Transactions, Credit, and Central Banking in a Model of Segmented Markets

Transcription

1 Transacions, Credi, and Cenral Banking in a Model of Segmened Markes Sephen D. Williamson Deparmen of Economics Universiy of Iowa Iowa Ciy, IA sephen-williamson@uiowa.edu hp:// February 2006 Absrac A segmened markes model is consruced in which ransacions are conduced using credi and currency. Goods marke segmenaion plays an imporan role, in addiion o he role played by convenional segmenaion of asse markes. An imporan novely of he paper is o show how he diffusion of a money injecion by he cenral bank depends no only on he ineracion of agens in exchanging money for goods, bu on he arrangemens for clearing and selemen of credi insrumens. The model permis open marke operaions, dayligh overdrafs, reserve-holding, and overnigh lending and borrowing, allowing us o consider a rich array of cenral banking arrangemens and heir implicaions.

2 . INTRODUCTION In his paper, we explore he implicaions of a racable segmened markes model wih credi and cash ransacions, and wih a rich array of possible cenral banking arrangemens. As in radiional segmened markes models, his model has limied paricipaion in paricular asse markes, bu a key elemen of he model is he segmenaion of goods markes. The model permis open marke operaions, consumer credi ransacions, dayligh overdrafs, reserve-holding, overnigh lending and borrowing, and clearing and selemen of consumer credi ransacions. In he model, he ransmission of moneary policy is in par deermined by he rae of diffusion of a cenral bank money injecion hrough he economy. In urn, his rae of diffusion depends firs on he paern of exchanges of cash and credi for goods, and second on he naure of arrangemens for clearing and selemen of credi insrumens. We explore he implicaions of differen vehicles for accomplishing a cenral bank money injecion, i.e. dayligh overdrafs and open marke operaions. Furher, we examine he implicaions of he paymen of ineres on reserves. This model builds on Williamson (2006), which is a pure-currency framework where injecions of ouside money ino he economy occur by way of lump-sum ransfers. An imporan feaure of ha model is ha here are wo kinds of households, hose who are conneced, and hose who are unconneced. Conneced households can rade on asse markes, while unconneced households canno. Furher, and his is a novely in ha model, conneced and unconneced households have proximiy o differen ses of goods markes, and his is criical o how moneary policy works. A Friedman rule for moneary policy is subopimal, and an anicipaed inflaion effec on nominal ineres raes ends o reinforce he liquidiy effec, so ha nominal ineres raes are more volaile han in convenional segmened markes models. Moneary shocks have small effecs on aggregae real quaniies, bu can have quaniaively imporan 2

3 disribuional effecs. This paper is relaed o he lieraure on asse marke segmenaion and moneary policy. One branch of he marke segmenaion lieraure is concerned wih he developmen of general equilibrium versions of Tobin (956) and Baumol (952). In hese models, some fracion of he populaion is engaged, or chooses o engage in, asse ransacions a any poin in ime, and hus cenral bank acions in asse markes will iniially direcly affec only his paricipaing populaion. A moneary injecion by he cenral bank causes a redisribuion of wealh which will in general resul in shor run changes in asse prices, employmen, oupu, and he disribuion of consumpion across he populaion. The firs models of his ype were consruced by Grossman and Weiss (983) and Roemberg (984). Laer conribuions include Alvarez and Akeson (997), and Alvarez, Akeson, and Kehoe (2002), Alvarez, Akeson and Edmond (2002), Chiu (2004) and Khan and Thomas (2005). Much, hough no all, of his lieraure focusses on asse pricing implicaions, and a key feaure of all hese models is ha all economic agens (paricipaing or no) purchase goods in a single goods marke. Anoher relaed class of models deals wih marke segmenaion in a represenaive household consruc, and includes work by Lucas (990), Fuers (992), and Chrisiano and Eichenbaum (995). Fuers s model, from which Chrisiano and Eichenbuam s is developed, obains a nonneuraliy of money hrough a cash-in-advance consrain faced by firms ha applies o he purchase of labor services. This is quie differen from wha occurs in Tobin-Baumol-ype models wih endogenous labor supply. Recen research in moneary heory is aimed a developing models of moneary economies ha capure heerogeneiy and he disribuion of wealh in a manner ha is racable for analyical and quaniaive work. One approach is o use a quasilinear uiliy funcion as in Lagos and Wrigh (2005), an approach ha, under some 3

