Breaking the New Keynesian Dichotomy: Asset Market Segmentation and the Monetary Transmission Mechanism

Transcription

1 Breaking he New Keynesian Dichoomy: Asse Marke Segmenaion and he Moneary Transmission Mechanism Rober G. King: Boson Universiy and NBER Julia K. Thomas: Federal Reserve Bank of Philadelphia and NBER February 2, 27 Absrac We develop a general framework o examine how he presence or absence of a moneary ransmission mechanism shapes aggregae responses o boh shocks and he e ecs of moneary policy. Our framework ness wo leading moneary models: a ex-book New Keynesian seing and a seing where small ransacions coss associaed wih adjusmens in households money balances lead o ime-varying asse marke segmenaion and an evolving disribuion of money across households. While each model allows for moneary nonneuraliies, hey di er sharply in heir predicions abou how real and nominal disubances, alongside moneary policy inervenions, ransmi hemselves hrough he economy. In he ex-book New Keynesian seing, he e ecive isolaion of a single condiion deermining aggregae money demand imposes a dichoomy prevening a nonrivial moneary ransmission mechanism, and i eliminaes any role played by he demand for money in he deerminaion of aggregae demand whenever he moneary auhoriy uses an ineres rae rule. As such, i raionalizes a narrow aenion o direc links beween ineres rae seing and objecives such as desired pahs for in aion and real aciviy in a wide range of curren discussions involving moneary policy. In his paper, however, we argue ha he simpliciy of he Keynesian dichoomy is no an ineviable or desirable feaure of a racable moneary model. The basic mechanism implying non-neuraliies in our second nesed model does no permi he dichoomy raised above, because here is no single isolaed condiion associaed wih money demand. Raher, households decisions regarding consumpion and labor supply in his model are inimaely a eced by boh heir individual money holdings and, hrough wages and ineres raes, he enire disribuion of money balances. This implies a rich moneary ransmission mechanism, in ha he level of aggregae demand depends crucially on moneary facors. When we examine a composie seing where he rm-side sluggish price adjusmen cenral hroughou New Keynesian moneary analysis is allowed o inerac wih he rich money demand mechanism implied by household-side invenory-heoreic porfolio managemen, we nd ha he resuling model is no only racable, bu also has very desirable properies from an empirical sandpoin. For example, when i is solved under a money sock rule, i implies a pah for he nominal ineres rae ha iniially declines in he face of a moneary expansion and reurns only gradually reurns o is seady-sae value, in keeping wih he liquidiy e ec

2 documened across a broad range of empirical sudies. By conras, in he sandard New Keynesian model, he same shock ineviably raises he nominal rae. Moreover, when our composie model is solved under a sandard ineres rae rule, here are much more proraced dynamic responses following shocks o moneary policy, as well as nonmonoone responses o real shocks. These desirable implicaions from our model come precisely because i is no possible o describe aggregae demand wihou reference o money demand; he disribuion of ransacions balances across individuals is an essenial par of he ransmission mechanism from moneary policy acions o real economic aciviy. Inroducion Mos small modern macroeconomic models used for concepual and quaniaive moneary policy analysis have he propery ha he demand for money is irrelevan o he deerminaion of aggregae demand, when he moneary auhoriy is using an ineres rae rule. This propery which we label he Keynesian dichoomy has a long hisory in macroeconomic analysis. Indeed, he earlies generaion of quaniaive moneary policy models, buil by various research eams in he 95s and 96s, did no even include a demand for money. 2 However, is role has been srongly reinforced by he currenly dominan se of macroeconomic models, in which i is a key ingredien. Furher, in a wide range of curren discussions of moneary policy heoreical, applied, and pracical his propery is used o raionalize a focus enirely on he links beween ineres rae seing and objecives such as desired pahs for in aion and real aciviy. argue ha he Keynesian dichoomy is no an ineviable or desirable feaure, as follows. In his paper, we Firs, in secion 2, we develop a composie moneary framework ha ness he ex-book fully ariculaed New Keynesian macroeconomic model as a special case. In his seing, here is moneary non-neuraliy in he shor run because rms face small menu coss of price adjusmen. In such models, he dynamic adjusmen process o real and nominal shocks is heavily in uenced by he fac ha some rms make price adjusmens quickly, while ohers do no, so ha here is an evolving disribuion of nominal prices. This disribuion implies ha adjusmens in he aggregae price level following moneary disurbances are gradualized, so ha nominal shocks have real consequences over he shor-run. In is mos common form, he Keynesian dichoomy is imposed wihin he model by assuming a role for money ha is self-conained, e ecively quaranined from oher variables in he model. Here, we begin by considering a simple and popular version in which real balances appear in he economy only as an addiively separable source of household uiliy. In secion 3., we use his model o display some key ideas and o obain a se of reference calculaions. Thereafer, we show how he perfec dichoomy of his model leads o complee irrelevance of money demand, indeed money iself, when moneary policy akes on an acive sabilizaion role implemened hrough Examples of hese models may be found in Clarida, Gali and Gerler (999), McCallum and Nelson (9xx), and Woodford (9xx). framework as an IS model. 2 Add hisorical references here. To sress he naure of moneary policy in his seing, Kerr and King (99x) describe he 2

