WORKING PAPER SERIES PERPETUAL YOUTH AND ENDOGENOUS LABOUR SUPPLY: A PROBLEM AND A POSSIBLE SOLUTION NO. 346 / APRIL 2004

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1 WORKING PAPER SERIES NO. 346 / APRIL 2004 PERPETUAL YOUTH AND ENDOGENOUS LABOUR SUPPLY: A PROBLEM AND A POSSIBLE SOLUTION by Guido Ascari and Neil Rankin

2 WORKING PAPER SERIES NO. 346 / APRIL 2004 PERPETUAL YOUTH AND ENDOGENOUS LABOUR SUPPLY: A PROBLEM AND A POSSIBLE SOLUTION 1 by Guido Ascari 2 and Neil Rankin 3 In 2004 all publicaions will carry a moif aken from he 100 banknoe. This paper can be downloaded wihou charge from hp:// or from he Social Science Research Nework elecronic library a hp://ssrn.com/absrac_id= Neil Rankin hanks he Universiy of Pavia and he European Cenral Bank for heir hospialiy while working on his paper. Responsibiliy for all opinions and errors is of course he auhors alone. 2 Diparimeno di Economia Poliica e Meodi Quaniaivi, Universiy degli Sudi di Pavia,Via S. Felice, 5, Pavia, Ialy, gascari@eco.unipv.i. 3 Deparmen of Economics, Universiy of Warwick, Covenry CV4 7AL, UK, n.rankin@warwick.ac.uk.

3 European Cenral Bank, 2004 Address Kaisersrasse Frankfur am Main, Germany Posal address Posfach Frankfur am Main, Germany Telephone Inerne hp:// Fax Telex ecb d All righs reserved. Reproducion for educaional and noncommercial purposes is permied provided ha he source is acknowledged. The views expressed in his paper do no necessarily reflec hose of he European Cenral Bank. The saemen of purpose for he Working Paper Series is available from he websie, hp:// ISSN (prin) ISSN (online)

4 CONTENTS Absrac 4 Non-echnical summary 5 1. Inroducion 7 2. Households wih uncerain lifeimes and wealh-independen labour supply General equilibrium in a flexible-price economy wih money and governmen deb Effecs of moneary and fiscal policy Conclusions 32 Appendices 34 References 36 European Cenral Bank working paper series 37 3

5 Absrac In he perpeual youh overlapping-generaions model of Blanchard and Yaari, if leisure is a normal good hen some agens will have negaive labour supply. We sugges a soluion o his problem by using a modified version of Greenwood, Hercowiz and Huffman s uiliy funcion. The modificaion incorporaes real money balances, so ha he model may be used o analyse moneary as well as fiscal policy. In a Walrasian version of he economy, we show ha increased governmen deb and increased governmen spending raise he ineres rae and lower oupu, while an open-marke operaion o increase he money supply lowers he ineres rae and raises oupu. JEL classificaion: D91, E63 Keywords: Blanchard-Yaari overlapping generaions, endogenous labour supply 4

6 Non-Technical Summary Building on work by Yaari, Blanchard (Journal of Poliical Economy 1985) showed how a simple model of overlapping generaions (OLG) in which agens have uncerain lifeimes could be consruced. The advanages of Blanchard s framework over oher OLG models are ha he average lengh of life of a ypical agen can be parameerised, and ha he special case of infinie lives is nesed wihin i. This is achieved while reaining elegance and simpliciy a he aggregae level. A key assumpion is ha an agen s probabiliy of deah is a consan, independen of he agen s age. This has caused he model someimes o be referred o as one of perpeual youh. The perpeual youh framework has found many useful applicaions. In paricular i implies ha Ricardian Equivalence does no hold, and so provides a fruiful srucure for analysing he effecs of governmen deb and deficis. So far i has mos ofen been used in he long-run conex of growh and capial accumulaion, bu i is also beginning o be used in he shor-run conex of business cycles. Here, changes in labour inpus are imporan, and hence i becomes desirable o model households as having endogenous labour supply. In his paper, we poin ou a poenial problem wih making labour supply endogenous in a perpeual youh model. The problem is ha some agens labour supply is hen likely o be negaive. Negaive labour supply makes no sense, and of course should be ruled ou by imposing a non-negaiviy consrain. However, if his is done, hen he fac ha he consrain is binding for some agens and no for ohers causes he perpeual youh model o lose is elegance and simpliciy a he aggregae level. The inuiive explanaion for why here is a negaive labour supply problem is as follows. Wih mos specificaions of agens preferences, leisure is a normal good, i.e. he wealhier he agen is, he more leisure she demands. In he perpeual youh model, he older he agen is, he more financial wealh she has. Moreover, he age disribuion is such ha 5

7 here is no upper bound on age. Therefore a any poin in ime here are always some arbirarily wealhy agens in he populaion. If leisure is a normal good, hen here are guaraneed o exis agens who wan o consume more leisure han is feasible given heir ime endowmen. Oherwise saed, here are bound o be some agens who wan o supply negaive amouns of labour. The main par of he paper is devoed o suggesing a soluion o his problem. Our proposal is o use a specificaion of households preferences adaped from Greenwood, Hercowiz and Huffman (American Economic Review, 1988) (GHH) which makes labour supply independen of wealh. We exend GHH by including real balances in he uiliy funcion. This is so ha he model can be used o address quesions of moneary policy, and no jus of fiscal policy. We firs explain he new preference srucure, derive is implicaions for aggregae relaionships in a perpeual youh environmen, and embed hese relaionships in a simple flexible-price, dynamic general equilibrium model. We hen explore he model s implicaions for some sandard fiscal and moneary policy experimens. We find ha, wih he proposed uiliy funcion, addiive separabiliy beween leisure and real balances canno be preserved. In consequence, policy changes which increase real balances also increase labour supply. This provides a channel for policy o affec oupu, even hough prices and wages are fully flexible. A permanen increase in he sock of governmen deb, for example, parially crowds ou real balances, because (given ha Ricardian Equivalence does no hold) i permanenly raises he real and nominal ineres rae. Thereby i also causes a conracion in oupu. On he oher hand, an open marke operaion o increase he money supply acs jus like a permanen decrease in he sock of governmen deb, and so raises oupu. The model hence exhibis a liquidiy effec of moneary policy. Thirdly, we show ha a permanen balanced-budge increase in governmen spending (which leaves he sock of deb unchanged) raises he ineres rae and conracs oupu. 6