4 circumsances, will lead o he resul ha economic agens opimally redisribue money balances uniformly among hemselves whenever hey have he opporuniy. Anoher approach is o use a represenaive household wih many agens, as in Shi (997), in which (also see Lucas 990) here can be redisribuions of wealh wihin he household during he period, bu hese disribuion effecs do no persis. Work by Williamson (2005) and Shi (2004) uses he quasi-linear-uiliy and represenaivehousehold approaches, respecively, o sudy some implicaions of limied paricipaion for opimal moneary policy, ineres raes, and oupu. Oher relaed work is Head and Shi (2003), and Head and Lapham (2005). In he model consruced here, each household consiss of a producer and a coninuum of consumers. The consumers purchase goods in differen markes, bu are more likely o buy from households of heir own ype (conneced or unconneced). Goods are purchased wih credi, and hese debs mus be seled wihin he period. However, some debs are seled more quickly han ohers. If deb is seled quickly, hen sales of goods can be used o finance oher purchases by he household wihin he period. Oherwise, if goods are sold by a household in exchange for credi insrumens ha do no sele quickly, hen he receips from hese sales canno be spen unil he following period. A conneced household can borrow and lend on bond markes (one-period and wihin-period), and he cenral bank also borrows and lends in hese markes. If a period is inerpreed as one day, hen he cenral bank can engage in acions ha can be inerpreed as he exension of dayligh overdrafs and inervenion in he overnigh credi marke. Conneced households hold ouside money as reserve accouns wih he cenral bank, and he cenral bank has he opion of paying ineres on hese reserves. Unconneced households canno borrow and lend in bond markes, and hey hold ouside money in he form of currency. In general, conneced and unconneced households sell goods a differen prices in 4

5 equilibrium. Furher, a consumer pays a premium in a goods purchase where he deb exchanged for he goods akes longer o clear. A moneary shock no only produces a liquidiy effec, buiaffecs relaive prices. Tha is, a posiive money shock ends o reduce he nominal ineres rae, increase he relaive price of goods sold by conneced households, and reduce he relaive price of goods exchanged for deb ha akes a long ime o clear. In Secion 2 we se up he model, while in Secion 3 we se up he opimizaion problems of households and show how o consruc an equilibrium. In Secion 4 we deermine he properies of an equilibrium. Secion 5 is a conclusion. 2. THE MODEL Thereisaconinuum ofinfiniely-lived households wih uni mass indexed by i [0, ]. Each household consiss of a producer and a coninuum of consumers wih uni mass, wih a consumer indexed by (i, j), wihj uniformly disribued on he inerval [0, ]. The preferences of household i are given by X Z β log(c i (j))dj v(n i ), () E 0 =0 0 where indexes ime, 0 <β<, c i (j) is he consumpion of consumer j who is a member household i, and n i is he labor supply of he producer from household i. Assume ha v( ) is wice coninuously differeniable and sricly convex, wih v 0 (0) = 0 and v 0 ( ) =. The producer can supply an unlimied quaniy of labor, and each uni of labor supplied yields one uni of he perishable consumpion good. Each household resides a a separae locaion. There is a fracion α of conneced households, where 0 <α<. Conneced households hold ouside money as reserve accouns wih he cenral bank, and also can rade on bond markes. Each conneced household has M0 unis of ouside money a he beginning of period 0. The remaining fracion α of households are unconneced, in ha hey hold ouside money in he 5