3 ineres rae argeing, he Taylor rule so frequenly analyzed hroughou modern moneary economics. This implies ha he iniially limied role of money in he model economy is e ecively eliminaed o accommodae a policy rule ha maps quie direcly from ineres rae seing o realized objecives for he pahs of in aion and oupu. As a resul, beween insrumen and goals, here is e ecively no moneary ransmission mechanism o complicae, or enrichen, our analysis of he model economy s dynamics. The framework we develop in secion 2 also houses, as a second special case, a exible-price model in which households face small ransacions coss of making adjusmens in he moneary balances which hey use o nance expendiure. In his class of models, he dynamic adjusmen process o real and nominal shocks is heavily in uenced by he fac ha some households make porfolio adjusmens quickly, while ohers do no. Because his seing yields an evolving nonrivial disribuion of money, he Keynesian dichoomy is forcefully broken. Here, he level of aggregae demand depends crucially on moneary facors. In his alernaive model, moneary non-neuraliy arises from ime-varying and heerogeneous money spending raes on he par of households, raher han from heerogeneous nominal prices on he par of rms. A ransacions-based role for money is imposed hrough he assumpion ha all goods and labor marke ransacions mus be conduced wih money. However, he environmen di ers from ha in a radiional cash-in-advance model in wo imporan respecs. Firs, households are able o adjus heir money holdings afer he resoluion of all uncerainy wihin a period, so hey canno be forced o hold an undesirable quaniy of money wihin he period purely as a resul of an unforeseen acion on he par of he moneary auhoriy. Second, however, his abiliy o adjus money balances is no wihou fricions, so households do no compleely undo he e ecs of a moneary shock hrough an immediae one-for-one adjusmen in he aggregae price level, as hey would in a perfec-foresigh cash-in-advance seing. Insead, households are assumed o face xed coss of ransferring wealh beween ineres-bearing asses and money. Given hese ransacions coss, here is endogenous asse marke segmenaion in ha households choose o access heir ineres income infrequenly. To implemen heir infrequen asse marke paricipaion, households carry invenories of money in excess of curren spending o nance heir spending over fuure daes. As a resul, mos households do no exhaus heir available money wihin any given period, which implies ha aggregae velociy in his economy deviaes from. Moreover, he aggregae spending rae varies over ime, because households are able o change boh he iming of heir paricipaion in asse markes in response o real and nominal shocks, as well as he individual spending raes hey adop given curren money holdings. I is well known ha marke segmenaion implies ha open marke operaions can have real e ecs, because hey direcly involve only a subse of households. The advanage of he endogenous version of he framework we examine is ha i allows endogenous changes in he fracions of households paricipaing in he asse markes i.e., changes in he exen of marke segmenaion over ime in response o aggregae disurbances. These changes can lead o long-lasing disrupions in he disribuion of household money holdings 3

4 ha lend added persisence o movemens in real and nominal variables. Our resuls for he endogenous segmenaion model under exible prices in secion 3.2 reveal ha changes in he disribuion of money can deliver powerful propagaion of shocks, yielding no only persisen movemens in oupu, employmen, in aion and real ineres raes following a money injecion, bu also nonmonoone responses in hese variables following a persisen shock o produciviy. Moreover, we illusrae how he convenien mapping from he moneary policy rule o he economy s resuling aggregae dynamics is desroyed in his seing. Money demand in his seing exhibis no dichoomy; raher, i is inexricably inerwoven wih oher variables in he economy. Thus, while ineres rae arges coninue o shape economic aciviy, here, he arges are hemselves in uenced by boh curren and fuure changes ha hey induce in money demand. A powerful moneary ransmission mechanism implies ha he e ecs of moneary policy can be boh complex and long-lived. Following our analyses of he wo disinc mechanisms in isolaion, we nex consider he implicaions of he full composie model conaining boh mechanisms, he evolving disribuion of nominal prices, alongside he rich disribuion of relaive money holdings across households. We show ha his model has some very desirable properies. When i is solved under a money sock rule, for one example, i has he implicaion ha he nominal ineres rae iniially declines in he face of a moneary expansion, while i ineviably rises in he sandard New Keynesian model. When i is solved under a sandard ineres rae rule, we observe much more proraced dynamic responses o moneary policy shocks, enirely due o he richer dynamics of aggregae demand. Mos economiss would view hese feaures as desirable implicaions of a macroeconomic model, bringing i closer o convenional viewpoins abou he implicaions of acual policy inervenions. However, hese implicaions come precisely because i is no possible o describe aggregae demand wihou reference o money demand: he disribuion of ransacions balances across individuals is a key par of he ransmission mechanism from moneary policy acions o real economic aciviy. 2 Composie Framework This secion presens a composie model conaining boh he disinguishing feaure of New Keynesian models of moneary economies, menu coss of price adjusmen, as well as he disinguishing feaure of Segmened Asse Markes models, small ransacions coss associaed wih households adjusmens in he division of heir wealh beween liquid versus illiquid asses. This framework ness boh he exbook New Keynesian environmen and he exible-price segmened asse markes environmen as special cases, so ha we can sudy he implicaions of each in isolaion before examining he dynamics of he full model. Our composie model embeds a ypical New Keynesian producion side of he economy, where he iming of monopolisically compeiive inermediae goods producers nominal price-seing is governed by a ( a) Calvo hazard, ogeher wih an endogenously segmened asse markes side of he economy, where households mainain invenories of money o nance heir spending across muliple periods, due o xed ransacions coss incurred when hey ransform heir less-liquid 4