8 1. Inroducion Building on Yaari (1965), Blanchard (1985) showed how a simple model of overlapping generaions (OLG) in which agens have uncerain lifeimes could be consruced. The advanages of Blanchard s framework over oher models of overlapping generaions (such as ha of Diamond (1965)) are ha he average lengh of life of a ypical agen can be parameerised, and ha he special case of infinie lives is nesed wihin i. This is achieved while reaining elegance and simpliciy a he aggregae level. A key assumpion in achieving his elegance is ha an agen s probabiliy of deah is a consan, independen of he agen s age. This has caused he model someimes o be referred o as one of perpeual youh (e.g. by Blanchard and Fischer, 1989). The perpeual youh framework has found many useful applicaions. Mos obviously, and like oher OLG models, i implies ha Ricardian Equivalence does no hold, and so i provides a fruiful srucure for analysing fiscal policy, in paricular he effecs of governmen deb and deficis. Many of hese applicaions (as in Blanchard s original 1985 paper iself) have been in he long-run conex of growh and capial accumulaion. However, as microeconomic foundaions have become more widely used also in shor-run models of business cycle flucuaions, a number of researchers have seen he advanage of deparing from he infiniely-lived agen (someimes called he represenaive agen ) assumpion in his conex oo. In he shor-run conex, changes in labour inpus are ypically a leas as imporan as changes in capial inpus in explaining oupu movemens, and hence i becomes desirable o model he labour marke, and as par of his, he labour supply, in some deail. In he long-run growh conex, labour supply is commonly reaed as exogenous and fixed by he populaion size. Tha is, households are assumed o obain no uiliy from leisure, and so hey supply heir given ime endowmens o he labour marke compleely inelasically. By conras in shor-run models i is desirable o assume ha households obain uiliy from leisure, so ha 7

9 heir opimal labour supply decision is hen in general a funcion of curren and fuure real wages, ineres raes, and wealh levels. In his paper, we poin ou a poenial problem wih making he labour supply decision endogenous in a perpeual youh model. The problem can be explained very simply: i is ha some agens labour supply is likely hen o be negaive. Negaive labour supply makes no sense, and of course should be ruled ou by imposing a nonnegaiviy condiion. However, if his is done, hen he fac ha he non-negaiviy consrain is someimes binding and someimes no causes he perpeual youh model o lose is elegance and simpliciy a he aggregae level. The inuiive explanaion for why here is a negaive labour supply problem is as follows. Wih mos specificaions of agens preferences, leisure is a normal good, i.e. he wealhier he agen is, he more leisure she demands, and herefore he less labour she supplies. In he perpeual youh model, he older he agen is, he more nonhuman wealh she has. Moreover, he age disribuion is such ha here is no upper bound on age: here are always some arbirarily old agens alive a any poin in ime. Therefore a any poin in ime here are always some arbirarily wealhy agens in he populaion, and, if leisure is a normal good, hese agens will wan o consume more leisure han is feasible given heir ime endowmen. Oherwise saed, such agens will wan o supply negaive amouns of labour. This problem is paricular o he perpeual youh model because of he lack of an upper bound on age. In a Diamond-ype model, wih n-period lives, hen while here may be a quesion of negaive labour supply, i is no guaraneed ha here are agens for whom he preferred labour supply is negaive. The main par of he presen paper is devoed o suggesing a soluion o his problem. Our premise is ha here are many poenial ineresing applicaions for a perpeual youh model wih endogenous labour supply, and ha i is hence desirable and imporan o have a version of his model which avoids he negaive labour supply problem. Indeed, here is a 8

10 small bu growing number of published papers which already make use of perpeual youh models wih endogenous labour supply (for example, Heijdra and Lighar (2000), Cavallo and Ghironi (2002), Smes and Wouers (2002)). These are all, a leas poenially, vulnerable o he negaive labour supply problem: i appears o have gone unrecognised in he lieraure so far. The objecion may a his poin be raised ha he problem has been oversaed. Even if leisure is a normal good, i does no follow from his alone ha very wealhy agens mus have negaive labour supply. For example, depending on preferences, labour supply may end asympoically o zero (or o some posiive lower bound) as wealh ends o infiniy. However, here i needs o be remembered ha a number of resricions are already placed on agens preferences by he requiremen of he perpeual youh model ha aggregae equivalens of individual demand and supply funcions should exis. In paricular, agens demand funcions for goods and leisure need o be linear in wealh. This ensures ha aggregae demands for consumpion and leisure can be wrien as funcions only of aggregae wealh (and of relaive prices), and ha hey are independen of he wealh disribuion across agens of differen ages. Such a condiion rules ou preferences which would make labour supply end asympoically o zero, since labour supply would hen be nonlinear in wealh. A furher objecion may be ha all ha is needed is o ake proper accoun of he non-negaiviy condiion on labour supply. However, his will inroduce a kink in he individual s labour supply as a funcion of wealh, and is hus iself a form of nonlineariy. I would hen no longer be possible o wrie aggregae labour supply as a funcion of aggregae wealh alone: here would exis a hreshold level of wealh above which an individual s labour supply would be zero, and in solving for he general equilibrium we would need o keep rack of he fracion of agens whose wealh exceeded he hreshold level. 9

