Discussion Paper No. 1035

WEALTH PREFERENCE
AND RATIONAL BUBBLES

Jean-Baptiste Michau
Yoshiyasu Ono
Matthias Schlegl

June 2018

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Wealth Preference and Rational Bubbles
Jean-Baptiste Michauy Yoshiyasu Onoz Matthias Schleglx
June 2018

Abstract
We consider a neoclassical economy where households derive utility from holding wealth. We show that, under some conditions, there can be rational bubbles.
Hence, we provide a microfoundation for bubbles that relies on a frictionless in…nitehorizon economy without any heterogeneity across households. While our bubbly
equilibria are very similar to those obtained by Tirole (1985) in an overlapping
generation economy, the underlying economics is di¤erent. Turning to public debt,
we show that Ponzi schemes can be sustainable. Hence, in general, the limit on theaccumulation of public debt by the government is not given by its no-Ponzi condition but, instead, by the representative household’s transversality condition. The
Ricardian equivalence must hold in any of our equilibria. Finally, in the presenceof money, the real equilibrium structure of the economy remains unchanged. We
carefully investigate the e¤ects of helicopter drops of money on the possibility ofPonzi schemes and of speculative hyperin‡ation or de‡ation.
Keywords: Ponzi scheme, Rational bubble, Wealth preference
JEL Classi…cation: E13, E44, G12

1

Introduction

David Hume, Adam Smith, David Ricardo, Karl Marx, Alfred Marshall, Max Weber,John Maynard Keynes, Frank Knight, and Irving Fisher were all convinced that, formany people, the accumulation of wealth is an end in itself (Steedman 1981, Zou 1994).
For instance, Marshall (1890) wrote "There are indeed some who …nd an intense pleasure
We are grateful to Edouard Challe, Jean-Michel Grandmont, Gerhard Illing, Sisira Jayasuriya andto seminar participants at LMU Munich for helpful comments and suggestions. Financial supports fromthe Joint Usage Research Center Program and JSPS KAKENHI Grants (JP15H05728 and JP22530178)
are gratefully acknowledged.
yEcole Polytechnique, France; jean-baptiste.michau@polytechnique.edu.
zInstitute of Social and Economic Research, Osaka University, Japan; ono@iser.osaka-u.ac.jp.
x
JSPS International Research Fellow, Osaka University; LMU University of Munich, Germany;
matthias.schlegl@econ.lmu.de.

1

in seeing their hoards of wealth grow up under their hands, with scarcely any thought for
the happiness that may be got from its use by themselves or by others.". However, since theemergence of neoclassical economics in the early 20th century, the standard assumption
has been that households only care about wealth because of the future consumptionthat it can buy. Over the past few years, there has been a growing body of researchthat has generalized the standard neoclassical framework by assuming that households
derive utility from holding wealth. This includes work on the business cycle (Michaillat
and Saez 2015), on secular stagnation (Ono 2015, Michau 2018), on the New Keynesianmodel (Michaillat and Saez 2018), on leverage dynamics (Kumhof, Rancière, and Winant2015), or on capital taxation (Saez and Stantcheva 2018). In this paper, we investigate
the asset pricing implications of the preference for wealth.
Asset prices are composed of two components: a fundamental and a bubble. A wellknown result (Tirole 1982) is that, in a frictionless economy with a …nite number ofin…nitely-lived households, bubbles are inconsistent with individual rationality. Hence,models of rational bubbles either rely on an overlapping generation (OLG) structure(Samuelson 1958, Tirole 1985) or on …nancial market imperfections (Woodford 1990,
Kocherlakota 1992). In this paper, we show that rational bubbles can exist in a frictionlesseconomy provided that in…nitely-lived households derive utility from holding wealth. This
theoretical possibility signi…cantly expands the scope for rational bubbles.
We consider a standard Ramsey economy with a …nite number of in…nitely-lived households, but growing population within each household. Households derive utility fromwealth, with a diminishing marginal utility of wealth. This raises their propensity to saveto such an extent that the steady state capital stock can exceed the golden rule level.1
In that case, there is a bubbly equilibrium that induces the steady state capital stock tocoincide with the golden rule.
While our bubbly equilibria are very similar to those obtained by Tirole (1985) in an
OLG economy, the underlying mechanism is fundamentally di¤erent. In a simple OLG
economy, the bubble acts as a coordinating mechanism across generations whereby theold of each cohort exchange the bubbly asset against consumption goods produced by theyoung of the following cohort.2 By contrast, in our framework, in…nitely-lived households(or dynasties) want to accumulate wealth without the intention of spending it. Hence,
they can be willing to pay more for an asset than its fundamental value, which could
make a bubble sustainable over time. We therefore provide an alternative narrative for1

We avoid using the term "dynamic ine¢ ciency" since, in our setup, if the capital stock is above
the golden rule and bubbles cannot exist (as in Section 3 below), then there is no Pareto improving
reallocation of resources.
2
Similarly, in the perpetual youth model à la Blanchard (1985) and Yaari (1965), the possibility ofrational bubble is due to the birth of new individuals who will subsequently buy some of the bubbly
asset (Weil 1989).

2the existence of rational bubbles.
In both cases, we cannot have an equilibrium where the bubble grows without bounds.
But the underlying reason is again di¤erent. In the OLG economy, this is due to the fact
that, if the bubble grows over time, then it eventually becomes larger than the income of
the young who buy it, which is clearly unfeasible. In our economy, as the bubble grows
to in…nity, the marginal utility of wealth converges to zero, which brings us back to thestandard neoclassical case where the transversality condition rules out bubbles.
Hence, in both cases, a bubbly steady state must be characterized by a bubble ofconstant size relative to population. Moreover, by arbitrage with other risk-free assets,
the bubble must grow at the real interest rate. Hence, in a bubbly steady state, theinterest rate must be equal to the population growth rate and the capital stock must
therefore be at the golden rule. Thus, the fact that the steady state bubble implements
the golden rule in both Tirole (1985) and in our economy is due to the stationarity
requirement and is largely independent of the economic mechanism leading to the very
existence of the bubble.
Note that, with constant marginal utility of wealth, rational bubbles can occur evenwhen the capital stock is below the golden rule. In that case, the interest rate is above thepopulation growth rate and the rational bubble grows without bounds. The insatiabilityof the preference for wealth implies that, even with an ever growing asset price, thetransversality condition is satis…ed.
In environments where rational bubbles are possible, we know that Ponzi schemes ofgovernment debt can be sustainable. Fundamentally, the only limit to government indebtedness is imposed by the representative household’s willingness to lend to it. Hence, in
general, the government budget constraint is not given by the no-Ponzi condition but instead by the household’s transversality condition. Note that, in the standard neoclassicalframework, the two conditions are equivalent to each other. But, this equivalence breaks
down when households have a preference for wealth.3 This makes it possible to obtain asteady state equilibrium that violates the government’s no-Ponzi condition. However, insuch cases, there must also exist an equilibrium where households do not want to lend to
the government beyond its no-Ponzi condition.
In any equilibrium of the economy, the Ricardian equivalence holds, i.e. the timing oftax collection is neutral. This is a fundamental di¤erence with other models of rational
bubbles. In models based on the OLG structure or on …nancial frictions, bubbles exist
to redistribute resources across people. Hence, the very existence of bubbles relies on the
non-Ricardian nature of these models. By contrast, our work shows that the existence of3

In the perpetual youth model, the equivalence between the government’s no-Ponzi condition and theaggregation of households’transversality conditions also fails to hold, but this is due to the birth of newindividuals.

3

rational bubbles is not fundamentally inconsistent with the Ricardian equivalence.
The presence of money does not a¤ect the real equilibrium structure of the economy.
The preference for wealth remains solely responsible for the possibility of Ponzi schemes,
while money-in-the-utility-function only a¤ects nominal variables. In the special casewhere money is the only asset supplied by the government, through helicopter drops, thegrowth rate of the nominal money supply simultaneously a¤ects the possibility of Ponzi
schemes, on the real side, and of speculative de‡ation, on the nominal side. But, any real
allocation of resources that can be implemented through helicopter drops of money canalso be implemented through an appropriate …scal policy.
Following Japan’s lost decade or the Great Recession in the U.S. and in the eurozone, there has been concerns that these economies were trapped into demand-driven
secular stagnation characterized by a permanently depressed interest rate.4 In thesecircumstances, asset and government debt prices can permanently depart from their fundamental values. And, indeed stock market valuations have been unusually high in the
U.S. while the Japanese debt-to-GDP ratio has exceeded 250% without triggering any
increase in long term interest rates. Our work shows that there is no need to invoke
either …nancial frictions or an overlapping generation structure (without altruistic links)
to reconcile these patterns with rational optimizing behavior.
Related Literature. In a seminal paper, Tirole (1982) has shown that a bubble cannot
exist if the following two conditions simultaneously hold: (i) there is a …nite numberof in…nitely-lived households, and (ii) the economy is frictionless. Naturally, modelsof rational bubbles rely on the violation of at least one of these two assumptions. By
contrast, we obtain rational bubbles, not by relaxing the two key assumptions of Tirole(1982), but instead by generalizing the standard neoclassical framework through theintroduction of a preference for wealth.
Samuelson (1958) was the …rst to establish the possibility of rational bubbles within
an OLG economy. The result was subsequently generalized by Wallace (1980). Whileboth had exclusively focused on monetary bubbles within endowment economies, Tirole(1985) characterized the set of bubbly equilibria within a real production economy à la
Diamond (1965). He established that bubbles can exist provided that, in their absence,the economy would be dynamically ine¢ cient.5 Importantly, Weil (1989) has shown that
it is the birth of new households with zero wealth, rather than the …niteness of life, thatis responsible for the possibility of bubbles.6
4

See Ono (2015) and Michau (2018) for models of secular stagnation with wealth in the utility.
Also, O’Connell and Zeldes (1988) derived very general existence conditions for rational bubbles andproved some equivalence results between rational bubbles, Ponzi schemes, and valuable …at money.
6
More fundamentally, Shell (1971) considered an example of a static economy with an in…nity ofhouseholds and of commodities. He showed that, despite the existence of complete markets, the decentralized equilibrium could be Pareto ine¢ cient. As this economy is isomorphic to an OLG economy, it
54

By arbitrage, a bubble must grow at the real interest rate. But, explosive bubbles
are not feasible. Hence, the interest rate must be smaller than the growth rate of theeconomy, i.e. the economy must be dynamically ine¢ cient. Empirically, however, it is
not clear that the capital stock is above the golden rule.7 Hence, to obtain a rational
bubble in a dynamically e¢ cient economy, a number of theories have been proposed toraise the social returns on investment above the interest rate. This allows the growthrate of the economy to simultaneously be smaller than the social returns on investment,
as required by dynamic e¢ ciency, and larger than the interest rate, as required to havea non-explosive bubble.
Thus, to generate a rational bubble in a dynamically e¢ cient economy, Saint-Paul(1992), Grossman and Yanagawa (1993), and King and Ferguson (1993) have introduced
endogenous growth within the OLG framework. The presence of externalities in capital
accumulation raise the social returns on investment above the interest rate. Alternatively,
Woodford (1990) and Azariadis and Smith (1993) introduced …nancial frictions in an
OLG economy. Again, the …nancial frictions raise the returns on investment above theinterest rate. Interestingly, in an OLG economy with …nancial frictions, a bubble canraise economic activity by reallocating resources from unproductive to productive agents(Martin and Ventura 2012) or from unproductive to productive time periods (Farhi and
Tirole 2012). In all these papers, bubbles are due to the OLG structure of the economy,
not to the presence of …nancial frictions.
An alternative strand of the literature builds on Scheinkman and Weiss (1986), Woodford (1990), Kocherlakota (1992, 2008), Santos and Woodford (1997), and Hellwig andLorenzoni (2009) who have shown that …nancial frictions alone can generate rational
bubbles. Importantly, this can reconcile bubbles with a …nite number of in…nitely-lived
individuals.8 Miao and Wang (2014, 2017) and Hirano and Yanagawa (2017) generalized
these results to production economies. Interestingly, Miao and Wang (2014, 2017) showedthat stock price bubbles generate a "collateral yield", which allows bubbles to grow at a
slower rate than the interest rate. Finally, Aoki, Nakajima, and Nikolov (2014) showedthat bubbles can result from uninsurable idiosyncratic risk, rather than from credit constraints. In their setup, the market incompleteness depresses the interest rate below thegrowth rate of the economy.
In the whole literature, rational bubbles are fundamentally due to heterogeneity acrossfollows that the possibility of dynamic ine¢ ciency and rational bubbles is due to the existence of anin…nite number of households and commodities, not to a market incompleteness. (See also Acemoglu
2009, chapter 9, for a discussion of how this relates to the fundamental welfare theorems.)
7
Abel, Mankiw, Summers, and Zeckhauser (1989) established empirically that the U.S. economy is
dynamically e¢ cient. More recently, Geerolf (2017) found that many industrialized economies are now
likely to be dynamically ine¢ cient.
8
In a recent assessment of the literature, Miao (2016) argued that "developing in…nite-horizon models
of bubbles is important for us to further understand asset bubbles both qualitatively and quantitatively".