5 asses ino money (and vice versa). Here, we describe rs he producion-side of he economy, hen he household-side, and follow nex wih he equilibrium condiions by which he wo are linked. Thereafer, we brie y deail he assumpions under which his environmen can also be used o examine he more ypical New Keynesian environmen, as well as he exible-price segmened asse markes environmen, as special cases. A nal good, Y, is produced by a perfecly compeiive represenaive rm using a coninuum of inermediae inpus, y (i), i 2 [; ]. We assume a consan elasiciy of subsiuion across R i inermediae goods in he aggregae producion of nal goods, Y = hy (i) di, where >. Each inermediae inpu is produced by a single monopolisic compeior using labor, n (i), as he sole facor of producion. All inermediae producers have he common echnology y (i) = z n (i), where z is a persisen aggregae produciviy shock and 2 (; ]. Following he discussion of rms here, we nex describe he households supplying hem wih heir labor. 2. Final goods rm In any dae, he nal goods rm solves he following problem, aking as given he nominal price level associaed wih nal goods, P, and he nominal prices of inermediaes, P (i). max P Y y(i);i2[;] subjec o Z P (i) y (i) di Z y (i) di Y. The rm s rs order condiion wih respec o any individual inpu i is easily rearranged o yield a demand funcion of he form P (i) y(i) = Y, () P which implies he price elasiciy of demand for each inermediae is. Nex, using he demand funcions above, we can calculae he nominal price level, P, represening his rm s cos of supplying one uni of nal oupu, as he following funcion of nominal inermediae prices: Z P P (i) di. (2) Finally, before leaving his secion, we de ne he relaive price of he i h inermediae good as p (i) P (i) P, so ha we may wrie he demand funcion i faces as d(p(i); Y ) = p(i) Y. 2.2 Inermediae goods rms Inermediae goods rms are (occasional) price-seing monopolisic compeiors. Given curren relaive price p(i) and he demand funcion deermined by () above, he ow pro of he i h such rm is (i) = p(i) Y e (y(p(i); Y ); w; z), where e (y; w; z) represens is cos of 5

6 producing y unis of oupu. Given he producion funcion y = zn, his oal cos funcion is: y(i) z y (i) e (y(i); w; z) = w, (3) z implying marginal cos w z. If his inermediae goods rm resided in a exible-pricing environmen allowing i o fricionlessly rese is nominal (and hence is relaive) price in every period, i would simply maximize is saic pro s in each period. In ha case, i would se is relaive price in each period as a familiar mark-up over is marginal cos of producion, wih he markup given by.3 However, in he New Keynesian producion environmen ha we consider here, inermediae inpus producers are able o change he nominal price of heir oupu only infrequenly, wih he consan probabiliy of such a change being feasible given by 2 (; ). Le he aggregae sae of he economy be K which evolves over ime according o K + = (z ; K ), an endogenous law of moion which we will solve. The problem of a price-seing rm currenly able o rese is price is given below. V P (z; K) = max P P d P P ; Y P e d P ; Y Z + V z ; K F z; dz Z + ( ) V P P ; z ; K F z; dz In he funcional equaion above, he rm s value is described in unis of marginal uiliy, so ha i is seen o discoun fuure pro s by he consan (household subjecive) discoun facor, raher han he sochasic discoun facor D u(c ; n ) D u(c; n). Moreover, when his rm resides in our composie model economy wih segmened asse markes, he relevan marginal uiliy will be ha of a household currenly acive in he asse markes (rading asses for money, hence for goods). Thus, will represen an acive household s marginal valuaion of curren pro s. marginal valuaion is, of course, dependen upon he aggregae sae, (z; K), as will be rue for Y, oal producion, alongside he real wage and he price level, w and P. Wih probabiliy ( This ), he rm now seing is price will be unable o do so again in he nex period. In ha case, given curren nominal price seleced as P, is relaive price will become P P. The value of a rm wih nominal price P ha is unable o re-se is price is given by he following. V P P ; z; K P = P d Z + P P ; Y P e d P ; Y V z ; K F z; dz Z + ( ) V P P ; z ; K F z; dz Given he consan-elasiciy demand funcion i faces, he opimal choice for he rm currenly selecing is nominal price will saisfy he rs-order condiion below, wih he Benvenise- 3 In he special case of linear producion mos commonly considered, he marginal cos iself would be consan wih respec o producion, implying p(i) = w. z 6

7 Scheinkman condiion delivering he expression for D V (p; z; K). + P d + ( ) P P D V P P ; z ; K df =, (4) D V (p; z; K) = + P P d + ( ) P P D V P ; z ; K df. (5) Repeaed subsiuions of (5) ino (4) yield a forward-looking condiion describing he curren nominal price seleced by he rm, P ;. The rm ses is curren price as a funcion of expeced fuure ineres raes (deermined by he pah of ) and fuure demand and marginal cos condiions summarized in he pahs of P; d; e, o saisfy: + P e P ; d + E X s= 2.2. Soluion o he price-seing problem [ ( )] +s + P +s e +S d +s. (6) +s P ; Using ime subscrips, le ;+s = +s denoe he raio of he aggregae price level in period + s relaive o ha in period. Also, recall ha he rm s demand in dae + s is given by h d s;+s = Y +s p s;+s, which may be alernaively expressed as d p; s;+s = Y +s Wih hese subsiuions in mind, we can move from equaion (6) o he following expression for he opimal relaive price se by an inermediae goods rm. P p ; = E s= [ ( )]s +s ;+s Y +s e +s ;+s i. Y +s h p; ;+s i + ;+s E P s= [ ( )]s +s ;+s Y +s ;+s Noice ha he expression above provides an explici soluion only when marginal cos, e +s, is consan in he rm s level of producion. More generally, when 6=, e +s is deermined as a funcion h i p;. of he rm s producion, Y +s ;+s In his case, we have e +s = w +s which may be subsiued ino (7) o obain p ; = P E s= [ ( )]s +s Y +s w +s z +s ;+s E P s= [ ( )]s +s Y +s ;+s making he soluion for he rm s relaive price direc. (7) z +s Y +s ( ) ;+s p ( ) ;, ( ) p ;, (8) If we replace he relaive price se by he rm in (8) wih is nominal counerpar, P ; = p ;, and simplify he resuling expression, we arrive a! ( )+ X P; E [ ( )] s +s Y +s +s = X E [ ( )] s +s Y w +s +s z s= s= +s P +s, 7