11 Our proposed soluion o he problem is o use a specificaion of households preferences which makes labour supply independen of wealh. The specificaion which we shall use is adaped from Greenwood, Hercowiz and Huffman (1988) (henceforh GHH ). I exends GHH in ha i incorporaes real balances ino he uiliy funcion. Our moive for his exension is ha, in keeping wih he aim of providing a framework which can poenially be applied o shor-run, business cycle, issues, we wan o be able o address quesions of moneary policy, and no jus of fiscal policy. Moreover we wan o creae a seing ino which nominal rigidiies can ulimaely be inroduced, which means ha money and nominal variables need o be brough ino he model. Wha he res of he paper does is o explain he new preference srucure, derive is implicaions for aggregae relaionships in a perpeual youh environmen, and hen embed his OLG srucure ino a simple flexible-price, dynamic general equilibrium model. The implicaions of his ype of economy for some sandard fiscal and moneary policy experimens are hen explored. Our aim is mainly o esablish he feasibiliy of consrucing a perpeual youh model wih endogenous labour supply which avoids he problem menioned above, and o show ha i provides a basis for a richer analysis of moneary and fiscal policy han is possible in infiniely-lived agen models. Our specific resuls concerning moneary and fiscal policy we inerpre no as final saemens abou he real world, bu as merely laying down some baseline properies agains which hose of a more fully-feaured business cycle model, conaining for example nominal rigidiies, could be compared. The main findings are ha, firs, moneary GHH preferences do make possible a racable OLG model which avoids he negaive labour supply problem. When we examine he effecs of policy, he cenral experimen of ineres concerns he impac of a sep increase in he sock of governmen deb, injeced via a emporary ax cu. Since Ricardian Equivalence does no hold, we would expec his o have real effecs, as is indeed he case. 10

12 We show ha i raises he real ineres rae, since, as in any ypical OLG model, i raises curren demand for goods relaive o fuure demand for goods, by redisribuing wealh from fuure o curren generaions. However, i migh have been hough ha here would be no impac on oupu, because, alhough governmen deb now adds o households ne wealh, labour supply in our model is independen of wealh, and hence he level of he only inpu o producion would be unchanged. In fac, we find ha oupu falls. The cause is he way money eners he model. I urns ou ha he only reasonable way o bring real balances ino he uiliy funcion is in such a manner ha real balances and consumpion are complemens. The rise in he real (and nominal) ineres rae depresses equilibrium holdings of real balances, which in our model lowers he marginal uiliy of consumpion and discourages labour supply. Alhough we do no argue ha such an effec is likely o be empirically large, we hink i is of a plausible naure and demonsraes an ineresing ineracion beween he fiscal and moneary sides of he economy. A second policy measure examined is an open marke operaion (OMO). In dynamic general equilibrium models wih infiniely-lived agens he mechanism by which he money supply is ypically changed is via a ax handou. This is acknowledged o be much less realisic han he alernaive of he OMO, bu here is no poin in modelling he laer, since an OMO can be decomposed ino a ax-financed money supply increase plus a ax-financed reducion in governmen deb, and he second has no addiional effec when Ricardian Equivalence holds. In he presen model, raising he money supply by a ax cu and by an OMO are no equivalen. The former urns ou o have no real effec - unsurprisingly, since prices are flexible. The laer is jus he reverse of he policy change described in he previous paragraph. Hence he presen model provides an example of how a money supply increase can lower he ineres rae (he liquidiy effec ) and also raise oupu, hrough he complemenariy mechanism menioned above. 11

13 The final policy shock we look a is a balanced-budge increase in governmen spending on goods and services. This is found o increase he real ineres rae and depress oupu. I hence crowds ou consumpion by more han 100%. The mechanism causing he increase in he ineres rae urns ou o be he difference in effec of he fall in consumpion on, on he one hand, he demand for money as a medium of exchange, and, on he oher hand, he demand for money as a sore of value. The disincion beween hese wo roles of money is somehing which lies a he hear of he model. Our mehod of soluion draws aenion o hese roles, in aemping o reveal he workings of he economy more inuiively. The srucure of he remainder of he paper is as follows. In Secion 2 we presen he uiliy funcion which we use, and derive individual and aggregae household behaviour from i; in Secion 3 we define he general equilibrium of he model and show how i is deermined; and in Secion 4 we sudy he effecs of a variey of moneary and fiscal policy measures. Secion 5 concludes. 2. Households wih Uncerain Lifeimes and Wealh-Independen Labour Supply We use a discree-ime version of Blanchard s (1985) overlapping-generaions srucure, as in Frenkel and Razin (1987). Agens have an exogenous probabiliy, q (0 < q 1), of surviving o he nex period. This is he only source of uncerainy in he model. Each period, 1-q new agens are born and 1-q die, so ha he populaion remains consan. The number of agens of age a alive in any period is hus (1-q)q a. This illusraes he poin made in he Inroducion, ha here are always some agens alive of any arbirarily chosen age: he crosssecion disribuion of he populaion by age is a declining geomeric disribuion, wih an unbounded suppor. Under hese assumpions, he oal populaion size is 1. The expeced uiliy over her remaining lifeime of an agen who is alive in period n and was born in period s n, is hence: 12

14 E U =Σ ( β q) n u 0 < β < 1 (1) n sn, = n s, where u s, is flow uiliy. The novely in our specificaion lies in he form proposed for u s,. We assume: u = ln( c m d( l )) 0 < δ < 1 (2) 1 δ δ s, s, s, s, where d(l s, ) is a funcion giving disuiliy of labour supply, wih d, d > 0. In a model wihou money, in order o make labour supply wealh-independen he erm inside he log operaor mus ake he form c s, - d(l s, ), as Greenwood, Hercowiz and Huffman (1988) show. Here we wish o include real balances, m s, M s, /P, in uiliy o represen he liquidiy services which money provides in helping households o buy goods. I is no plausible o inroduce m s, ino c s, - d(l s, ) in an addiively separable way, because his would make no only labour supply, bu also money demand, wealh-independen, and he laer seems especially unreasonable. Since money is used for buying goods, we combine m s, wih c s,. To preserve wealhindependence of labour supply we hen need o make he combined funcion linearhomogeneous in (c s,,m s, ), as done in (2). The agen has he following single-period budge consrain: Pc + M + B = (1/ q)[ M + (1 + i ) B ] + Wl +Π T (3) s, s, s, s, 1 1 s, 1 s, Here, P is he price level, W is he money wage, Π is profi receips from firms, T is a lumpsum ax, and M s,, B s, are he socks of money and governmen bonds he agen holds a he end of period. 1 Noe ha, as well as receiving he nominal ineres rae i on bonds, he agen receives an annuiy a he gross rae 1/q on her oal financial wealh, so long as she says alive. If she dies, on he oher hand, all her financial wealh passes o he insurance company. 1 Profi receips and axes, which are exogenous o he agen, are assumed o be he same for all agens irrespecive of age. 13

