MONETARY POLICY IN A CREDIT-IN-ADVANCE ECONOMY

Transcription

1 MONETARY POLICY IN A CREDIT-IN-ADVANCE ECONOMY WP nº 2001/20 Emanuel Leão Ouubro 2001 DOCUMENTO DE TRABALHO WORKING PAPER

2 Emanuel Leão is Auxiliary Professor a he Deparamen of Economics of he Insiuo Superior de Ciências do Trabalho e da Empresa (ISCTE) and Research Fellow of he Dinâmia (Cenro de Esudos sobre a Mudança Socioeconómica). The auhor has received helpful commens from Alan J. Suherland, Peer N. Smih, Michael R. Wickens, Moren O. Ravn, John P. Huon and paricipans a he Young Economiss Conference Oxford They are in no way responsible for he inerpreaions or any errors ha he paper may conain.

3 DINÂMIA CENTRO DE ESTUDOS SOBRE A MUDANÇA SOCIOECONÓMICA MONETARY POLICY IN A CREDIT-IN-ADVANCE ECONOMY Emanuel Reis Leão eccl@isce.p Index Absrac 2 Lis of Symbols 3 1. Inroducion 5 2. The Economic Environmen 6 3. The Typical Bank s Behaviour 8 4. The Typical Firm s Behaviour 9 5. The ypical Household s Behaviour The Marke Clearing Condiions The Compeiive General Marke Equilibrium assuming H homogeneous households, F homogeneous firms and L homogeneous banks plus Raional Expecaions Calibraion The Dynamic Properies of he Model The impac of echnological shocks Moneary Policy Conclusion 26 Appendix 27 References 31 Tables 32 Figures 34 ISCTE - INSTITUTO SUPERIOR DE CIÊNCIAS DO TRABALHO E DA EMPRESA Av. das Forças Armadas, Lisboa, PORTUGAL Tel Fax dinamia@dinamia.isce.p Inerne:

4 Absrac In he benchmark dynamic general equilibrium model wih money [Cooley and Hansen (1989)], money is supplied o he economy in a way which may be argued o be no very realisic. In his paper, we develop a dynamic general equilibrium model where bank loans are he source of money creaion. We show ha he same resuls follow as in Cooley and Hansen. In paricular, moneary policy has only very small real effecs in he model. Keywords: Dynamic General Equilibrium Models, Cash-in-Advance, Moneary Policy. JEL Classificaion: E17, E41, E52 (The full lis of JEL codes can be found a hp://

5 Lis of Symbols c ` n s B +1 = (1 + R ) B +1 z f CD HOUSEHOLD VARIABLES household s consumpion in real erms leisure household s supply of labour he amoun he household borrows a he beginning of period household s deb a he beginning of period (+1) % of rm f ha he household buys a beginning of period (-1) and sells a beginning of period % of bank l ha he household buys a beginning of period (-1) and sells a beginning of period amoun of checkable deposis he household decides o hold a he beginning of period shadow price (marginal uiliy of consumpion, in his model) ineremporal discoun facor FIRM VARIABLES y rm s oupu k rm s sock of capial n d rm s labour demand i rm s invesmen f nominal pro s of rm f in period ± per period rae of depreciaion of he rm s capial sock B s RES bank;l r req P W R Q f Q bank;l BANK VARIABLES bank s nominal supply of credi a he beginning of period bank s reserves in period nominal pro s of bank l in period required reserve raio PRICE VARIABLES price of physical oupu nominal wage rae nominal ineres rae beween he beginning of period and he beginning of period (+1) nominal price of rm f a beginning of period (amoun necessary o buy 100% of rm f ) nominal price of bank l a beginning of period (amoun necessary o buy 100% of bank l) SHOCK VARIABLES A rm s echnological parameer ¹ rae of growh of he level of reserves beween he beginning of period (-1) and he beginning of period

6 x seady-sae value of variable x bx he percenage deviaion of variable x from is seady-sae (in period ) E [:] expecaion condiional on informaion peraining o he beginning of period and earlier of he indicaed argumen H number of households F number of rms L number of banks b +1 = b s = p = w = q f = B+1 RES 1 Bs RES 1 P RES 1 W RES 1 Qf RES 1 q bank;l = Qbank;l RES 1 ¼ f = f RES 1 ¼ bank;l = bank;l RES 1 k = F H k n d = F H nd q f = F H qf q bank;l = L H qbank;l ¼ f = F H ¼f ¼ bank;l = L H ¼bank;l

7 1 Inroducion Cooley and Hansen (1989) add money o he dynamic general equilibrium model of Hansen (1985). The way money is fed ino he economy in he model of Cooley and Hansen can be summarized as follows. Households sarevery period wiha givenamounofmoney carriedover fromhe previous period and hey hen receive a lump-sum ransfer from he governmen (which, supposedly, has obained i from he cenral bank). This mechanism of injecing money ino he economy sands in sharp conras wih he fac ha, in modern economies, mos money has is origin in loans from commercial banks o households, rms or he governmen. In his paper, we add money o a zero growh version of he dynamic general equilibrium model presened in King, Plosser and Rebelo (1988). In he model we build, all money has is origin in loans ha he households obain from commercial banks. In our model here are only households, nonbank rms, commercial banks and a cenral bank (in his paper, we shall refer o nonbank rms simply as rms ; likewise, we shall refer o commercial banks simply as banks ). Banks make loans o households a he beginning of each period and households hen use he money obained in his way o buy consumpion goods from he rms. We have log-linearized he compeiive equilibrium around he seady-sae values of is variables and hen calibraed i using Poswar daa. Aferwards, we examined he response of he model o echnological shocks and o moneary policy shocks. Our simulaion experimens gave us he same paern of resuls ha was obained by Cooley and Hansen: ( i ) when money is supplied o he economy a a consan rae of growh, he behaviour of he real variables in he presence of echnological shocks is he same ha is obained using he same model wihou money; ( ii ) when money is supplied o he economy a a consan rae of growh, he acual rae of growh a which i is supplied has no in uence on he resuls we obain when he model is hi by echnological innovaions; ( iii ) when here are shocks o he rae of growh of he money supply, he behaviour of he real variables changes only slighly and mos of he impac goes o he price level. The srucure of he aricle is as follows. In secion 2, we describe he economic environmen: preferences, echnology, resource consrains and marke srucure. In secion 3, we describe he ypical bank s behaviour. In secion 4, we describe he ypical rm s behaviour. In secion 5, we describe he ypical household s behaviour. In secion 6, we wrie down he marke clearing condiions. In secion 7, we wrie he se of equaions ha describes he compeiive general marke equilibrium. In secion 8, we describe he calibraion of he model. In secion 9, we look a he response of he model o echnological shocks and o moneary policy shocks. In secion 10, we make an overview and conclusion. DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 5