5

agents. In the OLG framework, their is heterogeneity across cohorts; while under …nancial
frictions, their is heterogeneity between creditors and debtors. In both strands of theliterature, bubbles exist to transfer resources across individuals. This is clearly not thecase within our representative household model, which is why the Ricardian equivalenceholds. We therefore provide a new microfoundation for rational bubbles. This provides
a straightforward way of generating bubbles within standard representative household
frameworks.9
Finally, there is a small literature on wealth preferences and bubbles. Ono (1994,
chapter 11) was the …rst to show that, if households are characterized by an insatiable
preference for wealth (which he interprets as a preference for liquidity), then an asset
price can have a bubble component that diverges to in…nity. However, he only considered
an endowment economy, which implies that the bubble could not interact with capital
accumulation.
Kamihigashi (2008) independently proved a similar result for an endowment economy.
He then considered a production economy. However, by not allowing for depreciation ofcapital, he constrained the interest rate to be positive, which ruled out steady state
bubbles. Thus, the only possibility was to have an explosive bubble. Interestingly, heshowed that a bubble reduces capital accumulation under decreasing returns to capital,
but it fosters capital accumulation under increasing returns.
Importantly, both Ono (1994) and Kamihigashi (2008) could only obtain explosive
bubbles under an insatiable preference for wealth, i.e. under the assumption that themarginal utility of wealth converges to a strictly positive value as wealth tends to in…nity.
By contrast, Zhou (2016) showed that a steady state bubble on a zero-dividend asset ispossible when the marginal utility of wealth converges to zero as wealth tends to in…nity.
He considered a production economy and found that a bubble could only arise if thecapital stock is above the golden rule. However, he only established his results for a
speci…c parametric case for which he could derive an explicit analytical solution. We
instead consider the general case and allow for di¤erent growth rates of dividends. The
only substantive restriction that we impose is that the household’s utility function isadditively separable between consumption and wealth.10
Our paper is organized as follows. In Section 2, we brie‡y motivate the preference for
9

There has recently been a rising interest in macroeconomics for the monetary policy consequences of
bubbles (Gali 2014, 2017, Asriyan, Fornaro, Martin, and Ventura 2016, Allen, Barlevy, and Gale 2017,
Dong, Miao, and Wang 2017, Ikeda 2017, Hanson and Pahn 2017, Biswas, Hanson, and Phan 2018).
In many of these papers, especially those relying on an OLG structure, the dimensions of heterogeneity
necessary to generate a rational bubble are largely orthogonal to the issue of monetary policy underinvestigation.
10
Relying on parametric examples within an endowment economy, Airaudo (2017) showed that, without
additive separability, the preference for wealth can generate extremely complex, and even chaotic, asset
price dynamics.

6

wealth. The structure of our neoclassical economy is presented in Section 3. We introduce
an in…nitely-lived asset in Section 4, which generates the possibility of bubbles. Section 5
considers government debt. We investigate a nominal economy with money in the utility
function in Section 6. The paper ends with a conclusion. All proofs are relegated to theappendix.

2

Wealth Preference

Beyond the re‡ection of classical economists mentioned at the beginning of the introduction, di¤erent strands of the modern literature in economics do provide support for the
preference for wealth.
Empirically, many wealthy households consume persistently less than their permanent
income. Carroll (2000) argued that the most natural explanation is that these households
derive utility from holding wealth.11 Dynan, Skinner, and Zeldes (2004) also found thathouseholds with higher lifetime income paradoxically have higher lifetime saving rates.
More recently, Straub (2018) estimated that the propensity to consume out of permanent
income is around 0.7 and, therefore, much below 1. Such …ndings can in fact be used
to calibrate the utility derived by households from their holdings of wealth (Kumhof,
Rancière, and Winant 2015).
Bequest motives are important determinants of the savings behavior of in…nitely-lived
households (or dynasties). But, we know that pure altruism alone cannot account for theobserved pattern of bequests (Carroll 2000, Kopczuk 2009). It has therefore becomecommon to assume that parents derive utility from the amount of bequeathed wealth,which is usually interpreted as a "joy of giving" or as a "capitalist spirit". Importantly,
our work shows that rational bubbles or Ponzi schemes are a theoretical possibility withinthese dynastic economies.
This brings us to quantitative models of wealth inequalities. Incorporating a bequest
motive, i.e. utility from bequeathed wealth, is typically necessary to account for thehigh concentration of wealth that we observe empirically (De Nardi 2004, Cagetti and
De Nardi 2006, Benhabib, Bisin and Luo 2017). Piketty (2011) also relies on a modelwith a preference for wealth to show that the inheritance-to-income ratio is a rising
function of r g, consistent with the historical experience of France from 1820 to 2009.
Also, Kumhof, Rancière, and Winant (2015) showed that the preference for wealth ofhigh income households can account for the increase in debt leverage of low and middle
income households in the run-up to the Great Recession. In all these papers, the role of
11

Note that, to obtain our results, we only need that a fraction of households have a preference for
wealth. Indeed, with heterogeneous agents, thrifty households lend to impatient households until thelatter are borrowing constrained. Hence, the interest rate and asset prices are eventually determined by
the behavior of thrifty households, i.e. by those who have a preference for wealth.

7the preference for wealth is to raise the propensity to save of wealthy households, in linewith empirical observations.
Finally, Bakshi and Chen (1996) were the …rst to investigate the asset pricing implications of the preference for wealth. Focusing on risky assets, they showed that thepreference for wealth raises the impact of asset returns on households’ utility, thereby
raising the perceived riskiness of stocks. They found that the resulting model does a better job at explaining asset prices than the standard neoclassical model. However, Bakshi
and Chen (1996) did not consider the possibility of bubbles.
The preference for wealth is not only a plausible assumption according to classicaleconomists, it is also a serious candidate explanation for a number of empirical regularities. It is therefore a natural generalization of the standard neoclassical framework.12

3

Economy without In…nitely-Lived Assets

In this section, we present our framework and derive the equilibrium of the economy in
the absence of in…nitely-lived assets (and therefore in the absence of bubbles).

3.1

Households

Time is continuous. There is a mass 1 of in…nitely lived households. Population withineach household grows at rate n. The total population of the economy is therefore given
by Nt = ent . At any time t, the representative household consumes a quantity Nt ct of asingle consumption good and inelastically supplies labor Nt at wage wt . Let At denotethe household wealth, which is composed of physical capital Kt and risk-free bonds Bt :At = Kt + Bt :(1)

Firms pay Rt to rent capital from households. Capital depreciates at rate . Bonds
yield a return rt . Finally, in addition to the output produced from capital and labor,
the economy receives an exogenous endowment of goods zNt at each point in time. It is
given to households in a lump-sum fashion. This endowment is not important for now,
but will play a key role in the next section when we introduce an in…nitely-lived asset.
12

There is also a literature on status preference, according to which welfare is increasing in relative
wealth (see, for instance, Zhou 2016, Ono and Yamada 2018). However, investigating how the di¤erent
speci…cations of the relative wealth preference a¤ect the possibility of rational bubbles is beyond thescope of this paper.

8Household wealth therefore evolves according to:
A_ t = [Rt] Kt + rt Bt + wt Nt + zNt= rt At + wt Nt + zNt

c t Nt ;ct Nt + [Rt(2)

r t ] Kt :By arbitrage, the returns on capital must be equal to the returns on bonds. It follows
that:
rt = Rt:(3)
Let at = At =Nt , kt = Kt =Nt , and bt = Bt =Nt denote wealth per capita, capital per capita,
and bonds per capita, respectively. We therefore have:(4)

at = k t + b t :Hence, a_ t = A_ t =Nt

nat , which implies that:a_ t = (rtn) at + wt + z

ct :

(5)

Finally, households are subject to an intertemporal budget constraint that prevents themfrom running Ponzi schemes:
Rt

0 (rs

lim e

t!1

n)ds

at

0:

(6)

The representative household discounts the future at rate , with > n. At any
point in time, it derives utility u (ct ) from consuming ct , with u0 ( ) > 0, u00 ( ) < 0, andlimc!0 u0 (c) = 1. The household also derives utility (at ) from holding wealth at , with
0( ) > 0, 00 ( ) < 0, 0 (0) < 1, and limk!1 0 (k) = 0. The household’s intertemporalutility function is therefore given by:
Z 1

e (

n)t

[u (ct ) + (at )] dt:(7)0

The household maximizes its intertemporal utility (7) subject to its budget constraint(5) and (6) with a0 given. By the maximum principle, the solution to the household’s
problem is characterized by:
c_t
= rt
ct

0

(at )
+ 0
u (ct )

u0 (ct );u00 (ct ) ct

(8)together with the transversality condition:
lim e (

t!1

n)t 0

u (ct ) at = 0:

9

(9)

3.2

Firms

Firms rent capital Kt from households and employ labor Nt to produce output Yt using
a constant returns to scale neoclassical production function:
Yt = F (Kt ; Nt ) :(10)

They choose their demand for capital Kt and for labor Nt such as to maximize theirpro…ts:F (Kt ; Nt ) Rt Kt wt Nt :(11)
In equilibrium, each factor of production must be paid its marginal product:
Rt = FK (Kt ; Nt ) ;(12)

wt = FN (Kt ; Nt ) :(13)

Let yt = Yt =Nt denote production per capita and de…ne f (k) = F (k; 1) for any k. We
therefore have yt = f (kt ), Rt = f 0 (kt ), and wt = f (kt ) kt f 0 (kt ). The real interest rate rt
is equal to the rental cost of capital Rt net of depreciation , which implies rt = f 0 (kt ) .

3.3

Market Clearing

The goods market clearing condition imposes that total output must be equal to the sumof consumption and investment:
Yt + zNt = ct Nt +As k_ t = K_ t =Nthi
Kt + K_ t :(14)

nkt and Yt = Nt yt = Nt f (kt ), we must have:k_ t = f (kt ) + z

ct( + n) kt :Also, bonds are in zero net supply. Hence, in equilibrium, bt = 0 and at = kt .

10

(15)

3.4

Equilibrium

The equilibrium of the economy (ct ; kt )1
t=0 is fully characterized by:0(kt )u0 (ct )c_t= f 0 (kt )+ 0;ctu (ct )
u00 (ct ) ct
k_ t = f (kt ) + z ct ( + n) kt ;

(16)
(17)

n)t 0

(18)

k0 given.

(19)

lim e (u (ct ) kt = 0;

t!1

Note that, under the capital market clearing condition at = kt , the household’s noPonzi condition (6) is trivially satis…ed. It has therefore been omitted from the list ofequilibrium conditions.
A steady state equilibrium (c; k) is characterized by:
0(k) = [ +f 0 (k)] u0 (c) ;(20)

c = f (k) + z( + n) k:(21)

Let us now introduce an assumption that we will use throughout our analysis.
Assumption 1 The equation 0 (k + ) = [ +f 0 (k)] u0 (f (k) + z…nes as a decreasing function of k for all admissible values of k.

( + n) k) de-This is implicitly an assumption on the functional forms for u ( ), ( ), and f ( ). It
implies that an exogenous increase in wealth reduces the household’s propensity tosave and therefore reduces the corresponding steady state capital stock k. Not only is
it the natural economic scenario to expect, it is also technically a very mild condition.
Totally di¤erentiating this equation with respect to k reveals that Assumption 1 must be
satis…ed for any given function forms for u ( ), ( ), and f ( ) provided that is su¢ ciently
close to n.
Lemma 1 A steady state equilibrium always exists and, under Assumption 1, it must be
unique.
We shall now consider throughout our analysis that Assumption 1 is satis…ed (but we
explicitly refer to it in the proofs whenever it is being used).
Let r denote the steady state natural interest rate, de…ned as the steady state interest
rate of the economy with no in…nitely-lived asset. It is simply given by:
r = f 0 (k)
0

=

(k)

u0 (c)
11:(22)

This implies that r 2 ( ; ).
By (21), the golden rule capital stock k , which maximizes steady state consumption,is characterized by:
f 0 (k ) = + n:(23)
Thus, the steady state capital stock k is above the golden rule level k whenever:r = f 0 (k)< f 0 (k )

= n:

(24)

A su¢ ciently strong preference for wealth ( ) can easily result in a steady state equilibrium with r < n. In that case, households choose to accumulate more capital than under
the golden rule since, in the absence of alternative stores of value, they have no otherway to satisfy their preference for wealth. Note that, without a preference for wealth, wewould have r = > n. Hence, it is the preference for wealth that can induce the capitalstock to exceed the golden rule level.

