive process, hese coefficiens decline geomerically, and he relaive srenghs of he segmenaion effec and he expeced inflaion effec are he same a all horizons; hus he impulse response of nominal ineres raes has he same sign a all horizons. Relaive o he moving average coefficiens of he auoregressive process, hose of he long-memory process decline more rapidly a firs and more slowly laer. From (20) we see ha such a paern leads nominal ineres raes o decline a firs and hen rise laer. V. Exchange Raes Having demonsraed ha our segmened marke model can reproduce he major observed ineres rae responses o money injecions, we urn

24 Fig. 4. How our model responds o a money shock: Paerns implied by a long-memory process. a, Impulse responses of shor-erm nominal ineres raes and money growh. b, Ineres rae yield curve on shock s impac and one and hree quarers laer.

25 money 97 Fig. 5. Moving average coefficiens: auoregressive and long-memory processes now o exchange raes. Here he feaures we wan o reproduce are differen for counries wih differen raes of inflaion. In low-inflaion counries, real and nominal exchange raes have similar volailiy, are highly correlaed, and are persisen (see Mussa [1986] and our able 1 below). In high-inflaion counries, nominal exchange raes are subsanially more volaile han real exchange raes (see fig. 7, discussed below). The sandard model canno reproduce hese observaions. We develop a wo-counry version of our segmened markes economy ha can. A. A Two-Counry Economy Firs we develop a more sophisicaed represenaion of moneary policy han we used in he one-counry model. Earlier we explored he implicaions of he one-counry model only for he impulse responses o exogenous money shocks. Here we explore he model s predicions for some uncondiional momens of he daa, so we need o ake a firmer sand on he policy rule followed by he moneary auhoriy. As we documen below, in he daa, nominal ineres raes are subsanially

26 98 journal of poliical economy more persisen han money growh raes. To capure his, we model money growh as he sum of an exogenous componen and an endogenous componen ha offses a ype of money demand shock. Consider now a wo-counry, cash-in-advance economy ha exends he work of Lucas (1982). We refer o one counry as he home counry and he oher as he foreign counry. For simpliciy, we absrac from rade in goods across counries by having he households in each counry desire only he local good. Specifically, households in he home counry use he home currency, called dollars, o purchase a home good. Households in he foreign counry use he foreign currency, called euros, o purchase a foreign good. In he asse marke, households rade he wo currencies and dollar and euro bonds ha promise delivery of he relevan currency in he asse marke in he nex period, and he wo governmens inroduce heir currencies via open marke operaions. As before, each ransfer of cash beween he asse marke and any individual household in eiher goods marke has a real fixed cos of g. Households in he home counry choose o ransfer only he home currency, and hose in he foreign counry choose o ransfer only he foreign currency. In he asse marke, however, households may choose o hold heir wealh in bonds denominaed in eiher currency, and as a resul, he relaionship beween he nominal exchange rae and he prices of bonds denominaed in he wo currencies is consisen wih he sandard (covered) ineres rae pariy condiions. In order o generae a ype of money demand shock, we allow shocks o he disribuion of idiosyncraic endowmens in he wo counries. The densiies of he endowmens are now given by f(y; h ) and f(y ; h ), where h and h are i.i.d. shocks, boh wih mean h. Thus he aggregae shock is s p (m, m, h, h ), and s p (s 1,,s ) is is hisory. Le g(s ) denoe he densiy of he probabiliy disribuion over such hisories. We le home households rade a complee se of dollar-denominaed claims wih a world inermediary, and we le foreign households similarly rade euro-denominaed claims. The home governmen s bonds are dollar bonds, and is budge consrain is (1) as before. The foreign governmen s bonds are euro bonds, and is budge consrain is he obvious analogue. The world inermediary buys boh dollar- and euro-denominaed governmen bonds and rades in boh dollar and euro householdspecific coningen claims in order o maximize profis for each aggregae sae s. Lack of arbirage across currencies implies ha 1 1 q(s, s 1) p q (s, s 1)e(s )/e(s ). Here q and q are he prices for one- period dollar and euro bonds and e is he nominal exchange rae in erms of dollars per euro. We use his relaionship o solve for movemens in nominal exchange raes. To solve for he period 0 nominal exchange rae e 0, we need o choose