Inflation and Interest Rates with Endogenous Market Segmentation

Transcription

1 Inflaion and Ineres Raes wih Endogenous Marke Segmenaion Aubhik Khan Federal Reserve Bank of Philadelphia Julia K. Thomas Federal Reserve Bank of Philadelphia and NBER March 27 ABSTRACT We examine a moneary economy where households incur fixed ransacions coss when exchanging bonds and money and, as a resul, carry money balances in excess of curren spending o limi he frequency of such rades. As only a fracion of households choose o acively rade bonds and money a any given ime, he marke is endogenously segmened. Moreover, because households in our model economy have he abiliy o aler he iming of heir rading aciviies, he exen of marke segmenaion varies over ime in response o real and nominal shocks. We find ha his added flexibiliy can subsanially reinforce boh sluggishness in aggregae price adjusmen and he persisence of liquidiy effecs in real and nominal ineres raes relaive o ha seen in models wih exogenously segmened markes. We hank Michael Dosey, Chris Edmond, Rober King, Igor Livshis, Sylvain Leduc, Edward Nelson, Thomas Sargen, Pierre-Olivier Weill, seminar paricipans a Boson Universiy, The Federal Reserve Banks of New York and S. Louis, Sern, Universiy of Briish Columbia and conference paricipans a he 25 SED meeing and he 26 Canadian Macro Sudy Group for commens and suggesions. Thomas hanks he Alfred P. Sloan Foundaion and he Naional Science Foundaion gran # for research suppor. The views expressed here are hose of he auhors and no of he Federal Reserve Bank of Philadelphia or he Federal Reserve Sysem.

2 1 Inroducion There is a wealh of empirical research documening no only he comovemen of real and nominal series a higher frequencies, bu wha is widely acceped as persisen responses in real variables following nominal disurbances. We sudy such comovemens using a moneary model where households face fixed coss of ransferring wealh beween ineres-bearing asses and money. As a resul of hese ransacions coss, households infrequenly access heir ineres income and carry money balances in excess of curren spending, and paricipaion in asse markes is endogenously segmened. As is well known, marke segmenaion implies ha open marke operaions can have real effecs, because hey direcly involve only a subse of households. Our paper esablishes ha, when marke segmenaion is endogenized in a model where households hold invenories of money, changes in he fracion of households paricipaing in asse markes can add considerable persisence o movemens in boh nominal and real variables. Our work builds on an imporan lieraure ha sudies moneary policy in models wih exogenously segmened markes. 1 As in he work of Grossman and Weiss 1983, Roemberg 1984, and Alvarez, Akeson and Edmond 23, households in our model economy only periodically access he marke for ineres-bearing asses broadly inerpreed as markes for relaively high-yield asses and hey carry invenories of money inerpreed o include relaively low yield liquid asses. Noneheless, our model is closes in spiri o he endogenous segmenaion model of Alvarez, Akeson and Kehoe 22 in ha heerogeneous households acively choose when o adjus heir porfolios of bonds and money. 2 Our model is disinguished relaive o heirs by a disribuion of moneyhaevolvesacrossperiodsasmoshouseholdsholdmoneybalancesexceedingheircurren consumpion expendiures. Moreover, as in Alvarez, Akeson and Edmond 23, household spending raes raios of he value of curren consumpion o money holdings are lowes among households ha have recenly ransferred wealh held as bonds ino money, and hey rise wih he ime since such a ransfer has occurred. In such an environmen, a ransiory shock o money growh changes he disribuion of money holding across households wih differen spending raes, which can in urn lead o persisen movemens in inflaion raes. Unlike exogenous segmenaion environmens such as Alvarez, Akeson and Edmond 23, he exen of marke segmenaion varies over ime in our model economy, as he fracion of households choosing o paricipae in he asse markes responds o changes in he economy s sae. However, in conras o he endogenous segmenaion model of Chiu 25, our allowance for idiosyncraic differences across households implies ha his fracion remains nonrivial over ime, as does he disribuion of money. This disincion has imporan implicaions for he propagaion of nominal disurbances. Following an open marke operaion, endogenous changes in he iming of households acive paricipaion in asse markes can gradualize aggregae price adjusmen relaive o ha in a model wih exogenous marke segmenaion. Furhermore, such changes can subsanially increase he persisence of liquidiy effecs in boh real and nominal ineres raes. 3 Endogenizing access o he asse marke, and hus allowing for movemens in he fracion of 1 See Alvarez, Lucas and Weber 21 and he references herein. 2 Alvarez, Akeson and Kehoe 22 build on he work of Chaerjee and Corbae 1991 who sudy an economy where some households choose o pay a fixed cos o rade bonds. 3 One excepion o his is he exogenous segmenaion model of Williamson 25, where households are permanenly divided ino groups wih and wihou access o he asse markes. There, he assumpion ha households wih wihou such access prefer o rade among hemselves in he goods markes delivers a second ype of segmenaion ha can lead o persisen liquidiy effecs. 1