15 This is he acuarially fair insurance scheme which operaes in Blanchard s (1985) OLG srucure. We can rewrie he budge consrain in real erms o ge: i q c + m + v = v + wl + π τ. (4) s, s, s, s, 1 s, 1+ i 1+ r This inroduces he conceps of he real ineres rae and real financial wealh, defined as: P r (1 + i ), v [ M + (1 + i ) B ], (5) P q P s, 1 s, 1 1 s, while (w,π,τ ) are he values of heir upper-case counerpars deflaed by P. Maximising he funcion obained by combining (1) and (2), subjec o (4) for = n,,, we may derive he firs-order condiions for he problem. These ake he form: c ( c / m ) d( l ) = β (1 + r)[ c ( c / m ) d( l )], (6) δ δ s, + 1 s, + 1 s, + 1 s, + 1 s, s, s, s, c m s, s, 1 δ i = δ 1+ i (7) w = δ c m d l (8) 1 δ (1 ) ( s, / s, ) ( s, ) I is worh emphasising ha (6)-(8) are condiional, in he sense ha hey obviously only apply if he agen remains alive for he periods concerned. A firs poin which hey reveal is ha cerain choices will be he same for all agens irrespecive of birhdae, s. Since all agens face he same nominal ineres rae i, (7) shows ha all will choose he same raio of real balances o consumpion. (7) may be inerpreed as a demand-for-money funcion, in which he demand is proporional o consumpion and inversely relaed o he nominal ineres rae. I arises from he medium of exchange role of money, here capured by he presence of real balances in he uiliy funcion. For laer reference, i is helpful o define z as money demand per uni of consumpion, so ha (7) hen implies: δ 1+ i zs, ( ms, / cs, ) =. (9) 1 δ i 14

16 z s, is he same for all generaions, even hough m s, and c s, are no. Having noed his, we may hen observe from (8) ha labour supply will be he same for all agens. This, of course, is as inended: our uiliy funcion is designed o remove he effec of wealh on labour supply, and since financial wealh is he only respec in which agens differ, hey should hen all choose he same labour supply. The way in which he wealh effec on labour supply would normally manifes iself is by he presence of c s, in (8) (oher han hrough z s, ), bu i is clear ha in he presen case c s, is absen. On he oher hand, he presence of z s, inroduces a link beween he labour marke and he money marke, and is due o he lack of addiive separabiliy in he flow uiliy funcion menioned above. Turning o he choice of consumpion, (6) shows ha he usual Euler equaion for δ consumpion is now modified by subracion of he erm c / m ) d( l ) ( s, s, s, from c s, (dio for c s,+1 ). From wha has jus been seen, his erm is he same for all generaions s, and so independen of c s,. To inerpre i, noice ha he flow uiliy funcion can be rewrien as: u δ = ln[ cs, ( cs, / ms, ) d( ls, )] + δ ln( ms, / cs, ). (10) s, δ Hence c / m ) d( l ) ( s, s, s, acs like a shif of origin for consumpion. I shows a resemblance of our funcion o he Sone-Geary uiliy funcion, in which he new origin can hough of as a subsisence level of consumpion. For fuure reference, we label he surplus of consumpion over is subsisence level as adjused consumpion, a s, : a s, s, ( s, s, s, δ c c / m ) d( l ). (11) As poined ou in he Inroducion, an essenial requiremen of he perpeual youh model is ha he agen s consumpion be linear in his lifeime wealh. To derive consumpion as a funcion of lifeime wealh, we combine he agen s firs-order condiions wih his ineremporal budge consrain. The laer is obained by inegraing (4) from n o infiniy and imposing a No Ponzi Game condiion, resuling in: 15

17 Σ n = nq αn, [ cs, + ( i /(1 + i )) ms, ] = vs, n 1 + hs, n ωs, n (12) where h n s, n Σ = nq αn, [ wls, + π τ ], α (for > n; α n,n 1) n, (1 + rn ) (1 + rn + 1)...(1 + r 1) Thus h s,n is human wealh, i.e. he discouned presen value of labour income and profis minus axes; while ω s,n is oal wealh, comprising human and nonhuman, or financial, wealh. Since we have seen ha l s, is he same for all age cohors, i is clear ha human wealh is also he same for all. Financial wealh, v s,n-1, on he oher hand, generally increases wih age. Combining (12) wih repeaed applicaions of (6) and (7), we arrive a: δ c s, z s, i= 0, + i + i + i i 1 δ d( l ) = (1 δ )(1 βq)[ ω Σ q α (1 δ ) z d( l )] (13) where we have dropped s subscrips on (z,l ), for he reasons explained. (13) says ha an agen s adjused consumpion is a consan fracion of his oal wealh ne of he presen value of fuure subsisence consumpion levels 2. The imporan poin is ha consumpion is sill linear in wealh. This means ha aggregae consumpion can be wrien as he same funcion of aggregae wealh and ha he disribuion of aggregae wealh across agens is immaerial. So far we have derived he behaviour of he choice variables of an individual household born in period s. We urn now o look a he behaviour of aggregae household variables. For any variable x, he relaionship of aggregae o individual values is s x Σs= (1 q) q xs, =, where he lack of an s subscrip indicaes an aggregae value. 3 Moreover, since he 2 In fac, hese are he subsisence levels of full consumpion, where full consumpion is consumpion plus he foregone ineres on money balances, fcs, cs, + ( i /(1 + i)) ms,. This explains he presence of (1-δ) s While his is also rue for asse holdings M s,, B s,, for v s, he relaionship is Σ s = ( 1 q ) q vs, = (1/ q) v. This is because our definiion of v s, in (5) includes he annuiy payou in he definiion of v s,, and his, being a pure redisribuion from hose who die o he survivors, does no apply o aggregae holdings. 16