8 ( )+ + which when divided hrough by P = P, brings us o he following equaion. below, P; ( )+ E X s= = E! [ ( )] s P+s +s Y +s X s= [ ( )] s +s Y +s w +s z +s P+s Finally, o make he resuling price seing rule recursive, we de ne he hree variable sysem P; ( )+ = E X = s= E [ ( )] s P+s +s Y +s X s= [ ( )] s +s Y +s w +s z +s. P+s = (9) where he de niions of and imply P = Y + ( ) = Y w z P P E + () + ( ) P E + () These las hree equaions, (9), () and (), fully summarize opimal price-seing in he model. Nex, we urn o consider he growh rae of he aggregae price level in his seing, as well as aggregae employmen demand Aggregae price level From equaion (2) above we know ha he aggregae price level is an inegral of nominal prices se by inermediae goods rms ha implies P = R (i). In his period, fracion of rms se heir nominal price o p ; = P ;. Of he remaining fracion of rms, fracion se heir nominal price las period o P ;. Based on hese observaions, we can wrie P = P ; + ( ) P ; + ( )2 P ; : : : + ( )s P ; s + : : : (2) Subsiuing ino his equaion he lagged version of iself, we arrive a a law of moion for he aggregae price level = P; + ( ) P, from which we may obain an expression for in aion raes, P P; = + ( )!. (3) 8

9 Noice ha his expression links he in aion rae o he nominal price se by inermediae goods rms ha are able o change heir price his period. Equaions (9) - () and (3) joinly deermine P ; ;, and P given (Y ; w ; ). In general, when seady sae in aion is no zero, noice ha hese variables will be funcions of he disribuion of employmen across rms Aggregae employmen demand Le n s; denoe he labor demanded by an inermediae goods rm ha las se is price s periods ago. Recalling he expendiure funcion e (y; w; z) = w, and noing his rm s curren relaive price is P ; s, we may wrie is labor demand as a funcion of aggregae producion; Y n s; = P; s z. Noing also he consan fracion of rms able o se heir prices in each y(i) z dae, we know ha a fracion of rms are choosing prices his period, while ( ) rms las se heir prices one period ago, and ( ) 2 las did so wo periods ago. Wih hese observaions in mind, we can express aggregae employmen demand as N D = P s= n s; ( ) s, and more speci cally as: N D = Y z X s= ( ) s P; s. To make he expression above recursive, we de ne he (ime + ) sae variable N = P s= ( )s P ; s and noice ha N P; = N D = Y z + ( ) P 2 N : (4) N. (5) P; Equaions (4) and (5) deermine he evoluion of oal employmen demand, given, he inverse of he lagged in aion rae P 2, N, aggregae oupu, Y and he echnology shock z. 2.3 Households and endogenous asse marke segmenaion In our composie moneary model, he represenaive household ypically residing in a New Keynesian moneary environmen is replaced by a nonrivial disribuion of households who di er in heir curren money holdings, bu who ensure heir bond holdings by pooling heir idiosyncraic risk period-by-period wihin a family compromised of hem all. This family of households is described below, and is based on ha derived from he individual households lifeime uiliy maximizaion problems in Khan and Thomas (26). Wha disinguishes any given household in he economy is is own hisory of xed ransacions coss drawn a he sar of each period from a disribuion G(). 9 Wihin he period, he

10 household may only exchange is asses in he bond markes for money in is bank accoun (required for all consumpion purchases) upon paymen of is xed ransacions cos. Thus, he household underakes such a rade, becoming an acive rader in he asses markes, only if is curren xed cos is su cienly low. Moreover, households di er more and more over ime as hose who have encounered relaively high ransacions coss, and hus foregone asse marke rades, for many periods can see heir real balances furher and furher eroded relaive o hose currenly or recenly acive in rading. A convenien aspec of he endogenously segmened asse markes environmen ha we adop here is ha, as shown by Khan and Thomas, he heerogeneiy among households is quie manageable. Firs, all households las acively rading in he asse markes a some common dae, say j periods in he pas, ener ino he curren period e ecively idenical in ha (a) he relevan di erences across households are limied o heir money balances, given perfec insurance in he bond marke, and (b) once he decision regarding wheher or no o pay he curren ransacions cos and become an acive rader has been made for he curren period, all households ha las raded bonds for money or vice-versa a he same ime have he same curren balances, and hus make he same curren decisions. Thus, we can convenienly rack he disribuion of households by grouping hem ogeher according o heir ime-since-las acive as raders and he common real money balances wih which every member of any one such group will have exied he previous period. Moreover, given he assumpion of a nie upper bound on he disribuion of ransacions cos, his disribuion is summarized by he populaion fracions and money holdings of a nie number of ime-since-acive groups, because all households evenually rade when hey nd heir money holdings su cienly far from heir desired, or arge, real balances. Much of he household side of our economy is a quie sraighforward exension of he endowmen economy exposied by Khan and Thomas (26), so we will (for now) be brief in summarizing he problem o follow. Tha said, we mus ake some care in describing he iming and dispersemen of households wage and pro income ino heir individual bank accouns and he family bond marke, or brokerage, accoun. We assume ha all such incomes are paid nominally a he very end of a period, so hey canno be used unil he subsequen period. We are also, for now, agnosic abou he fracions of hese incomes paid ino he households individual bank accouns ( N and ) versus hose paid ino he family brokerage accoun, ( N and ). Le w and denoe he real wage and real aggregae pro s from dae. Consider a household enering ino his period as a member of ime-since-acive group j having worked n j; hours las period. This household will receive a real paymen a he sar of dae of [ N (w n j; )+ ] ino is bank accoun and [( N )(w n j; )+( ) ] ino he family brokerage accoun. To be more precise abou hese incomes, we will represen he curren-ype-j household s real wage earnings wih which i ended he previous period as e j w n j; in he problem ha follows. The oher wealh speci c o a household as i eners he period is he real value of he money i (deliberaely) saved in is bank accoun from he period. Le m j represen, for a household currenly of ype j, is real money savings a he end of he

