Perpetual Youth and Endogenous Labour Supply:

A Problem and a Possible Solution*

Guido Ascari§

University of Pavia

Neil Rankin

University of Warwick

This version: March 2006

Abstract

In the “perpetual youth” overlapping-generations model of Blanchard and Yaari, if leisure is a “normal” good then some agents will have negative labour supply. We suggest a solution to this problem by using a modified version of Greenwood, Hercowitz and Huffman’s utility function. The modification incorporates real money balances, so that the model may be used to analyse monetary as well as fiscal policy. In a Walrasian version of the economy, we show that increased government debt and increased government spending raise the interest rate and lower output, while an open-market operation to increase the money supply lowers the interest rate and raises output.

JEL codes: E, E63

Keywords: overlapping generations, uncertain lifetimes, endogenous labour supply, wealth effect, monetary and fiscal policy

* We are grateful to two anonymous referees for their helpful comments. Neil Rankin thanks the University of

Pavia and the European Central Bank for their hospitality while working on this paper. Responsibility for all

opinions and errors is of course the authors’ alone.

§ Corresponding author: Guido Ascari, Dipartimento di economia politica e metodi quantitativi, Faculty of

Economics, University of Pavia, Via San Felice, 7 – I-27100 – Pavia, ITALY; Email: gascari@eco.unipv.it; Tel.

+39 0382 986211; Fax +39 0382 304226.

1

1. Introduction

The Blanchard (1985)-Yaari (1965) model of overlapping generations (OLG) has become

a standard tool of macroeconomic analysis. The advantages of Blanchard’s framework over

other models of overlapping generations, such as that of Diamond (1965), are that the average

length of life of a typical agent can be parameterised and that the special case of infinite lives

is nested within it. This is achieved while retaining elegance and simplicity at the aggregate

level. A key assumption is that an agent has a constant probability of death, independent of

age. This has caused the model sometimes to be referred to as one of “perpetual youth” (e.g.

by Blanchard and Fischer, 1989). The perpetual youth framework has found many useful

applications. Most obviously, and like other OLG models, it implies that “Ricardian

Equivalence” does not hold, and so it provides a fruitful structure for analysing fiscal policy,

in particular the effects of government debt and deficits.

Most of the applications to date (as in Blanchard’s original 1985 paper itself) have been

in the “long-run” context of growth and capital accumulation, assuming exogenous labour

supply. The Blanchard-Yaari structure is however also very relevant for studying the “short-

run” effect of fiscal policy in models of business cycle fluctuations, including the question of

the interaction between monetary and fiscal policy. Over a period of five years or less,

changes in labour inputs are typically at least as important as changes in capital inputs in

explaining output movements, and hence it becomes desirable to model the labour market,

and as part of this, the labour supply. Hence in short-run models it is usually assumed that

households obtain utility from leisure, so that their optimal labour supply decision is then in

general a function of current and future real wages, interest rates, and wealth levels.

In this paper, we point out a potential problem with making the labour supply decision

endogenous in a perpetual youth model: namely, some agents’ labour supply is likely then to

be negative. The intuitive explanation for why there is a “negative labour supply” problem is

2

as follows. With most specifications of agents’ preferences, leisure is a “normal” good, i.e. the

wealthier the agent is, the more leisure he demands, and therefore the less labour he supplies.

In the perpetual youth model, the older the agent is, the more financial wealth he has.

Moreover, the age distribution is such that there is no upper bound on age: there are always

some arbitrarily old agents alive at any point in time. Therefore at any point in time there are

always some very wealthy agents in the population, and, if leisure is a normal good, these

agents will want to consume more leisure than is feasible given their time endowment.

Otherwise stated, such agents will want to supply negative amounts of labour. This problem is

particular to the perpetual youth model because of the lack of an upper bound on age. In a

Diamond-type model with n-period lives, on the other hand, while there may still be a

question of negative labour supply, it is not guaranteed that there are agents for whom the

preferred labour supply is negative.

Given the importance of the Blanchard-Yaari model as a central tool for

macroeconomists, the main part of the present paper is devoted to suggesting a solution to this

problem. In doing this, it needs to be remembered that, first, a number of restrictions are

already placed on agents’ preferences to permit us to derive the aggregate equivalents of

individual demand functions. In particular, agents’ demand functions for goods and leisure

need to be linear in wealth. This ensures that aggregate demands for consumption and leisure

can be written as functions only of aggregate wealth (and of relative prices), and that they are

independent of the wealth distribution across agents of different ages. Such a condition rules

out preferences which would make labour supply tend asymptotically to zero, since labour

supply would then be nonlinear in wealth. Second, simply to take proper account of the non-

negativity condition on labour supply, without changing preferences, would create a “kink” in

the individual’s labour supply as a function of wealth, and thus would introduce another form

of nonlinearity. It would then no longer be possible to write aggregate labour supply as a

3

function of aggregate wealth alone: there would exist a threshold level of wealth above which

an individual’s labour supply would be zero, and in solving for the general equilibrium we

would need to keep track of the fraction of agents whose wealth exceeded the threshold level.