4

In…nitely-Lived AssetTo allow for the possibility of rational bubbles, let us now introduce an in…nitely-lived
asset into this economy. At any time t, a fraction e t of the exogenous endowment zNt
accrues to the owners of the single in…nitely-lived asset of the economy, where0.
t
The remaining fraction, 1 e , is given to the representative household in a lump-sum
fashion, as in the previous section.
Thus, relative to the size of the economy, the dividends shrink at rate . This could, forinstance, result from gradually depreciating technologies or patents. Three benchmarks
are of particular interest. If = 0, then the dividends grow at the same speed as the
economy. If = n, then the dividends are constant in real terms, as in Tirole (1985).
Finally, in the limit as tends to in…nity, the asset does not yield any dividends and istherefore intrinsically worthless.13
Let Qt denote the price of the asset. The returns on the ownership of the asset consistof the dividends zNt e t = ze(n )t and of the capital gains Q_ t . By arbitrage, these
returns must be equal to the returns on risk-free bonds:
ze(n

)t

+ Q_ t = rt Qt :Let qt = Qt =Nt denote the price per capita of the asset. As q_t = Q_ t =Nt
13(25)
nqt , we must

If …rms are in…nitely lived, then equity can be an in…nitely-lived asset. However, under a constant returns to scale production function, the dividends are equal to zero. Equity would therefore be
intrinsically worthless. The equilibrium with equity would be identical to the equilibrium with = 1.

12have:
q_t = (rt

n) qt

ze

t

(26)

:

Household wealth at consists of physical capital kt , risk-free bonds bt , and of the asset
with value qt . Thus, in equilibrium:(27)

at = k t + b t + q t :The net supply of bonds is equal to zero. So, in equilibrium, bt = 0. As the asset can be
freely disposed of, we must always have qt 0. Hence, with at = kt + qt , the household’s
no-Ponzi condition (6) is trivially satis…ed. The equilibrium of the economy (ct ; kt ; qt )1
t=0
is therefore fully characterized by:0(kt + qt )u0 (ct )c_t= f 0 (kt )+;ct
u0 (ct )u00 (ct ) ct
k_ t = f (kt ) + z ct ( + n) kt ;q_t = (f 0 (kt )

t

(29)

;

(30)

u (ct ) [kt + qt ] = 0;(31)

k0 given.

(32)

n)qt

ze

n)t 0

lim e (

t!1

(28)

Before we characterize the equilibria of the economy, it is useful to decompose the asset
price qt into a fundamental component t and a bubble component t . The fundamental
component is the solution to:_ t = (f 0 (kt )

t

n) t

ze

;

n)du

T = 0:0t (f (ku )n+ )du(33)

subject to the no-bubble boundary condition:
lim e

T !1

RT
t

(f 0 (ku )(34)This immediately implies that:t = ze

t

Z 1
t

e

Rs

ds:

(35)

The fundamental price of the asset is equal to the present value of the dividend stream
generated by the asset. The bubble component is the solution to:_ t = (f 0 (kt )n) t :(36)

By arbitrage, the bubble component must grow at the rate of interest net of population
13

growth. It must therefore be equal to:
Rt

0

0 (f (ku )
t = e

n)du

(37)

0:

Thus, any bubble must be indexed by 0 . If 0 = 0, there is no bubble; if 0 > 0, there isa positive bubble; and if 0 < 0 there is a negative bubble. Finally, the price of the asset
is simply given by:
qt = t + t :
(38)

4.1

Steady StateA steady state equilibrium is characterized by ct = c and kt = k for all t. We must also
have rt = r for all t, with r given by:
r = f 0 (k)

(39)

:

Before characterizing the steady state equilibria of the economy, we establish a numberof properties of the asset price in steady state.
Is it possible to have a steady state with an ever growing asset price? The answer isno.
Lemma 2 There cannot be a steady state equilibrium with an ever growing asset priceqt .
As the asset price becomes in…nitely large, the marginal utility of wealth converges to zeroand, by the consumption Euler equation (28), the interest rate converges to . Hence,by the asset pricing equation (30), the asset price must asymptotically grow at rate
n. But, under such rapidly growing household wealth, the transversality condition(31) imposes an in…nitely high demand for consumption. This is ruled out by the resource
constraint (29). Hence, in steady state, we must have qt = q for all t.
Before characterizing the steady state equilibria of the economy, we establish an important property of the economy with in…nitely-lived assets.
Lemma 3 The steady state interest rate r cannot be smaller or equal to n.

The fundamental price of the asset is not well de…ned when the growth rate of rents
is larger than the interest rate net of population growth r n. We must therefore haver >n. This implies that the existence of capitalizable rents ze t , no matter how
small, constrains the steady state interest rate of the economy to be above n.
In the presence of a bubble, the asset price can either be above or below its fundamental value. However, the following lemma rules out negative bubbles.
14

Lemma 4 Under the steady state interest rate r > n
bubble.

, there cannot be a negative

In steady state, the bubble component grows at rate r n, while the fundamental component grows at rate
. Hence, the bubble component grows at a higher rate than thefundamental component. It follows that a negative bubble would grow so fast that theasset price would eventually become negative. But, this is not possible as the asset canbe freely disposed of. Note that, if there cannot be a negative bubble in steady state,then there also cannot be a negative bubble on any trajectory converging to a steadystate.14
Let us now characterize the steady state equilibria (c; k; q) of the economy. From (28)and (29), we must have:0(k + q) = [ +c = f (k) + z

f 0 (k)] u0 (c) ;(40)( + n) k:(41)

Regarding the third equation characterizing the steady state, (30), we must distinguish
two cases: = 0 and > 0.
When = 0, the dividends grow at the same speed as the economy. In that case, by
(30):
z
:
(42)
q= 0
f (k)
n
Thus, when = 0, the price of the asset must be equal to its fundamental value. In
equilibrium, we must have q 2 [0; 1). Hence r = f 0 (k)> n, which implies thatthe steady state capital stock must be smaller than under the golden rule. The following
proposition refers to r, which was de…ned in the previous section as the steady stateinterest rate of the economy without in…nitely-lived assets.
Lemma 5 When = 0, there always exists a unique steady state equilibrium, which ischaracterized by (40), (41), and (42). It must be bubble-less and it must satisfy r >max fr; ng.
If the rent z could not be capitalized, then we would be back to the previous section witha steady state interest rate equal to r. The fact that, here, the rent z can be capitalized
raises household wealth by q, which increases the demand for consumption and reduces
the accumulation of capital. This raises the interest rate r above r.
Why is it impossible to have a bubbly steady state when = 0? Recall that bubbles
always grow at rate r n. But, we have just seen that the fundamental value of the asset
14

From the expression for the bubble component (37), if the bubble is negative at some point in time,then it must always be negative.

15

is only well de…ned if r = f 0 (k)> n (which is also implied by Lemma 3). Thus, abubble would grow forever, which by Lemma 2 is not possible.
Let us now turn to the case where > 0. From (30), in steady state, we must eitherhave:
f 0 (k)
= n;
(43)
or:
q = 0;(44)

The …rst case corresponds to a bubbly equilibrium, which results in a constant asset priceper capita; while the second case corresponds to the asset being worth its fundamental
value, which asymptotically converges to zero whenever > 0.
When f 0 (k)= n, a bubble induces the interest rate to be equal to the rate of population growth and, hence, the capital stock to be equal to the golden rule level. However,by Lemma 4, a bubble must always raise the price of the asset above its fundamental
value. This must reduce the household’s propensity to save and therefore decrease thecapital stock. Hence, the bubbly equilibrium can only occur if the steady state capitalstock in the economy without the in…nitely-lived asset is larger than under the golden
rule, i.e. if r < n.
Lemma 6 When > 0, there (asymptotically) exists a unique bubbly steady state equilibrium if and only if r < n. It is characterized by (40), (41), and (43) and it satis…esr = n.
The steady state equilibrium only exists asymptotically as ze t is only equal to zero in
the limit as t tends to in…nity.
Note that, in the bubbly steady state, the asset price qt must be constant over time.
This requires r = n and a capital stock equal to the golden rule. The steady state
asset price q is therefore entirely determined by the preference for wealth through theconsumption Euler equation (40).
In the alternative case, where q = 0, the equilibrium is unchanged from the previous
section without any in…nitely-lived asset. However, as shown by Lemma 3, the very
existence of a single in…nitely-lived asset prevents the interest rate from being smaller or
equal to n
. Hence, this equilibrium is only meaningful when r > n. Indeed, while
is irrelevant in the limit as t tends to in…nity, we must have r > n
for any …nite
value of t.
Lemma 7 When > 0, there (asymptotically) exists a unique bubble-less steady state
equilibrium, which is characterized by (40), (41), and (44). It satis…es r = r. Thisasymptotic steady state is economically meaningful if and only if r > n.
16

The existence and uniqueness of this equilibrium trivially follows from Lemma 1 of theprevious section.
When the two steady states simultaneously exist, they can be Pareto ranked.
Lemma 8 When > 0 and r 2 (n; n), the bubbly steady state equilibrium is Pareto
superior to the bubble-less steady state.
The bubble allows households to have a store of value, such as to satisfy their preference for wealth, without over-accumulating capital. Here, as in a dynamically ine¢ cient
overlapping generation economy, steady state bubbles are perfectly e¢ cient.15
Let us now investigate the stability properties of these steady state equilibria.

4.2

Stability

To determine the local stability properties of the steady state equilibria, we linearize thesystem of di¤erential equations (28), (29), and (30) around these steady states. Let us…rst investigate the stability of the steady state equilibrium when = 0.
Lemma 9 When = 0, the unique steady state equilibrium is locally saddle-path stable.
For a given k0 , there is a unique trajectory converging to the steady state. This trajectoryis bubble-less.
As there cannot be a bubble along the trajectory converging to the steady state, the asset
must always be worth its fundamental value.
When > 0, the system of di¤erential equations is not homogenous and not autonomous, due to the ze t term in (30). However, as this term tends to zero over time,it does not fundamentally change the stability analysis.
Lemma 10 When
> 0, the ze t term in (30) does not modify the local stability
analysis of the linearized system of di¤erential equations.
Thus, when > 0, the local stability of a steady state equilibrium is determined by the
eigenvalues of the transition matrix of the linearized system of di¤erential equations, as
would be the case in the absence of the ze t term.
When > 0, one steady state equilibrium is bubbly, the other is not. The bubbly
steady state exists whenever r < n.
15

If wealth at is constrained to be equal to the physical capital stock kt , as in Section 3, then the …rst
fundamental theorem applies and the equilibrium allocation is Pareto e¢ cient. However, if wealth canalso consist of …nancial wealth, as in this section, then welfare is directly a¤ected by asset prices and the…rst fundamental theorem does not hold. This is another fundamental di¤erence with the OLG economy,
where the failure of the …rst fundamental theorem is due to an in…nity problem (see footnote 6).

17

Lemma 11 When > 0, the bubbly steady state equilibrium is locally saddle-path stable.
For a given k0 , there is a unique trajectory converging to the steady state. This trajectoryis bubbly.
Let us now turn to the stability of the bubble-less steady state, which exists wheneverr>n.
Lemma 12 Whenstable.