11 P previous period. These savings imply real balances of m j a he sar of his period. in he household s bank accoun I is a convenien cion o model he family as deermining which households acively rade in he bond markes in any period and which do no, wih his family implemening he acual allocaion ha arises from households individually implemening heir sae-coningen lifeime uiliy maximizaion plans, given households access o complee insurance in heir bond marke accouns. Given his alernaive view of household decision making, wihin any paricular imesince-acive group of households, j, we can isolae he maximum ransacions cos ha he risksharing family will be willing o pay from he family bond accoun on behalf of a ype j household o allow i o reurn o he family accoun and replenish or shed money balances. Given he common disribuion from which ransacions coss are drawn, if ha hreshold cos associaed wih households of ype j is denoed T j, hen we have he fracion of group j becoming acive is j = G( T j). More generally, we can hink of he family choosing he fracions of each group ha will become acive j wih knowledge of he associaed hreshold cos. Alongside his choice, he family selecs he real balances wih which all currenly acive households will leave he family accoun. Finally, owards a convenien summary of how he disribuion of household money balances evolves over ime, we denoe he fracion of all households enering he curren period as members of ime-since-acive group j as j, and le J represen he maximum number of periods before which any currenly acive household will again be acive. This implies ha he disribuion of households over ime-since-las acive may be racked according o a vecor [ ; :::; J ]; where, P for j = 2; :::; J, j = j ; ( j ; ) and = J j ; j ;. Noe ha members of group his period were acive las period, and hus made heir consumpion, labor supply and money savings decisions wihin ha period as members of ime-since-las acive group Family sae vecor and consrains Given he remarks above, we have he following predeermined family sae variables enering ino dae : [f j ; m j ; e j g J ; ; ]. In fac, he nal sae variable is super uous. Purely for convenience, we use o summarize oal new real income deposied ino he family brokerage accoun a he end of dae, as a resul of fracions ( N ) and ( ) of wage and pro income. This will have real value in his period. Obviously, i can be consruced from he oher family sae variables, bu is presence helps o reduce he lengh of some equaions ha follow. The family s ow budge consrain requires is oal real new brokerage income, plus real money savings and las-period real bank accoun earnings ha reurn o he family brokerage accoun wih currenly acive households (or adjusors), ogeher wih is new real balances arising from any moneary injecion, cover oal real balances ha i sends he curren adjusors back ou wih, as well as he associaed oal adjusmen coss. This consrain, lised jus below, will be

12 insered wih he muliplier. + where '( j ) j j [m j + N e j + ] + M m o G R( j ) xg(x)dx. j j + j '( j ), The consrain deermining he family s end-of-dae real new income for he brokerage accoun is nex, and will be given he muliplier H. ( ) + ( 2 N )w 4 j j n + j ( 3 j )n j 5 + The consrains re ecing he evoluion of he ime-since-acive disribuion, jus below, will have mulipliers q and fq j g J. j j ;+ j ( j ) j+;+ for j = ; :::; J The family acs o maximize he weighed sum of households uiliies, wih each household s period uiliy ow being u(c; n). As such, he bank accoun consrains, and bank balance evoluion, facing individual households are relevan o he family. Firs, consumpion for adjusors and for nonadjusors, respecively, will be consrained by m m ;+ c [m j + N e j + ] m j+;+ c j There are hree hings o noe regarding he above consrains. Firs, m is unique relaive o he oher m variables, in ha (a) i is a curren choice variable raher han a predeermined sae and (b) i is curren-dae real balances. Second, he choices of m j+;+, for j = ; :::; J are subjec o non-negaiviy consrains. objecive. Third, hese will be subsiued direcly ino he family s The nal se of consrains links households curren labor inpus o he real labor income i is eniled o a he end of he period, for (nominal) deposi a he sar of nex period. be enered ino he family problems wih mulipliers j+;+ r j, for j = ; :::; J ; w n j e j+;+ for j = ; :::; J. These will Family Problem The family akes as given, in each dae, M, he oal supply of real balances exising a he end of, alongside curren aggregae oal facor produciviy, z, and he currenly growh 2