Our proposed solution to the problem, therefore, is to use a specification of households’

preferences which makes labour supply independent of wealth. The specification which we

use is adapted from Greenwood, Hercowitz and Huffman (1988) (henceforth “GHH”). It

extends GHH in that it incorporates real balances into the utility function. Our motive for such

an extension is that, in keeping with the aim of providing a framework which can potentially

be applied to short-run, business-cycle, issues, we want to be able to address questions of

monetary policy, and not just of fiscal policy.

Indeed, there is a growing number of papers extending the already large literature on

optimal monetary policy to optimal monetary and fiscal policy. Many of these papers (e.g.,

Beetsma and Jensen (2003), Benigno and Woodford (2003), Schmitt-Grohe and Uribe (2003),

Correia et al. (2003)) use the representative agent framework for reasons of simplicity.

Others, following Marini and van der Ploeg (1988), already make use of perpetual youth

models to study the interrelation of monetary and fiscal policy but retain the assumption of

exogenous labour supply (e.g., Cushing (1999), Leith and Wren-Lewis (2000)). Finally a few

published papers in the literature do use the Blanchard-Yaari model with endogenous labour

supply (e.g., Heijdra and Ligthart (2000), Cavallo and Ghironi (2002), Smets and Wouters

(2002)).1 These last are, at least potentially, vulnerable to the negative labour supply problem:

it appears to have gone unrecognised in the literature so far.

Our premise is that it is desirable and important to have a version of a perpetual youth

model with endogenous labour supply which avoids the negative labour supply problem. Our

1 The Blanchard-Yaari structure has also recently been employed in open-economy monetary macro models, without looking at the effects of fiscal policy, because the OLG structure ties down a country’s long-run net foreign asset position (see Ghironi (2000)).

4

aim in this paper is mainly to establish the feasibility of constructing such a model, and to

show that it provides a basis for a richer analysis of monetary and fiscal policy than is

possible in infinitely-lived agent models. What the rest of the paper does is to explain the new

preference structure, derive its implications for aggregate relationships in a “perpetual youth”

environment, and then embed this in a simple flexible-price, dynamic general equilibrium

model. The implications of this type of economy for some standard fiscal and monetary

policy experiments are then explored. Despite its simplicity, our model demonstrates an

interesting interaction between the fiscal and monetary sides of the economy. The roles of

money as, on the one hand, a medium of exchange, and, on the other hand, a store of value, lie

at the heart of the model. Our method of solution draws attention to these roles, in attempting

to reveal the workings of the economy more intuitively.

The structure of the remainder of the paper is as follows. In Section 2 we present the

utility function which we use, derive individual and aggregate household behaviour from it,

and we define the general equilibrium of the model. In Section 3 we study the effects of a

variety of monetary and fiscal policy measures. Section 4 concludes.

2. A Model with Uncertain Lifetimes and Wealth-Independent Labour Supply

2.1 Households

We use a discrete-time version of Blanchard’s (1985) overlapping-generations structure,

as in Frenkel and Razin (1987). Agents have an exogenous probability, q (0 < q ≤ 1), of

surviving to the next period. This is the only source of uncertainty in the model. Each period,

1-q new agents are born and 1-q die, so that the population remains constant. The number of

agents of age a alive in any period is thus (1-q)qa. This illustrates the point made in the

Introduction, that there are always some agents alive of any arbitrarily chosen age: the cross-

5

section distribution of the population by age is a declining geometric distribution, with an

unbounded support. Under these assumptions, the total population size is 1.

The expected utility over her remaining lifetime of an agent who is alive in period n and

was born in period s ≤ n, is hence:

, ,( )t nn s n t n s tE U q uβ∞ −

== Σ 0 < β < 1 (1)

where us,t is flow utility. The novelty in our specification lies in the form proposed for us,t. We

assume:

1, , , ,ln( ( ))s t s t s t s tu c m d lδ δ−= − 0 < δ < 1 (2)

where d(ls,t) is a function giving disutility of labour supply, with d′, d′′ > 0. In a model without

money, in order to make labour supply wealth-independent the term inside the log operator

must take the form cs,t - d(ls,t), as GHH show. Here we wish to include real balances, ms,t ≡

Ms,t/Pt, in utility to represent the liquidity services which money provides in helping

households to buy goods. It is not plausible to introduce ms,t into cs,t - d(ls,t) in an additively

separable way, because this would make not only labour supply, but also money demand,

wealth-independent, and the latter seems especially unreasonable. Since money is used for

buying goods, we combine ms,t with cs,t. To preserve wealth-independence of labour supply

we then need to make the combined function linear-homogeneous in (cs,t,ms,t), as done in (2).