3 households acively adjusing heir nominal balances a any ime, implies larger movemens in individual households spending raes following a shock o he money supply. When ransacions coss are high, and hus he mean ime beween acive rades is long, households end o hold relaively large invenories of money and have lower average spending raes. If, in addiion, he maximum ime beween rades is significanly longer han is he mean ime, households on average reurn o he asse markes wih subsanial remaining balances. We find ha, in such circumsances, he persisence in inflaion ha is implied by he exogenous segmenaion model is reduced, as households no currenly paricipaing in he asse markes sharply raise heir spending raes following an open marke operaion. Conversely, when he mean ime beween asse marke rades is no as long, so ha households have higher average spending raes or when he mean and maximum imes beween such rades are similar, so ha households on average reurn o he bond marke wih lile remaining money, endogenous changes in he disribuion of households increase persisence in he inflaion response beyond ha in he exogenous segmenaion model and, moreover, lead o persisen changes in ineres raes. As in he many sudies in moneary economics ha have preceded us, several empirical relaionships involving money, ineres raes and prices moivae our work. Firs, shor-erm real ineres raes are negaively correlaed wih expeced inflaion. Barr and Campbell provide direc evidence for his using U.K. daa involving inflaion-indexed bonds. Second, VAR sudies consisenly have found evidence of liquidiy effecs; expansionary open marke operaions appear o reduce shorerm nominal ineres raes, a leas in he shor-run. See, for example, Leeper, Sims, and Zha 1996 and Chrisiano, Eichenbaum, and Evans Finally, he general price level appears o adjus slowly o nominal shocks. This finding is widely suppored by he VAR lieraure as, for example, in he sudies of Leeper, Sims, and Zha 1996, Chrisiano, Eichenbaum, and Evans 1999, and Uhlig 24. Moreover, King and Wason 1996 show ha, a business cycle frequencies, he price level is posiively correlaed wih lagged real oupu. Addiional evidence for he slow adjusmen of he price level, discussed in Alvarez, Akeson and Edmond 23, is provided by he paern of shor-erm movemens seen beween he raio of money o consumpion and velociy. The correlaion beween he raio of money M2 o consumpion PCE and he corresponding measure of velociy is.89 for HP-filered monhly daa. The mos common heoreical approach o addressing his empirical evidence involves models where nominal prices are sicky a he firm level. While here are several open issues involving he viabiliy of models wih sicky prices, we discuss one ha direcly moivaes our work. In heir recen paper, Dosey and King 25 resolve a long-sanding issue for his lieraure by developing an S,s model of nominal price seing ha is consisen wih empirical esimaes of he persisence in inflaion. In he process, hey discover ha heir model predics a rise in shor-erm nominal ineres raes following a persisen shock o money growh raes. The sae-of-he-ar micro-founded menu cos model canno address he liquidiy effec described by Milon Friedman 1968, The iniial impac of increasing he quaniy of money a a faser rae han i has been increasing is o make ineres raes lower for a ime han hey would oherwise have been. Wheher or no we choose o view his as a criical shorcoming of he paradigm, here is a more subsanive disagreemen beween Friedman s view of he real effecs of moneary policy and ha underlying sicky price models. Friedman viewed changes in he velociy of money as cenral in deermining he mechanics of movemens in oupu, employmen and prices following an expansionary open marke operaion. Consider his esimony o he House of Commons Selec Commiee in 1979;... he iniial effec of a change in moneary growh is an offseing movemen in velociy, followed by 2

4 changes in he growh of spending iniially manifesed in oupu and employmen, and only laer in inflaion. Friedman, This view is firmly rejeced by sicky price models where he velociy of money, and indeed real balances hemselves, are almos enirely irrelevan o he predicions of he model. Changes in velociy are a he hear of he shor-erm nonneuraliy exhibied by models wih segmened asses markes. Moreover, he endogenous segmenaion model of Alvarez, Akeson and Kehoe 22 succeeds in generaing liquidiy effecs and in reproducing he negaive relaion beween real ineres raes and anicipaed inflaion. The Alvarez, Akeson and Edmond 23 invenory-heoreic model of money wih exogenous segmenaion separaely delivers sluggish adjusmen of he price level, and hence persisen inflaion responses o nominal shocks. Drawing upon elemens of each of hese frameworks, we develop an endogenous segmenaion model of money ha simulaneously succeeds wih regard o boh ses of regulariies. Moreover, as menioned above, changes in he number of households choosing o exchange bonds and money can subsanially reinforce he effecs of marke segmenaion. Following a ransiory shock o he money growh rae, such changes prolong responses in inflaion and he real ineres rae. When shocks o money growh are persisen, hey lead o far more sluggish price adjusmen and more persisen liquidiy effecs in boh nominal and real ineres raes. Alernaively, when we examine real shocks wih moneary policy following a Taylor rule, our economy generaes persisence in he responses of inflaion and ineres raes alogeher absen under fixed marke segmenaion. Finally, in versions of our model wih endogenous producion, we find ha persisen echnology shocks can lead o non-monoone responses in employmen and oupu. Finally, our paper offers an independen heoreical conribuion in formally esablishing how he resuls of Alvarez, Akeson and Kehoe 22 may be exended o a model where here are persisen differences across households. Here, such exension is necessary, because cash-in-advance consrains do no always bind so ha households carry invenories of money, hereby ransmiing he effecs of emporary idiosyncraic differences across periods. By assuming a full se of sae-coningen nominal bonds ha allow risk-sharing across households, we ensure ha hese differences across households are persisen, bu no permanen. Because households mus pay fixed ransacions coss o access heir bond holdings, he presence of sae-coningen bonds in our economy does no lead o full insurance; households ha are ex-ane idenical diverge over ime as idiosyncraic realizaions of shocks drive differences in heir money and bond holdings. Noneheless, we prove ha, whenever a heerogeneous group of households eners he bond marke a he same ime, all previous differences among hem are eliminaed. As a resul, our model economy exhibis limied memory. Exploiing his propery, we are able o apply he numerical approach o solving generalized S,s models developed by King and Thomas forhcoming in a seing where he consumpion and savings decisions of heerogeneous risk-averse households are direcly influenced by nonconvex coss. While his approach has been applied previously in solving models where risk-neural producion unis face idiosyncraic fixed coss of adjusing heir prices or facors of producion as in Dosey, King and Wolman 1999 and Thomas 22, his is o our knowledge he firs applicaion involving heerogeneiy among households. 4 We hank Ed Nelson for bringing his o our aenion. 3