13 rae of he aggregae money supply,, as well as he resuling prices and pro s, w,,, a all daes. Is choice variables are summarized by, where h i f j g J ; f j+;+g J j= ; fn jg J j= ; fe j+;+g J j= ; m ; fm j+;+ g J j= ; +. Le he relevan aspecs of he aggregae sae be summarized by [ M ; z ; ]. Noing ha he supply of aggregae real balances evolves as M = M ( + ), and ha oal ransacions G JP R( j ) coss paid wihin a period are j '( j ), where '( j ) xg(x)dx, and ' ( j ) = G ( j ) = T ( j ), we may express he family s opimizaion problem handled on behalf of he economy s households as follows. V f j ; m j ; e j g J ; ; ; M ; z ; = max u(m m ;+ ; n ) Z + zx V + j j P j ( j )u [m j + N e j + ] m j+;+ ; n j f j;+ ; m j;+ ; e j;+ g J ; +; ; M ; z + ; + m + F j j [m j + N e j + ] A m o j j 2 +H 4( ) + ( N j j n + j ( 2 + q 4 j j + r E ciency condiions j j h w n ;+ e ;+ i j ( [z ; ]d[z + ; + ] 3 j '( j ) 5 3 j )n j A + 5 h i q j j ( j ) j+;+ j )r j h w n j e j+;+ i # Following some minor inermediae algebra, we arrive a he e ciency condiions associaed wih he household-side of our economy. Firs, we have he m rs order condiion: D u(c ; n ) =. (6) Nex, we have he rs order condiions wih regard o he m j+;+ choices: P h i D u(c j ; n j ) = E j+;+ + + ( j+;+ )D u(c j+;+ ; n j+;+ ), (7) + for j = ; :::; J 2, 3

14 hough he analogous rs order condiion for m J;+ is replaced by m J; =, because non-negaiviy binds on his choice in he presence of posiive nominal ineres raes. D u(c J ; ; n J ; ) = E + + is insead: m J; =. (8) Nex, here are he n j rs order condiions, followed by he e j+;+ rs order condiions. D 2 u(c j ; n j ) = ( N )w H + w r j ; for j = ; :::; J (9) P h i r j = E N j+;+ + + ( j+;+ )D u(c j+;+ ; n j+;+ ), for j = ; :::; J 2, and, + r J ; = E + N + However, noice ha he righ-hand-side in each equaion above maches is counerpar from equaions 7 and 8 perfecly, bu for he N. The j rs order condiions are: So, for convenience, we wrie hese as r j = N D u(c j ; n j ), for j = ; :::; J 2, and (2) r J ; = E + N + (provided n J ; ). (2) [u(c ; n ) u(c j ; n j )] (22) h P i + m j + N e j + m ' ( j ) +H ( N )w [n n j ] + [q q j ] =, for j = ; :::; J. Nex, we have he j+;+ condiions, followed by he rs order condiion wih respec o +. q j = E j+;+ u(c ;+ ; n ;+ ) + ( j+;+ )u(c j+;+ ; n j+;+ ) (23) + + h P + j+;+ m j+;+ + N e j+;+ + +H + ( N )w + j+;+ n ;+ + ( i j+;+ m ;+ '( j+;+ ) j+;+ )n j+;+ + j+;+ q ;+ + ( j+;+ )q j+;+ #, for j = ; :::; J 2, and q J ; = E u(c ;+ ; m ;+ n ;+ ) + + h P + m J;+ + N e J;+ + # i '() + H + ( N )w + n ;+ + q ;+ (24) 4

15 H = E + (25) + A se of addiional essenial condiions governing he behavior on he household side of he economy are he consrains already lised above; hese are reieraed here only for + m + j j [m j + N e j + ] A = m o j j + j '( j ) (26) + = ( ) + ( N j j n + j ( ;+ = j )n j A (27) j j (28) j+;+ = j ( j ), for j = ; :::; J (29) e j+;+ = w n j, for j = ; :::; J (3) c = m m ;+ (3) c j = [m j + N e j + ] m j+;+, for j = ; :::; J (32) Finally, before specifying he naural condiions ha connec households and rms in he equilibrium of our model, i is useful o de ne some household aggregaes. Aggregae labor supply, consumpion and oupu demand, and demand for real balances are as follow. M N S = n C = c Y D = C + D = j j + j ( j j + j ( j '( j ) j )n j j )c j j j [c + m ;+ ] + j ( j )[c j + m j+;+ ] + j '( j ) 5

16 2.4 Marke clearing The equilibrium sequence of wages, w, aggregae price levels,, and nominal ineres raes, i = E + +, ensure ha he opimizing choices made by rms and households clear he markes for real balances (bonds), nal oupu, and labor in each period. The sequence of relevan marke clearing condiions are as follow. M ( + ) = Y D = Y N S = N D D M Finally, before leaving his secion, i is useful o lis he predeermined sae variables of his economy. These are: fm j g J, f j; e j g J,,, M, 2, and N. 2.5 Special case models Our wo special case models may be described quie simply. Firs, when examining he ex-book New Keynesian model, we drop he enire discussion of households in secion 2.3 above, replacing he disribuion of households here wih insead a represenaive household ha direcly values real balances as a source of uiliy. In he case of he exible-price segmened asse markes model, we insead drop he enire discussion of inermediae inpu producers wih Calvo price seing opporuniies, and insead consider aggregae producion of a perfecly compeiive represenaive nal goods rm Tex-book New Keynesian case In his special case model, he represenaive household s lifeime uiliy maximizaion problem is X max E U M (C ; N ) + V = subjec o C + B + + i + M w N + B + M +, given (B ; M ). If we denoe he LaGrange muliplier for he household s problem by, he problem above hen leads o he following household rs-order condiions. D U (C ; N ) = w D U (C ; N ) = D 2 U (C ; N ) + = E + i = DV 6 P + M + E + +