The agent has the following single-period budget constraint:

, , , , 1 1 , 1 ,(1/ )[ (1 ) ]t s t s t s t s t t s t t s t t tPc M B q M i B W l T− − −+ + = + + + + Π − (3)

Here, Pt is the price level, Wt is the money wage, it is the nominal interest rate, Πt is profit

receipts from firms, Tt is a lump-sum tax, and Ms,t, Bs,t are the stocks of money and

government bonds the agent holds at the end of period t.2 Following the actuarially fair

insurance scheme in Blanchard (1985), the agent receives an “annuity” at the gross rate 1/q on

6

his total financial wealth, so long as he stays alive. If he dies, on the other hand, all his

financial wealth passes to the insurance company. The first-order conditions for the

intertemporal problem of the agent take the form:

, 1 , 1 , 1 , 1 , , , ,( / ) ( ) (1 )[ ( / ) ( )]s t s t s t s t t s t s t s t s tc c m d l r c c m d lδ δβ+ + + +− = + − , (4)

,

,

11

s t t

s t t

c im i

δδ−

=+

, (5)

1, , ,(1 ) ( / ) ( )t s t s t s tw c m d lδδ − ′= − (6)

where 1+rt ≡ (1+it)Pt/Pt+1 is the real interest rate and, unless otherwise defined, lower case

letters indicate real variables.

It is worth emphasising that (4)-(6) are conditional, in the sense that they obviously only

apply if the agent remains alive for the periods concerned. A first point which they reveal is

that certain choices will be the same for all agents irrespective of birthdate, s. Since all agents

face the same nominal interest rate it, (5) shows that all will choose the same ratio of real

balances to consumption. (5) is a standard demand-for-money function. It arises from the

“medium of exchange” role of money, captured by the presence of real balances in the utility

function. For later reference, it is helpful to define zt as money demand per unit of

consumption, so that (5) then implies:

, , ,1( / )

1t

s t s t s tt

iz m ci

δδ

+≡ =

−. (7)

zs,t is the same for all generations, even though ms,t and cs,t are not. Having noted this, we then

observe from (6) that labour supply will be the same for all agents. This, of course, is as

intended: our utility function is designed to remove the effect of wealth on labour supply, and

since financial wealth is the only respect in which agents differ, they should then all choose

the same labour supply. It is also important to note that the presence of zs,t introduces a link

2 Profit receipts and taxes, which are exogenous to the agent, are assumed to be the same for all agents

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between the labour market and the money market, and that is due to the lack of additive

separability in the flow utility function mentioned above.

Turning to consumption, (4) shows that the usual Euler equation for consumption is now

modified by subtraction of the term )()/( ,,, tststs ldmc δ from cs,t (similarly for cs,t+1). As just

seen, this term is the same for all generations s, and so independent of cs,t. To interpret it, first

notice that flow utility can be rewritten as:

)/ln()]()/(ln[ ,,,,,,, tststststststs cmldmccu δδ +−= . (8)

Hence )()/( ,,, tststs ldmc δ acts like a “shift of origin” for consumption. This brings out a

resemblance of our function to the Stone-Geary utility function, in which the new origin can

be thought of as a “subsistence” level of consumption. For future reference, we label the

surplus of consumption over its subsistence level as “adjusted” consumption, as,t:3

)()/( ,,,,, tststststs ldmcca δ−≡ . (9)

Moreover, we define:

1,,, −+≡ tststs vhω ; ][ ,,0, itititsititti

its lwqh +++++∞= −+≡ τπαΣ ;

11

11

1, )1...()1()1( −

−+−

+−

+ +++≡ itttitt rrrα (and αt,t ≡ 1) ; [ ]t

tsttsts PBiM

qv 1)1(1

1,11,1, −−−− ++≡ .

Here ωs,t is total real wealth, comprising real human (hs,t) and “nonhuman”, or financial,

wealth (vs,t-1). Since we have seen that ls,t is the same for all age cohorts, “human wealth” (i.e.

the discounted present value of labour income and profits minus taxes) is also the same for all.

Financial wealth, on the other hand, generally increases with age. We can then write the

agent's lifetime budget constraint as:

, , , ,[ ( /(1 )) ]t nt n n t s t t t s t s nq c i i mα ω∞ −=Σ + + = (10)

irrespective of age.

8

This has been obtained by re-writing (3) in real terms, integrating it from n to infinity

applying a ‘No Ponzi Game’ condition, and then using the wealth definitions just given.

As pointed out earlier, an essential requirement of the perpetual youth model is that the

agent’s consumption be linear in his lifetime wealth. Standard manipulations deliver:

)]()1()[1)(1()( 1,0,, itititt

iitsttts ldzqqldzc +

−+

−+

∞=

− −Σ−−−=− δδ δαωβδ (11)

where we have dropped s subscripts on (zt,lt), for the reasons explained. (11) says that an

agent’s adjusted consumption is a constant fraction of his total wealth net of the present value

of future subsistence consumption levels.4 The important point is that consumption is still

linear in wealth. This means that aggregate consumption can be written as the same function

of aggregate wealth and that the distribution of aggregate wealth across agents is immaterial.