5 2 Model We begin wih an overview of he model. Thereafer, we proceed o a more formal descripion of households problems, followed by he descripion of a financial inermediary ha sells households claims coningen on boh aggregae and individual saes. Nex, we show ha here is an equivalen, bu more racable, represenaion of households lifeime opimizaion problems, given heir abiliy o purchase such individual-sae-coningen bonds alongside he fac ha hey are ex-ane idenical. Proofs of all lemmas are provided in he appendix. 2.1 Overview The model economy has hree ses of agens: a uni measure of ex-ane idenical households, a perfecly compeiive financial inermediary, and a moneary auhoriy. Each infiniely-lived household values consumpion in every dae of life, wih period uiliy uc, and i discouns fuure uiliy wih he consan discoun facor β, whereβ, 1. In each period, households receive a common endowmen, y. This endowmen varies exogenously over ime, as does he growh rae of he aggregae money supply, µ. Defining he dae realizaion of aggregae shocks as s =y,µ, we denoe he hisory of aggregae shocks by s =s 1,...,s, and he iniial-period probabiliy densiy over aggregae hisories by g s. Households have wo means of saving. Firs, hey have access o a complee se of saeconingen nominal bonds. These are purchased from a financial inermediary described below, and are mainained in ineres-bearing accouns ha we will refer o as households brokerage accouns, following he language of Alvarez, Akeson and Edmond 23. Nex,heyalsosave using money, which hey mainain in heir bank accouns and use o conduc rades in he goods marke. 5 Households have he opporuniy o ransfer asses beween heir wo accouns a he sar of each period; his occurs afer he realizaion of all curren shocks, bu prior o any rading in he goods marke. As such, i is exposiionally convenien o refer o each period as consising of wo subperiods ha we will erm ransfer-ime and shopping-ime, alhough nohing in he environmen necessiaes his approach. There are hree iner-relaed fricions leading households o mainain money in heir bank accouns. Firs, as in a sandard cash-in-advance environmen, households canno consume heir own endowmens. Each household consiss of a worker and a shopper, and he worker mus rade he household endowmen for money while he shopper is purchasing consumpion goods. As a resul, he household receives he nominal value of is endowmen, P s y s, only a he end of he period afer curren goods rade has ceased. 6 We assume ha hese end-of-period nominal receips are deposied across heir wo accouns, wih fracion λ paid ino bank accouns and he remainder ino brokerage accouns. Second, as all rades in he goods marke are conduced wih money, each household s consumpion purchases are consrained by he bank accoun balance i holds when shopping-ime begins. Noe ha, absen oher fricions, each household would, in every period, simply shif from is brokerage accoun ino is bank accoun exacly he money needed o finance curren consumpion 5 When allowed o sore money in heir brokerage accouns, households never do so given posiive nominal ineres raes paid on bonds. Thus, we simplify he model s exposiion here by assuming ha money is held only in bank accouns and verify ha nominal raes remain posiive hroughou our resuls. 6 While his worker-shopper arrangemen may appear sark in an endowmen economy, i is less so if one envisions ha each household s endowmen is one of a uni measure of differeniaed inpus ha ener a consumpion aggregaor wih idenical weighs o produce he single good consumed by all. 4

6 expendiure no covered by he bank accoun paycheck from he previous period. There is, however, a hird fricion ha prevens his, leading households o deliberaely carry money across periods; his is he assumpion ha hey mus pay fixed coss each ime hey ransfer asses beween heir wo accouns. Given hese fixed coss, households mainain socks of money o limi he frequency of heir ransfers, and hey follow generalized S,s rules in managing heir bank accouns. Transfer coss are fixed in ha hey are independen of he size of he ransfer; however, hey vary over ime and across households. Here, we subsume he idiosyncraic feaures ha disinguish households direcly in heir fixed coss by assuming ha each household draws is own curren ransfer cos, ξ, from a ime-invarian disribuion Hξ a he sar of each period. Becausehiscosdrawinfluences a household s decision of wheher o underake any ransfer, and hence is curren consumpion and money savings, each household is disinguished by is hisory of such draws, ξ =ξ 1,...,ξ, wih associaed densiy h ξ = hξ 1 h ξ. As will be seen below, households are able o insure hemselves in heir brokerage accouns hrough he purchase of nominal bonds coningen on boh aggregae and individual exogenous saes. 2.2 Households A he sar of any period, given dae-even hisory s,ξ, a household s brokerage accoun asses include nominal bonds, B s,ξ, purchased in he previous period a price q s,ξ,aswell as he fracion of is income from he previous period ha is deposied here, 1 λp s 1 y s 1. The remainder, he paycheck, λp s 1 y s 1, is deposied ino he household s bank accoun and supplemens is money savings here from he previous period, A s 1, ξ 1. Given his sar of period porfolio and is curren fixed cos, he household begins he period by deermining wheher ornooransferassesacrossiswoaccouns. Denoinghehousehold ssar-of-periodbank balance by M s 1, ξ 1,where M s 1, ξ 1 A s 1, ξ 1 + λps 1 y s 1, 1 he relevan feaures of his choice are summarized in he char below. z z brokerage accoun wihdrawal shopping-ime bank balance s,ξ =1 x s, ξ + Ps ξ M s 1, ξ 1 + x s, ξ s,ξ = M An acive household is indicaed by z s,ξ =1. s 1, ξ 1 In his case, he household selecs a nonzero nominal ransfer x s,ξ from is brokerage accoun ino is bank accoun and has M s 1, ξ 1 + x s, ξ available in is bank accoun a he sar of he curren shopping subperiod. Here, he household s curren fixed cos applies, so P s ξ is deduced from is nominal brokerage wealh. Alernaively, he household may choose o underake no such ransfer, seing z s,ξ =and remaining inacive. In ha case, i eners ino he shopping subperiod wih no change o is sar-of-period bank and brokerage accoun balances. Each household chooses is sae-coningen plan for he iming and size of is accoun ransfers z A s, ξ and c s, ξ, o maximize is expeced discouned lifeime uiliy, β 1 u cs, ξ h ξ g s dξ ds, 2 =1 s,ξ and x s, ξ, and is bond purchases, money savings and consumpion B s,s +1, ξ,ξ+1 s ξ, 5