17 We may slighly rearrange he las condiion using ha above i, and also noe ha he absence of any xed ransacions coss implies ha he household s only use for oupu is oward consumpion, so ha Y D = C. This leaves us wih he following four equaions replacing he series of condiions from secion 2.3. D U Y ; N S = (33) w D U Y ; N S = D2 U Y ; N S (34) + = E (35) + i P + M DV = D U Y ; N S i (36) + i The rs equaion deermines he household s sochasic discoun facor, and (as his household is always acive in asse markes), i is consisen wih he use of as he marginal uiliy of consumpion for acive households in he composie model above. The second equaion deermines he real wage. The las equaion may be used o deermine he nominal ineres rae given he equilibrium supply of real balances Flexible-price Endogenous Marke Segmenaion case In his special case model, a represenaive nal goods rm replaces he descripion of all rms in secions 2. and 2.2 above. This perfecly compeiive rm hires labor from he households of secion 2.3, and produces nal oupu direcly via he Cobb-Douglas producion funcion Y = zn, where 2 (; ). Maximizing is pro ows, period-by-period, he rm solves max z N D arriving a he obvious rs order condiion, N D w N D, w = z N D (37) and achieves pro s in each period of: (i) = ( )z N D : (38) 3 Resuls This secion examines he dynamics of each model following a real shock when moneary policy plays an acive role. In each of he gures below, we consider our model economies responses o a persisen rise in aggregae oal facor produciviy (wih z = :9), in he case where changes in he rae of money growh are dicaed by he moneary auhoriy s pursui of speci c sabilizaion goals. Here, he moneary auhoriy follows a Taylor rule in responding o deviaions in in aion and oupu relaive o heir rend levels. This rule akes he speci c form i = i + 2:[ ] + :35[y y ]. 7

18 In each environmen we sudy here, he rise in produciviy on is own implies a rise in available oupu. Taken on is own, in he absence of any response in he money growh rae, his rise implies a rise in he demand for real balances, and hus a fall in he in aion rae o accommodae i. Given he Taylor rule, his requires a fall in he nominal ineres rae. Indeed, given he acive policy rule we assume here, he nominal ineres rae mus fall by more han he fall in in aion o imply a fall in he real ineres rae. A he same ime, however, he policy rule also pus weigh on he oupu gap. Thus, if he produciviy shock leads o raised aggregae oupu, hen he drop in he nominal rae is less han would oherwise be implemened. 3. Responses in a exbook New Keynesian model In gure, we begin by considering he New Keynesian model in he absence of he Taylor rule above. Here, he supply of nominal balances grows a a consan rae, una eced by he produciviy shock. By brie y summarizing his economy s response absen he Taylor rule, we may make more ransparen how lile role money demand - indeed money iself - plays in he deerminaion of he dynamics of real variables wihin his model. Following he rise in produciviy, all inermediae inpu producers enjoy a drop in heir marginal cos schedules. In response, hose rms ha are currenly able o adjus heir nominal price will do so, lowering heir prices. This immediaely lowers he aggregae price level. Moreover, given reduced relaive prices, hese rms see a rise in demand for heir goods, and hus raise heir oupu and employmen. However, a large fracion of rms (wo-hirds in he example here), are no currenly able o rese heir nominal prices and hus see heir real prices rise. These rms all experience a fall in he demand for heir goods, which implies below-usual oupu and employmen among hem. Because hey are such a large fracion of he economy, we see aggregae oupu and employmen acually fall wih he rise in produciviy. Of course, over ime, hese e ecs are gradually reversed as more and more rms are able o reduce heir nominal prices in response o he shock and hus raise heir own producion. Noice, however, ha our enire discussion of he responses in real variables in he cener of gure has avoided any reference o household money demand. This is because here is only a single equaion in he model re ecing he demand for money, he rs order condiion in 36. This condiion serves only o deermine wha changes will occur in he nominal ineres rae in response o he shock. In his paricular example, he fall in he price level caused by currenly price-seing rms implies roughly he same decline in he marginal uiliy associaed wih he enjoymen of real balances as he decline in he marginal uiliy of consumpion ha arises from he decline in labor inpu ha accompanies he fall in consumpion. As a resul, he nominal ineres rae has only a iny posiive adjusmen o clear he money marke of is exra real balances. The discussion so far foreshadows how, once we add he Taylor rule ino he environmen above, we will fully encouner he New Keynesian dichoomy. Wih he Taylor rule in place, he single equaion associaed wih money demand is eliminaed, as i becomes enirely irrelevan. In his case, he moneary auhoriy no longer ses a nominal ineres rae o clear he marke for nominal 8