So far we have looked at variables pertaining to an individual born in period s. We now

turn to look at aggregate variables. For any variable xt, the relationship of aggregate to

individual values is tsstt

st xqqx ,)1( −−∞= −Σ= , where absence of an “s” subscript indicates an

aggregate value.5 (Note that, since the population size is 1, aggregate and average values

coincide.) As regards the behaviour of zt and lt, we have already shown that individual choices

are independent of age, whence zt = zs,t, ls,t = lt, for all s. For ct, as further noted, (11) can be

applied, where ωs,t is replaced by its aggregate counterpart, ωt. It is also useful to derive an

aggregate counterpart of the individual Euler equation for consumption, (4), namely:

tttttttt vqqldzcrldzc )1/1)(1)(1()]()[1()( 111 −−−−−+=− −+

−++ βδβ δδ . (12)

3 (8) implies that individual’s preferences are undefined for consumption values below the “subsistence” level. Hence as,t needs to be positive for all s,t in equilibrium. In the Appendix we prove this for a particular case and provide a more general discussion. 4 In fact, these are the subsistence levels of “full” consumption, where full consumption is consumption plus the foregone interest on money balances, , , ,( /(1 ))s t s t t t s tfc c i i m≡ + + . This explains the presence of (1-δ)-1. 5 While this is also true for asset holdings Ms,t, Bs,t, for vs,t the relationship is .)/1()1( , tts

stts vqvqq =−Σ −

−∞= This is because our definition of vs,t includes the annuity payout in the definition of vs,t, and this, being a pure redistribution from those who die to the survivors, does not apply to aggregate holdings.

9

(12) shows that the growth rate of aggregate adjusted consumption is negatively related to

aggregate financial wealth. Intuitively, the reason for this is the “generational turnover

effect”.6 Between t and t+1, some already-existing agents are replaced by newborn agents. If

the newborn have lower consumption than the average consumption of those who die, this

compositional effect reduces aggregate consumption between the two periods. Now, as seen,

an agent’s consumption is increasing in his financial wealth; moreover, agents are born with

zero financial wealth. Therefore the newborn do indeed have lower consumption than the

average of the rest of the population, and it is a random sample of the latter who die. This can

be brought out by manipulating (12) to obtain:

⎥⎦

⎤⎢⎣

⎡−−

−++=−−

−

−

−+

−++

)()(

)1()1()(

)( ,111

ttt

ttttt

ttt

ttt

ldzcldzc

qrqldzcldzc

δ

δ

δ

δ

β (13)

which shows the (gross) rate of growth of aggregate adjusted consumption to be a weighted

average of the rate of growth of individual adjusted consumption (i.e. β(1+rt)) and the ratio of

the adjusted consumption of the newborn to that of the average citizen.

2.2 Firms

The representative firm has an increasing, concave production function, )( tt lfy = , and

so maximises profits where:

)( tt lfw ′= . (14)

We neglect capital as an input for simplicity, to enable us to focus on the household side of

the model. If f(.) is strictly concave then the firm will make positive profits, which we assume

are paid out equally to all households as lump-sum dividends.7

6 This effect is so named by Heijdra and Ligthart (2000). 7 In dynamic models it might be argued that profits would be driven to zero by firm entry, but we shall abstract from this factor here. Even with a fixed number of firms, a more complete treatment of profits would introduce an equity market such that profits would be the return on equity, considered as an additional asset alongside government bonds and money. The present assumption, in which firms are treated as if state-owned, enables us to avoid this complication, even if it does slightly alter the equilibrium.

10

2.3 Government

The government has a single-period budget constraint of the form:

1 1 1 1( ) ( ) ( )t t t t t t t t tP g i B B B M Mτ − − − −− + = − + − (15)

where gt denotes government spending on firms’ output. We can re-write this in real terms as:

])/([])1([ 1111 −−−− −++−=− ttttttttt mPPmbrbg τ (16)

in which bt ≡ Bt/Pt. There are many policy regimes which could be considered, but we wish to

focus on the simplest types of monetary and fiscal policy experiment. Hence we will treat gt

and Mt as exogenous and constant over time, except for the possibility of once-and-for-all

changes. Government debt will be treated as a third independent policy instrument, leaving τt

to balance the budget as a residual. There is more than one way to fix the stock of government

debt exogenously. Here we choose to set it in real terms, inclusive of interest. Hence the debt

policy instrument is taken to be:

ttt brb )1( +≡′ . (17)

Under this assumption one unit of tb′ is like an indexed treasury bill: it is a promise of one

unit of goods in one period’s time.

2.4 General Equilibrium

The equilibrium in the labour market is given by (6) and (14), so that:

δδ ttt zldlf )1/()()( −′=′ . (18)

This equation implicitly determines lt, and thus yt, as a function of zt. Hence there is a positive

relationship between employment and the level of real balances per unit of consumption

(since f′′ < 0, d′′ > 0).8 The cause of this is complementarity between consumption and real

balances in households’ utility: note that the cross-partial derivative in (2), ucm, is positive.

8 Note that in a non-monetary version of our economy, in which we would therefore have δ = 0, (18) would tie down lt by itself. The non-monetary version of the economy thus has the “classical dichotomy” property, in the sense that employment is independent of monetary and fiscal policy.