7 subjec o he sequence of consrains in 3-6. B s,ξ + 1 λ P + s 1 y s 1 [ x s, ξ + P s ] ξ z s,ξ 3 s +1 ξ +1 q s, s+1,ξ+1 B s,s +1, ξ,ξ+1 ds +1 dξ +1 M s 1, ξ 1 + x s, ξ z s,ξ s P s c s,ξ + A s, ξ 4 M s, ξ 5 A s, ξ + λp s y A s, ξ 6 Equaion 3 is he household s brokerage accoun budge consrain associaed wih hisory s, ξ, and requires ha expendiures on new bonds ogeher wih any ransfer o he bank accoun and associaed fixed cos no exceed curren brokerage accoun wealh. Nex, he bank accoun budge consrain in equaion 4 requires ha he household s money balances enering he shopping subperiod cover is curren consumpion expendiure and any money savings for nex period. 7 Money balances for nex period, in 5, are hese savings ogeher wih he bank paycheck received afer compleion of curren goods rade. Equaion 6 prevens he household from ending curren rade wih a negaive bank balance; hus, aken ogeher wih he resricion in 4, i imposes cash-in-advance on consumpion purchases. Finally, in addiion o his sequence of consrains, we also impose he No-Ponzi condiion: lim s ξ q s,ξ B s,ξ ds dξ. 7 Following he approach of Alvarez, Akeson and Kehoe 22, we find i convenien o model risk-sharing by assuming a perfecly compeiive financial inermediary ha purchases governmen bonds wih payoffs coningen on he aggregae shock and, in urn, sells o households bonds wih payoffs coningen on boh he aggregae and individual shocks. In paricular, given aggregae hisory s, he inermediary purchases governmen-issued coningen claims B s,s +1 a price q s,s +1, and i sells hem across households as claims coningen on individual ransfer coss, ξ +1. Noe ha, as households cos draws are no auocorrelaed, he price of any such claim is q s,s +1, ξ +1, independen of he individual hisory ξ. s,s +1, he inermediary selecs is aggregae bond purchases, B s,s +1 For each individual bond sales, Bs,s +1, ξ,ξ +1, o solve max q s,s +1,ξ +1 Bs,s +1,ξ,ξ +1 hξ dξ dξ +1 q s,s +1 B s,s +1, and 8 subjec o: ξ +1 ξ B s,s +1 Bs,s +1,ξ,ξ +1 hξ hξ +1 dξ dξ ξ +1 ξ 7 In hose periods when a household is acive, i has a single unified budge consrain, B s,ξ + P s 1 y s 1 + A s 1 1,ξ P s [ ξ + c s ],ξ + A s,ξ + q s,s +1,ξ +1 B s,s +1,ξ +1,ξ ds+1dξ +1. s +1 ξ +1 6

8 The consrain in 9 requires ha, for any s,s +1, he inermediary mus purchase sufficien aggregae bonds o cover all individual bonds held agains i for ha aggregae hisory. Given s +1 occurs, fracion hξ +1 of he households wih hisory ξ o whom i sells such bonds will realize ha sae and demand paymen. As shown in Lemma 1 below, he financial inermediary s zero profi condiion immediaely implies ha he price of any individual bond associaed wih s +1,ξ +1 is simply he produc of he price of he relevan aggregae bond and he probabiliy of an individual household drawing he ransfer cos ξ +1. Lemma 1. The equilibrium price of sae-coningen bonds issued by he financial inermediary, q s,s +1, ξ +1, is given by q s,s +1,ξ +1 = q s,s +1 h. By assuming an iniial period hroughou which households are perfecly idenical, we allow hem he opporuniy o rade in individual-sae-coningen bonds a a ime when hey have he same wealh and face he same probabiliy disribuion over all fuure individual hisories. In his iniial period, he governmen has some ousanding deb, B, haisevenlydisribuedacross households brokerage accouns, and i repays his deb enirely by issuing new bonds. Households receive no endowmen, draw no ransfer coss and do no value consumpion in his iniial period. Raher, hey simply purchase sae-coningen bonds for period 1 subjec o he common iniial period brokerage budge consrain: B ξ +1 Bs1, ξ1qs1hξ 1dξ1ds1. s1 ξ 1 Following he proof of Lemma 1, secion B of he appendix shows ha he period budge consrain above can be combined wih he sequence of consrains in 3 o yield he following lifeime budge consrain common o all households. B =1 where q s q s 1 q s 1,s 2 q q [ s h ξ zs,ξ x s,ξ + P s ] ξ P s λ y s dξ ds, 1 s 1, s. Finally, we assume ha he moneary auhoriy is subjec o he sequence of consrains, B s s +1 q s,s +1 B s, s +1 ds +1 = M s M s 1, 11 requiring ha is curren bonds be covered by a combinaion of new bond sales and he prining of new money. This sequence of consrains, alongside equilibrium in he money marke, immediaely implies ha households aggregae expendiures on new bonds in any period is exacly he difference beween he aggregae of heir curren bonds and he change in he aggregae money supply: M s M s 1 = B s q s,s +1 hξ+1 Bs, s +1,ξ,ξ +1 hξ dξ dξ +1 ds A risk sharing arrangemen Three aspecs of he environmen described above may be exploied o simplify our soluion for compeiive equilibrium: i households are ex-ane idenical, ii fixed ransfer coss are independenly and idenically disribued across households and ime and iii households have access o a 7