19 balances given reduced demand for hese balances and a xed money growh rae. Insead, he auhoriy ses a money growh rae o achieve is argeed nominal ineres rae direcly. Household money demand, irrelevan above in all bu he deerminaion of nominal variables, now loses even ha abiliy o a ec he economy s dynamics, as will be clear in our discussion of gure 2 below. In gure 2, we examine he e ecs of he same persisen produciviy shock in our New Keynesian model, bu his ime in he presence of acive ineres rae argeing. In his case, he economy avoids he somewha paradoxical resul above, in ha we now see aggregae producion and employmen rise wih he rise in produciviy. How did he Taylor rule do his? If we ignore policy for a momen, we know from he discussion above ha he shock reduces boh he aggregae price level and aggregae oupu. Thus, he Taylor rule dicaes a fairly sizeable drop in he nominal ineres rae, which is implemened by a rise in he rae of money growh relaive o seady-sae. The acive policy response on he par of he moneary auhoriy sharply miigaes he fall in in aion resuling from he shock relaive o ha seen in gure. Moreover, precisely because i limis he resuling immediae fall in he aggregae price level, he moneary response also overcomes he unforunae real implicaions of Calvo-price-sickiness following he produciviy shock. Now, hose rms unable o adjus heir nominal prices see a lesser rise in heir relaive price, and hus a smaller drop in he demand for heir goods. Wih he resuling lesser fall in producion among hese rms, he raised producion and employmen among hose rms ha are able o rese heir prices dominaes in he real aggregae responses shown in he cener panel. As a resul, we see more familiar impulse responses ha migh arise in an economy wih no money and no price sickiness. In sum, we see ha he only role of money in he dynamics following he produciviy shock is as he insrumen hrough which he ineres rae rule is implemened, and he moneary auhoriy is able o compleely re-shape hese dynamics wih no concern for aspecs oher han oupu and in aion hemselves. 3.2 Responses in a richer model of money demand The model wih endogenously segmened asse markes gives rise o far more complex aggregae dynamics han we observed in he secion above. Despie he Taylor rule replacing he money marke clearing equaion, many condiions associaed wih individual households money demands remain, and a ec he resuling equilibrium. I is hese remaining condiions ha imply a moneary ransmission mechanism, and ha break he simple mapping from he policy rule o he resuling aggregae dynamics ha we saw in he exbook New Keynesian model considered above. Figure 3 illusraes his economy s aggregae dynamics following he persisen shock o oal facor produciviy. Here, we see clear di erences in boh he ampliude and persisence of aggregae quaniy and price responses relaive o hose seen in Figure 2. Broadly speaking, hese di erences arise from wo disinguishing elemens of he environmen wih endogenous asse marke segmenaion. Firs, here is a nonrivially rising adjusmen hazard ha re ecs how 9

20 he fracions of households underaking bank ransfers varies wih he ime since heir las such ransfer. Second, and more criically o he responses in gure 3, here is he fac ha his hazard is alered boh by he aggregae produciviy shock iself, and by he resuling moneary policy response. As oupu increases wih he shock o echnology, here is a rise in he demand for real balances o purchase his addiional oupu, which, given unchanged money supply, reduces he price level relaive o is rend. The moneary auhoriy responds according o he Taylor rule, lowering he nominal ineres rae. Wih he resuling fall in he reurn on bonds, households have a furher incenive o shif some of heir less-liquid asses ino money for use in purchasing consumpion over he near-erm. In oher words, he lowered reurn on bonds compounds he direc e ec of he raised demand for real balances oward purchasing he addiional oupu, and hus provides households a srong incenive o reurn early o he asse markes. Given an unchanged disribuion of associaed ransacions coss, his implies a rise in he fracion of households acively rading bonds for money, relaive o he economy s average behavior. In oher words, boh he shock iself and he policy response o his shock lead o below-average asse marke segmenaion - i.e., greaer household paricipaion in he asse markes han usual. Time-varying marke segmenaion in his moneary environmen propagaes he e ecs of he iniial shock, beyond he simple mechanics of he Taylor rule, hrough changes in he iming of households ransfer aciviies in daes following he shock s impac. Following a subsanial iniial rise, he overall measure of acively rading households falls below is seady-sae value for a number of periods, despie persisenly high aciviy raes across groups, j. This is because large iniial rises in hese raes shif he household disribuion o imply higher han usual money balances for he mean household in subsequen daes, hereby reducing is incenive o ransfer funds from he brokerage accoun. Thus, in daes following he shock, he measure of acive households is su cienly below average ha each such household receives an above-average ransfer of real balances in equilibrium. In response, hese households increase heir curren consumpion spending relaive o rend. Combining heir above-average consumpion demand ogeher wih ha among oher groups of households acive in he asse markes since he shock s impac, he disribuional e ecs of he shock ulimaely drive furher increases in oupu and employmen. As a resul, we observe a non-monoone response in hese series. This arises enirely from households abiliy o choose when o rade bonds for money, and o change he iming of hese rades in response o changes in income, prices, wages, and ineres raes. I does no exis when we suppress such choices. 3.3 Responses in he composie model This secion combines he key elemens of each model above o examine he dynamics of he composie model. Resuls are TBA. 2

21 4 Conclusion TBA 2

22 Figure. Pure Sicky Price Model: percen deviaions in produciviy and inflaion Taylor shock P)/P(-) z() percen deviaions in aggregae oupu, employmen and real wage Y() N() w() x -8 deviaion in nominal ineres rae 6 i() dae

23 Figure 2. Sicky Price Model wih a Taylor Rule: percen deviaions in produciviy and inflaion. Taylor shock.5 P)/P(-) z() percen deviaions in aggregae oupu, employmen and real wage Y() N() w() x -3 deviaion in nominal ineres rae dae i()

24 Figure 3. Segmened Markes Model wih a Taylor Rule: percen deviaions in produciviy and inflaion deviaion from ss deviaion from rend deviaion from rend Taylor shock P)/P(-) z() percen deviaions in aggregae oupu, employmen and real wage Y() N() w() x -3 deviaion in nominal ineres rae dae i()