6 form of divisible fia currency and do no rade on bond markes. Each unconneced household has M0 2 unis of ouside money in period 0. There is an absence-of-double-coincidence problem in his economy. Each household produces a disinc good, and he good ha is produced by he producer in a household is no consumed by any of he consumers in ha household. A he beginning of he period, each consumer in he household receives a preference shock, which deermines he good ha consumer wishes o consume during he period. Each consumer hen ravels o he locaion of he household ha produces his or her desired good, purchases some quaniy of ha good (deails of he exchange will be described laer), consumes, and hen reurns home. A given consumer canno visi more han one locaion, in addiion o his or her home locaion, during a period. For a consumer in a conneced household, he probabiliy ha he consumer s desired good during he curren period will be one sold by a conneced household is ( α)π and he probabiliy ha he desired good will be sold by an unconneced household is ( α)π. For a consumer from an unconneced household, he probabiliy of ha he consumer s desired good during he period is produced by an unconneced household is απ, and he probabiliy is απ ha he consumer s desired good is produce by a conneced household. For any consumer, he probabiliy disribuion for desired goods, condiional on he desired good being produced by a conneced or unconneced household, is uniform. These meeing probabiliies guaranee ha he flows of consumers going from conneced o unconneced households, and from unconneced o conneced households, are equal each period. The parameer π governs he ineracion beween conneced households and unconneced households as groups. Tha is, if π<hen he populaion of consumers arriving a a conneced locaion will have a greaer proporion of consumers from conneced locaions han would be observed arriving a an unconneced locaion, and similarly for unconneced locaions. If π =, hen he populaion of consumers is idenical across locaions when 6

7 consumers go shopping. A he beginning of period, consumers receive heir preference shocks, and hen purchase goods from he appropriae households wih credi. Tha is, consumers exchange IOUs for goods, and he IOUs are seled during he period. For he purposes of clearing and seling IOUs, here are N clearinghouses, indexed by k =, 2,..., N, and each clearinghouse has households as members. Clearinghouse membership N is deermined a random, so ha each clearinghouse has a mass of households as N members, and for a given clearinghouse he fracion of conneced member households is α. Le γ N. When consumers arrive o purchase goods from a producer in a household, he producer can deermine in which cases he consumer s and producer s clearinghouses are he same, and in which cases hese clearinghouses are differen. The law of large numbers implies ha each household will be selling o a fracion γ of consumers who have he same clearinghouse membership, and o a fracion γ whose clearinghouse membership is differen. In general, goods will be sold a differen prices o he wo differen groups of consumers, so ha here are effecively wo differen markes for goods on which each individual household producer sells. Afer households receive IOUs in exchange for he goods hey have produced, he IOUs are sen o he appropriae clearinghouse; ha is, an IOU issued by a paricular household goes o he clearinghouse of which ha household is a member. Thus, all of he IOUs issued by a household s consumers will a his poin find heir way back o he household s clearinghouse, and will represen debis on he household s accoun wih is own clearinghouse, and he household will have received some IOUs from oher household s ha are also members of is clearinghouse, and hese IOUs will consiue credis on he household s accoun wih is clearinghouse. The household will also have submied IOUs o clearinghouses of which i is no a member, and will herefore have credis agains hese oher clearinghouses. A his poin, selemen 7

8 among clearinghouses akes place, and conneced households and he cenral bank rade asses. The sequence of evens is as follows. Firs, conneced households and he cenral bank rade asses, hen each household seles is accoun wih is own clearinghouse in ouside money. For conneced households his involves a ransfer of reserve balances, and for unconneced households a ransfer of currency. Nex, each household seles wih all oher clearinghouses in a similar manner. In he asse marke on which conneced households and he cenral bank rade, here are hree asses: reserve balances, wihin-period nominal bonds, and one-period nominal bonds. In period, a wihin-period bond sells for one uni of reserve balances and is a claim o r unis of reserve balances a he end of he period, while a oneperiod bond sells for one uni of reserve balances in period and pays off R + unis of reserve balances in period +. One inerpreaion of hese arrangemens is ha a period is one day, borrowing by a household wihin he period is a dayligh overdraf wih he cenral bank, and overnigh borrowing and lending can be accomplished hrough combinaions of wihin-period and one-period borrowing and lending. The key consequences of hese paymens arrangemens can be summarized in he consrains faced by households, which will differ somewha depending on wheher he household is conneced or unconneced. We will consider equilibria where prices depend only on he mehod of paymen and wheher he seller of he good is a conneced or unconneced household. Le p and q denoe he prices a which a producer from a conneced household sells when paymen is made wih an IOU ha is cleared hrough he household s clearinghouse, or anoher clearinghouse, respecively. Similarly, p 2 and q 2 are he prices a which an unconneced household sells. The firs 8