9 complee se of sae-coningen claims in heir brokerage accouns. In his secion, we show how hese assumpions allow us o move o a more convenien represenaion of households problems. In paricular, exploiing he common lifeime budge consrain in 1 above, we will move from he household problem saed in secion 2.2 o consruc he equivalen problem of an exended family ha manages all households bonds in a join brokerage accoun, and whose period-by-period decisions regarding bond purchases and accoun ransfers implemen he sae-coningen lifeime plan seleced by every household. In doing so, we ransform our somewha inracable iniial problem ino somehing o which we can apply he King and Thomas 25 approach for solving aggregae economies involving heerogeneiy arising due o S,s policies a he individual level. Money as he individual sae variable: A complee se of sae-coningen claims in he brokerage accoun allows individuals o insure heir bond holdings agains idiosyncraic risk; hese shocks only affec heir bank accouns. Alernaively, an individual s money balance fully capures he cumulaive effec of his hisory of idiosyncraic shocks. In Lemma 2, we prove ha prior o households draws of curren ransfer coss, all differences across hem as hey ener ino any period are fully summarized by heir sar-of-period money balances. Lemma 2. Given M independen of he hisory ξ 1. s 1, ξ 1, he decisions c s,ξ,a s, ξ, x s, ξ and z s,ξ are This resul is fairly inuiive. Given ha each ξ comes from an i.i.d. disribuion, a household s draw in any given period does no predic is fuure draws, and hus direcly affecs only is asse ransfer decision in ha one period. While his cerainly affecs curren shopping-ime money balances, and hence consumpion, is only fuure effec is in deermining he money balances wih which he household will ener he subsequen period, given he household s abiliy o insure iself in is brokerage accoun by purchasing bonds coningen on boh aggregae and individual shocks. In proving his resul, we show ha he soluion o he original household problem from secion 2.2, given he lifeime consrain in 1, is idenical o he soluion of an alernaive problem where households pool risk period-by-period by each commiing o pay he economywide average of he oal ransfers and associaed fixed coss incurred across all acive households in every period, irrespecive of he iming and size of heir own porfolio adjusmens. I is immediae from his ha households bond holdings may be modelled as independen of heir individual hisories ξ. Thus, wihin every period, he disinguishing feaures affecing any household s decisions can be summarized enirely by is sar-of-period bank balance, M s 1, ξ 1, and is curren ransfer cos, ξ. Households as members of ime-since-acive groups: Our nex lemma esablishes ha, wihin any period, all households ha underake an accoun ransfer will selec boh a common consumpion and a common end-of-period bank balance; hence hey begin he subsequen period wih he same bank and brokerage accoun balances. Lemma 3. s, ξ in which z s,ξ = 1, c s,ξ,a s,ξ and M s,ξ are independen of ξ. For any To undersand his resul, recall ha household brokerage and bank accouns are joined in periods when hey choose o adjus heir porfolios, and all are idenical when hey make heir saeconingen plans in dae. Given his, in selecing heir consumpion for such periods, households 8

10 equae heir appropriaely discouned marginal uiliy of consumpion o he muliplier on he lifeime brokerage budge consrain from 1, which is common o all households. Nex, in selecing wha porion of heir shopping-ime bank balances o reain afer consumpion hence heir nex-period balances, households equae he marginal uiliy of heir curren consumpion o he expeced reurn on a dollar saved for he nex period weighed by heir expeced discouned marginal uiliy of nex-period consumpion. Given common inflaion expecaions and he common curren consumpion of acive households, his implies ha acive households also share in common he same expeced consumpion for nex period. Thus, all currenly acive households exi his period andenerhenexperiodwihcommonmoneyholdings. Noe ha he resuls of Lemmas 2-3 combine o imply ha, wihin any period, households ha underake balance ransfers all ener shopping-ime wih he same bank balance, make he same shopping-ime decisions, and hen ener he nex period as effecively idenical. Moreover, of his group of currenly acive households, hose households ha do no underake an accoun ransfer again in he nex period will coninue o be indisinguishable from one anoher as hey ener shopping-ime, and hence will ener he subsequen period wih common bank and brokerage accoun balances, and so forh. In oher words, any household ha was las acive a some paricular dae is effecively idenical o any oher household las acive a ha same dae. This is useful in our numerical approach o solving for compeiive equilibrium, since i allows us o move from idenifying individual households by heir curren money holdings o insead idenifying each household as a member of a paricular ime-since-acive group, wih all members of any one such group sharing in common he same sar-of-period money balances. Given he above resuls, we may rack he disribuion of households over ime hrough wo vecors, one indicaing he measures of households enering he period in each ime-since-acive group, [θ j, ], j = 1, 2,..., and he oher soring he balances wih which members of eachofhese curren groups exied shopping-ime in he previous period, [A j 1, 1 ]. From he laer, he curren sar-of-period balances held by members of each group are rerieved as M j = A j 1, 1+λP 1y 1, where P 1 represens he previous period s price level, and y 1 he common endowmen of he previous period. Households wihin any given sar-of-period group j hadonopayheirfixed coss move ogeher ino he curren shopping subperiod wih heir saring balances M j. Across all sar-of-period groups, hose households ha do pay o underake a bank ransfer will ener he curren shopping subperiod in ime-since-acive group wih common shopping-ime balances, M,, which we refer o as he curren arge money balances. Threshold ransfer rules: Finally, we esablish ha households follow hreshold policies in deermining wheher or no o ransfer asses beween heir brokerage and bank accouns. Specifically, given is sar-of-period money balances, each household has some maximum fixed cos ha i is willing o pay o underake an accoun ransfer and adjus is balances o he curren arge. Lemma 4. For any s, ξ 1, A = { ξ z s,ξ = } 1 is a convex se bounded below by. As our preceding resuls imply ha all members of any given sar-of-period group j are effecively idenical prior o he draws of heir curren ransfer coss, his las resul allows convenien deerminaion of he fracions of each such group underaking accoun ransfers, and hus he shopping-ime disribuion of households. Define he hreshold cos T ξj as ha fixed cos ha leaves any household in ime-since-acive group j indifferen o an accoun ransfer a dae. Householdsinhe group drawing coss a or below T ξj pay o adjus heir porfolios, while oher members of he 9