18 populaion size is 1, aggregae and average values coincide. As regards he behaviour of z and l, we have already shown ha individual choices are independen of age, whence z = z s,, l s, = l, for all s. For c, as furher noed, (13) can be applied, where ω s, is replaced by is aggregae counerpar, ω. I is also useful o derive an aggregae counerpar of he individual Euler equaion for consumpion, (6). We do his by firs deriving a relaionship of ω +1 o ω bringing in aggregae financial wealh, v, and hen we use he aggregae version of (13) o link oal wealh levels o consumpion levels. Afer some manipulaion, we obain: c 1) v. (14) δ z d( l ) δ = β (1 + r )[ c z d( l )] (1 δ )(1 βq)(1/ q (14) shows ha no only is he growh rae of aggregae adjused consumpion posiively relaed o r - as is he case for individual adjused consumpion - bu i is also negaively relaed o aggregae financial wealh. Inuiively, he reason for his is he generaional urnover effec. 4 Beween and +1, some already-exising agens are replaced by newborn agens. To he exen ha hose who have jus been born have lower consumpion han he average consumpion of hose who die, his composiional effec will reduce aggregae consumpion beween he wo periods. Now, as seen, an agen s consumpion is increasing in her financial wealh, and moreover, agens are born wih zero financial wealh. Therefore he newborn do have lower consumpion han he average of he res of he populaion, and i is a randomly chosen sample of he laer who die. 3. General Equilibrium in a Flexible-Price Economy wih Money and Governmen Deb We now combine he OLG household srucure as described in Secion 2 wih some simple assumpions abou firms and he governmen and sudy heir ineracion in a Walrasian 4 This effec is highlighed by Heijdra and Lighar (2000). 17

19 environmen. The represenaive firm has an increasing, concave producion funcion, y = f ( l ), and so maximises profis where: The governmen has a single-period budge consrain of he form: w = f l ). (15) τ ( P( g ) + i B = ( B B ) + ( M M ) (16) where g denoes governmen spending on firms oupu. We can re-wrie his in real erms as: g τ = b (1 + r ) b ] + [ m ( P / P ) m ] (17) [ in which b B /P. There are many policy regimes which can be considered, bu we wish o focus on he simples ypes of moneary and fiscal policy experimen. Hence we will rea g and M as exogenous and consan over ime, excep for he possibiliy of a once-and-for-all change in heir values. The level of governmen deb will be reaed as a hird independen policy insrumen, leaving τ o balance he budge as an endogenous residual. There is more han one way o fix he sock of governmen deb exogenously: i can be se in real or nominal erms, exclusive or inclusive of curren ineres obligaions. Here we choose o se i in real erms, inclusive of ineres. Hence he deb policy insrumen is aken o be: b ( 1+ r ) b. (18) Under his assumpion a governmen bond is like an indexed reasury bill: i is a promise of one uni of goods in one period s ime. To deermine he general equilibrium, we begin in he labour marke. Equaing he supply wage as given by (8) o he demand wage as given by (15), we have: f ( l ) = d ( l ) /(1 δ ) z. (19) δ This equaion implicily deermines l (and hus y ) as a funcion of z. Noe ha in a nonmoneary version of our economy, in which we would herefore have δ = 0, (19) would ie down l by iself. The non-moneary version of he economy hus has a naural rae propery, 18

20 in he sense ha employmen is independen of moneary and fiscal policy. However, when money is included, here is a posiive relaionship beween he level of real balances per uni of consumpion and employmen (as can be seen from (19), recalling f < 0, d > 0). The classical dichoomy beween real and moneary secors does no hold here. The cause of his is complemenariy beween consumpion and real balances in households uiliy, associaed wih he fac ha he cross-parial derivaive in (2), u cm, is posiive. This indicaes ha he higher he individual s consumpion, he more useful are real balances o he agen, which makes inuiive sense given ha he purpose of money is o faciliae ransacions. In (19), hen, a rise in z (or m, a given c) raises he marginal uiliy of consumpion and so makes i worhwhile for he agen o work an exra hour. In oher words, higher real balances simulae labour supply. This ype of effec has also been highlighed in he shopping ime approach o he demand for money (see, e.g., Walsh, 1998). In ha approach, real balances are assumed o economise on he agen s shopping ime, raher han provide uiliy. An increase in real balances releases ime for work or leisure, and so generally simulaes labour supply. Turning nex o he goods marke, he equilibrium condiion is: y = c + g. (20) Having seen ha y can be found as an implici funcion of z, i is hen clear ha c can be expressed as an implici funcion of (z,g ). Moreover, in he special case where g = 0, c obviously coincides wih y and is hen likewise a funcion of z alone. I is helpful for wha follows o adop specific funcional forms for he producion and disuiliy-of-work funcions. Thus, suppose ha f(.) and d(.) are consan-elasiciy funcions: f l = l σ ( ) 0 < σ 1, (21) d ( l ) = l ε 1. (22) ε η Using hese in (19), we obain explici soluions for l and y as funcions of z : 19

21 l, (23) 1/( ε σ ) δ /( ε σ ) = [ σ (1 δ ) / ηε ] z y. (24) σ /( ε σ ) δσ /( ε σ ) = [ σ (1 δ ) / ηε ] z We nex consider equilibrium in he asse markes. The aggregae Euler equaion, (14), relaes aggregae adjused consumpion growh o he real ineres rae and he aggregae sock of financial wealh, where he laer is given by: v. (25) = ( 1+ r ) b + M / P + 1 Under a policy of holding b (see (18)) and M consan over ime a values b and M, his equals: v. (26) = b + m+1 Using he definiion of adjused consumpion (11) (in is aggregae version), we can hence wrie (14) as: a β 1+ r ) a (1 δ )(1 βq)(1/ q 1)( b m ). (27) + 1 = ( We now proceed o develop his ino a difference equaion which will be he cenral equaion of he model. We will iniially assume ha here is no governmen spending. This means, as seen, ha no only l and y, bu also c, can be wrien as funcions of z alone. Moreover he same is rue of m, since m z c. As regards he endogenous variables in (27), we can hence see ha a can be expressed as jus a funcion of z, and a +1 and m +1 jus as funcions of z +1. This leaves 1+r. 1+r (1+i )P /P +1, and i is simply relaed o z by (9). P /P +1 can furhermore be eliminaed as m +1 /m, so ha 1+r is seen also o depend only on z and z +1. We hence reduce he model o an implici firs-order difference equaion in z. To make he sages of his ransformaion a lile more explici, firs use he subsiuions for 1+r o rearrange (27) as: a m = β 1 a b (1 δ )(1 βq)(1/ q 1) 1+ 1 δ /(1 δ ) z m m. (28)