9 consrain faced by he household is a finance consrain, which is [ ( α)π][γp c +( γ)q d ] (2) +( α)π[γp 2 c 2 +( γ)q 2 d 2 ]+b + + f i m + p x + R b τ In consrain (2), c denoes he consumpion of consumers from he conneced household who buy from anoher conneced household wih a clearinghouse in common and p is he money price of hose goods, while c 2 is wha is consumed by he consumers who buy from an unconneced household wih a clearinghouse in common, a he price p 2. Similarly d and d 2 denoe wha is consumed by consumers from a conneced household who buy from conneced and unconneced households respecively, bu do no share a clearinghouse wih he seller, and q and q 2 are he corresponding prices. As well, b is he quaniy of one-period nominal bonds acquired by he household in period, f is he quaniy of wihin-period nominal bonds purchased by he household, and m is he household s beginning-of-period money balances. Here, i denoes he gross nominal ineres rae on reserve balances held from he end of period o he beginning of period + (i.e. overnigh). Finally, x is he quaniy of goods sold by he household s producer o consumers who are members of he same clearinghouse, and τ is a nominal lump-sum ax paid o he governmen. Thus, consrain (2) saes ha oal household expendiure on goods and nominal bonds mus be financed by he money balances wih which he household begins he period, plus he IOUs acquired from consumers who share a clearinghouse wih he household. A conneced household mus saisfy is budge consrain, which is [ ( α)π][γp c +( γ)q d ] (3) +( α)π[γp 2 c 2 +( γ)q 2 d 2 ]+b + + f + m + i m + p x + q y + R b + r f τ τ 2 9

10 In consrain (3) m +is he quaniy of money carried by he household ino he nex period, y ishequaniyofgoodssoldoconsumerswhodonohaveaclearinghouse in common wih he household, r f denoes he oal nominal payoff on wihin-period bonds, and τ 2 is a nominal lump-sum ransfer paid o he governmen. A key feaure of he environmen is ha income earned by he household from he sale of goods for IOUs which clear on he second round of selemen canno be spen unil he following period. Tha is, consrain (2) is a ype of cash-in-advance consrain. Similarly, an unconneced household faces he finance consrain απ[γp c 2 +( γ)q d 2 ]+( απ)[γp 2 c 22 +( γ)q 2 d 22 ] m 2 + p 2 x 2, (4) and he budge consrain απ[γp c 2 +( γ)q d 2 ]+( απ)[γp 2 c 22 +( γ)q 2 d 22 ]+m 2 + m 2 +p 2 x 2 +q 2 y 2. (5) Noe ha, in conras o he conneced household, he unconneced household does no rade bonds, receives no governmen ransfers, and does no receive ineres on is ouside money balances, which are in he form of currency. There are a number of resricions in his model on he ypes of asses ha can be raded, and who can rade hese asses. Firs, coningen claims markes are absen. As we will see, in he equilibria ha we examine, all conneced households will be idenical, and all unconneced households will be idenical, bu conneced and unconneced households would wan o rade coningen claims if hey could. However, he only ineracion beween hese households is in meeings beween consumers and households. I is assumed ha anonymiy holds in hese meeings, i.e. as agens canno be idenified, coningen claims conracs canno be enforced. Furher, clearinghouses are no able o verify paricular coningencies and so hey canno ac as coningen-claim inermediaries. Second, unconneced households canno rade 0

11 bonds, and bonds (a leas some of hem) canno be used in selemen. Since bonds mus be purchased wih reserve balances, he payoffs are in reserve balances, and unconneced households canno hold reserve balances, hese households herefore canno hold bonds eiher. We assume ha he governmen prohibis he issue of liabiliies ha have he feaures of governmen-issued fia currency, and so bonds canno be inermediaed and used in ransacions. Consrain (2) could be relaxed if a household s clearinghouse would accep he IOUs of non-member households in selemen. However, we assume ha a clearinghouse does no have he means o verify he auheniciy of IOUs oher han he ones issued by is members. To normalize, we will suppose ha he iniial aggregae quaniy of ouside money is uniy, i.e. αm0 +( α)m0 2 =, and here are iniially no ousanding governmen bonds. As well, all ineres on governmen bonds in periods, 2,... is financed by he lump-sum ransfers, so ha he aggregae quaniy of nominal governmen liabiliies is fixed a uniy forever. Our principal concern is in deermining he effecs of changes in he he composiion of he governmen s deb, i.e. he effecs of moneary policy. Le M (M 2 ) denoe he sock of money per capia supplied o conneced (unconneced) households a he beginning of period, B he quaniy of one-period governmen bonds per conneced household mauring in period, and F he quaniy of wihin-period governmen bonds mauring in period. The governmen s budge consrain is hen αm++( α)m + 2 = i αm +( α)m 2 αb + +R αb +(r )αf ατ ατ 2, (6) where B 0 =0. The lump-sum axes ha financeineresonhegovernmendebare levied in such a way as o have no disribuional consequences, ha is τ =(R )B +(i )M (7)