8 2 The Economic Environmen This is a closed economy model wih no governmen. There are H homogeneous households, F homogeneous rms, L homogeneous banks and a cenral bank. Firms and banks are owned by he households. As a consequence, boh he rms pro s and he banks pro s are disribued o households (he shareholders) a he end of each period. There is only one (homogeneous) physical good produced in his economy which we denoe physical oupu. There are wo possible uses for his oupu: i can eiher be consumed or used for invesmen (i.e., used o increase he level of he sock of capial). In our model, he only source of money is loans from commercial banks o households. A he beginning of each period, commercial banks obain reserves from he cenral bank which allow hem o make loans o households. In making loans o households, a he beginning of each period, commercial banks creae deposis which did no exis before. Households pay he amoun borrowed a he end of he same period hereby desroying he deposis which had been creaed a he beginning of he period. Le us now concenrae on he way he household s demand for credi is modelled. When we sar hinking abou modelling household borrowing, a key issue ha we have o deal wih is he household s iniial deb posiion. If period 0 is he period where our analysis of he economy is saring (he presen dae, for example) and he household has been living for some periods before arriving a he beginning of period 0, wha deb should we assume ha she carries from he previous period? In his aricle, we propose using a speci c iniial condiion o describe he household s iniial deb posiion. In he Appendix we show ha he iniial condiion we use is he iniial condiion which naurally arises when we hink he economy back o is iniial momen. Inany case, oher iniial condiions canandshouldbe ried. Wha we cannodo is avoid modelling he iniial deb posiion of he household. The iniial condiion on he ypical household s opimizaion problem ha we use is: a he beginning of he period where our analysis of he economy is saring (period 0), he ypical household owes he banks an amoun which equals he sum of he wage earnings and dividend earnings ha she received a he end of he previous period. Afer paying he deb o he banks, he household is lef wih nohing (he household also owns shares bu, because households are all alike, shares are no raded in equilibrium) and mus herefore borrow again from he banks o nance her consumpion expendiure during he period ha is jus beginning (period 0). The marke clearing condiions of he model ogeher wih he de niions of pro s imply ha he amoun he household borrows a he beginning of period 0 is such ha when she arrives a he end of period 0 she will again be lef wih nohing (afer paying he deb o he banks). The consequence is ha he household will need o borrow again from he banks a he beginning of period 1 in order o nance her consumpion expendiure during his period. And his sory is repeaed every period ino he fuure. All his is achieved by he inroducion of he iniial condiion ha we have jus menioned. DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 6

9 The complee descripion of moneary ows among economic agens is as follows. A he beginning of each period, he ypical household borrows from he banks he amoun ha she needs in order o be able o buy consumpion goods from he rms during he period ha is beginning. Loans obained from a bank ake he form of checkable deposis. During he period, he ypical household spends hese checkable deposis buying consumpion goods from he rms. A he end of he period, he household receives back from he rms hese checkable deposis (as wage paymens and dividend paymens). Then, he household pays he banks ineres on he amoun borrowed a he beginning of he period (he amoun of ineres due is paid by reducing he amoun of checkable deposis ha he household owns a he banks). However, since banks are owned by he households and households are all alike, he household immediaely receives back he amoun of ineres paid o he banks (in he form of bank dividends). Aferwards, he household pays he banks he principal of he deb conraced a he beginning of he period. The srucure of he model is such ha, afer all hese paymens, he household is lef wih nohing and mus herefore borrow again from he banks a he sar of he new period. Since all ransacions in our economy are nanced by bank loans, i can be labelled a pure credi economy (as de ned by Wicksell). I is imporan o emphasize ha he precise credi srucure of he model depends on he iniial condiion we choose. I is he iniial condiion ha ransforms a cash-in-advance economy ino a speci c credi-in-advance economy. We nex examine he ypical household s preferences, he echnology available in he economy (producion funcion and capial accumulaion equaion), he resource consrains ha exis in a given period and he marke srucure. Le us suppose ha we are a he beginning of period 0 and ha households, rms and banks are considering decisions for periods wih = 0; 1; 2; 3;:::. Le us sar by describing he preferences of he ypical household. The ypical household seeks o maximize lifeime uiliy. Uiliy in period is given by u(c ;`) where c is he ow amoun of consumpion and ` is he amoun of leisure enjoyed in ha period. The funcion u(:;:) has he usual =1 properies. A he beginning of period 0, he household maximizes U 0 = E 0 u(c ;`) where is a discoun facor (0 < < 1) ha re ecs a preference for curren over fuure consumpionleisure bundles. Applicaion of he operaor E 0 [:] yields he mahemaical expecaion, condiional on complee informaion peraining o he beginning of period 0 and earlier, of he indicaed argumen. Le us now describe he echnology available in he economy: producion funcion and capial accumulaion equaion. Each rm s producion funcion is described by y = A F(k ;n d ) where y is he physical oupu of he rm, A is a echnological parameer, k is he rm s sock of capial and n d is he rm s labour demand in period. Capial accumulaion is described by k +1 = (1 ±)k + i where i is he ow of invesmen in period and ± is he per-period rae of depreciaion of he sock of capial which is assumed o be consan and belonging o he closed inerval [0,1]. P =0 DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 7