22 We may now show ha, wih he consan-elasiciy funcions (21)-(22), m a = ε z ε ( 1 δ ) σ, (29) m. (30) σ /( ε σ ) [ ε (1 δ ) σ ] /( ε σ ) + 1 = [ σ (1 δ ) / ηε ] z + 1 Subsiuing (29) and (30) ino (28) yields: z σ ε (1 δ )(1 βq)(1/ q 1) ηε ε σ [ ε (1 δ ) σ ] /( ε σ β + 1+ b z + 1 = 1 ε (1 δ ) σ σ (1 δ ) δ /(1 δ ) z 1 ) + (31) Thus we arrive a he law of moion for he economy. Since z is a non-predeermined variable, (31) mus be solved in a forward-looking manner. This means ha, for a unique non-explosive soluion (or, a leas, locally unique, disregarding soluions which may be generaed by he nonlineariy of he equaion), we need (31) o have a unique, locally unsable, seady sae. In his case z jumps sraigh o is seadysae value. We herefore proceed o sudy he seady-sae version of (31). Seing z = z +1 = z, muliplying hrough by z/β, and subracing 1 from boh sides, we have: δ /(1 δ ) z δ /(1 δ ) = ε (1 δ )(1/ βq 1)(1 q) ηε (1/ β 1) + z + b ε (1 δ ) σ σ (1 δ ) σ ε σ z δσ ε σ. (32) Noe ha he lef-hand side (LHS) also gives he value of i, and is jus he inverse of he money demand funcion, (9). This is ploed in Figure 1 as he ME curve, since i gives a relaionship beween z and i which derives from he medium of exchange role of money. ME is clearly negaively sloped, wih horizonal and verical asympoes a i = 0 and z = δ/(1-δ), respecively. Turning o he righ-hand side of (32), firs noe ha in he seady sae, inflaion is zero, and hence i = r. The RHS can be inerpreed as giving he value of r consisen wih various 21

23 levels of financial wealh per uni of adjused consumpion, v/a, where v/a is in urn relaed o z. To see his, consider he seady-sae version of (27), rearranged as an expression for r: r = ( 1/ β 1) + (1 δ )(1/ βq 1)(1 q) v / a. (33) (33) shows ha in he seady sae here is a posiive relaionship beween he real ineres rae and he level of financial wealh per uni of adjused consumpion. To undersand his inuiively, noe ha he role of financial wealh as a whole, from households poin of view, is as a sore of value. By accumulaing or decumulaing v s,, a household is able mach he ime profile of is labour income (i.e. w l + π τ ) o he desired ime profile of is consumpion. 5 The greaer is r, he seeper will be he desired growh pah of consumpion and he lower he iniial level, so households will be accumulaing financial wealh more rapidly. Across he populaion as a whole, herefore, a higher r is associaed wih a higher demand for financial wealh (in relaion o he average level of consumpion). The RHS of (32) furher links v/a o z. This link arises firsly because he numeraor of z, i.e. m, is one of he componens of v (= b + m), bu also because he oher endogenous erms, c and a, can be relaed o z using he labour and goods marke linkages. In view of is derivaion from he sore-of-value role of financial wealh, we label he RHS of (32) as he SV curve. In he special case b = 0, i is clear ha he SV curve is jus an upward-sloping sraigh line, wih a verical inercep a r = 1/β - 1. The equilibrium of he model in he absence of any governmen deb is herefore a poin A in Figure 1. A his poin, he real and nominal ineres raes are equal, and he level of real balances per uni of consumpion, z, is he value consisen boh wih he demand for real 5 In fac, decumulaion will never be done willingly, in an equilibrium of he perpeual youh model. I occurs only when he agen dies. A his poin decumulaion happens abruply, as all he agen s financial wealh passes o he insurance company. 22

24 balances as a medium of exchange, and wih he demand for real balances as a sore of value. We can see from he diagram ha in general r exceeds 1/β-1, he pure ime preference rae. Noice ha when q = 1, i.e. when agens live forever, he SV line is horizonal. The real ineres rae is hen idenical o he pure ime preference rae - a sandard feaure of represenaive agen models. The inroducion of overlapping generaions, i.e. he reducion of q below 1, pivos upwards he SV line and hence raises he real ineres rae, moving he equilibrium from poin C o poin A. Figure 1 Noice also ha, even when q < 1, he equilibrium real ineres rae would sill be 1/β-1 if i were no for he exisence of money. If we le δ (he exponen on real balances in he uiliy funcion) end o zero, hen he verical asympoe of he ME curve shifs lef unil i coincides wih he verical axis, and he ME curve shrinks in owards is asympoes, so ha he equilibrium in Figure 1 hen occurs a he verical inercep of he SV line. Thus, in he absence of any financial asses, he presence of overlapping generaions makes no difference o he ineres rae. I is only when money or (as we discuss below) governmen deb is inroduced ha he real ineres rae is raised above he pure ime preference rae. The reason 23

25 for his is ha when r exceeds 1/β-1, he individual wans o choose a rising lifeime profile of consumpion and hus, on average, posiive holdings of financial wealh. By conras when r = 1/β-1, he individual jus consumes her labour income period by period and does no ry o accumulae wealh. The exisence of a posiive supply of financial wealh herefore requires r > 1/β-1 in order ha agens willingly hold his wealh. Conversely, in he absence of a sock of financial wealh, r = 1/β-1 is needed in order ha he demand for wealh be zero. When b = 0 we can in fac solve for z explicily. (32) can be arranged as he following quadraic equaion in z: ε 1 δ )(1/ βq 1)(1 q) z ε (1 δ ) σ ( 2 1 δε (1/ βq 1)(1 q) 1 δ + 1 = 0 (1 ) z. (34) β ε δ σ β 1 δ This has wo soluions. Since he coefficien on z 2 and he consan erm have opposie signs, one soluion mus be negaive and he oher posiive. Obviously he relevan soluion is he posiive one, and his is he soluion depiced in Figure 1. A negaive value of z does no seem o make sense. Moreover he negaive soluion would be associaed wih a negaive value of i, as could be seen if we exended Figure 1 ino he negaive quadran, and his also does no make sense. Despie his, we can gain some insigh ino why he equaions end o generae wo seady-sae soluions by looking again a he case δ = 0. δ = 0 means ha here is no demand for money as a medium of exchange. (34) hen yields he following soluions for z: z 1 βq( ε σ ) = 0 or 1. (35) β (1 βq)(1 q) ε Inuiively, z = 0 is he righ soluion in his case, and i is puzzling ha here is any oher soluion a all. Noice, however, ha when δ = 0 he economy is in principle similar o Samuelson s (1958) OLG model of a moneary economy, in which money sill poenially has a role o play as a pure sore of value. Here i should be recalled ha Samuelson found ha money would only be valued in equilibrium if he resource allocaion in he economy before 24