12 and τ 2 =(r )F (8) The governmen chooses i,b +,F, τ, and τ 2 a he beginning of period, possibly in a random fashion. The gross ineres raes R and r are hen marke-deermined, and (6), (7), and (8) hen deermine he oal quaniy of aggregae ouside money in period +on he lef-hand side of (6). 3. OPTIMIZATION AND EQUILIBRIUM In his secion, our goals are o characerize he soluion o he households opimizaion problems, and impose equilibrium condiions. For a conneced household, given he household s objecive funcion () and is consrains (2) and (3), and assuming an inerior soluion (which we mus have in equilibrium), inraemporal opimizaion gives p c = q d = p 2 c 2 = q 2 d 2 = v0 (x + y ) p = λ + v0 (x + y ) q = r v 0 (x + y ), where λ denoes he muliplier associaed wih he household s finance consrain (2). In (9), he log uiliy funcions of consumers imply ha he household will equalize expendiures across he household s consumers a he opimum. This will give us considerable mileage in he analysis. Ineremporal opimizaion by a conneced household gives and v 0 (x + y ) q p c = βi + E µ = βr + E µ p +c + p +c + q (9), (0). () Similarly, for unconneced households, given (), (4), and (5), he analogs of (9) 2

13 and (0) are p c 2 = q d 2 = p 2 c 22 = q 2 d 22 v 0 (x 2 + y 2 ) q 2 = v0 (x 2 + y 2 ) p 2 = βe µ p 2 +c 22 + = λ 2 + v0 (x 2 + y 2 ), (2) q 2. (3) Nex, in equilibrium he marke clears for goods sold in conneced locaions for IOUs ha clear early, γ{[ ( α)π]c +( α)πc 2 } = x, (4) for goods sold in conneced locaions for IOUs ha clear lae, ( γ){[ ( α)π]d +( α)πd 2 } = y (5) and for goods sold in unconneced locaions for IOUs ha clear early and lae, respecively, Finally, asse markes clear, ha is γ[απc 2 +( απ)c 22 ]=x 2, (6) ( γ)[απd 2 +( απ)d 22 ]=y 2. (7) B = b,f = f,m = m,m 2 = m 2. (8) 4. ANALYSIS In his secion, given he characerizaion of a compeiive equilibrium from he previous secion, we obain some resuls concerning prices and he effecs of open marke operaions. Firs, noe from (9) and (2) ha, p i = q i if and only if λ i =0and p i <q i if and only if λ i > 0, for i =, 2. Therefore, a consumer will pay a premium if he or she purchases goods wih IOUs ha clear lae, if and only if he finance consrain 3

14 binds. Tha is, so long as here is a binding finance consrain, hen all income received by he household from he sale of goods for early-clearing IOUs is spen in he curren period. However, income received from selling goods in exchange for lae clearing IOUs canno be spen unil he following period. Therefore, he household will demand a premium o accep a lae-clearing IOU. As well, from (9) we have r = q, (9) p where r is he gross nominal ineres rae on wihin-period bonds. Therefore, he wihin-period nominal ineres rae is greaer han zero if and only if consumers pay a premium when hey make a purchase wih a lae-clearing IOU, i.e. if and only if he finance consrain binds for conneced households. We will assume for now (and check his laer) ha he finance consrains (2) and (4) always bind. Then, leing z (z 2 ) denoe nominal expendiure in period by a conneced (unconneced) household, and given (2), (4), (6), (7), (8), and (8), we ge z = p x + M + B B + F (20) z 2 = p 2 x 2 + M 2. (2) From (9), (2), (4), and (6), nominal expendiures in conneced and unconneced locaions, respecively, where paymen is in early-clearing IOUs, is given by p x = γ{[ ( α)π]z +( α)πz 2 }, (22) p 2 x 2 = γ[απz +( απ)z 2 ] (23) Then, subsiuing in (20) and (2) for p x and p 2 x 2 using (22) and (23), and solving for z and z 2, we obain ( γ)z = [ γ( απ)][m + B B + F ]+( α)πγm 2 [ γ( π)], (24) 4