10 The resource consrains are as follows. Each rm sars period wih a sock of capial k which is pre-deermined [which was deermined a he beginning of period ( 1)]. In oher words, he sock of capial ha will ener he producion funcion in period canno be changed by decisions aken during period. Each household has an endowmen of ime per-period which is normalized o be equal o one by an appropriae choice of unis. This amoun of ime can be used o work or o res. Therefore, we can wrie n s + ` = 1 where n s is he household s supply of labour during period : There is a legal consrain which implies ha he oal amoun of credi ha each commercial bank can o er canno exceed a cerain amoun. This may be seen as a resricion se by he exisence of a required reserve raio ogeher wih a xed amoun of reserves made available o each bank. Therefore, he maximum amoun of credi ha each bank can lend is denoed by 1 r req RES where RES is he oal reserve endowmen of each bank and r req is he required reserve raio. In his model, reserves pay no ineres. Le us now describe he marke srucure. There are ve markes: he goods marke, he labour marke, he bank loans marke, he marke for rms shares and he marke for banks shares. We assume ha each household behaves as a price-aker, each rm behaves as a price-aker and each bank also behaves as a price-aker. Prices are perfecly exible and adjus so as o clear all markes in every period. 3 The Typical Bank s Behaviour In his model, each commercial bank receives an endowmen of reserves from he cenral bank a he beginning of period which is denoed RES. We assume ha he commercial bank pays no ineres o he cenral bank because of his amoun of reserves i receives. I is also assumed ha commercial banks have no coss of operaion. On he oher hand, commercial banks operae under a fracional reserve sysem (wih he required reserve raio denoed r req ). The combinaion of a xed amounof reserves made available o each bank wih a requiredreserve raio means ha here is an upper bound on he amoun of credi ha he bank can supply in each period. Alernaively, we can hink of his upper bound on he amoun of credi ha each bank can supply as resuling from some capial requiremen raio such as he Basle Capial Accord, also known as Cook Raio. In he model, we shall reason as if he upper bound on he banks abiliy o supply credi were he resul of a required reserve raio. The alernaive moivaion of his upper bound as resuling from a capial requiremen raio has been menioned because i may be more appealing for economiss living in counries where here is no required reserve raio or where his raio is very low. Wih he presen assumpions (no coss of obaining reserves and no coss of operaion), as long as he lending rae is always sricly posiive, banks will always wan o supply he maximum amoun of credi ha hey can. Therefore, he supply of credi by each bank in period is jus he xed amoun 1 r req RES. To see his mahemaically, we rs noe ha, under he presen assumpions, he pro s of bank l in period are given by bank;l = R B s where R is he nominal DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 8

11 ineres rae (lending rae) beween he beginning of period and he beginning of period ( + 1) and B s is he nominal amoun of credi supplied by he bank. The pro s earned by each bank during period are disribued o households a he end of he period in he form of dividends. Each bank maximizes he Value of is Asses (VA), i.e., he expeced discouned value of is sream of presen and fuure dividends. Therefore, when we are a he beginning of period 0, he ypical bank s opimizaion problem is =1 1 MaxV A = E 0 B s 1+R 0;+1 R B s ha P =0 s.. B s 1 r req RES for = 0,1,2,3,... Given he economic environmen we are working wih, we hink i is appropriae o assume (1 + R 0;+1 ) = (1 + R 0 )(1 + R 1 )(1 + R 2 ):::(1 + R ) for = 0; 1; 2;::: Noe ha we are a he beginning of period 0. Therefore, because dividends are only disribued a he end of he period, we discoun period 0 dividends by muliplying hem by 1=(1 + R 0 ), we discoun period 1 dividends by muliplying hem by 1=[(1 + R 0 )(1 + R 1 )],... As long as he nominal ineres rae is always sricly posiive, he soluion o he bank s opimizaion problem will obviously be B s = 1 r req RES (1) for = 0; 1; 2;::: We assume ha he endowmen of reserves grows over ime [he rae of growh of he level of reserves beween he beginning of period (-1) and he beginning of period is denoed ¹ ]. Hence, B s will also grow over ime. Noe ha he nominal pro s of bank l in period can be wrien as 4 The Typical Firm s Behaviour bank;l = 1 r req RES R (2) The focus of he aricle is on modelling he demand for bank credi by households and he supply of credi by commercial banks. Hence, he behaviour of rms is modelled in a sandard way. In nominal erms, he pro s of rm f in period are given by income from he sale of oupu minus he wage bill minus invesmen expendiure f = P A F(k ;n d ) W n d P [k +1 (1 ±)k ] (3) where P is he nominal price of goods in period and W is he nominal wage in period. The rm pays wages o households a he end of he period. The pro s earned by each rm during DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 9

12 period are disribued o households a he end of he period in he form of dividends. Each rm f (f = 1; 2;:::;F) maximizes he Value of is Asses (VA), i.e., he expeced discouned value of he sream of presen and fuure dividends. Therefore, when we are a he beginning of period 0; he ypical rm s opimizaion problem is =1 P 1 Max V A = E 0 n d ;k 1+R 0;+1 f +1 where =0 f = P A F(k ;n d ) W n d P [k +1 (1 ±)k ] for =0,1,2,... There is also an iniial condiion for he capial sock, he sandard ransversaliy condiion for he capial sock and non-negaiviy consrains. 5 The Typical Household s Behaviour In his secion we presen he ypical household s problem wrien in a cash-in-advance form which was learned from Lucas (1982), i.e., using a porfolio allocaion consrain and a cash-in-advance consrain. The way loans work in his model is as follows. We have menioned ha R denoes he nominal ineres rae beween he beginning of period and he beginning of period (+1). A he beginning of period, he household borrows from banks he amoun B+1 1+R. This means ha he household receives B+1 1+R moneary unis a he beginning period and ha she will have o pay B +1 1+R (1 + R ) = B +1 moneary unis a he end of period [beginning of period ( + 1)]. Hence, B +1 denoes he deb he household has a he beginning of period (+1). The way shares work in his model is as follows. Q f is he nominal price ha he household would have o pay o buy 100% of rm f a he beginning of period. z f is he percenage of rm f [i.e. he share of rm f] ha he household bough a he beginning of period (-1) and sells a he beginning of period. z f +1 is he percenage of rm f ha he household buys a he beginning of period. These percenages are measured as a number belonging o he closed inerval [0,1]. Therefore, z f Q f is he nominal value of he shares of rm f ha he household sells a he beginning of period. On he oher hand, z f +1 Qf is he nominal amoun ha he household spends buying shares of rm f a he beginning of period. The shares of banks work in he same way: Q bank;l bank l a he beginning of period. is he nominal price ha he household would have o pay o buy 100% of is he percenage of bank l ha he household bough a he beginning of period (-1) and sells a he beginning of period. +1 is he percenage of bank l ha he household buys a he beginning of period. Le CD denoe he amoun of checkable deposis ha he household decides o hold a he beginning of period. Le us now wrie he porfolio allocaion consrain and he cash-in-advance consrain ha capure he srucure of his model s consumer problem. A he beginning of period, he household DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 10