11

Thus, the higher the agent’s consumption, the more useful are real balances to him, which

makes sense given that the purpose of money is to facilitate transactions. In (18), then, a rise

in z (or in m, at given c) raises the marginal utility of consumption and so makes it worthwhile

for the agent to work an extra hour. In other words, higher real balances stimulate labour

supply. While we would not expect this effect to be empirically powerful, its robustness at a

theoretical level is indicated by the fact that a very similar mechanism also arises in the

“shopping time” approach to the demand for money (see, e.g., Walsh, 1998).9

It is helpful for what follows to adopt specific functional forms for the production and

disutility-of-work functions. Thus, suppose that f(.) and d(.) are constant-elasticity functions:

σtt llf =)( 0 < σ ≤ 1, (19)

( ) ( / )t td l lεη ε= ε ≥ 1. (20)

Using these in (18), we obtain explicit solutions for lt and yt as functions of zt:

)/()/(1]/)1([ σεδσεηδσ −−−= tt zl ; )/()/(]/)1([ σεδσσεσηδσ −−−= tt zy . (21)

(For this, we need that ε and σ are not both equal to one.)

The equilibrium condition in the goods market is ttt gcy += . Having seen that yt can be

found as an implicit function of zt, it is then clear that ct can be expressed as an implicit

function of (zt,gt).

Finally, we consider equilibrium in the asset markets. Assume a policy of holding tb′ and

Mt constant over time at values b′ and M. The aggregate stock of real financial wealth, i.e.

1/)1( +++= ttttt PMbrv , then becomes:

1++′= tt mbv . (22)

We can hence write (12) as (using also (9), in its aggregate version):

9 In that approach, real balances are assumed to economise on the agent’s shopping time rather than provide utility. An increase in real balances releases time for work or leisure, and so generally stimulates labour supply.

12

))(1/1)(1)(1()1( 11 ++ +′−−−−+= tttt mbqqara βδβ . (23)

Next, assuming for the moment that there is no government spending, we can substitute out

the endogenous variables in (23) to reduce the model to the following implicit first-order

difference equation in zt:10

[ (1 ) ]/( )1

1

1 (1 )(1 )(1/ 1) 1(1 ) (1 ) /(1 )t

t t

q q b zz z

σε σ

ε δ σ ε σε δ β η βε δ σ σ δ δ δ

−− − − −+

+

⎡ ⎤⎛ ⎞− − − ⎢ ⎥′+ + =⎜ ⎟⎢ ⎥− − − − −⎝ ⎠⎢ ⎥⎣ ⎦

. (24)

This difference equation is the central equation of the model.

Since zt is a non-predetermined variable, (24) must be solved in a “forward-looking”

manner. This means that, for a unique non-explosive solution (or, at least, locally unique,

disregarding solutions which may be generated by the nonlinearity of the equation), we need

(24) to have a unique, locally unstable, steady state. In this case zt jumps straight to its steady-

state value. Indeed, since there are no other state variables (such as the capital stock) driving

the dynamics, the whole economy attains its new steady state immediately following a

shock.11 It is straightforward to construct a phase diagram to show the steady state of (24) is

indeed locally unstable. We therefore proceed to study the steady-state version of (24). Setting

zt = zt+1 = z, multiplying through by z/β, and subtracting 1 from both sides, we have:

/(1 ) (1 )(1/ 1)(1 )(1/ 1)/(1 ) (1 ) (1 )

q q z b zz

σδσε σ

ε σδ δ ε δ β ηβδ δ ε δ σ σ δ

− −−

⎡ ⎤⎛ ⎞− − − − ⎢ ⎥′= − + + ⎜ ⎟⎢ ⎥− − − − −⎝ ⎠⎢ ⎥⎣ ⎦

. (25)

Note that the left-hand side (LHS) of (25) equals i. The LHS function is just the inverse

of the money demand function, (5). This is plotted in Figure 1. We label it the “ME” curve,

10 To see how, note that, with no government spending, lt, ct, mt (of which at is a function - recall (9)) can all be written as functions of zt alone. Hence at can be expressed as just a function of zt, and at+1 and mt+1 just as functions of zt+1. In the case of 1+rt, which equals (1+it)Pt/Pt+1, it is simply related to zt by (5), and Pt/Pt+1 can be eliminated as mt+1/mt, so that 1+rt is also seen to depend only on zt and zt+1. 11 More generally the level of government debt could also be a state variable. Here this is not the case because we consider a simple policy regime in which the level of government debt is held constant, rather than evolving endogenously over time.

13

since it gives a relationship between z and i which derives from the “medium of exchange”

role of money.

The right-hand side (RHS) of (25) can be interpreted as giving the value of r consistent

with various levels of demand for financial wealth per unit of adjusted consumption, v/a,

since v/a can be shown to be proportional to z. To see the former, first recall that in the steady

state inflation is zero and hence i = r. Next, note that the RHS of (25) derives partly from the

steady-state version of (23):

avqqr /)1)(1/1)(1()1/1( −−−+−= βδβ . (26)

(26) shows that in the steady state there is a positive relationship between the real interest rate

and the level of financial wealth per unit of adjusted consumption. The origins of this

relationship lie in the fact that financial wealth as a whole, from an individual’s point of view,

is a “store of value”. By accumulating or decumulating tsv , , an individual is able match the

time profile of his “labour” income to the desired time profile of his consumption.12 The

greater is r, the steeper will be the desired growth path of consumption, so (recalling the

positive relationship between individual consumption and financial wealth in (11)), the agent

will be accumulating financial wealth more rapidly over his lifetime. Since he has zero

financial wealth at birth, this means that his average level of financial wealth over his lifetime

will be higher, for the same average level of consumption, than it would be if r were smaller.