> 0, the bubble-less steady state equilibrium is locally saddle-path

When r > n, for a given k0 , there is a unique trajectory converging to the steadystate. This trajectory is bubble-less.
When r < n, for a given k0 , there is a continuum of values of q0 that are consistent
with convergence to the steady state.
Recall that the bubble component grows at rate f 0 (kt )n, which is approximately
equal to r n in the neighborhood of the steady state. Hence, when r > n, any bubble
must grow as it approaches the steady state. But the steady state is bubble-less. Hence,
when r > n, there cannot be a bubble along the trajectory converging to the steadystate. Conversely, when r < n, a bubble must shrink as it approaches the steady state.
In that case, there is a continuum of bubbly trajectories, with the magnitude of bubbles
becoming vanishingly small as the economy converges to the bubble-less steady state.
Putting all these results together yields the following proposition.
Proposition 1 Focusing on equilibrium paths that converge to a steady state, we havethe following possibilities for any given value of k0 :
When

= 0:

There exists is a unique equilibrium. It is bubble-less and the interest rate converges
to r, which satis…es r > max fr; ng.
When

> 0:

If r 2 (n; 1), there exists a unique equilibrium. It is bubble-less and the interest
rate converges to r.
If r 2 (n; n), there exists two steady state equilibria: one bubbly with r = n;
the other bubble-less with r = r.
– There exists a unique value of q0 such that the economy converges to the bubbly
steady state.
18

– There is a continuum of values of q0 such that the economy converges to the
bubble-less steady state. These equilibria are asymptotically bubble-less.
If r 2 ( 1; n], there exists a unique equilibrium. It is bubbly and the interest
rate converges to n.
Thus, bubbles can only occur when r < n and > 0. Tirole (1985) derived a similar
result within an OLG economy for the special case where = n. As discussed in the
introduction, despite this similarity, the underlying mechanism generating the possibilityof bubbles is fundamentally di¤erent under the two approaches. In the OLG economy,
the bubble is a coordination device that transfers resources across cohorts, while in our
economy the bubble is a store of value that allows the representative household to satisfy
its preference for wealth.
When > 0 and r 2 (n; n), there is a continuum of equilibrium values of q0 .
While we cannot characterize them analytically, Figure 1 shows that these values satisfy
q0 2 [ 0 ; q^0 ] where 0 is the fundamental value of the asset. When q0 = q^0 , i.e. thehighest feasible asset value, then the economy converges to the bubbly steady steadystate (dashed line). For all q0 2 [ 0 ; q^0 ), the economy converges to the bubble-less steadystate. In fact, when q0 = 0 , the asset is worth its fundamental value throughout (solideline); while, when q0 2 ( 0 ; q^0 ), the bubbly component is strictly positive but becomes
arbitrarily small as the economy converges to the bubble-less steady state (dashed-dottedlines). This pattern has been veri…ed for all our numerical simulations.
Proposition 1 relies on local stability results, which is why we have only characterized
equilibria that converge to a steady state. In the special cases where = +1, we
can use a three dimensional phase diagram to rule out alternative equilibrium paths(since the q_t = 0 hyperplane is now time invariant). Figure 2 shows the main candidate
trajectories. The solid lines correspond to equilibrium trajectories, while the dashed lines
are trajectories that violate some equilibrium condition. The steady state equilibria arelocated at the intersection of the c_t = 0, k_ t = 0, and q_t = 0 hyperplanes. Note that,when = +1, we have q_t = (f 0 (kt )n) qt . Thus, the q_t = 0 hyperplane has two
0
components: f (kt )= n and qt = 0. The former corresponds to the bubbly steadystate, the latter to the bubble-less steady state. The trajectories that do not converge
to a steady state equilibrium must either end up with: (i) kt = 0 in …nite time, whichmust violate the consumption Euler equation (28); or (ii) kt = k where k is given by
f k + z = ( + n) k, which violates the transversality condition (as would be the casein the absence of in…nitely-lived assets).16
16

Note that Figure 2 is a three-dimensional equivalent to Figure 1 of Tirole (1985). Tirole does notneed the consumption dimension since, in the Diamond model, aggregate consumption is a function of
the capital stock and of the asset value.

19

Figure 1: Equilibrium dynamics when

> 0 and r 2 (n; n).

A similar exercise could be performed when = 0. However, when 2 (0; +1), thesystem of di¤erential equation is not homogeneous and not autonomous, which prevents
the construction of a simple three dimensional phase diagram. However, numerical simulations show that alternative equilibrium paths would also be ruled out in that case.
Thus, for any 2 [0; +1), all the equilibrium paths converge to one of the steady statesof Proposition 1, which therefore are globally saddle-path stable.
So far, we have assumed a decreasing marginal utility of wealth. Note that, in thespecial case of a constant marginal utility of wealth, the dynamics of consumption andcapital, given by (28) and (29), are independent of the asset price qt . Thus, the economy
converges to the bubble-less steady state, where r = r. However, when r n, thiscannot be an equilibrium (by Lemma 3). Hence, in that case, there is no equilibrium.17
17
In this economy, under a downward price or wage rigidity, there could exist a secular stagnation
equilibrium (Ono 2015, Michau 2018). If the frictionless neoclassical equilibrium does not exist, thenthe secular stagnation equilibrium can be the unique equilibrium of the economy. Interestingly, here
the secular stagnation equilibrium could be made unique by the requirement that r > n, while in
Michau (2018) it could be made unique by an in‡ation ceiling that also imposes a lower bound on thereal interest rate. See Schlegl (2018) for a careful analysis of secular stagnation in the = 0 case.

20

Figure 2: Phase diagram when= +1 and r < n.

Conversely, when r > n, any positive bubble can be an equilibrium.18 When r > n,these bubbles must be growing forever. This is sustainable as the insatiability of thepreference for wealth implies that, in equilibrium, there is a zero marginal propensity toconsume out of wealth.

5

Government Debt

Let us now return to the situation of Section 3, where there is no in…nitely-lived real
asset. Thus, the exogenous rents zNt can no longer be capitalized. However, we now
introduce government debt.
The government makes an initial transfer of wealth b0 > 0 to each individual. It isthen subject to the standard debt accumulation equation:b_ t = (rt

n) bt

18

t;

(45)

With a constant marginal utility of wealth, rt must eventually converge to r. Thus, the fundamental
value of the asset must either converge to a positive constant (when = 0) or to zero (when > 0),
while the growth rate of the bubble component must converge to r n. But r < , which implies thatthe transversality condition (31) is always satis…ed; even when qt tends to in…nity.

21

where t denotes a lump-sum tax imposed on households at time t. Let 't denote thepresent value of taxes from time t onwards:
't =

Z 1

e

t

Rs

n)du

t (ru

(46)

s ds:

Throughout our analysis, we treat the …scal resources t , and therefore 't , as exogenously
given. It is common to impose a no-Ponzi condition on government debt:
lim e

t!1

Rt

0 (rs

n)ds

bt

(47)

0:

It simply requires that, at any time t, the present value of lump-sum taxes is larger or
equal to the current debt level. It can therefore equivalently be written as:
bt

(48)

't :

However, we shall show that this condition is excessively restrictive. Indeed, the government’s ability to borrow is only constrained by households’willingness to lend to it. Thisis determined by their transversality condition. To prove this result, we shall characterize
the equilibrium of the economy without imposing the government’s no-Ponzi conditionand show that, in some cases, Ponzi schemes can be sustainable.
Note that, without a preference for wealth, the economy would converge to a steadystate where rt = . In that case, the household’s transversality condition entails thatthe household’s no-Ponzi condition is binding. By Walras’ law, this implies that the
government’s no-Ponzi condition is also binding. Without a preference for wealth, thereis therefore no loss of generality in imposing the government’s no-Ponzi condition.
Households consider that their wealth is equal to their asset holdings at net of thepresent value of taxes 't . Indeed, if this was not the case, the government could arti…cially increase households’welfare by making a huge lump-sum payment that would
subsequently be o¤set by a huge lump-sum tax. We therefore maintain that, from awelfare perspective, e¤ective household wealth is equal to at 't . The utility of therepresentative households is therefore given by:
Z 1

e (

n)t

[u (ct ) + (at't )] dt;

(49)

ct ;

(50)

0

and its budget constraint by:
a_ t = (rtn) at + wt + z

together with the no-Ponzi condition limt!1 e

22

Rt

0 (rs

t
n)ds

at

0.

Throughout our analysis, we assume that the government’s no-Ponzi condition iseither binding or violated, i.e. bt 't . In such cases, under the capital market clearing
condition at = kt + bt , the household’s no-Ponzi condition (6) must be satis…ed.19 We cantherefore omit the household’s no-Ponzi condition from the set of equilibrium conditions.
The equilibrium of the economy (ct ; kt ; bt )1
t=0 is characterized by:0(kt + bt 't )u0 (ct )c_t= f 0 (kt )+;ct
u0 (ct )u00 (ct ) ct
k_ t = f (kt ) + z ct ( + n) kt ;b_ t = (f 0 (kt )lim e (

t!1

(51)
(52)

t;

(53)

u (ct ) [kt + bt ] = 0;(54)

k0 given.

(55)

n) bt

n)t 0

If the government chooses to satisfy its no-Ponzi condition, then bt = 't . In that case,
the Ricardian equivalence holds and …scal policy cannot a¤ect the equilibrium of theeconomy. The equilibrium paths of ct and kt are the same as in Section 3.
Can the government choose to violate its no-Ponzi condition? To answer this question,
we need to determine the maximum sustainable level of government debt. Let ~bt = bt 't
denote the magnitude of the Ponzi scheme at time t. The following lemma allows us tosimplify our analysis.
Lemma 13 Any allocation (ct ; kt )1
t=0 that can be implemented under the …scal policy
1~'t .
( t ; bt )t=0 can also be implemented under (0; ~bt )1
t=0 , where bt = bt
Thus, without loss of generality, we could set t = 't = 0 and replace bt by ~bt . The
resulting equations are identical to the ones that we had in the previous section in thespecial case where = 1. Government debt now plays the same role as the asset price inSection 4. Why is government debt mathematically identical to an intrinsically worthless
in…nitely-lived asset? Government bonds are …nitely lived, but they can be rolled overforever. Moreover, the (per capita) returns on such bonds are equal to rt n, which isthe same as the returns to the bubble component of an in…nitely-lived asset.
However, a crucial di¤erence with the in…nitely-lived asset is the choice of initialdebt. Indeed, in Section 4 the initial price q0 of an asset is determined by market forces,
while here the initial level of government debt b0 is, at least partly, determined by thegovernment.
Rt

The de…nition of 't , given by (46), implies thatRlimt!1 e 0 (rs n)ds 't =R 0 (this is formally showntt
below in the proof of Lemma 13). Thus, limt!1 e 0 (rs n)ds at = limt!1 e 0 (rs n)ds [kt + (bt 't ) +'t ] 0, since kt 0 and bt 't .
19

23

Let b0 denote the initial transfer of wealth to households that the government intends
to make. We assume that b0'0 . This transfer is …nanced by issuing a quantity
b0 of government bonds. Importantly, only a level '0 of debt can be backed by …scal
resources. Let b0 denote the amount of resources that households are willing to lend to
the government, i.e. the quantity of bonds they are willing to buy. Note that, we cannot
have b0 < '0 , since the government never fails to raise debt backed by …scal resources.
We must therefore always have b0 2 '0 ; b0 .
Let us focus on the interesting case where b0 > '0 , i.e. the government would liketo run a Ponzi scheme. This naturally results in multiple equilibria. If b0 < b0 , thegovernment fails to sell the desired quantity of bonds. In that case, its initial transfer ofwealth to households and its initial debt must both be equal to b0 , instead of b0 . Indeed,
not a single household is willing to lend beyond b0 if it does not expect others to do so.
In fact, there is a continuum of equilibria indexed by b0 . If b0 = '0 , then the equilibrium does not feature a Ponzi scheme; while, if b0 > '0 , then it does. While themultiplicity of equilibria is similar to the case of the in…nitely-lived asset, a key di¤erenceis that the government can reduce the range of equilibria through its choice of b0 . However, the government cannot eliminate the equilibrium with no Ponzi scheme, i.e. theequilibrium with b0 = '0 .
From Lemma 13, we can readily apply Proposition 1 with = +1 to characterize
the equilibria of the economy. We just need to consider that b0 is su¢ ciently high suchas not to rule out any equilibrium.
Proposition 2 Focusing on equilibrium paths that converge to a steady state, we havethe following possibilities for any given value of k0 :If r > n, there exists a unique equilibrium. The no-Ponzi condition must be binding,i.e. b0 = '0 , and the interest rate converges to r.
If r < n, there exists two steady state equilibria: one with a Ponzi scheme wherer = n; the other without a Ponzi scheme where r = r.
– There exists a unique value of b0 > '0 such that the economy converges to thesteady state with a Ponzi scheme.
– There is a continuum of values of b0 such that the economy converges to thesteady state without a Ponzi scheme. While the no-Ponzi condition is satis…edin steady state, the government is running a Ponzi scheme whenever b0 > '0 .
There cannot be a Ponzi scheme when r > n as bt would tend to in…nity, which would beinconsistent with the equilibrium conditions (as shown by Lemma 2). Conversely, when24the steady state capital stock without a Ponzi scheme is higher than under the golden
rule, debt can be rolled over forever and a Ponzi scheme can therefore be sustainable.
When r < n, there is a continuum of equilibrium values of b0 . They satisfy b0 2 ['0 ; ^b0 ]^b0 . When b0 = ^b0 ,for some constant ^b0 > '0 , where we implicitly assume that b0
the economy converges to the steady state with a Ponzi scheme. When b0 2 ['0 ; ^b0 ), theeconomy converges to the steady state where the no-Ponzi condition is binding. However,when b0 2 ('0 ; ^b0 ), the no-Ponzi condition is violated, even though bt converges to 't .
These possibilities could be seen on a three dimensional phase diagram.
With a constant marginal utility of wealth, there is no limit to government indebtedness. Indeed, in that case, the level of government debt does not a¤ect the behavior ofhouseholds. Hence, the initial transfer of wealth b0 from the government to households
can be arbitrarily large without altering the real allocation of resources in the economy.
The Ricardian equivalence states that, for a given present value of government expenditures (which are here normalized to zero), the timing of tax collection is neutral.
Lemma 13 implies that, for a given initial transfer b0 to households, two di¤erent tax
policies ( t )1
t=0 that result in the same present value of taxes '0 implement the same
allocation of resources. Hence, for a given value of ~b0 and therefore a given path ofR~bt = ~b0 e 0t (rs n)ds , i.e. along a given equilibrium trajectory, the timing of tax collection
does not matter. This immediately entails the following corollary.
Corollary 1 Along any equilibrium trajectory, the Ricardian equivalence holds.
Along any equilibrium trajectory, households expect a certain time path for the level ofdebt net of the present value of taxes and the government cannot be expected to deviate
from it.
However, when r < n, there exists a continuum of equilibrium trajectories. Can thegovernment induce the economy to jump to a di¤erent equilibrium trajectory at time t > 0
through a surprise change in the present value of taxes 't ? The government can always
raise 't , i.e. decrease the magnitude of its Ponzi scheme. However, the government can
only decrease 't , i.e. increase the magnitude of its Ponzi scheme, if households expect
the new policy to be sustainable over time. Once again, there are multiple equilibria
due to the coordination of expectations across households. Of course, from an ex-ante
perspective, once the economy is on a given equilibrium trajectory, a surprise change in't must be a zero probability event.20
20