13 faces he following porfolio allocaion consrain (CD 1 P 1 c 1 ) + W 1 n s 1 + z f f 1 + z f Q f + l=l + bank;l l=l 1 + Q bank;l B + B +1 = 1 + R = l=l z f +1 Qf + +1 Q bank;l + CD (4) The rs erm on he lef-hand side of he equaion, (CD 1 P 1 c 1 ), denoes checkable deposis no spen during he previous period. The second erm denoes he household s wage earnings received a he endofperiod (-1) [beginning of period] inreurn for work e orsupplied during period (-1). These wage earnings are received inhe formof checkable deposis (ransferred from he rms accouns ino he household s accoun). The hird erm denoes he amoun of dividends received from he F rms a he end of period (-1) corresponding o shares of rms bough by he household a he beginning of period (-1). These dividends are received in he form of checkable deposis (ransferred from he rms accouns ino he household s accoun). The fourh erm corresponds o he amoun ha he household receives a he beginning of period from selling he shares of rms she had bough a he beginning of period (-1). This amoun is received in he form of checkable deposis. The fh erm denoes he amoun of dividends received from he L banks a he end of period (-1) corresponding o shares of banks bough by he household a he beginning of period (-1). These dividends are received in he form of checkable deposis. The sixh erm corresponds o he amoun ha he household receives a he beginning of period from selling he shares of banks she had bough a he beginning of period (-1). This amoun is received in he form of checkable deposis. The sevenh erm subracs he amoun ha he household uses o pay he deb conraced from commercial banks a he beginning of period (-1). This paymen is made by desroying par of he checkable deposis ha he household owns in is curren accoun. The eighh erm adds he amoun received from he new loan ha he household obains from he banks a he beginning of period. This amoun is received in he form of new checkable deposis creaed by he banks. Inshor, he lef-hand side ofhe equaion gives he oal amoun of checkable deposis ha he household owns a he beginning of period. A he beginning of period, he household uses he whole of his amoun in he following way: she buys shares of rms and of banks and she keeps he res as checkable deposis (he erms on he righ-hand side of he equaion). As a way of modelling he fac ha shares are less liquid han deposis, we assume ha he amoun spen buying shares a he beginning of period canno be used o buy consumpion goods during period : shares bough a he beginning of period can only be sold a he end of period. Hence, in deciding he amoun ha she will keep as checkable deposis, CD, he household mus be aware ha in DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 11

14 order o buy consumpion goods during period, she can only use checkable deposis. This is wha he cash-in-advance consrain (which follows) saes in a very clear way. This cash-in-advance consrain is P c = CD (5) Noe ha he cash-in-advance consrain is no wrien as an inequaliy because checkable deposis are dominaed in erms of reurn by oher asses (shares, in his case). This being so, i wouldn be opimal for he household o hold an amoun of checkable deposis greaer han he amoun she needs o buy consumpion goods during he period. Le us now show ha he porfolio allocaion consrain (equaion 4) and he cash-in-advance consrain (equaion 5) ogeher imply a budge consrain similar o he budge consrains we can nd in RBC models. Since he cash-in-advance consrain (equaion 5) is always binding, i mus have been binding in period ( 1): Therefore, we have P 1 c 1 = CD 1. Using his equaliy and equaion 5 in he porfolio allocaion consrain (equaion 4), and hen rearranging he equaion, we obain W 1 n s 1 + z f f 1 + l=l z f Q f + = B + P c + bank;l l=l z f +1 Qf + l=l 1 + Q bank;l + B +1 = 1 + R +1 Q bank;l (6) This equaion is he consumer s budge consrain. This equaion simply saes ha he oal amoun of money he household obains a he beginning of period [wage earnings, dividend earnings from rms, money received from selling he shares of rms bough a he beginning of period (-1), dividend earnings from banks, money received from selling he shares of banks bough a he beginning of period (-1), and he amoun she borrows from he banks a he beginning of period ] mus be equal o he amoun he household spends a he beginning or during period [paymen of he deb conraced from banks a he beginning of period (-1), consumpion expendiure, purchase of shares of rms and purchase of shares of banks]. We can summarize by saying ha he porfolio allocaion consrain and he cash-in-advance consrain ogeher imply he consumer s budge consrain. Le us now normalize he household s budge consrain (he purpose of he normalizaion is o wrie he model in erms of variables ha are consan in he seady-sae). We can do ha by dividing boh sides of he consrain by RES 1. Afer doing his, we rearrange he equaion and hen de ne he following new variables: ¹ = RES RES 1 1; p = q bank;l = Qbank;l RES 1 ;b +1 = P RES 1 ;w = B+1 RES 1 ;¼ f = Afer all hese seps, we arrive a W RES 1 ;q f = f RES 1 ;¼ bank;l Qf RES 1 ; = bank;l RES 1 DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 12