Aggregating across the population, therefore, a higher r is associated with a higher aggregate

demand for financial wealth in relation to consumption. Note that generational turnover, i.e. q

< 1, decouples an individual’s rate of wealth accumulation from the aggregate rate of wealth

accumulation. The latter is zero in a steady state, so, without this decoupling, for a steady

state to exist the individual’s rate of financial wealth accumulation would also have to be

12 In fact, decumulation will never be done willingly, in an equilibrium of the perpetual youth model. It occurs only when the agent dies. At this point decumulation happens abruptly, as all the agent’s financial wealth passes to the insurance company.

14

zero, which would require r always to equal 1/β-1. In view of its derivation from the store-of-

value role of financial wealth, we label the RHS of (25) as the “SV” curve.

Figure 1

Consider now the implications for the equilibrium value of r. First, notice that when

agents live forever (q = 1), the SV line is horizontal at r = 1/β-1. The real interest rate is then

identical to the pure time preference rate - a standard feature of “representative agent” models.

In Figure 1 the equilibrium is in this case at point C. Second, we can see from the diagram

that, more generally, when q < 1, r exceeds 1/β-1. In the special case where b′ = 0, the SV

curve is just an upward-sloping straight line with a vertical intercept at 1/β-1. The equilibrium

of the model is therefore at point A. At point A the equilibrium level of real balances per unit

of consumption, z, is the value consistent both with the demand for real balances as a medium

of exchange, and with the demand for real balances as a store of value.13 When b′ > 0

(discussed in more depth in Section 3), the SV curve shifts up further, to SV′, so that r > 1/β-1

holds a fortiori. Third, there is nevertheless one case in which r = 1/β-1 still holds even when

13 Note that what determines the equilibrium level of real balances is not the supply of nominal balances. Changes in the latter (if they were implemented by lump-sum taxes or transfers rather than by sales or purchases of government debt, and were distributed in proportion to an agent’s initial money holdings) would be neutral in

15

q < 1. This is when there are no financial assets at all. This case would arise if we set b′ = 0

and also let δ (the exponent on real balances in the utility function) tend to zero. The vertical

asymptote of the ME curve would then shift left until it coincided with the vertical axis, while

the curve itself would shrink in towards its asymptotes, so that in the limit equilibrium would

occur at the vertical intercept of the SV line. This shows that, in the absence of all financial

assets, the presence of overlapping generations makes no difference to the interest rate. It is

only when real balances or government debt is introduced that the real interest rate is raised

above the pure time preference rate. The reason for this is that the existence of a positive

stock of real financial wealth requires r > 1/β-1, in order that agents willingly hold this wealth

to finance a rising lifetime profile of consumption. Conversely, in the absence of a stock of

real financial wealth, r = 1/β-1 is needed in order that the demand for wealth be zero.14,15

3. Effects of Monetary and Fiscal Policy

3.1 A change in government debt

We first test “Ricardian Equivalence”. Assume that the government runs a very

temporary budget deficit, so that b′ is raised from zero to a positive value by a tax cut lasting

for one period only. In Figure 1, a positive b′ causes an upward shift of the SV curve to SV′

(see (25)) and the economy hence jumps to the new steady state at point B . As can be seen,

the interest rate (real and nominal) increases, and the level of real balances per unit of

consumption declines. An increase in b′, for a given z, increases the level of the interest rate

necessary for equilibrium in the market for financial wealth. However, this reduces demand

this flex-price economy. Rather, the level of real balances is determined by the demand for real balances, through the latter’s effect on the general price level. 14 Interestingly, the case of no real balances and no government debt is the one used in the recent open-economy literature where the Blanchard/Yaari structure is employed to make the steady-state stock of net foreign assets determinate (e.g., Smets and Wouters (2000)). 15 The statement that introducing real balances raises the real interest rate is not inconsistent with the statement that money is “neutral” when introduced through lump-sum cash handouts. The former is a change in agents’ preferences, and it is these, not the supply of money, which determine the equilibrium level of real balances, as already noted.

16

for money as a medium of exchange and so motivates a portfolio shift. Associated with the

fall in z is a reduction in output and employment. Government debt thus has real effects on the

economy. As expected, Ricardian Equivalence does not hold. It is easy to verify that this is

due to overlapping generations, because if we set q = 1 then the SV curve becomes horizontal

at 1/β-1 and it is clearly then unaffected by changes in b′.

Intuitively, Ricardian Equivalence fails because, although the current tax cut must be

matched by future tax increases equal in present value to the cut in order to finance the extra

interest payments on the higher government debt, when q < 1 some of these tax increases will

fall on agents yet to be born. Hence the currently living - the beneficiaries of the tax cut - do

not face the full future tax increase, and they perceive their lifetime wealth to have increased.

At an unchanged interest rate they would want to save some of the tax cut in order to pay the

higher future taxes but, unlike when q = 1, they would not want to save all of it. Thus at the

old interest rate the demand for financial assets rises by less than the supply (the latter being

the extra bonds on the market). The interest rate must therefore increase in order to restore

equilibrium. This is a standard result. In our model, the rise in the interest rate also has a

negative effect on labour supply and thus on output. If the wealth effect on labour supply had

not been eliminated, we might have suspected that the reason for the fall in labour supply was

the increase in perceived lifetime wealth. In fact, the mechanism is instead the fall in real

money balances induced by the rise in the interest rate. This then has a disincentive effect on

labour supply via the complementarity of real balances and consumption in preferences.