15 ( γ)z 2 = απγ[m + B B + F ]+{ γ[ ( α)π]} M 2 (25) [ γ( π)] In (24) and (25), noe ha M + B B + F is he quaniy of ouside money available o a conneced household a he beginning of period afer he he governmen makes asse rades, while M 2 is he quaniy of ouside money available o an unconneced household. Then, (24) shows ha he nominal expendiures by a conneced household on goods purchased wih IOUs ha clear lae, ( γ)z, equals a weighed average of he quaniies of ouside money available o conneced and unconneced households. Tha is, when consumers from an unconneced household make purchases from a conneced household, some of he receips of he conneced household are available o spend during he period. Since nominal expendiure by unconneced households will end o increase wih M 2, herefore nominal expendiure by conneced households will end o increase wih M 2 as well, as we see in (24). Noe ha he weigh on M 2 in equaion (24) is increasing in ( α)π, he fracion of consumers buying from a conneced household who come from an unconneced household. As well, he weigh on M 2 in equaion (24) is increasing in γ, he fracion of goods ransacions ha are seled early in he period. Equaion (25) shows a similar relaionship o (25) for an unconneced household raher han a conneced one. Nex, we can deermine he quaniies of money per household in each locaion in period +. Given ha he finance consrains (2) and (4) bind, from (2)-(5), (6), (7), (8), and (8), we ge M + = q y + F, M 2 + = q 2 y 2. Therefore, from (9), (2), (5), (7), (24), and (25), we obain M+ = M + B B + ( α)π [M + B B + F M 2 ], (26) [ γ( π)] 5

16 M+ 2 = M 2 + απ [M + B B + F M 2 ]. (27) [ γ( π)] Equaions (26) and (27) show ha, if he quaniy of ouside money available o spend per conneced households is greaer han he quaniy available per unconneced households, hen money will flow from conneced o unconneced households, and vice-versa. Noe, in equaions (26) and (27) ha money flows are larger he larger is π and he larger is γ. Tha is, π and γ deermine he speed of diffusion of an ouside money injecion by he cenral bank, which can occur eiher hrough an open marke purchase (B B + ) or a dayligh overdraf ( F ). The parameer π governs he degree o which households purchase goods from oher households of he same ype. Noe from (26) and (27) ha, if π =, hen diffusion occurs in one period, ha is M + = α(m + B B + )+( α)m 2 +( α)f = M 2 + F The parameer γ ishefracionofgoodsransacionsvolumehaclearsearlyinhe period, so ha an increase in γ also speeds diffusion. Seing γ =in (26) and (27) gives he same resul as seing π =, i.e. diffusion occurs in one period. However, he economy wih γ =isonewhereousidemoneyisnoneededasamediumof exchange. Now, o solve for an equilibrium, firs le ψ and ψ 2 denoe oal nominal expendiure on he goods produced by a conneced and unconneced household, respecively. From (24) and (25) we ge ψ = [ ( α)π]z +( α)πz 2 (28) = [ γ( π) ( α)π](m + B B + F )+( α)πm 2 ( γ)[ γ( π)] ψ 2 = απz +( απ)z 2 (29) = απ(m + B B + F )+[ γ( π) απ]m 2 ( γ)[ γ( π)] 6