15 w 1 n s 1 + z f 1 + ¹ 1 ¼ f ¹ 1 l=l z f q f + ¼ bank;l l=l ¹ 1 q bank;l + b +1 = 1 + R = b + p c ¹ 1 l=l z f +1 qf + for = 0,1,2,3, q bank;l We sugges using he following iniial condiion concerning he household s deb posiion a he beginning of period 0 B 0 = W 1 n s 1 + l=l z f 0 f bank;l 1 (7) This iniial condiion simply saes ha he household begins period 0 wih a deb which equals he sum of he wage earnings, dividend earnings from rms and dividend earnings from banks ha she receives here because of he hours she worked during period ( 1) and because of he shares of rms and of banks she bough a he beginning of period ( 1). Noe ha period 0 is no he period where he household s life sars bu raher he period where our analysis of he economy sars (he household has been living for some periods and we cach her in period 0 and ry o model her behaviour). In he Appendix, we show ha his iniial condiion is he iniial condiion which naurally arises when we hink back o is iniial momen a closed economy wihou governmen and where rms don borrow. This iniial condiion can also be normalized by dividing boh sides of he equaion by RES 1 giving b 0 = w 1 n s ¹ ¹ 1 z f 0 ¼ f l=l ¹ 1 0 ¼ bank;l ¹ 1 Consequenly, a he beginning of period 0 he household is looking ino he fuure and acs in a way ha can be described as follows Max c, `, n s, b+1, zf +1, zbank;l +1 s.. w 1 n s ¹ + 1 z f =1 P E 0 =0 ¼ f ¹ 1 u(c ;`) l=l z f qf + ¼ bank;l l=l ¹ 1 q bank;l + b +1 = 1 + R = b + p c ¹ 1 l=l z f +1 qf + +1 q bank;l n s + ` = 1 DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 13

16 b 0 = w 1 n s ¹ ¹ 1 for = 0,1,2,3,... z f 0 ¼ f l=l ¹ 1 0 ¼ bank;l ¹ 1 There are also iniial condiions on holdings of shares [ assuming marke clearing in he shares marke in period ( 1), hese iniial condiions will be z f 0 = 1 H and zbank;l 0 = 1 H ransversaliy condiion on he paern of borrowing and non-negaiviy consrains. ], a sandard The bes way o summarize he formal srucure of he consumer s opimizaion problem presened in his secion is o say ha i adds a speci c iniial condiion o a cash-in-advance forma. As will be shown in secion 7, he iniial condiion we add is such ha, in equilibrium, i makes he cash-in-advance srucure become a credi-in-advance consrain in period 0: since i creaes a siuaion where he household sars period 0 wih zero ne wealh, he iniial condiion makes he household s need o obain cash in order o buy consumpion goods become synonymous wih a need o borrow from a bank. Wih Raional Expecaions he credi-in-advance consrain is propagaed o all fuure periods. 6 The Marke Clearing Condiions Wih H homogeneous households, F homogeneous rms and L homogeneous banks, he marke clearing condiions for period 0 are as follows. In he goods marke, he condiion is Hc 0 + Fi 0 = Fy 0, c 0 + F H [k 1 (1 ±)k 0 ] = F H A 0F(k 0 ;n d 0) (8) In he labour marke, he condiion is Hn s 0 = Fn d 0, n s 0 = F H nd 0 (9) In he bank loans marke, he condiion is H B R 0 = L B s 0, (10) De ning a new variable b s =, H (B 1=RES 1 ) 1 + R 0 = L B s 0 RES 1 Bs RES 1 his las equaion can be rewrien as b R 0 = L H bs 0 (11) The marke clearing condiion in he shares marke is ha each rm and each bank should be compleely held by he households (he only owners of shares in his model). Since households DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 14

17 are all alike, each holds an equal share of each rm and an equal share of each bank. Therefore, he marke clearing condiions in he shares marke are Hz f 1 = 1, zf 1 = 1 H (12) and H 1 = 1, 1 = 1 H (13) 7 The Compeiive General Marke Equilibrium assuming H homogeneous households, F homogeneous rms and L homogeneous banks plus Raional Expecaions To obainhe sysemha describes he compeiive general marke equilibrium, we pu ogeher in a sysem he ypical household s rs order condiions (ha give, in an implici way, he household s demand and supply funcions), he ypical rm s rs order condiions (ha give, in an implici way, he rm s demand and supply funcions), he ypical bank s supply of credi equaion and he marke clearing condiions. The nex sep in solving his model is o normalize he ypical bank s supply of credi equaion (equaion 1) by dividing boh sides of he equaion by RES 1. Then, we normalize he ypical rm s rs-order condiions by dividing boh sides of he equaions by RES 1. We hen assume Raional Expecaions and use a Cerainy Equivalence argumen. Afer all hese seps and if we also assume ha he producion funcion is homogeneous of degree one and de ne he following new variables k = F H k, n d = F H nd, ¼ f = F H ¼f, q f = F H qf, ¼ bank;l = L H ¼bank;l, q bank;l = L H qbank;l we can wrie he sysem describing he Compeiive General Marke Equilibrium assuming H homogeneous households, F homogeneous rms and L homogeneous banks plus Raional Expecaions and Cerainy Equivalence as u 1 (c ; 1 n s ) = p (14) u 2 (c ; 1 n s ) = w 1 + ¹ E [ +1 ] (15) DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 15

18 = E [ +1 ] 1 + R 1 + ¹ (16) Ã q f ¼ f h = E [ +1 ] + E q 1 + ¹ +1i! f (17) q bank;l = E [ +1 ] Ã ¼ bank;l h + E 1 + ¹ q bank;l +1 i! (18) b R = p c (19) p A F 2 (k ;n d ) = w (20) E [A +1 ]F 1 k+1 ;E n d p (1 ±) = (1 + E [R +1 ]) E [p +1 ] (1 + ¹ ) (21) b s = 1 r req (1 + ¹ ) (22) c + k +1 (1 ±)k = A F(k ;n d ) (23) n s = n d (24) b R = L H bs (25) DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 16