The effects of government debt in this OLG model with an endogenous supply of labour

to some extent resemble the better known effects of debt in OLG models with an endogenous

supply of capital (Diamond (1965), Blanchard (1985)). In both types of model, the real

interest rate is increased, output is reduced and government debt “crowds out” another asset

from private portfolios. The difference is that it is capital which is crowded out in the

17

Diamond and Blanchard models, but real money balances, here. Although, unlike physical

capital, real balances are not an input to production, they are nevertheless positively linked to

labour supply, and so they act indirectly like an input to production.

3.2 An open-market operation (OMO)

In practice, a central bank injects money into the economy mainly through asset-swap

operations, in which bonds are bought or sold in exchange for money. If, for example, the

central bank wants to increase the money supply by ∆Mt in period t, then (15) implies that it

needs to change the stock of debt in public hands by t tB M∆ = −∆ , i.e. it needs to purchase (in

absolute value) this quantity of debt on the open market. Equivalently, it needs to change tb′

by 1((1 ) / )t t ti P M+− + ∆ . The easiest way to study the impact of an OMO is to break it into two

separate operations: an increase in Mt financed by a cut in τt; and a reduction in tb′ financed

by an increase in τt, such that, overall, τt is unchanged. Now, since the first of these operations

leaves b′ unchanged, it causes no alteration in the equilibrium condition (25). This way of

increasing Mt is just the standard “helicopter drop” method, and it has no real effects, merely

causing an equi-proportional rise in all nominal variables. The second of the operations

reduces b′, and so is just the reverse of the government debt increase studied above. It can

therefore be considered as a movement from B to A in Figure 1, which lowers the real and

nominal interest rates and raises z. We hence find that an increase in the money supply

through an OMO does have real effects: by raising z, which stimulates labour supply, it

expands output and employment. The analysis here moreover shows how a “liquidity effect”

of monetary policy can be generated, i.e. a fall in the interest rate, which is hard to get in

18

many dynamic general equilibrium models. Our model explains this fall by the associated

reduction in the bond stock and the lack of Ricardian Equivalence.16

3.3. A change in government spending

A tax-financed increase in spending is represented by an increase in gt at unchanged

b′.17 With a positive level of government spending, equation (24) now contains (gt,gt+1) as

additional exogenous variables. The steady-state equation (25) is then modified to:18

/(1 ) (1 )(1/ 1)(1 )(1/ 1) ( , )/(1 ) (1 )

q q z z gz

δ δ ε δ ββδ δ ε δ σ

− − − −= − + Γ

− − − −,

where

1

(1 ) (1 )( , ) 1 1(1 ) (1 )

z g z g

σδσε σ

ε σσ δ σ δ ε ηε ε ε δ σ σ δ

−

− −−

⎡ ⎤⎛ ⎞− − ⎢ ⎥Γ ≡ − + − ⎜ ⎟⎢ ⎥− − −⎝ ⎠⎢ ⎥⎣ ⎦

. (27)

As before, Figure 2 plots the LHS and RHS of the equation. The LHS is unaltered, and so

represented by the same ME curve as in Figure 1. As regards the RHS, first note that if g = 0,

then Γ(.) = 1 for all z, and the SV curve is just an upward-sloping straight line with intercept

1/β-1 (as in Figure 1). When g > 0, the RHS is instead described by the curve SV′ in Figure 2.

This is because the function Γ(.) is then decreasing in z but always greater than one, tending to

one as z tends to infinity and to infinity as z tends (from above) to some strictly positive value.

16 The idea that an OMO may be non-neutral goes back to Metzler (1951). Overlapping generations have also recently been used to provide a model of the liquidity effect by Benassy (2003). 17 We can also consider a debt-financed increase in spending. However, a spending increase cannot be debt-financed forever, since the debt stock would explode; hence such a policy is most naturally broken into a combination of a balanced-budget spending increase and a separate debt increase, where the latter has already been analysed (see 3.1). For simplicity, in this subsection we assume b′ = 0. 18 Note again that the lack of any predetermined variables, such as the capital stock or endogenous government debt, makes the money-consumption ratio, zt, jump straight to its steady state value. So again there are no transitional dynamics and we just need to look at the steady state to characterize the behaviour of the model.

19

Figure 2

From this it follows that the effect of an increase in government spending, starting from

zero, is to move the economy from point A to point B. It therefore raises r and lowers z. The

fall in z means that output and employment also fall. At a more intuitive level, the interest rate

rise is driven by the fact that the demand for money as a store of value is proportional to

adjusted consumption, whereas the demand for money as a medium of exchange is

proportional to actual consumption. By crowding out consumption, the rise in g thus reduces

the demand for money as a store of value by more than it reduces the demand for money as a

medium of exchange, and this means the interest rate has to rise, to restore equality between

them. Moreover, these impacts on r and z are clearly associated with the presence of

overlapping generations, because if q = 1 the SV curve is horizontal and unaffected by a

change in g.