11 group do no. Thus, wihin each group j, he fracion of is members shifing asses o reach he curren arge bank balance is given by α j Hξ T. j Each such acive household underakes a ransfer x j = M, M j,, and he oal ransfer cos paid across all members of he group are H 1 α j θ j ξh ξ dξ. A family problem: Collecing he resuls above, and assuming ha aggregae shocks are Markov, we may re-express he lifeime plans formulaed by individual households as he soluion o he recursive problem of an exended family ha manages he join brokerage accoun of all households and acs o maximize he equally-weighed sum of heir uiliies. In each period, given he saring disribuion of households summarized by {θ j,a j } and he curren price level P,he family selecs he fracions of households from each ime-since-acive group o receive accoun ransfers, α j, and hence he disribuion of households over ime-since-acive groups a he sar of nex period, θ, he shopping-ime bank balance of each acive household, j M, achieved by ransfers from he family brokerage accoun, as well as he consumpion and money savings associaed wih members of each shopping-ime group, c j and A j+1 respecively, o solve he problem in below. In solving his problem, he family akes as given he curren endogenous aggregae sae K = [{θ j, A j },P 1y 1, M 1 ], and i assumes he fuure endogenous sae will be deermined by a mapping Ϝ ha i also akes as given; K = ϜK,s. In equilibrium, K is consisen wih he family s decisions. V {θ j,a j }; K,s=max θ j [α j u c +1 α j u c j ] + β j=1 subjec o: j=1 α j θ j [M M j ] + P j=1 θ j[ H ] 1 α j ξh ξ dξ s V {θ j,a j }; K,s gs, s ds 13 M M λ P 1y 1 14 M j = [A j + λp 1y 1 ], for j> 15 M j P c j + A, j+1 for j 16 A j+1, for j 17 α j θ j θ 1 18 j=1 θ j 1 α j θ, j+1 for j > 19 Recall from equaion 12 ha money marke clearing in each period requires ha he aggregae of households curren bonds less heir expendiures on new bonds mus equal he change in he aggregae money supply. By imposing his equilibrium condiion, we may use equaion 14 o represen he family s budge consrain requiring ha is join brokerage asses cover all curren ransfers o acive households and associaed fixed coss, as well as all bond purchases for he nex period. Nex, equaion 15 idenifies he sar-of-period money balances associaed wih each ime-since-acive group j, and represen he bank accoun budge and cash-in-advance consrains ha apply o members of each shopping-ime group. Finally, equaions describe he evoluion of households across groups over ime. In 18, he oal acive households shopping in group in he curren period is he populaion-weighed sum of he fracions of households 1

12 made acive from each sar-of-period group, and hese households move ogeher o begin he nex period in ime-since-acive group 1. In 19, households in any given ime-since-acive group j ha are inacive in he curren period will move ino he nex period as members of group j Soluion Recall ha we imposed money-marke clearing in formulaing he family s problem above. As such, we can rerieve equilibrium allocaions as he soluion o by appending o ha problem he goods marke clearing condiion needed o deermine he equilibrium price level aken as given by he family: y = c θ j α j + θ j 1 α j c j + ] θ j=1 j=1 j=1 j[ H 1 α j xhxdx. 2 Equaion 2 simply saes ha, wihin each period, he curren aggregae endowmen mus saisfy oal consumpion demand across all acive and inacive households ogeher wih he economywide fixed coss associaed wih accoun ransfers. In he resuls o follow, we absrac from rend growh in endowmens, and we assume ha money supply is increased a rae µ in he economy s seady-sae. Thus, he seady-sae is associaed wih inflaion a rae µ and a saionary disribuion of households over real balances described by [θ, a ], where θ = {θ j } and a ={a j }, wih a j A j P 1. As any given household ravels ouward across ime-since-acive groups, i finds is acual real balances for shopping ime, a + y /1 + µ, falling furher and furher below arge shopping balances; hus, he maximum j fixed cos i is willing o pay o underake an accoun ransfer rises. Given a finie upper suppor on he disribuion of fixed ransfer coss, his implies ha no household will delay aciviy beyond some finie maximum number of periods, which we denoe by J. Thus, he wo vecors describing he disribuion of households are each of finie lengh J. In solving he seady-sae of our economy, we isolae J as ha group j by which α j is chosen o be 1. Having arrived a he ime-since-acive represenaion described above, we are now almos in a posiion o follow King and Thomas 25 in applying linear mehods o solve for our economy s aggregae dynamics local o he deerminisic seady-sae. Two deails remain. Firs, as he linear soluion does no allow for a changing number of ime-since-acive groups, we mus resric J o be ime-invarian. Thus, we assume ha, for all, α J, = 1, andwehenverifyhaα j,,1, for j =1,..., J 1, is seleced hroughou our simulaions. Second, we assume ha, in every dae, all households ha ener shopping in ime-since-acive group J 1 compleely exhaus heir money balances; a J, =. Given ha any such household will underake an accoun ransfer wih cerainy a he sar of he nex period, his assumpion is consisen wih opimizing behavior so long as we verify ha nominal ineres raes are always posiive. 8 In parameerizing our model, we se he lengh of a period o one quarer, and we choose he seady-sae inflaion rae µ o imply an average annual inflaion a 3 percen. Period uiliy is iso-elasic, uc = c1 σ 1, wih σ = 2, and we selec he subjecive discoun facor β o imply an 1 σ average annual real ineres rae of 3 percen. The seady-sae aggregae endowmen is normalized 8 Given posiive nominal raes, if a J, > ever were o occur, he family could have improved is welfare by reducing he arge balances given o acive households a dae J 1 and increasing is bond purchases a ha dae o finance increased ransfers o a subsequen group of acive households for whom he non-negaiviy consrain would evenually bind. 11