19 z f +1 = 1 H (26) +1 = 1 H (27) ¼ f = p A F(k ;n d ) w n d p k+1 (1 ±)k (28) ¼ bank;l = L H 1 r req (1 + ¹ )R (29) for = 0; 1; 2; 3;::: Equaions have heir origin in he ypical household s rs order condiions. Equaion 19 is he credi-in-advance consrain which resuls from combining he household s budge consrain wih he iniial condiion and hen using he marke clearing condiions from he shares marke. Le us be more precise abou his. We sar by showing ha if we add our speci c iniial condiion o he household s period 0 budge consrain, in equilibrium we obain a credi-in-advance condiion for period 0. In order o prove his, we sar by wriing he household s budge consrain (equaion 6) for period = 0 W 1 n s 1 + z f 0 f 1 + l=l z f 0 Qf bank;l l=l Q bank;l 0 + B R 0 = = B 0 + P 0 c 0 + l=l z f 1 Qf Q bank;l 0 Using he iniial condiion before normalizaion (equaion 7) in his las equaion, we obain B 1 = P 0 c R 0 l=l Q f 0 (zf 1 zf 0 ) + Q bank;l 0 ( 1 0 ) Using he marke clearing condiions in he rms shares marke and in he banks shares marke in periods ( 1) and 0 (which are z f 0 = 1 H, zf 1 = 1 H, zbank;l 0 = 1 H and zbank;l 1 = 1 H ) his las equaion becomes B R 0 = P 0 c 0 (30) DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 17

20 which means ha, in equilibrium, he household will have o borrow a he beginning of period 0 an amoun equal o he amoun i wans o spend buying consumpion goods during period 0 (i.e., in equilibrium, here is a credi-in-advance consrain for period 0). Dividing boh sides of his equaion by RES 1, we obain equaion 19 for period = 0. If we assume Raional Expecaions, his credi-in-advance consrain is propagaed o all fuure periods (periods 1,2,3,...). Le us show why his happens. By assuming Raional Expecaions we inroduce all fuure budge consrains of he household and all fuure marke clearing condiions ino he srucure of he mahemaical represenaion of his economy. Hence, we can reason saring from period 0 and going successively ino every fuure period in he following way. We rs wrie he following auology B 1 = B R 0 (1 + R 0 ), B 1 = B R 0 + B R 0 R 0 This auology simply says ha he household s deb a he beginning of period 1 is equal o he principal borrowed a he beginning of period 0 plus ineres on i. Using he credi-in-advance consrain for period 0 which we have jus derived (equaion 30), his las equaion can be wrien as B 1 = P 0 c 0 + B R 0 R 0 Using he marke clearing condiion in he goods marke in period 0 (equaion 8), we obain B 1 = F H P0 A 0 F(k 0 ;n d 0) P 0 [k 1 (1 ±)k 0 ] + B R 0 R 0 Using he de niion of pro s of rm f (equaion 3) in period 0, his becomes B 1 = F H h i f 0 + W 0n d 0 + B 1 R 0, 1 + R 0, B 1 = W 0 F H nd 0 + F H f 0 + B R 0 R 0 Wih he marke clearing condiion in he labour marke (equaion 9), we obain B 1 = W 0 n s 0 + F H f 0 + B R 0 R 0 Using he marke clearing condiion from he bank loans marke (equaion 10), we obain B 1 = W 0 n s 0 + F H f 0 + L H Bs 0 R 0 Using equaion 1, his las equaion becomes B 1 = W 0 n s 0 + F H f 0 + L H 1 r req RES 0R 0 Using he de niion of pro s of bank l in period 0 (equaion 2), we obain DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 18

21 B 1 = W 0 n s 0 + F H f 0 + L H bank;l 0 Finally, using he marke clearing condiions in he shares marke, his can be wrien as B 1 = W 0 n s 0 + l=l z f 1 f bank;l 0 (31) Noe ha his equaion is idenical in form o he iniial condiion we propose using for period 0 (equaion 7) bu wrien one period ahead. In oher words, i is an iniial condiion for period = 1. This is an ineresing propery of he iniial condiion we propose using: if we assume i holds a he beginning of period 0, hen he srucure of he model will reproduce i auomaically ino he following periods. Combining equaion 31 wih he household s budge consrain for period = 1, we obain B R 1 = P 1 c 1 which is a credi-in-advance consrain idenical in form o he credi-in-advance consrain for period 0 which we have already obained (equaion 30) bu wrien one period ahead. In oher words, i is a credi-in-advance consrain for period 1. Dividing boh sides of his equaion by RES 0, we obain equaion 19 for period = 1. = 2. If we repea he whole reasoning we will also obain a credi-in-advance consrain for period And if we successively repea he whole reasoning, we will obain a credi-in-advance consrain for all fuure periods. The inuiion behind his propagaion of he credi-in-advance consrain is as follows. The iniial condiion for he beginning of period 0 forces he household o borrow from he banks a he beginning of period 0 he amoun she wans o spend buying consumpion goods from he rms during period 0. In equilibrium, he amoun of consumpion desired by he household is equal o oupu per household minus invesmen per household (his follows from he marke clearing condiion in he goods marke). Using he de niion of rm s pro s and he marke clearing condiions in he labour marke and in he rm shares marke, i is easy o show ha oupu per household minus invesmen per household is equal o he sum of wage earnings and rm dividend earnings of each household. Hence, wha he household wans o borrow a he beginning of period 0, is an amoun which is equal o he sum of her wage earnings and rm dividend earnings. Therefore, a he end of he period 0 (beginning of period 1) he household s deb will be his amoun plus ineres on i. However, since he ineres paid o banks is equal o he dividends he banks will pay he household, he oal deb of he household a he end of he period 0 (beginning of period 1) can be expressed as he sum of wage earnings and rm dividend earnings and bank dividend earnings of each household. Bu his household s deb a he beginning of period 1 is jus wha he iniial condiion had forced he household s deb o be a he beginning of period 0. DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 19