26 he inroducion of money was dynamically inefficien, i.e. if r < n was saisfied. Oherwise, he level of real balances required o generae a moneary seady sae would be negaive, corresponding o a desire by agens o be borrowers raher han lenders. Negaive real balances are of course impossible, so in ha case money would no be valued. The δ = 0 case of our model is like he second case: if a supply of financial wealh were assumed absen from our economy, he equilibrium real ineres rae would have o be 1/β-1, as noed above. This is greaer han he populaion growh rae, n, (namely zero), so ha he equilibrium of our economy prior o inroducing money is dynamically efficien. When a supply of money is inroduced, we hen find, as Samuelson did, ha he only non-zero real value of his sock consisen wih seady sae condiions is negaive. So far we have shown ha he perpeual youh model wih moneary GHH preferences appears o possess a seady sae which is economically relevan. However, wo furher quesions mus be asked abou i. I is easies o answer hese iniially if we coninue o work wih he benchmark case where b = 0. The firs quesion is wheher he seady sae is locally unsable. I is sraighforward o invesigae he dynamics of he difference equaion (31) using a phase diagram approach. We do his in Appendix A, where we show ha he seady sae is indeed locally unsable. The second quesion arises because he uiliy funcion (2) resrics he se of consumpion values for which an individual s preferences are well defined. As (10) makes clear, hey are undefined for consumpion values below he subsisence level. The quesion is herefore wheher a s, is posiive for all s, in he seady sae. Now, since 1+r > 1/β, an individual s consumpion grows as long as she says alive (see (6)), which means ha her oal wealh, and wihin his, her financial wealh, also grow (see (13)). Thus we see ha he agens mos a risk of having negaive adjused consumpion are he newborn, who have zero financial wealh. Hence if we can show ha a, > 0 in he seady 25

27 sae, hen a s, > 0 is assured for agens of all ages, -s. In Appendix B we invesigae he sign of a, in he seady sae. We are able o prove ha i is indeed posiive. 4. Effecs of Moneary and Fiscal Policy A key propery which we expec he model o possess is lack of Ricardian Equivalence, so o es his i is naural o consider, as he firs policy experimen, he effec of a once-and-for-all increase in governmen deb. Suppose, hen, ha b is raised from zero o a posiive value by a ax cu which lass for one period only. Oherwise saed, he governmen runs a emporary budge defici. If he change occurs in period, he only difference which his makes o he law of moion (31) is ha b now becomes posiive. The economy hus jumps immediaely o is new seady sae, as given by (32) wih posiive b. In Figure 1, his is shown as an upward shif of he SV curve o SV. I is clear ha he inroducion of a posiive b adds a new componen o he RHS of (32), a componen which considered by iself is a decreasing funcion of z, and which has he horizonal and verical axes as is asympoes. I herefore shifs up he SV curve and makes i non-linear. The effec of he increase in he deb is hence o move he economy o poin B in Figure 1. As can be seen, he ineres rae (real and nominal) increases, and he level of real balances per uni of consumpion decreases. Associaed wih he fall in z is a reducion in oupu and employmen. Governmen deb hus has real effecs in his economy, i.e. Ricardian Equivalence does no hold. I is easy o verify ha his is due o overlapping generaions, because if we se q=1 hen he SV curve becomes horizonal a 1/β-1, and i is clearly unaffeced by b in his case. The reasons for he real effecs when q < 1 are familiar ones: alhough oday s ax cu is mached by permanenly higher fuure axes of equal presen value, hese being necessary o enable he governmen o pay he ineres on he permanenly higher 26

28 deb, he currenly-alive generaions who benefi from he ax cu do no bear all he burden of he fuure increases, since some of he increases fall on agens no ye born. Hence he currenly alive feel wealhier and aemp o increase heir consumpion spending. To he exen ha oupu canno increase, he real ineres rae has o rise o choke off demand. In fac, oupu falls. If he wealh effec on labour supply had no been eliminaed, we migh have suspeced ha his was he explanaion for he fall in oupu, wih he increase in perceived lifeime wealh encouraging households o demand more leisure. However, as is by now clear, he mechanism is insead he fall in real money balances, and is effec on labour supply via he complemenariy of money and consumpion. The reducion in real balances is induced boh by he increase in demand for goods, which raises he general price level in he face of an unchanged money supply; and by he increase in he nominal ineres rae, which reduces demand for money as a medium of exchange and so moivaes a porfolio shif. In summary, he effecs of governmen deb in his OLG model wih an endogenous supply of labour are no dissimilar o he beer known effecs of deb in OLG models wih an endogenous supply of capial (Diamond (1965), Blanchard (1985)). In boh ypes of model, he real ineres rae is increased and oupu is reduced. Moreover, in boh ypes of model, governmen deb crowds ou anoher asse from privae porfolios. In Diamond and Blanchard i is physical capial which is crowded ou, whereas here real money balances, m, are crowded ou. Alhough, unlike physical capial, real balances are no direcly an inpu o producion, hey are neverheless linked o such an inpu, namely labour, by heir effec on labour supply; and hence heir reducion is likewise par of he causal chain whereby oupu is reduced. 6 6 A furher quesion is wha happens as we le he level of deb end o infiniy. Inuiively, here mus exis some maximum susainable level of governmen deb: for example, Rankin and Roffia (2003) analyse his in Diamond s (1965) economy. In he presen model, he maximum occurs where he adjused consumpion of he newborn is driven o zero. 27