As with the effects of government debt, the effects of a balanced-budget increase in

government spending in our perpetual youth model with endogenous labour supply somewhat

resemble those in Blanchard’s (1985) perpetual youth model with an endogenous capital

stock. In both models, the real interest rate rises and output falls.

20

4. Conclusions

In this paper we have drawn attention to a problem which is likely to be encountered as

soon as one tries to extend the “perpetual youth” model of overlapping generations to

incorporate endogenous labour supply, namely the problem that some agents will have

negative labour supply. We have proposed a solution, which is to use a form of preferences

adapted from Greenwood, Hercowitz and Huffman (1988). We chose a utility function which

includes real money balances in order to generate an OLG framework with potential for

monetary as well as fiscal analysis. To demonstrate the logical coherence of the framework,

we have shown how it permits a general equilibrium to exist in a simple Walrasian economy

where the only input to production is labour. To demonstrate its usefulness, we applied it to

the study of three types of fiscal and monetary policy shock. The interest of the policy effects

lies in the fiscal-monetary interaction which occurs. An increase in government debt lowers

output because it crowds out real balances, and the latter is a disincentive to labour supply. An

open market operation to increase the money supply lowers the interest rate, i.e. has a

liquidity effect, because it reduces the stock of government bonds in circumstances where the

latter are "net wealth". Finally, an increase in government spending raises the interest rate

because it reduces the demand for money as a store of value by more than it reduces the

demand for money as a medium of exchange.

One criticism which might be made of our model is that, although it avoids the problem

of negative labour supply, it is still unrealistic in implying that individuals of, say, 80 years of

age work the same hours as thosed aged, say, 30. In fact, empirical evidence suggests that

hours of work between the ages of 25 and 60 are fairly ‘flat’ (see, e.g. Andalfatto, Ferrall and

Gomme, 2000). Our model is thus not seriously inconsistent with the evidence as regards the

normal period of working life. It is true that it does not allow for other aspects of the life

cycle, such as education in early years or retirement in later years, but this is an inevitable cost

21

of the stylised ‘perpetual youth’ set-up. These must be weighed against the many advantages

of the latter in permitting a tractable general equilibrium model. Another feature which may

be thought unsatisfying is that consumption of an individual tends to infinity as the individual

ages. This, however, is just another facet of the highly stylised demographic assumption of no

upper bound on age: to remove it, we would have either to drop the assumption of constant

probability of death or to adopt preferences which make consumption non-linear in wealth,

both of which would make aggregation and hence a simple solution impossible.

Although, here, we have embedded our model of households only in a very simple flex-

price economy, we hope that it may also prove useful in more sophisticated representations of

short-run business cycle behaviour, for example economies which incorporate market

imperfections such as imperfect competition and nominal rigidities. Since its key feature is

that Ricardian Equivalence does not hold, it should offer greater scope for the modelling of

fiscal policy issues involving government debt and deficits. Although we have carried out

some basic analysis of these issues here, we see this as just a benchmark against which more

realistic analyses could be compared.

22

Appendix

Here we show that as,t is positive for all s,t in a steady state in which b′ = g = 0, that is,

consumption is always above the “subsistence” level for any agent. First note that, since 1+r >

1/β, an individual’s consumption grows as long as he stays alive (see (4)), which means that

his total wealth, and within this, his financial wealth, also grow (see (11)). Thus we see that

the agents most “at risk” of having negative adjusted consumption are the newborn, who have

zero financial wealth. Hence if we can show that at,t > 0 in the steady state, then as,t > 0 is

assured for agents of all ages, t-s. We prove that at,t is indeed positive. To determine its sign,

we first use (11) to express it as:

⎥⎦

⎤⎢⎣

⎡−

−++

−−−= −− )()1(1

1)1)(1( 1,, ldz

qrrhqa tttt

δδβδ .

Here we have set vt,t-1 = 0 (see main text) and have assumed that the economy is in a steady

state. In a steady state, human wealth is given by:

)(1

1, π+

−++

= wlqr

rh tt

where τ is zero since b′ = g = 0 in the present case. Now, wl + π = y = c, so we then have:

[ ])()1(1

1)1(, ldzcqr

rqa ttδδβ −−−

−++

−=

The sign of at,t therefore depends on the sign of (1-δ)c - z-δd(l). Appealing to the constant-

elasticity production and disutility-of-work functions and thus using (19)-(21), this can be

expressed as:

(1 ) ( ) [ (1 ) / ] (1 )(1 / )c z d l zσ δσ

δ ε σ ε σδ σ δ η δ σ ε− − −− − = − − − .

The RHS is unambiguously positive, which hence implies at,t > 0.

We can generalise this argument to the case b′ > 0. There is an upper bound on b′ above

which at,t becomes negative. This upper bound hence determines the maximum sustainable

23

level of government debt. This is an issue of interest for its own sake, but it is beyond the

scope of the present paper to pursue it.

Our demonstration above is for the case of a steady state. However, since the economy

jumps immediately to a new steady state under our particular policy experiments, it also

covers the case of the ‘transition path’ to the steady state.

24

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