22 The previous paragraph can be summarized as follows. The iniial condiion forces households o borrow a he beginning of period 0 an amoun which is equal o he value of consumpion. Consumpion is equal o oupu less invesmen. Oupu less invesmen is equal o wages plus rm dividends (his follows from he de niion of rm pro s). Hence, wha he household wans o borrow a he beginning of period 0 is an amoun which is equal o he value of wages plus rm dividends. This implies a deb a he end of period 0 equal o wages plus rm dividends plus ineres on his amoun. Since he ineres paid o banks corresponds o bank dividends, we can conclude ha he household s deb a he end of period 0 is equal o wages plus rm dividends plus bank dividends. Bu his household s deb a he end of period 0 (beginning of period 1) is jus wha he iniial condiion had forced he household s deb o be a he beginning of period 0. Hence, he sory is repeaed in he following period and beyond ha ill he end of ime. Equaions 20 and 21 have heir origin in he ypical rm s rs order condiions. Equaion 22 has is origin in he ypical bank s supply of credi equaion. Equaions are he marke clearing condiions. Equaion 28 resuls from muliplying he ypical rm s normalized pro funcion by (F=H). Equaion 29 resuls from muliplying he ypical bank s normalized pro funcion by (L=H). We have 2 exogenous variables (A and ¹ ) and 16 endogenous variables. The speci c uiliy and producion funcions used were u(c ;`) = lnc + Á ln` and A F(k ;n d ) = A (k ) 1 n d. 8 Calibraion In order o sudy he dynamic properies of he model, we rs log-linearize each of he equaions in he sysem around he seady-sae values of he variables. The log-linearized sysem was hen calibraed. To calibrae he log-linearized sysem we used he following parameers. Wih he speci c uiliy funcion we are using we obain Elasiciy of he MU of consumpion wih respec o consumpion -1 Elasiciy of he MU of consumpion wih respec o leisure 0 Elasiciy of he MU of leisure wih respec o consumpion 0 Elasiciy of he MU of leisure wih respec o leisure -1 where MU denoes Marginal Uiliy. From he U.S. daa, we obain value source Invesmen share of oal expendiure in he s.s. (i=y) Barro (1993) Labour s share ( ) 0.58 King e al. (1988) Labour supply in he seady-sae (n s ) 0.2 King e al. (1988) Nominal ineres rae in he seady-sae (R) FRED Rae of growh of he money supply in he s.s. (¹) Barro (1993) As usual we ake he Poswar average as represening he seady-sae value. The las wo values in he able are per-quarer values. The nominal ineres rae used was he Bank Prime DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 20

23 Loan Rae. We have used daa from he Federal Reserve Economic Daa (FRED) o compued he average quarerly value of his rae for he period The value used o calibrae he seady-sae rae of growh of he money supply ( ) was he average quarerly in aion rae for he period The argumen for using his value is as follows. Combining equaions 19; 22 and 25 above, which corresponds o equaing he supply and he demand for money, we obain L H 1 r req (1 + ¹ ) = p c, L H 1 r req RES RES 1 = P RES 1 c,, L 1 r req RES = P H c (32) Since c is consan in he seady-sae, his equaion implies ha in he seady-sae he rae of growh of he money supply [he rae of growh of (L 1 r req RES )] is equal o he rae of growh of P. In oher words, since velociy is consan in he model (i is equal o one and his is because each uni of money is only used once during he period) and consumpion is consan in he seady-sae, we have a seady-sae where he rae of growh of he money supply is equal o he rae of growh of he price level. We can herefore use he average in aion rae from he daa o calibrae he seady-sae rae of growh of he money supply in our model. The advanage of calibraing he seady-sae rae of growh of he money supply in his way can be saed as follows. When we se a value for ¹ in his model, we are also seing a value for he model s seady-sae in aion rae (as we have jus seen). Hence, if we wan o calibrae he seady-sae nominal ineres rae in his model using he average value of he nominal ineres rae obained from he U.S. daa, he only way o obain a value for he seady-sae real ineres rae in he model equal o he average value i akes in he U.S. daa is o force he seady-sae in aion rae in he model o be equal o he average value i akes in he daa. This requires making he seady-sae rae of growh of he money supply in he model (¹) equal he average in aion rae obained from he daa. The values in he preceding able imply consumpion share of oupu (c=y) per-quarer rae of depreciaion of he capial sock (±) household s discoun facor ( ) Wih his calibraion, he model in his chaper has exacly he same seady-sae values of physical oupu, consumpion, invesmen, labour e or, real wage and real ineres rae as would be obained in a zero growh version of he RBC model presened in King, Plosser and Rebelo (1988) calibraed wih our parameers. This means ha we have buil an economy wih money which has exacly he same seady-sae values of he key real variables ha we would obain in a benchmark economy wihou money. DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 21

24 9 The Dynamic Properies of he Model The response of he log-linearized model o shocks in he exogenous variables (A and ¹ ) can be obained using he King, Plosser and Rebelo (1988) mehod. We rs look a he impac of shocks in he rms echnological parameer. Aferwards, we look a he impac of shocks in he rae of growh of he money supply. 9.1 The impac of echnological shocks We nex examine he resuls of wo experimens which use shocks in he rms echnological parameer: impulse response experimen and sochasic simulaion. In order o perform hese wo experimens, we assumed ha he echnological parameer evolves according o ^A = 0:9 ^A 1 + " (33) where ^A denoes he % deviaion of A from is seady-sae value and " is a whie noise Impulse Response The rs experimen we carried ou was a sandard impulse response experimen: a 1% shock in he ypical rm s echnological parameer. This exercise was performed assuming a consan rae of growh of he money supply (no moneary policy shocks). The resuls are ploed in gures 1 o 9. The rs imporan hing o noice is ha, in spie of he fac ha ours is a moneary economy, i is capable of reproducing he key resuls ha he King, Plosser and Rebelo (1988) paper is able o reproduce. Firs: consumpion, invesmen and hours of work are procyclical. Second: consumpion is less volaile han oupu and invesmen is more volaile han oupu. These are very well documened sylized facs abou he Unied Saes economy [references on his include Kydland and Presco (1990) and Backus and Kehoe (1992)]. We have checked he acual numbers and concluded ha he impulse response resuls for he real variables ^y ; ^c ;^{ ; ^n s, he real wage and he capial sock are exacly he same as would be obained in a zero growh version of he model presened in King, Plosser and Rebelo (1988) calibraed wih our parameers. As already menioned, he model in his chaper has exacly he same seady-sae values of physical oupu, consumpion, invesmen, labour e or, real wage and real ineres rae as he zero growh version of he model presened in King, Plosser and Rebelo (1988). This means ha in his paper we have a model wih money which no only has he same seady-sae values of he key real variables as he zero growh version of he model wihou money presened in King, Plosser and Rebelo (1988) bu also reacs in exacly he same way o an exogenous echnological shock. DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 22