We are assuming here that the government is fully committed to the new …scal policy, provided
that it is feasible. This policy is therefore perceived as credible by households. Alternatively, if thegovernment cannot credibly commit to a …scal policy, then the equilibrium trajectory is determined byhouseholds’beliefs about future …scal policy. Note that these beliefs can be a¤ected by the government’s…scal policy announcements. Investigating the strategic interactions between government announcements
and households’beliefs is beyond the scope of our analysis.

25

The fact that our economy is consistent with the Ricardian equivalence is a fundamental di¤erence with other models of rational bubbles (or Ponzi schemes). As we haveseen in the introduction, in models based on the OLG structure or on …nancial frictions,bubbles exist to redistribute resources across people. Hence, the very existence of bubbles builds on the non-Ricardian nature of these models. Interestingly, this also implies
that, in these models, the government can implement the bubbly allocation of resources
through a lump-sum redistribution policy that mimics the bubble. For instance, in a
dynamically ine¢ cient OLG economy, the government can always implement the bubbly
allocation by establishing a Pay As You GO (PAYGO) social security system. By contrast, in our economy, the government cannot implement the bubbly allocation throughlump-sum transfers.21
If the government can roll over debt, why can’t some households do the same? Intheory, that’s possible. However, in…nitely-lived households are in fact dynasties. And,in practice, a fundamental constraint on the allocation of resources is that heirs cannot
be forced to inherit debt. Thus, in…nitely-lived dynasties cannot roll over debt forever.
Only the government can do so! This justi…es our assumption that households cannot
violate their no-Ponzi condition.

6

Money

Let us now investigate the properties of a classical monetary economy with a preference
for wealth. To generate a demand for money, we follow the money-in-the-utility-function
approach à la Sidrauski (1967). This allows us to show that, despite apparent similarities,
the preference for wealth is fundamentally di¤erent from money in the utility function.
We introduce a new …nancial asset, money, within the framework of the previous
section. Money is a nominally risk-free asset that yields a zero nominal return. Let M0denote the initial supply of money in the economy. The money supply increases over timeat rate ! t . The nominal quantity of money at any time t is therefore given by:
Rt

Mt = e 0 !s ds M0 :Let Pt denote the price level at time t and(56)

t the in‡ation rate. Thus, the real supply of

21

The government can rely on distortionary taxes to implement the capital stock and the consumption
level of the bubbly steady state. However, the government cannot force the implementation of a Ponzi
scheme and, without the corresponding increase in wealth, this policy would clearly reduce e¢ ciency.

26

money per capita mt is equal to:
mt =

Mt
;
Pt Nt
Rt

= e 0 (!ssn)ds M0

P0

(57)

:

Thus, the evolution of the money supply over time satis…es:m_ t = (! t

t

(58)

n) mt :

Newly supplied money is revenue to the government, whose real debt level therefore
evolves according to:b_ t = (rt n) bt! t mt :(59)
tCombining the previous two equations yields:b_ t + m_ t = (rtn) [bt + mt ]t

it m t ;(60)

where it denote the nominal interest rate, which is related to the real interest rate rt
through the Fisher identity it = rt + t .
By issuing money rather than bonds, the government economizes the nominal interest
rate, which is the di¤erence between the real returns rt on bonds andt on money.
Thus, from a …scal point of view, the nominal interest rate is a tax on households’money
holdings. So, the present value of taxes 't from t onwards is now equal to:
't =

Z 1

Rs

t (ru

e

t

n)du

( s + is ms ) ds:(61)

If the government does not run a Ponzi scheme, then it must be able to redeem both its
supply of bonds and of money. Thus, the government’s no-Ponzi condition is given by:
lim e

t!1

Rt

0 (rs

n)ds

(bt + mt )

0;

(62)

or, equivalently, by:
bt + m t

't :

(63)

This implies that the magnitude of a Ponzi scheme is now equal to:~bt = bt + mt

't :

(64)

By the government liability accumulation equation (60) and the de…nition of the present27value of taxes (61), we have::~bt = (rtn) ~bt ;(65)

regardless of monetary and …scal policy. As in the previous section, we focus on cases
where the government’s no Ponzi condition is either binding or violated, i.e. ~bt 0.
Real household wealth per capita is now given by:(66)

at = k t + b t + m t ;where mt denotes the real money balances per capita. We have:a_ t = (rtn) at

it mt + wt + z

t

ct ;

(67)

where it mt is the real opportunity cost of holding money rather than bonds or capital.
We now introduce money in the utility function by assuming that the representative
household derives utility h (mt ) from holding real money balances mt , with h0 ( ) > 0,h00 ( ) < 0, limm!0 h0 (m) = 1, and h0 (m) = 0 for all m m. At m, the household is satiated with real money balances and does not derive any utility from holding more money
for transaction purposes. The household’s intertemporal utility function is therefore given
by:
Z
1

e (

n)t

[u (ct ) + h (mt ) + (at't )] dt:

(68)

0

The utility from real money balances h (mt ) corresponds to a preference for liquidity,
which is conceptually di¤erent from the preference for wealth (at 't ).
The household’s wealth net of the present value of taxes is equal to at 't = kt + bt +mt 't = kt + ~bt . Thus, ~bt measures the extent to which government assets bt + mt net ofthe present value of taxes 't represent a source of wealth for the representative household.
If the government’s no-Ponzi condition holds, then ~bt = 0 and the household’s net wealthis simply equal to kt . If the government runs a Ponzi scheme, then the household’s net
wealth rises to kt + ~bt .22
The solution to the household’s problem is characterized by the same equations as
before, i.e. the consumption Euler equation (8) and the transversality (9) condition,
together with a money demand equation:h0 (mt ) = it u0 (ct ) :
22

(69)

Note that the household’s present value of taxes 't is a function of its money holdings mt . However,
we assume that the representative household takes its net wealth from government assets ~bt as given.
In other words, the household determines its money demand independently of its e¤ect on ~bt . While
we can solve the alternative case, where the household reduces its future money holdings such as to feel
wealthier today, it seems behaviorally implausible to us.

28Note that this money demand equation trivially implies that the nominal interest rate it
cannot be negative, which is the zero lower bound.
The household’s transversality condition can be written as limt!1 e ( n)t u0 (ct ) [kt +~bt + ' ] = 0. If ' is …nite, which must be the case in equilibrium, we must have
t
t
Rt
(r
n)du
u
't = 0 and therefore limt!1 e ( n)t u0 (ct ) 't = 0.23
limt!1 e 0
The real equilibrium of the economy is characterized by:"c_t= f 0 (kt )ct+

k_ t = f (kt ) + z

0(kt + ~bt )u0 (ct )#u0 (ct );u00 (ct ) ct

(70)

( + n) kt ;(71)n) ~bt ;n)t 0
u (ct ) [kt + ~bt ] = 0;

(72)
(73)

k0 given.

(74)

ct

:

~bt = (f 0 (kt )lim e (

t!1

It immediately follows that the real equilibria of the economy, and hence the possibilityof Ponzi schemes, are still characterized by Proposition 2 from the previous section. The
presence of money does not a¤ect the equilibrium structure of the economy, regardless of
the actual choice of monetary and …scal policy.
The government can only a¤ect the equilibrium of the economy though its choice ofthe maximum magnitude of the Ponzi scheme, which we have denoted b0 in the previous
section. In equilibrium, the government is assumed to be committed to a Ponzi scheme
Rtof a given magnitude ~bt = ~b0 e 0 (rs n)ds , with b0 2 0; b0 , that households believe tobe sustainable over time.24 It is therefore natural to consider that lump-sum taxes t
are set such that, for any path of real money holdings mt , the magnitude of the Ponzi
scheme remains unchanged.25 This implies that money is completely neutral. The pathof real money holdings mt is jointly determined by the money supply equation m
_t =
(! t
n) mt and by the money demand equation h0 (mt ) = [f 0 (kt )+ t ]u0 (ct ).
t
Substituting the latter into the former yields:
m
_t=

!t

n + f 0 (kt )h0 (mt )u0 (ct )

mt ;

(75)

Any path of real money holdings is feasible, provided that mt remains positive. This
includes speculative de‡ation, where mt tends to in…nity, which is not ruled out by the23

Formally, this is shown in the proof of Lemma 13.
If b0 < b0 , then the Ponzi scheme of magnitude b0 was not necessarily perceived as sustainable byhouseholds.
1
25
Alternatively, we could consider that, given the monetary and …scal policy ( t ; ! t )t=0 , the Ponzi
scheme can only be of the equilibrium magnitude for a unique feasible path of mt . In that case, all otherfeasible paths of mt are ruled out by this equilibrium.
24

29

household’s transversality condition since money is e¤ectively redeemed by …scal policy.
Alternatively, we can consider that the government sets the lump-sum tax t suchthat bt is equal to zero at all times. This policy implies b0 = 0 and t = ! t mt , i.e. therevenue from money creation is redistributed lump-sum to households. The governmente¤ectively implements helicopter drops of money, which are negative whenever ! t < 0.
Money is therefore the only asset issued by the government, which corresponds to an
important benchmark of the literature in monetary economics.
In the remainder of this section, we investigate the equilibrium possibilities underhelicopter drops of money, assuming a constant growth rate ! of the money supply. We…rst need to compute the magnitude ~bt of the Ponzi scheme as a function of the money
growth rate !.
Lemma 14 Under helicopter drops of money, i.e. b0 = 0 andof the Ponzi scheme is given by:~bt = mt

8
>
< 1 R
1
't =
mt e t (! iu )du
>
:
0

t =

! t mt , the magnitude

if ! > limt!1 it
if ! = limt!1 it :if ! < limt!1 it(76)

Recall that taxes on real money balances are equal to it mt , while lump-sum transfers
to households amount to !mt . Thus, if ! > limt!1 it , then transfers are asymptotically larger than taxes, resulting in an in…nitely large Ponzi scheme. Conversely, if! < limt!1 it , transfers are asymptotically lower than taxes, implying that the government eventually redeems the money supply, which entails 't = mt and ~bt = 0. Finally,in a steady state equilibrium where ! = i, taxes and transfers are equally large resulting
in a zero present value of taxes and, hence, in a Ponzi scheme equal to the outstanding
stock of real money balances.
Clearly, the magnitude of the Ponzi scheme must be consistent with the real equilibrium structure of the economy. For instance, we cannot have an equilibrium withan in…nitely large Ponzi scheme, which rules out any equilibrium with ! > limt!1 it .
We therefore need to investigate how the real and the nominal sides of the economy are
related to each other.
On the real side, the equilibrium paths of ct , kt , and ~bt are still determined by Proposition 2. The economy must either converge to the no-Ponzi steady state characterizedby ~b = 0 and r = r or to the Ponzi steady state with ~b > 0 and r = n. For simplicity,
we now consider that the real economy is in steady state. However, this does not impose
any constraint on the nominal variables mt , it , and t .
The following proposition gives the equilibrium possibilities under helicopter drops ofmoney, assuming that the real economy is in steady state.
30