17 Then, from (0), (22), (28), and since q y ( γ) =p x γ, we ge and (0) and (22) gives v 0 (x + y )y γ µ = βi + ψ E, (30) z+ v 0 (x + y )x γ Similarly, for unconneced households, we obain and (0) and (22) gives v 0 (x 2 + y 2 )y 2 γ v 0 (x 2 + y 2 )x 2 γ = ψ. (3) z µ = βψ 2 E, (32) z+ 2 = ψ2, (33) z 2 and (32) and (33) solve for x 2 and y 2. Furher, from (9) and () nominal ineres raes are deermined by µ = βz E, (34) R + z+ and r = R +. (35) i + A moneary policy is a sochasic process for {B +,F,i + } =0 given B 0 =0and saisfying and M + B B + F > 0 i + R + for all, which hen deermines a sochasic process for {M,M 2 } = given M 0 and M 2 0 from (26) and (27). Then, we can use (24), (25), (28), and (29) o deermine {z,z 2,ψ,ψ 2 }, which is an exogenous sochasic process. Then, he mahemaics involved in he soluion is very simple. Equaions (30) and (3) solve joinly for x 7

18 and y, equaions (32) and (33) solve joinly for x 2 and y 2 (34) solves for R + and (35) solves for r, all for =0,, 2,.... Then, he consumpion allocaion is given by c ij = x j z i ψ j, for i, j =, 2, d ij = y j z i ψ j, for i, j =, CONCLUSION We have consruced a racable model where here are alernaive paymens arrangemens for purchasing goods, and where he cenral bank can use differen vehicles o injec ouside money ino he privae economy. All goods are purchased wih IOUs, bu some IOUs clear more quickly han ohers. As a resul, ouside money is useful in seling debs. However, only conneced households can borrow and lend on bond markes and hold ouside money as reserve balances wih he cenral bank. Because of goods marke segmenaion, prices are in general differen in differen markes, and a cenral bank money injecion will affec relaive prices in he shor run across goods markes. Furher, consumers pay a premium in a goods purchase if he IOU wih which he good is purchased does no clear quickly. Thus, in a paricular marke, prices depend on he paymen insrumen used, and relaive prices in any given marke change in response o a money injecion. Theraeofdiffusion of a money injecion hrough he economy deermines he persisence in he effecs of moneary policy and is quaniaive iniial impac. The rae of diffusion increases as he probabiliy ha differen ypes of households rade increases, and i change wih he arrangemens for clearing and seling credi insrumens. 8

19 REFERENCES Alvarez, F. and Akeson, A Money and Exchange Raes in he Grossman- Weiss-Roemberg Model, Journal of Moneary Economics 40, Alvarez, F., Akeson, A., and Kehoe, P Money, Ineres Raes, and Exchange Raes wih Endogenously Segmened Markes, Journal of Poliical Economy 0, Baumol, W The Transacions Demand for Cash: An Invenory Theoreic Approach, Quarerly Journal of Economics 66, Chiu, J Endogenously Segmened Asse Marke in an Invenory Theoreic Model of Money Demand, working paper, Universiy of Wesern Onario. Chrisiano, L. and Eichenbaum, M Liquidiy Effecs, Moneary Policy, and he Business Cycle, Journal of Money, Credi, and Banking 27, Cooley, T. and Hansen, G The Inflaion Tax in a Real Business Cycle Model, American Economic Review 79, Grossman, S. and Weiss, L A Transacions-Based Model of he Moneary Transmission Mechanism, American Economic Review 73, Head, A. and Lapham, B Limied Paricipaion, Marke Power and Inflaion Dynamics, working paper, Queen s Universiy. Head, A. and Shi, S A Fundamenal Theory of Exchange Raes and Direc Currency Trades, Journal of Moneary Economics 50, Lagos,R.,andWrigh,R AUnified Framework for Moneary Theory and Policy Analysis, Journal of Poliical Economy 3,

20 Lucas, R Liquidiy and Ineres Raes, Journal of Economic Theory 50, Lucas, R Inflaion and Welfare, Economerica 68, Roemberg, J A Moneary Equilibrium Model wih Transacions Coss, Journal of Poliical Economy 92, Shi, S A Divisible Model of Fia Money, Economerica 65, Shi, S Liquidiy, Ineres Raes, and Oupu, working paper, Universiy of Torono. Tobin, J The Ineres-Elasiciy of Transacions Demand for Cash, Review of Economics and Saisics 38, Williamson, S Search, Limied Paricipaion, and Moneary Policy, forhcoming, Inernaional Economic Review. Williamson, S Moneary Policy and Disribuion, working paper, Universiy of Iowa. 20