25 Le us ry o explain why he resuls obained so far wih our model (resuls obained wih a consan rae of growh of he money supply) seem o be he same as he resuls we would obain wih a zero growh version of he nonmoneary economy presened in King, Plosser and Rebelo (1988) calibraed wih our parameers. We can summarize our descripion of he moneary ows among economic agens, done in he inroducion o his aricle, as follows. A he beginning of each period, commercial banks ell he nonbank economic agens: Take his money, use i for your ransacions and, a he end of he period, give i back o us. We can say his because, alhough households acually have o pay ineres on he amoun ha hey borrow from he banks, hey hen receive he ineres paid back in he form of bank dividends (his happens because banks are owned by he households and households are all alike). This means ha he fac ha households have o borrow o nance heir expendiure does no involve an income e ec. I seems ha, wih he speci c uiliy funcion we are using and in he absence of moneary shocks, inroducing money in he model in he way we did does no involve a subsiuion e ec eiher. Wih he speci c uiliy funcion we are using (log-linear uiliy funcion) and in he absence of moneary shocks, he ex-ane real ineres rae is given by he same expression in he model of his paper and in he zero growh version of he model presened in King, Plosser and Rebelo. Also imporan o explain he fac ha we obain he same resuls wih his model as we obain wih he zero growh version of he model presened in King, Plosser and Rebelo is he fac ha in his model commercial banks provide money o he economy wihou spending resources (banks have no coss of operaion). The resuls in gures 1-9 were obained wih ¹ = 0: I is imporan o poin ou ha changes in he seady-sae rae of growh of he money supply used in he calibraion (changes in ¹) do no a ec he impulse response resuls obained wih echnological shocks as long as we adjus he seady-sae nominal ineres rae so as o make he seady-sae real ineres rae remain he same in our calibraion (noe ha changing ¹ implies changing he seady-sae rae of in aion). In oher words, when money is supplied o he economy a a consan rae, he acual rae a which i is supplied has no in uence on he resuls we obain wih echnological shocks. The same is rue in Cooley and Hansen s model Sochasic Simulaion The second experimen we carried ou was a sochasic simulaion exercise where only echnological shocks were considered (he supply of money was assumed o grow a a consan rae). The resuls are shown in able 1. These resuls are he same ha are obained in he same model wihou money [i.e., wih a zero growh version of he model in King, Plosser and Rebelo (1988) calibraed wih our parameers]. Cooley and Hansen (1989) also conclude ha when only echnological shocks are considered and he money supply grows a a consan rae, he saisics summarizing he behaviour of he real variables are he same as would be obained in he same model wihou money [i.e., in he model of Hansen (1985)]. Inour model, he correlaionbeween derended prices and derendedoupu is represened by he correlaion beween he deviaions of he normalized price of physical oupu from is DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 23

26 seady-sae and he deviaions of physical oupu from is seady-sae. As can be seen in able 1B, he value we obained for his correlaion in he sochasic simulaion experimen was 0:59. In Cooley and Hansen s model, he correlaion beween he price level and real oupu ha hey obained when here are only echnological shocks and he rae of growh of he money supply is consan was 0:87. There are a number of relaively recen empirical sudies concerning he relaionship beween prices and oupu. Since we have calibraed our model wih measures compued using U.S. Poswar daa, we shall concenrae only on empirical sudies abou he U.S. and only in ha period. Cooley andhansenuse quarerly daa for he U.S. (he periodconsidered is 1955:3-1984:1). All ime series were seasonally adjused, logged and derended. The correlaion of he implici GNP De aor wih real GNP hahey obainis 0:53. The correlaion of he Consumer Price Index wih real GNP ha hey obain is 0:48. Using deviaions from rend and U.S. quarerly daa for he period , Kydland and Presco (1990) repor a correlaion coe cien of he cyclical deviaions of he implici GNP De aor wih he cyclical deviaions of real GNP equal o 0:55. They also repor a correlaion coe cien of he cyclical deviaions of he Consumer Price Index wih he cyclical deviaions of real GNP equal o 0:57. In he sudy by Cooley and Ohanian (1991), he simple conemporaneous correlaion beween derended oupu and derended prices is 0:67 for he period 1948:2-1987:2; i is 0:69 for he period 1954:1-1973:1 (a sample period beginning immediaely afer he Korean War and erminaing before he rs oil shock); and i is 0:87 forhe period1966:1-1987:2 (a period of highaverage in aion rae). Using Hodrick-Presco lered daa he resuls are 0:57, 0:36 and 0:68, respecively. Oher sudies ha quoe similar resuls include Backus and Kehoe (1992), Smih (1992), Chada and Prasad (1994) and King and Wason (1996). These sudies seem o indicae ha he price level is counercyclical. This corresponds o he resuls from our impulse response experimen and sochasic simulaion experimen. I is easy o undersand why our model produces hese simulaion resuls. Combining equaions 19; 22 and 25 above, which corresponds o equaing he supply and he demand for money, we obain p c = L H 1 r leg (1 + ¹ ) Since we are no considering moneary shocks (¹ = ¹), when c moves upward (as a consequence of he posiive echnological shock), p mus move in he opposie direcion. 9.2 Moneary Policy We nex examine he e ec of a emporary increase in he rae of growh of he money supply on he behaviour of boh nominal and real variables (in he model). In order o perform his experimen, we assumed ha money shocks follow he following process ^¹ = 0:481 ^¹ 1 +» 0:01538 (34) DINÂMIA-Cenro de Esudos sobre a Mudança Socioeconómica 24