Proposition 3 Under helicopter drops of money, i.e. b0 = 0 andt =! t mt :The Ponzi steady state with ~b > 0 and r = n can only arise if r < n and ! =h0 (~b)=u0 (c) where ~b is the solution to 0 (k + ~b) = (n) u0 (c). In that case, mt =~b > 0.
The no-Ponzi steady state with ~b = 0 and r = r can arise under the following
circumstances, where m is de…ned by h0 (m) = [! + r n] u0 (c) whenever !
(r n):– Constant real money balances (i.e. mt = m for all t) if and only if r > n and!
(r n);– Speculative hyperin‡ation (i.e. m0 < m, with m = 1 if ! <limt!1 mt = 0) if and only if limx!0+ xh0 (x) = 0;(rn), and– Speculative de‡ation (i.e. m0 > m and limt!1 mt = 1) if and only if r > n
and ! 2 [ (r n); 0).
By lemma 14, the Ponzi steady state can only arise with mt = ~b > 0 and ! =i = h0 (~b)=u0 (c). There is therefore a unique growth rate of the money supply consistent
with the Ponzi steady state. Intuitively, the real side of the economy determines theequilibrium magnitude of the Ponzi scheme, equal to ~b, while money demand determines
the corresponding nominal interest rate, equal to h0 (~b)=u0 (c), and therefore the requiredmoney growth rate !.
Under the no-Ponzi steady state, the magnitude ~b of the Ponzi scheme must be equal tozero. By Lemma 14, this requires limt!1 h0 (mt ) =u0 (c) = limt!1 it > !. On the nominal
side, three di¤erent types of equilibria can be consistent with the no-Ponzi steady state.
First, there is a nominal steady state, where real money balances are constant overtime. By the money supply equation (58), in‡ation must be equal to ! n. This
steady state can only exist if the corresponding nominal interest rate is positive, i.e. if
i = r + = r + ! n 0. Also, the condition limt!1 it > ! simpli…es to r > n.
Second, there is a speculative hyperin‡ation equilibrium, where real money balancesmonotonically converge to zero and the nominal interest rate to in…nity. The requirement
limt!1 it > ! is therefore trivially satis…ed. The only other requirement is that realmoney balances must remain positive. By the dynamic equation for real money balances(75), this requires limx!0+ xh0 (x) = 0 as, otherwise, mt = 0 would imply m_ t < 0.
Finally, there is the possibility of speculative de‡ation, where real money balances
diverge to in…nity and the nominal interest rate reaches the zero lower bound as soon
as mt m. First, this requires the initial real money balances to be higher than in thenominal steady state, i.e. m0 > m. The steady state value m must therefore be …nite,
31

i.e. !
(r n). Also, the condition limt!1 it > ! simpli…es to ! < 0. Obviously, thetwo requirements, ! < 0 and !
(r n), can only be jointly satis…ed when r > n.
Interestingly, we know that in a classical monetary economy, speculative in‡ation requires limx!0+ xh0 (x) = 0 (Obstfeld and Rogo¤ 1983), while speculative de‡ation requires! < 0 (Brock 1975; Buiter and Sibert 2007). Hence, in the absence of Ponzi schemes, thepreference for wealth preserves the usual properties of monetary economies. This should
not be surprising as the classical monetary economy of the literature corresponds to thespecial case where the preference for wealth is always equal to zero.
We have a dichotomy between the real side of the economy, where the preference for
wealth is responsible for the possibility of Ponzi schemes, and the nominal side, wheremoney-in-the-utility-function determines the dynamics of real money balances and ofother nominal variables. This shows that, despite apparent similarities, the preference
for wealth and money-in-the-utility-function are fundamentally of a di¤erent nature.
Nonetheless, when money is the only …nancial asset supplied by the government,
the size of the money-Ponzi scheme must be consistent with the real equilibrium of
the economy. Hence, helicopter drops of money can restrict the range of equilibrium
possibilities. In particular, there is a unique growth rate of the money supply consistent
with the Ponzi steady state. But, importantly, any real allocation of resources that canbe implemented under helicopter drops of money can also be implemented under anappropriate …scal policy.

7

Conclusion

In this paper, we have shown that rational bubbles can exist in a frictionless neoclassicaleconomy, provided that in…nitely-lived households have a preference for wealth. Hence,an OLG structure or …nancial frictions are not necessary for the existence of rational
bubbles. Not only does this …nding broadens the scope for rational bubbles, it also
provides a convenient micro-foundation to introduce bubbles within a representative agent
economy.
Turning to public debt, we have shown that Ponzi schemes can be sustainable under
the same circumstances that make bubbles possible. Hence, in general, the sustainabilityof public debt is not determined by the no-Ponzi condition but, instead, by the representative household’s transversality condition. Also, we have reconciled the existence ofrational bubbles with the Ricardian equivalence.
All these insights survive in a monetary economy, where the preference for wealth
only a¤ects the real side of the economy, while money-in-the-utility-function only a¤ects
the nominal side. However, when money is the only …nancial asset supplied by thegovernment, the growth rate of the nominal money supply severely limits the scope for32

Ponzi schemes.
Bubbles are particularly likely to occur in environments with persistently low interest
rates. This exactly corresponds to the type of circumstances where the economy could
also su¤er from seculars stagnation, i.e. from a persistent lack of demand.26 As shown by
Ono (2015) and Michau (2018), introducing money and a downward wage rigidity within
our framework would be su¢ cient to obtain a secular stagnation equilibrium. In further
research, we therefore intend to investigate the interaction between secular stagnation,
rational bubbles, and Ponzi schemes.

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37

A

Proof of Lemma 1

The steady state equilibrium is jointly characterized by equations (20) and (21). The…rst of these two equations can be written as:
u0 (c) =

0

+

(k)
:
f 0 (k)~Let k~ be de…ned as f 0 (k)= . Thus, this equation de…nes c as a continuous and~ +1). Also, c tends to 0 as k tends to k~strictly increasing function of k for all k 2 (k;from above. Let us call the resulting function c1 .
^ + z = ( + n)k^ and f 0 (k ) = + n, respectively.
Let k^ and k be de…ned as f (k)The other equilibrium relationship, (21), de…nes c as a continuous function of k, which is^ Also, c is equal to 0 when k is equal to k.
^
increasing on [0; k ) and decreasing on (k ; k].
Let us call this function c2 .
~Note that f 0 (k )= n < = f 0 (k), which implies that k~ < k . Also, by^ < + n = f 0 (k ). Thus, k~ < k < k.
^ We
de…nition of k^ and k , we trivially have f 0 (k)~ +1) with lim ~ c1 (k) = 0 andhave shown that c1 is continuous and increasing on (k;k!k+^ with c2 (k)^ = 0. An equilibrium must therefore exist
c2 is positive and continuous on [0; k]~ k).
^
on (k;
Uniqueness of the equilibrium requires the slope of c1 to exceed the slope of c2 at eachintersection of both lines in the k c diagram. We have:
@c1
@k

00
@c2
(k)
f 00 (k)= 00+@ku (c) [ +f 0 (k)] u00 (c) [ +0(k)f 0 (k)]2[f 0 (k)(k) u0 (c) f 00 (k) [u0 (c)]2+ 00[f 0 (k)n] ;u00 (c) 0 (k)u (c) 0 (k)00(k) u0 (c)f 00 (k) u0 (c) u00 (c) 0 (k) 01+ 00[f (k)=
00 (k)
u00 (c) 0 (k)(k) u0 (c)
=

n] ;

00

n] :

From Assumption 1, we can compute d =dk for = 0. It exactly corresponds to theterm in bracket, and we know by Assumption 1 that it must be strictly negative. We
must therefore have:@c1 @c2> 0;@k@kfor all k such that c1 (k) = c2 (k). The equilibrium must therefore be unique.

38

B

Proof of Lemma 2

As the asset price qt tends to in…nity, the marginal utility of wealth 0 (kt + qt ) tends tozero. Hence, by (28), the interest rate f 0 (kt )tends to .27 The asset pricing equation(30) implies that, as qt tends to in…nity, the price of the asset must eventually growat rate f 0 (kt )n and, therefore, at rate
n. But, this is inconsistent with thetransversality condition (31).

C

Proof of Lemma 3

The asset pricing equation (30) under the steady state interest rate r can be written as:
q_t = (r

n)qt

ze

t

:

Integrating this equation from time 0 to t yields:de (r n)s qs=
ds

e (r n)s ze s ;Z t(r n)teq t q0 = z
e (r n+ )s ds;
Z 0t
qt = e(r n)t q0 ze te(r n+ )(t s) ds:

(C1)

0

This implies:(r n)tqt = eq0= e(r n)t q0
ze t
=
r n+

Z t

e(r n+ )(t s);r n+0e(r n+ )t 1
;
ze t
r n+
z
+ e(r n)t q0
:
r n+
ze

t

d

We can therefore express the asset price as:
qt = etrzn++ e(r n+ )t q0

r

z
n+

:

(C2)

Thus, if r n + < 0, then qt tends to ze t = (r n + ), which is negative. But, as theasset can be freely disposed of, its price can never be negative. Hence, we cannot have27

An alternative would be to have ct tending to zero and c_t =ct tending to a negative constant. However,from (29), this can only be consistent with a steady state equilibrium such that k = k where k is then = f0 k
n < 0.
solution to f k +z = ( + n) k. However, this steady state implies that f 0 (k)By the asset pricing equation (30), this is inconsistent with an ever increasing asset price qt .

39

r<n

. Also, if r = n, then, from (C1), we have:
qt = e(r n)t q0
= e

t

[q0

ze

t

t;

zt] ;

which must also eventually be negative. Hence, we also cannot have r = nD.

Proof of Lemma 4

By (C2), if q0 < z= (r n + ) with r > n, then qt must eventually be negative.
But, as the asset can be freely disposed of, its price can never be negative. Thus, q0
z= (r n + ), which implies by (C2) that qt
z= (r n + ) for all t. The asset pricecannot be below its fundamental value.

E

Proof of Lemma 5

The steady state equilibrium with = 0 is jointly characterized by (40), (41), and (42).
Equation (42) immediately implies that any steady state equilibrium must be bubble-less.
Also, as q must be positive, we must have f 0 (k)> n, i.e. r > n. Let k~ and k be~de…ned by f 0 (k)= and f 0 (k )= n, respectively. Note that, asn, we musthave k~ k .
Substituting (42) into (40) yields:
u0 (c) =1+

f 0 (k)0k+z
f 0 (k)n:This equation de…nes c as a continuous and strictly increasing function of k for all k 2~ k ). Also, c tends to 0 as k tends to k~ from above. In addition, as k tends to k from(k;below, the asset price q tends to in…nity and hence 0 tends to zero. Therefore, c tends
to in…nity. Let us call the resulting function c1 .
^ + z = ( + n)k,
^ where we trivially have k^ > k . Equation
Let k^ be de…ned as f (k)(41) de…nes c as a continuous function of k, which is increasing on [0; k ) and decreasing^ Also, c is equal to 0 when k is equal to k.
^ Let us call this function c2 .
on (k ; k].
An equilibrium is characterized by an value of k that satis…es c1 (k) = c2 (k). We have~ k ) with lim ~ c(k) = 0
therefore shown that c1 is continuous and increasing on (k;k!k+^ with c2 (k)^ = 0. An
and limk!k c(k) = 1. Also, c2 is positive and continuous on [0; k]~ k ).
equilibrium must therefore exist on (k;We know that q > 0. Assumption 1 immediately implies that the equilibrium capitalstock with q > 0 is smaller than in the corresponding equilibrium with q = 0. Thus,40

r > r.
Uniqueness of the equilibrium requires the slope of c1 to exceed the slope of c2 at eachintersection of both lines in the k c diagram. We have:
@c1
@k

00
@c2
(k + q)= 00
1@ku (c) [ +f 0 (k)]=zf 00 (k)[f 0 (k)n]2f 00 (k) 0 (k + q)
[f 0 (k)
n] ;
+
2
00
0
u (c) [ +f (k)]
00(k + q) u0 (c)f 00 (k) u0 (c)1
0 (k + q)
00 (k + q)
u00 (c)0(k + q) 0
u00 (c)[f (k)n]+ 00(k + q) u0 (c)00(k + q) u0 (c)zf 00 (k):0 (k + q)
u00 (c)[f 0 (k)n]2Compared to the corresponding expression of Lemma 1, we have a new term (the lastterm), which is positive. Hence, under Assumption 1 (which guarantees that the term in
the main bracket is negative), we still have:
@c1
@k

@c2
> 0;
@k

for all k such that c1 (k) = c2 (k). The equilibrium must therefore be unique.