13 o 1, and he fracion of he endowmen paid o household bank accouns which may be inerpreed as household wages is λ =.6, corresponding o labor s share of oupu. Holding hese parameers fixed, we will consider several alernaive assumpions regarding he disribuion of he fixed coss ha cause marke segmenaion in our model, as we discuss below. We begin o explore our model s dynamics in secion 4 hrough a series of examples involving he response o a money injecion ha, once observed, is known o be perfecly ransiory. There, we absrac from shocks o he endowmen o sudy he effecs of a moneary shock in isolaion, and o isolae hose aspecs caused by he endogenous changes in he degree of marke segmenaion ha disinguish our model. We consider each of hree examples disinguished only by he disribuion of fixed ransfer coss, beginning wih a baseline case where his disribuion is uniform on he inerval o B. There, we se he upper suppor a B =.25 o imply ha he maximum ime ha any household remains inacive is J = 6 quarers. For individual households, he resul is a 4.82 quarer average duraion beween accoun ransfers. In he aggregae, his calibraion resuls in a seadysae velociy of 1.9, which corresponds o he U.S. average over he pas decade. 9 In our second example, we raise he maximum ransfer cos o imply an aggregae velociy maching he U.S. poswar average, a 1.5. Reaining he assumpion ha ransfer coss are disribued uniformly, his implies a mean household inaciviy duraion of 7 quarers and a subsanially longer maximum inaciviy spell, a 1 quarers. This large difference beween a household s average expeced period of inaciviy versus he maximum such spell will be seen o have imporan qualiaive implicaions for he model s aggregae dynamics. Thus, in our hird example, we will move o consider a more flexible cos disribuion under which aggregae velociy again averages 1.5, bu mean and maximum duraions are close a 9.55 and 1 quarers, respecively. Following our emporary money growh shock examples, we will move in secion 5 o examine he model s aggregae dynamics under more realisic assumpions abou moneary policy. Firs, we will consider he response o a persisen rise in he money growh rae. There, we will assume ha money growh follows a mean-zero AR-1 process in logs wih persisence.57, as consisen wih he finding of Chari, Kehoe and McGraan 2. 1 Nex, in a second se of resuls, we will consider he response o a persisen shock o he real endowmen in an environmen where changes in he rae of money growh are dicaed by he moneary auhoriy s pursui of specific sabilizaion goals. In ha case, he common household endowmen will follow a persisen lognormal process, logy =ρ logy 1 + ε,ε n, σ 2 ε, wih ρ =.9 and σε =.7, and he moneary auhoriy will follow a Taylor rule in responding o deviaions in inflaion. In he endowmen economy, we assume ha he Taylor rule places zero weigh on deviaions in oupu, and is hus: i = i + 1.5[π π ]. A version of he model wih producion, where he Taylor rule does respond o changes in oupu, isdiscussedinsecion For comparabiliy, we follow Alvarez, Akeson and Edmond 23 in our measures of money and velociy. As in heir paper, money is broadly defined as he sum of currency, checkable deposis, and ime and savings deposis. They show ha he opporuniy cos of hese asses, relaive o shor-erm Treasury securiies, is subsanial and, as a whole, no very differen from ha of M1. Nex, velociy is compued as he raio of nominal personal consumpion expendiures o money. 1 The persisence of he moneary measure used o calibrae our model is acually subsanially higher, a.93 over he sample period 1954:1 o 23:1. Since our resuls are no qualiaively changed, we use he Chari, Kehoe and McGraan M1-based value for comparabiliy. 12

14 4 Examples 4.1 Seady-sae Before examining is responses o shocks, i is useful o begin wih a discussion of household porfolio adjusmen iming in our model s seady-sae. We firs consider how each of our hree examples relaes o he available micro-evidence provided by Vissing-Jørgensen 22. Using he Consumer Expendiure Survey, Vissing-Jørgensen compues ha he fracion of households ha acively bough or sold risky asses socks, bonds, muual funds and oher such securiies, beween one year and he nex ranges from.29 o.53 as a funcion of financial wealh. 11 For a direc comparison wih each version of our quarerly model, we compue he seady-sae uncondiional probabiliy ha a household will underake acive rade wihin one year as J j=1 θ j θ j+4, wih θ j+4 = for j > J We find ha he fracion of households acively rading in an average year is.78 in our baseline example, which is quie high relaive o he Vissing-Jørgensen daa. This may be explained in par by he fac ha he ransfer coss in his example are calibraed o mach aggregae velociy over only he pas decade, when ransacions coss were presumably lower han in her sample period. When we insead calibrae o mach aggregae velociy over he poswar period in our second example wih higher ransacions coss, he fracion of households rading annually falls o.55, slighly above he empirical range. Our mos successful example wih regard o his evidence is he hird, where high ransacions coss are drawn from a disribuion implying he same poswar aggregae velociy, bu longer expeced episodes of inaciviy. There, he model predics an average annual fracion of households conducing rades well wihin he empirical range, a.42. We canno compare our examples mean inaciviy duraions o ha implied by he Vissing- Jørgensen daa wihou making some assumpion abou he shape of he empirical hazard. If one assumes ha he probabiliy of an acive rade is consan from quarer o quarer in he daa, hen he range repored above implies a mean duraion of household inaciviy ranging from 7.5 o 13.8 quarers. Recall ha he mean duraion of inaciviy in our baseline example is only 4.8 quarers, while ha in our second example involving high ransacions coss is 7 quarers. This again suggess ha he frequencies of acive rades implied by hese wo versions of our model are, if anyhing, high relaive o he daa. However, our hird example wih boh high maximum and mean inaciviy spells exhibis an average duraion wihin he range implied by he daa, a 9.55 quarers. Thus, we will sudy his hird case as we move o examine our model s dynamic resuls in secion 5. We confine our remaining discussion of he model s seady-sae o ha arising under our baseline parameers, as he qualiaive aspecs ha we will emphasize hold across all of our examples. Here, wih boh he aggregae endowmen and he money growh rae fixed a heir mean values, 11 The CEX inerviews abou 45 households each quarer, and each household is inerviewed five imes, wih financial informaion gahered in he final inerview only. Vissing-Jørgensen 22 limis her sample o 677 households ha held risky asses boh a he ime of he fifh inerview and one year earlier. She finds ha he probabiliies of buying or selling risky asses do no significanly change when he sample, spanning , is spli ino subsamples according o inerview daes. 1 2 For example, in any dae of our model s seady-sae, here are θ1 households enering he period in ime-sinceacive group 1. Afer one year, a he sar of period +5, θ 1 θ 5 of ha original group have underaken a leas one rade. Thus, he fracion of hem ha have raded wihin a year is θ 1 θ5. The overall fracion rading wihin θ1 one year is he populaion-weighed sum of hese fracions across each saring group, j =1,..., J. 13