29 A second policy experimen on which our model can shed ligh is an open-marke operaion (OMO) o change he money supply. As poined ou in he Inroducion, he sandard way of modelling a money supply change in dynamic general equilibrium models is o assume ha is counerpar is a ax cu. In pracice, however, he main mechanism open o a cenral bank is an asse-swap operaion, in which bonds are bough or sold in exchange for money. If, for example, he cenral bank wans o increase he money supply by M in period, hen (16) implies ha i needs o change he sock of deb in public hands by B = M, i.e. i needs o purchase (in absolue value) his quaniy of deb on he open marke. Equivalenly, i needs o change b by ((1 + i) / P+ 1) M. The easies way o sudy he impac of an OMO is o break i ino wo separae operaions, namely an increase in M financed by a cu in τ, and a reducion in b financed by an increase in τ, such ha, overall, τ is unchanged. Now, since he firs of hese operaions leaves b unchanged, i causes no aleraion in he equilibrium condiion (32). This way of increasing M is jus he sandard helicoper drop mehod, and i has no real effecs, simply causing an equi-proporional rise in all nominal variables. The second of he operaions reduces b, and so is jus he reverse of he governmen deb increase sudied above. I can herefore be considered as a movemen from B o A in Figure 1, which lowers he real and nominal ineres raes and raises z. We herefore find ha an increase in he money supply hrough an OMO does have real effecs: by raising z, which simulaes labour supply, i expands oupu and employmen. The analysis here moreover shows how a liquidiy effec of moneary policy can be generaed. In many dynamic general equilibrium models, an increase in he money supply is associaed wih no change, or even wih a rise, in he nominal ineres rae; whereas he usual view is ha in realiy, as in he IS-LM model of he exbooks, moneary expansion causes he nominal ineres rae o fall. Our model explains his fall by he associaed reducion in he bond sock and he lack of Ricardian Equivalence. Therefore, boh on he grounds of he way i allows us 28

30 o represen moneary conrol, and on he grounds of is effec on he economy, he model has some advanages for he modelling of moneary policy. 7 Le us lasly resore governmen spending. A ax-financed increase in spending is represened by an increase in g a unchanged b. (We can also consider a deb-financed increase in spending. However, a spending increase canno be deb-financed forever, since he deb sock would explode; hence such a policy is mos naurally broken ino a combinaion of a balanced-budge spending increase and a separae deb increase, where he laer has already been analysed.) In he labour marke, (19) holds wheher or no g = 0, so here are sill unique posiive relaionships beween l and z, and y and z. Tha is, (23) and (24) remain valid if he producion and disuiliy-of-work funcions ake heir consan-elasiciy forms. The relaionship (28), beween a +1 /m +1 and a /m, also sill applies. However, m /a can no longer be expressed as a funcion of z alone, since in linking m /a o z we need o use he goods marke clearing condiion (20), in which g now drives a wedge beween y and c. In consequence m /a becomes a funcion of g as well as of z (cf. (29)). Similarly m +1 becomes a funcion of g +1 as well as of z +1 (cf. 30)). We can neverheless proceed o derive a firs-order difference equaion in z as he cenral equaion of he model, as we did before (cf. (31)). This equaion now conains (g,g +1 ) as addiional exogenous variables. For breviy, we shall no presen hese seps here; insead we advance o he seady-sae equaion, he counerpar of (32). I is simples o sudy his when b = 0, when i is given by: δ /(1 δ) ε(1 δ)(1/ β q 1)(1 q) = (1/ β 1) + z Γ( zg, ), z δ /(1 δ) ε (1 δ) σ where σ δσ σ(1 δ) σ(1 δ) ε ηε ε σ ε σ Γ(, z g) z g ε ε ε (1 δ) σ σ(1 δ) 1 (36) 7 The broad idea ha a money supply change implemened hrough an open-marke operaion may be nonneural is no new - i goes back a leas o Mezler (1951). Anoher recen demonsraion of i in an OLG model is provided by Benassy (2003). 29

31 As before, we seek o plo he LHS and RHS of (36) as funcions of z (see Figure 2). The LHS is unaffeced by he presence of g, and so is represened by he same downward-sloping ME curve as in Figure 1. As regards he RHS, firs noe ha he new erm Γ(z,g) equals 1 if g = 0. The RHS, depiced as he SV curve, is hen jus an upward-sloping sraigh line wih inercep 1/β-1, as in Figure 1. When g > 0, Γ(z,g) is clearly greaer han one. We can also see ha i is decreasing in z, ha i ends o 1 as z ends o infiniy, and ha i ends o infiniy as z falls owards some sricly posiive lower bound. The curve SV, which akes is appearance from he produc of z and Γ(z,g), herefore has he U-shape depiced in Figure 2. Is asympoes are he SV line associaed wih g = 0, and a verical line a some sricly posiive value of z. From his i follows ha he effec of an increase in governmen spending, saring from zero, is o move he economy from poin A o poin B. I herefore raises r and lowers z. The fall in z means ha oupu and employmen also fall. These impacs on r and z are clearly associaed wih he presence of overlapping generaions, because if q = 1 he SV curve is horizonal and unaffeced by a change in g. To undersand beer why he ineres rae has o increase, suppose ha i were o remain he same. The demand for z, which we can read off from he ME curve, would hen be unchanged, and oupu would hence be unchanged oo. In his case c would have o fall by he amoun of he increase in g, i.e. here would be 100% crowding ou of consumpion. A unchanged z, a fall in c means an equiproporional fall in demand for real balances, m (recall m zc). However, o keep he marke for financial wealh in equilibrium a he old value of r, he fall in m (remember m = v, since b = 0) mus be mached by an equiproporional fall in adjused consumpion, a, as (33) reminds us. Bu a falls by proporionally more han c in he case posied, since i equals c ne of subsisence consumpion. Therefore he rise in g reduces he demand for money as a sore of value by more han i reduces he demand for money as a medium of exchange, and his fac means he 30