F

Proof of Lemma 6

For any > 0, the fundamental value of the asset converges to zero. Hence, a steadystate with q > 0 must be bubbly.
If a bubbly steady state equilibrium exists, it must be characterized by (40), (41),and (43). Equation (43) uniquely determines k; then equation (41) uniquely determinesc; and then …nally equation (40) uniquely determines q.
To prove that such an equilibrium exists, it is su¢ cient to show that q > 0 (since, from
Lemma 3 and 4, there cannot be a steady state with q < 0 and, if q = 0, then there is no~bubble). Let k~ and k be de…ned as f 0 (k)= and f 0 k= r (thus, k is the capitalstock under Lemma 1, i.e. without the in…nitely-lived asset). As r < , we must havek~ < k. Combining (40) and (41) yields 0 (k+q) = [ + f 0 (k)]u0 (f (k)+z ( +n)k). This28
de…nes q a positive and continuous function of k with q(k) = 0 and limk!k+~ q(k) = +1.
28

If there is a unique value of r (i.e. there is a unique solution to 0 (k) = [ +f 0 (k)]u0 (f (k) + z~~ k). If there( + n)k)), then there is no k 2 (k; k) such that q (k) = 0. Thus, q (k) > 0 for all k 2 (k;are multiple values of r, then the argument applies when taking the largest value of r, i.e. the smallest
value of k.

41

We always > n. If in addition n > r, then the equilibrium value of k given by (43)~ k). The corresponding value of q determined by (40) and (41) is given bysatis…es k 2 (k;q (k) and it satis…es q (k) > 0. This proves the existence of a unique bubbly steady state
equilibrium whenever r < n.
Let us now consider the case where rn. Under Assumption 1, q is a strictly
decreasing function of k. But, q(k) = 0. Hence, the existence of an equilibrium withq > 0 requires k < k or, equivalently, r > r. But, if rn, then this requires r > n,which is inconsistent with the asset pricing equation (43). Thus, under Assumption 1,
there cannot exist a bubbly steady state equilibrium whenever r n.

G

Proof of Lemma 7

If a bubble-less steady state equilibrium exists, it must be characterized by (40), (41),and (44). By Lemma 1, such an equilibrium always exist and, under Assumption 1, it
must be unique. This equilibrium must trivially satisfy r = r.
Finally, by Lemma 3, if r n, then for any …nite value of time t there cannot be
an equilibrium with r = r. In that case the (asymptotic) steady state is not economicallymeaningful (since it can only exist in the limit as t tends to in…nity).

H

Proof of Lemma 8

From (40), (41), and (43), the bubbly steady state cB ; k B ; q B is characterized by:
f 0 kB= n;cB = f k B + zn=0( + n) k B ;kB + qB:
u0 (cB )From (40), (41), and (44), the non-bubbly steady state c; k; 0 is characterized by:
0

=

k
;
u0 (c)

c=f k +z

( + n) k:

f0 k

As k B is the golden rule capital stock, when r < n, we clearly have c < cB and k > k B .
From the above equations, we have:0kB + qB = (n) u0 cB < (n) u0 (c) =

42

n
f0 k

0

k :

f0 k

As

n > 0, we have:

>

f0

n
k

< 1:

Hence:
0kB + qB <0

k :This establishes that k B + q B > k. It immediately follows that:k B + q B > u (c) +u cB +Ik :Proof of Lemma 9

Linearizing the system (28), (29), and (30) around a steady state (k ; c ; q ) yields:
1 0
f 0 (k )
ni
k_ t
h
B
00 (k +q )B
C B 00c@ c_t A = @ f (k ) + u0 (c )
"(c )
q_t
f 00 (k ) q0

where " (c) =that f 0 (k )

1
0 (k

+q )
u0 (c )

0

1 0
z
k_ t
q
h
i
00 (k +q )
B
C B
c
00
B
=
f
(k
)
+
@ c_t A @u0 (c )"(c )q_t
f 00 (k ) q
The eigenvalues
2

x

k
CB t
00 (k +q )cC
u0 (c ) "(c ) A @ ct
qt
f 0 (k )
n

1
k
C
c A;
q

cu00 (c) =u0 (c). The steady state is characterized by (42), which implies
n = z=q . Hence:

0

z
q

10

0

1,
0

2 , and(k + q )u0 (c )

1

10

0

k
CB t
00 (k +q )cC
u0 (c ) "(c ) A @ ct
z
qt
q

0 (k

+q )
u0 (c )

0

3 of the matrix are the solution to:

x +

z
q

00

(k + q )c0
u (c )" (c )
00(k + q ) c
f 00 (k ) q = 0:
u0 (c ) " (c )f 00 (k ) +x

The characteristic equation of the matrix can be written as:(xx3 + ( 1 +

1
k
C
c A:
q

1 ) (x
2

2+

3) x

1+

2+

2 ) (x( 1 2+

where:3 =013 ) = 0;3+

23) x +(k + q )z+ 2 > 0:0u (c )q43

1

2

3 = 0;

We also have:z
1 2 3 =
q

"

z
q

0

"
(k + q )+ f 00 (k ) +u0 (c )

00

(k + q )u0 (c )

1(q )2
f 00 (k )
z

!#

#
c
:
" (c )

Assuming that the steady state equilibrium is unique, we know (from the proof of Lemma5) that in the neighborhood of the steady state:
@ct
@ct
:
>
@kt c_t =0@kt k_ t =0
Indeed, for a given value of qt , the c_t = 0 locus is steeper than the k_ t = 0 locus. The
c_t = 0 locus is given by (40). Implicitly di¤erentiating this equation at the steady stateyields:000(k + q )@ct(k + q ) 00
f 00 (k ) =+:2 u (c )00
u (c )@kt c_t =0[u (c )]
Thus:
@ct
=
@kt c_t =0=

u0 (c )
f 00 (k ) +0 (k + q )u0 (c )
f 00 (k ) +0 (k + q )00(k + q ) u0 (c );u0 (c )
u00 (c )00
c(k + q ):0
u (c )" (c )

Similarly, the k_ t = 0 locus is given by (41). Implicitly di¤erentiating it at the steadystate yields:z@ct= f 0 (k )n= :@kt k_ t =0q
Combining these results implies that:u0 (c )
f 00 (k ) +0 (k + q )00(k + q )cz> ;0u (c )" (c )qor equivalently:
f 00 (k ) +00c(k + q )+u0 (c )" (c )

0(k + q ) z< 0:
u0 (c ) q

This implies that:1

2

3 < 0:

Thus, the left-hand side of the characteristic equation is equal to 1 2 3 < 0 if x is
equal to zero and it tends to in…nity if x tends to minus in…nity. Hence, there exists atleast one real value of x smaller than zero such that the left-hand side is equal to zero, i.e.
the matrix has at least one negative real eigenvalue. Let 1 be this negative eigenvalue.
44

As 1 < 0, we must have 2 + 3 > 0 and 2 3 > 0. Hence, if 2 and 3 are real,then we must have 2 > 0 and 3 > 0. If 2 and 3 are complex, then, as their sum isreal, they must be complex conjugate of one another. Moreover, as 2 + 3 > 0, theirreal part must be positive.
Finally, we need to show that there cannot be a bubble along a trajectory converging
to the steady state. Recall that the bubble component must grow at rate f 0 (kt )n,which is positive in the neighborhood of the steady state. Thus, a bubble would be
growing as it approaches the steady state. But, the steady state is bubble-less. Thus, atrajectory converging to the steady state must also be bubble-less.

J

Proof of Lemma 10

The system of di¤erential equations linearized around a steady state (k ; c ; q ) can be
written as:u_ t = Aut + vt ;T

where ut = (kt k ; ct c ; qt q )T and vt = 0; 0; ze t . Let D be a diagonal matrixof eigenvalues and P a matrix of eigenvectors of A, i.e. AP = P D. Let xt be de…ned as
xt = P 1 ut . We therefore have:x_ t = P 1 u_ t = P 1 Aut + P 1 vt = P 1 AP xt + P 1 vt = Dxt + P 1 vt .
Since D is a diagonal matrix, we obtain three independent linear di¤erential equations.
Each of these di¤erential equations is of the following form:
x_ 1 (t) =1 x1 (t) +

1e

t

,

where x1 (t) is the …rst element of the vector xt , 1 is an eigenvalue, andIntegrating this linear di¤erential equation yields:
de
e

1t

x1 (t)1t

x1 (t)=dt1e

x1 (0) =1

x1 (t) = e 1 t x1 (0) +1( 1 + )t

Z t

Z 0t
0

45

;

e ( 1 + )s ds;

e ( 1 + )s ds :1 is a constant.

If

1+

6= 0, which should generically be the case29 , then:x1 (t) = e 1 t x1 (0) +1e ( 1 + )t
1+

1

e

1

= e 1 t x1 (0) +

1+

;
1

t

1+

:

Thus, the second term tends to zero, while the …rst term has the same structure as forthe solution to a homogeneous and autonomous system of linear di¤erential equations,i.e. it consists of a constant determined by initial conditions multiplied by e 1 t . Hence,
the local stability of our system of di¤erential equations u_ t = Aut + vt is determined by
the eigenvalues of the matrix A, as if the system was homogenous and autonomous.30

K

Proof of Lemma 11

Linearizing the system (28), (29), and (30) around a steady state (k ; c ; q ) yields:
1 0
f 0 (k )
ni
k_ t
h
B
00 (k +q )B
C B 00c@ c_t A = @ f (k ) + u0 (c )
"(c )
q_t
f 00 (k ) q
0

1
0 (k

+q )
u0 (c )

0

10

0

k
CB t
00 (k +q )C @ ctu00 (c )A0qt
f (k )
n

1 0
1
k
0
C B
C
c A+@ 0 A ;q
ze t

where " (c) = cu00 (c) =u0 (c). The bubbly steady state is characterized by (43), i.e.
f 0 (k ) = + n. Also, by (40), we have 0 (k + q ) =u0 (c ) =[f 0 (k )]=n.
The linearized system can therefore be simpli…ed to:
1 0
0
k_ t
h
i
00 (k +q )
C B
B
c
00
B
@ c_t A = @ f (k ) + u0 (c )
"(c )
q_t
f 00 (k ) q
0

The eigenvalues

1,

2 , andx2 (x)

x f 00 (k ) +

n

1
n
0

0

10

kt
00 (k +q ) C B
C @ ctu00 (c )
A
qt
0

1
1 0
0
k
C B
C
c A+@ 0 A :ze tq

3 of the matrix are the solution to:00(k + q )c+0
u (c )" (c )

00(k + q ) 00
f (k ) q = 0:
u00 (c )The left-hand side of this expression is a continuous function of x, which is negative whenx = 0 and tends to in…nity as x tends to minus in…nity. Hence, there exists at least one
real value of x smaller than zero such that the left-hand side is equal to zero, i.e. thematrix has at least one negative real eigenvalue. Let 1 be this negative eigenvalue.
29

Note that does not enter the matrix A from which the eigenvalue 1 is derived.
Recall that, for a given set of initial conditions k0 , c0 , and q0 , a trajectory is said to be stable if ut ,and therefore xt , tend to zero as time tends to in…nity.
30

46

The characteristic equation of the matrix can be written as:(xx3 + ( 1 +

2+1 ) (x3) x22 ) (x( 1 2+

13 ) = 0;3+

23) x +

1

2

3 = 0;

where:
1+
1

2

3 =

2+
00

3 =

n > 0;(k + q ) 00
f (k ) q < 0:
u00 (c )As 1 < 0, we must have 2 + 3 > 0 and 2 3 > 0. Hence, if 2 and 3 are real, thenwe must have 2 > 0 and 3 > 0. If 2 and 3 are complex, then, as their sum is real,
they must be complex conjugate of one another. Moreover, as 2 + 3 > 0, their realpart must be positive.