15 six groups of households ener ino each period, wih hese groups corresponding o he number of quarers ha have elapsed since members las accoun ransfer. As any individual household moves hrough hese groups over ime, is real money balances available for shopping fall furher and furher below he arge value, 2.936, given boh inflaion and is expendiures subsequen o is las ime acive. To correc his widening disance beween acual and arge real balances, he household becomes increasingly willing o incur a fixed ransfer cos. This implies ha he hreshold cos separaing acive households from inacive ones rises wih households ime-since-acive. Thus, as ransfer coss are drawn from a common disribuion, he fracion of households exhibiing curren aciviy in Table 1 rises across sar-of-period groups. Table 1: Deerminaion of seady-sae shopping-ime disribuion ime-since-acive group sar-of-period populaions fracion currenly acive shopping-ime real balances n/a shopping-ime populaions In figure 1A, we plo he seady-sae disribuion of households across groups as hey ener shopping-ime from he final row of able 1. Corresponding o he rising fracions of acive households shown above, he dashed curve reflecing he measures of households in each shopping-ime group monoonically declines across groups. The solid curve in he figure illusraes he raios of real consumpion expendiure relaive o real balances, individual velociies, associaed wih he members of each shopping-ime group. Because households are aware ha hey mus use heir curren balances o finance consumpion no only in he curren period bu also hroughou subsequen periods of inaciviy, individual spending raes rise across groups in response o a declining expeced duraion of fuure inaciviy. Currenly acive households, hose households in group, face he longes poenial ime before heir nex balance ransfer, and hus have he lowes individual velociies. By conras, households currenly shopping in group 5 will receive a ransfer wih cerainy a he sar of he nex period; hus, individual velociy is 1 for members of his las group. Two aspecs disinguishing our endogenous segmenaion model will be relevan in is responses o shocks below. Firs, on average, a household s probabiliy of becoming acive monoonically rises wih he ime since is las acive dae, as seen above. Second, hese probabiliies change over ime as shocks influence he value households place on adjusing heir bank balances. To isolae he imporance of hese wo elemens below, we will a imes conras he responses in our economy o hose in a corresponding economy ha has neiher. In ha oherwise idenical fixed duraion model, he iming of any household s nex accoun ransfer is cerain and is no allowed o change wih he economy s sae. Consisen wih our endogenously segmened economy, where households mean duraion of inaciviy is 4.8 quarers, households in he corresponding fixed duraion model are allowed o underake ransfers exacly once every 5 quarers. Figure 1B displays he seady-sae of he fixed duraion model. There, households ener every period evenly disribued across 5 ime-since-acive groups. Throughou groups 1 hrough 4, fracion of each group s members are allowed o underake accoun ransfers, while fracion 1 of he members of group 5 are auomaically made acive. Thus, 2 percen of households ener ino shopping in each ime-since-acive group hrough 4, and his shopping-ime disribuion remains 14

16 fixed over ime. As in our model wih endogenously imed household porfolio adjusmens, here oo individual velociies monoonically rise wih ime-since-acive and hi 1 in he final shopping group. However, given is lesser maximum duraion of inaciviy 5 quarers here versus 6 in he endogenous segmenaion model, households in he fixed duraion economy exhibi somewha higher spending raes hroughou he disribuion relaive o hose in panel A. 4.2 Money injecion: a baseline example In his and he following secion, we begin our sudy of he endogenous segmenaion economy s dynamics using wo examples designed o illusrae is underlying mechanics. Here, we examine heeffecsofanunanicipaedoneperiodriseinhemoneygrowhrae. Fixed duraion model. For reference, we begin in figure 2 wih an examinaion of he aggregae response in he fixed duraion model, where he fracions of acive households across groups are fixed and dicaed by α FD = [ 1]. 13 As seen in he op panel, he aggregae price-level rises only halfway a he dae of he money supply shock, wih he remaining price adjusmen saggered across several subsequen periods. This inflaion episode coninues unil hose households who were acive a he shock dae have raveled hrough all ime-since-acive groups and are once again acive, a he sar of dae 6. The aggregae price-level adjuss gradually in his exogenously segmened markes economy for precisely he reasons explained by Alvarez, Akeson and Edmond 23. Open marke operaions ha injec money ino he brokerage accouns mus be absorbed by acive households. 14 However, as hey will be unable o access heir brokerage accouns again for 5 periods, hese households reain large invenories of money relaive o heir curren consumpion spending. As noed above, heir spending rae is he lowes among all households in he economy. Consequenly, oal nominal spending does no rise in proporion o he money supply, and a rise in he share of money held by acive households leads o a rise in aggregae real balances. Equivalenly, in his endowmen model, velociy falls. Formally, in a fixed duraion model, given any fixed number of ime-since-acive groups J, aggregae velociy may be expressed as he sum of wo erms, one associaed wih he common velociy of currenly acive households and one associaed wih he velociies of inacive households across heir respecive groups: V = 1 J M v + J 1 1 M j v j. 21 M J M j=1 From his equaion, i is clear ha he rise in relaive money holdings of acive households mus reduce aggregae velociy, so long as individual velociies do no rise much in response o he shock. As seen in he boom panel of figure 2, in our fixed duraion example, half of he money injecion is absorbed by an iniial fall in aggregae velociy. As households ha were acive a he ime of he shock ravel hrough ime-since-acive groups in subsequen periods, heir spending rae rises, pulling aggregae velociy back up. During his episode nominal spending rises faser han 1 3 Figures in his and he subsequen secion reflec he effecs of a emporary.1 percenage poin rise in he money growh rae. Given ha our model is solved linearly, we have re-scaled all responses o correspond o a 1 percenage poin rise for readabiliy. 1 4 No household unable o shif asses from is brokerage accoun ino is bank accoun will accep he addiional money, given he rae-of-reurn dominance implied by posiive nominal ineres raes. 15