L

Proof of Lemma 12

The bubble-less steady state is characterized by (44), i.e. q = 0. Hence, the linearized
system of di¤erential equations around this steady state is:
1 0
0
k_ t
h f (k ) 00 in
B
C
B
(k )
c
00
@ c_t A = B@ f (k ) + u0 (c ) "(c )q_t
00The eigenvalues

1,

2 , and[f 0 (k )

x]

2

n

1

10

0

0 (k )
u0 (c )k
CB t
C @ ctAqtn

00 (k )
u00 (c )f 0 (k )

0

1
1 0
0
k
C
C B
c A+@ 0 A :ze tq

3 of the matrix are the solution to:

0

(k )
0
u (c )

x + [f 0 (k )nx] f 00 (k ) +

00

(k )

u0 (c )c
= 0:
" (c )

This can be simpli…ed to:[f 0 (k )nx] [f 0 (k )

n

x]

0

(k )

u0 (c )x
+ f 00 (k ) +

47

00

c
(k )
u0 (c ) " (c )= 0;

[f 0 (k )

x] x2

n

But, by (40), f 0 (k )[f 0 (k )

n

+
x] x2

f 0 (k )

0

n+

0

(k )
x + [f 0 (k )u0 (c )0(k )
u0 (c )
00
c
(k )
= 0:
+ f 00 (k ) + 0
u (c ) " (c )n](k ) =u0 (c ) = . Thus:(n) x + [f 0 (k )

n]

0

(k )
u0 (c )00

+ f (k ) +

00

(k )

u0 (c )c
" (c )= 0:

One eigenvalue is equal to f 0 (k )n = r n. Thus, let 1 = r n. The other two
eigenvalues 2 and 3 are the solutions to the quadratic equation x2 (n) x + = 0,
where we need to determine the sign of the constant .
Assuming that the steady state equilibrium is unique, we know (from the proof of
Lemma 1) that in the neighborhood of the steady state:
@ct
@ct
>
:
@kt c_t =0@kt k_ t =0
Indeed, for qt = 0, the c_t = 0 locus is steeper than the k_ t = 0 locus. The c_t = 0 locus isgiven by (40). Implicitly di¤erentiating this equation at the steady state yields:
f 00 (k ) =0(k )(k ) 00@ct:+
u (c )
20
u (c )@kt c_t =0[u0 (c )]
00

Thus:
00
@ct
u0 (c ) 00(k ) u0 (c )f (k ) + 0;= 0@kt c_t =0(k )u (c ) u00 (c )00
u0 (c ) 00
(k )
c
= 0
f (k ) + 0:(k )u (c ) " (c )Similarly, the k_ t = 0 locus is given by (41). Implicitly di¤erentiating it at the steadystate yields:@ct= f 0 (k )n:@kt k_ t =0
Combining these results implies that:00u0 (c ) 00
(k )
c
f
(k
)
+
> f 0 (k )0 (k )0u (c ) " (c )48n;or equivalently:
f 00 (k ) +

00

(k )
c
+ [f 0 (k )0u (c ) " (c )n]0(k + q )< 0:
u0 (c )

This implies that 2 and 3 are the solutions to the quadratic equation x2 (n) x+ =
0 withn > 0 and < 0. Hence, these two eigenvalues are real, with 2 negative and3 positive.
If r > n, two eigenvalues are positive and one is negative. Hence, for a given value ofk0 , there must be a unique trajectory converging to the steady state.
If r < n, one eigenvalues is positive and two are negative. In that case, for a given
value of k0 there is a continuum of values q0 that are consistent with convergence to thesteady state.31 For each pair k0 and q0 there is at most one value of c0 that is consistent
with convergence to the steady state.
Finally, we need to show that, when r > n, there cannot be a bubble along a trajectory
converging to the steady state. Recall that the bubble component must grow at rate
f 0 (kt )n, which is equal to r n at the steady state. Thus, a bubble must be
growing in the neighborhood of the steady state. But, the steady state is bubble-less.
Thus, when r > n, a trajectory converging to the steady state must be entirely bubbleless.

M

Proof of Lemma 13

To prove the lemma, we need to show that replacing the …scal policy ( t ; bt )1
t=0 by the11~~policy (0; bt )t=0 with bt = bt 't would leave the allocation (ct ; kt )t=0 unchanged.
The equilibrium is characterized by (51), (52), (53), (54), and (55). In the Euler
equation (51), we can immediately replace bt 't by ~bt . The capital accumulation equation(52) is independent of …scal policy.
By de…nition of 't , given by (46), we have:'_ t = (rt

n) 't

31

t:

With two negative eigenvalues, an alternative would be that, for a given value of k0 , there is aunique value of q0 consistent with convergence to the steady state and, for this value of q0 , there is acontinuum of values of c0 consistent with convergence to the steady state. However, if this was the case,the transition matrix (from which the eigenvalues are derived) would have a special structure, which is
not the case.

49

The debt accumulation equation (53) can therefore be written as::~bt = b_ t'_ t ;= [(rt= (rtn) btn) ~bt ;

t]

[(rt

n) 't

t] ;

where rt = f 0 (kt ).
Integrating the above di¤erential equation for 't from time t to in…nity yields:
lim e

T !1

RT
t

(ru n)du

'T

Z 1

't =

Rs

t (ru

e

t

n)du

s ds:

Thus, if 't is …nite, then by de…nition of 't in (46) we must have:
RT

lim e

t

T !1

(ru n)du'T = 0:Also, the consumption Euler equation (51) can be written as:
d ln [u0 (ct )]
=
dt

0

rt +

(kt + bt 't ):u0 (ct )Integrating this di¤erential equation from time zero to t yields:
Rt

Hence:
lim e

t!1

Rt

0(

ru )du 0

0 (k +~
u bu )
u0 (cu )ru

u0 (ct ) = u0 (c0 ) e 0

Rtu (ct ) = u0 (c0 ) lim e

0

t!1

du

:

0 (k +~
u bu )duu0 (cu )u0 (c0 ) :The transversality condition (54) is:
lim e

i
~
u (ct ) kt + bt + 't = 0:
h

n)t 0

(

t!1

But:
lim e (

t!1

n)t 0

u (ct ) 't =

lim e

t!1

Rt

n)du

0 (ru

= 0 u0 (c0 ) lim e
t!1

't

Rt
0

lim e

t!1

0 (k +~
u bu )duu0 (cu )= 0:
The transversality condition can therefore be expressed as:
lim e

t!1

(

h

i
~
u (ct ) kt + bt = 0:

n)t 0

50

Rt

0(

;

ru )du 0

u (ct ) ;

Let us …nally show that there cannot be an equilibrium with an in…nite value of 't .
If 't = 1, then ~bt = bt 't = +1.32 This implies 0 (kt + ~bt ) = 0. From the above
consumption Euler equation, we have:Rt

u0 (ct ) = u0 (c0 ) e 0 (ru )du:The household’s transversality condition is:n)t 0

lim e (u (ct ) [kt + bt ] = 0:

t!1

It can therefore be simpli…ed to:
lim e

t!1

Rt

0 (ru

n)du

[kt + bt ] = 0:

Integrating the government liability accumulation equation (60) from t to in…nity yields
RT
RT
Rt
limT !1 e t (ru n)du bT = bt 't or, equivalently, limT !1 e 0 (ru n)du bT = e 0 (ru n)du~bt =1. This implies that the household’s transversality cannot be satis…ed when 't = 1.33
~ 1
We have been able to replace ( t ; bt )1
t=0 by (0; bt )t=0 without changing any of the equations that characterize the equilibrium of the economy. Hence, the allocation (ct ; kt )1
t=0
must remain unchanged.

N

Proof of Lemma 14

Under helicopter drops of money, we have bt = 0 andvalue of taxes is therefore given by:
't =

Z 1
t

=

e

Z 1
t

Rs

t (ru

e

n)du

Rs

t (ru

32

t =

!mt for all t. The present( s + is ms ) ds;

n)du

(!

is ) ms ds:

Recall that, throughout our analysis, we exclusively focus on cases where the no-Ponzi condition iseither binding or violated, i.e. ~bt 0.
33
Note that, even if we impose the transversality condition limt!1 e ( n)t u0 (ct ) [kt + ~bt ] = 0, insteadof limt!1 e ( n)t u0 (ct ) [kt + bt ] = 0, equilibria where ~bt tends to in…nity are ruled out (since u0 (ct )asymptotically grows at rater while ~b grows at rate r n). There is therefore no need to impose
the original household’s transversality condition limt!1 e ( n)t u0 (ct ) [kt + bt ] = 0 to rule out such
equilibria.

51

The money supply equation (58) implies that:
Z 1

't =

t

=

mt

=

mt

e

Z 1

Zt 1

Rs

Rs

t (ru

n)du

(!

is ) e t (! iu )du ds;(!

is ) mt e t (!

u

n)du

ds;

Rs

Rs

d e t (! iu )du :tThus, for any given path of it , we must have:
8
>
<

1
't =
mt 1
>
:
mt

R1

et

(! iu )du

if ! > limt!1 it
if ! = limt!1 it :if ! < limt!1 it

This immediately implies that:~bt = mt

O

8
>
< 1 R
1
't =
mt e t (! iu )du>:0

if ! > limt!1 it
if ! = limt!1 it :if ! < limt!1 it

Proof of Proposition 3

From the equilibrium conditions (70), (71), and (72), the steady state of the economy(c; k; ~b) must be jointly characterized by 0 (k + ~b) = [ +f 0 (k)] u0 (c), c = f (k) + z( + n) k, and either f 0 (k)= n or ~b = 0. The steady state real interest rate satis…es
r = f 0 (k).
In steady state, for any growth rate ! of the money supply, the path of real money
balances mt must be consistent with these equations together with the magnitude of the
Ponzi scheme ~bt given by Lemma 14 and the dynamic equation for real money balances(75), which in steady state is:
m
_t
=!
mt

h0 (mt )+ru0 (c)n:(O1)

Importantly, real money balances cannot be negative. Let m be de…ned by h0 (m) =[! + r n] u0 (c) whenever !
(r n). From (O1), there are three possibilities for the
dynamics of real money balances:
Constant real money balances: mt = m.
– This is only possible if !

n) since, otherwise, m cannot be de…ned.

(r52

Speculative hyperin‡ation: m0 < m (with m = 1 if ! <0.

(r n)) and limt!1 mt =– This is only possible if limx!0+ xh0 (x) = 0 since, otherwise, mt = 0 implies by(O1) that m_ t < 0, which violates the mt 0 condition.
Speculative de‡ation: m0 > m and limt!1 mt = 1.
– This is only possible if !

n) since, otherwise, m cannot be de…ned.

(r

Let us now investigate the conditions under which these possibilities can arise when theeconomy is in steady state equilibrium.
We …rst consider the no-Ponzi steady state, where ~b = 0 and r = r. From equation(76) of Lemma 14 together with the money demand equation h0 (mt ) = it u0 (c), we musthave limt!1 h0 (mt ) =u (c) > !. Equilibrium paths of mt are fully characterized by (O1)and mt0, together with limt!1 h0 (mt ) =u (c) > !. Thus, we just need to check thatthe above three possibilities are consistent with limt!1 h0 (mt ) =u (c) > !.
Constant real money balances: mt = m where m is the solution to h0 (m) =[! + r n] u0 (c) implies that limt!1 h0 (mt ) =u (c) = h0 (m) =u0 (c) = ! + r n,which must be greater than !. This can only be an equilibrium under the additional requirement that r > n.
Speculative hyperin‡ation: limt!1 mt = 0 implies that limt!1 h0 (mt ) =u (c) = +1,
which is always greater than !. This is always an equilibrium, with no additional
requirement.
Speculative de‡ation: limt!1 mt = 1 implies that limt!1 h0 (mt ) =u (c) = 0, whichmust be greater than !. This can only be an equilibrium under the additional
requirement that ! < 0.
We now consider the Ponzi steady state, where ~b > 0 and r = n. From Proposition
2, we know that it is only feasible when r < n. From Lemma 14 together with the
R1
h0 (mu )du!
u0 (c)
andmoney demand equation h0 (mt ) = it u0 (c), we must have ~b = mt e t
0
0
0
0
34
limt!1 h (mt ) =u (c) = !, where ~b is the solution to (k + ~b) = (n) u (c). Using
34

From Proposition 2, we know that ~b > 0 whenever r < n.

53

the dynamics of real money balances (O1) with r = n, we have:
R1~b = mt e t!

= mt e t

mu

R1 m
_u

h0 (mu )u0 (c)

du

du

;

;

R1

= mt e t d ln(mu ) ;= mt lim eln(mT =mt ) ;=T !1

lim mT :T !1

Combining this with the requirement that limt!1 h0 (mt ) =u0 (c) = ! gives ! = h0 (~b)=u0 (c).
Thus, when ! = h0 (~b)=u0 (c), any path of real money balances satisfying limt!1 mt = ~bis consistent with the Ponzi steady state. But, by (O1) with r = n, the path of real money
balances is given by:
h0 (~b) h0 (mt )
m
_t
= 0
:
mt
u (c)
u0 (c)
Clearly, the only solution consistent with limt!1 mt = ~b > 0 is to have mt = ~b for all t.

54