# On the Ramsey Equilibrium with heterogeneous consumers and endogenous labor supply

Stefano Bosi, Thomas Seegmuller

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UNIVERSITÉ PARIS 1
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Documents de Travail du
Centre d'Economie de la Sorbonne

CES
Working Papers

<figure><img src="image_2.png" /></figure>

On the Ramsey Equilibrium with Heterogeneous
Consumers and Endogenous Labor Supply

Stefano BOSI, Thomas SEEGMULLER

2007.03

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CENTRE NATIONAL
DE LA RECHERCHE
SCIENTIFIQUE

Maison des Sciences Économiques, 106-112 boulevard de L'Hôpital, 75647 Paris Cedex 13
http://ces.univ-paris1.fr/cesdp/CES-docs.htm

ISSN : 1955-611X
<figure><img src="image_2.png" /></figure>

UNIVERSITÉ PARIS 1
- PANTHÉON - SORBONNE -

Documents de Travail du
Centre d'Economie de la Sorbonne

CES
Working Papers

<figure><img src="image_3.png" /></figure>

On the Ramsey Equilibrium with Heterogeneous
Consumers and Endogenous Labor Supply

Stefano BOSI, Thomas SEEGMULLER

2007.03

<figure><img src="image_4.png" /></figure>

CENTRE NATIONAL
DE LA RECHERCHE
SCIENTIFIQUE

Maison des Sciences Économiques, 106-112 boulevard de L'Hôpital, 75647 Paris Cedex 13
http://ces.univ-paris1.fr/CESPublicat.htm
ISSN en cours d'attribution
# On the Ramsey Equilibrium with Heterogeneous Consumers and Endogenous Labor Supply*

Stefano BOSI†and Thomas SEEGMULLER‡

July 19, 2007

**Abstract**

In this paper, we address the stability issue, stressing the role of labor supply, in a Ramsey model with heterogeneous households subject to borrowing constraints. Making labor supply endogenous leads us to prove the existence of two kinds of steady state: the one where everybody supplies labor, the other where only the most patient agent refrains from working. Focusing on the latter and going beyond models with inelastic labor supply, we show how preferences of impatient agents affect the saddle-path stability and the occurrence of endogenous cycles. When their elasticity of intertemporal substitution in consumption exceeds one, instability and cycles are less likely, requiring lower degrees of capital-labor substitution. Conversely, elasticity values below one promote the emergence of fluctuations.

JEL classification: C62, D30, E32.

Keywords: Saddle-path stability, endogenous cycles, heterogeneous agents, endogenous labor supply, borrowing constraint.

<footnote>*We would like to thank Robert Becker and Cuong Le Van for helpful comments. All remaining errors are our own. This work was supported by French National Research Agency Grant (ANR-05-BLAN-0347-01).</footnote>

<footnote>†Corresponding author. EQUIPPE (Université de Lille 1) and EPEE (Université d'Evry). Département d'Economie, Université de Lille 1, Bât. SH2, Cité Scientifique, 59655, Villeneuve d'Ascq Cedex, France. Tel: (33) 3 20 33 70 12. E-mail: stefano.bosi@orange.fr.</footnote>

<footnote>‡CNRS and Université Paris 1, Paris School of Economics, Centre d'Economie de la Sorbonne, 106-112, bd. de l'Hôpital, 75647, Paris Cedex 13, France. Tel: (33) 1 44 07 81 99. Fax: (33) 1 44 07 82 31. E-mail: seegmu@univ-paris1.fr.</footnote>
# On the Ramsey Equilibrium with Heterogeneous Consumers and Endogenous Labor Supply<fnref n="1" />

Stefano Bosi

EQUIPPE, Département d'économie,
Université de Lille 1, Bât. SH2, Cité Scientifique,
59655 Villeneuve d'Ascq Cedex, France.

Tel: (33) 3 20 43 70 12

EPEE, Université d'Evry.

E-mail: stefano.bosi@univ-lille1.fr.

Thomas Seegmuller

CNRS and Centre d'Economie de la Sorbonne,
Paris School of Economics, Université Paris 1,
106-112 Boulevard de l'Hôpital 75647 Paris Cedex 13, France.

Tel: (33) 1 44 07 81 99

Fax: (33) 1 44 07 82 31

E-mail: seegmu@univ-paris1.fr

<footnote n="1">¹This work was supported by French National Research Agency Grant (ANR-05-BLAN-0347-01).</footnote>
# 1 Introduction

Growth theorists are usually confronted with the question of convergence of economic systems. In neoclassical models of capital accumulation, commonly addressed questions are whether economies converge to the same long-run equilibrium, how convergence takes place and whether is monotonic, how fundamentals affect the stability properties of equilibrium.<fnref n="1" />

The most influential growth model is undoubtedly Ramsey which is characterized, in its basic version, by a representative infinite-lived agent, exogenous labor supply, saddle-path stability, equilibrium uniqueness and optimality. In order to add a degree of realism, economic literature has not only introduced various kinds of market imperfections, but also considered elastic labor supply and agents' heterogeneity.

On the one side, several contributions have focused on the convergence of capital accumulation and distribution in the long run when consumers are heterogeneous and labor supply is inelastic. Heterogeneous discounting promotes a concentration of capital in the hands of the most patient agent. Unlike the case with a representative agent, consumers' heterogeneity accounts for borrowing transactions. These transactions allow impatient agents to consume more today and work more tomorrow to refund the debt: their consumption asymptotically vanishes (Le Van and Vailakis (2003)). Under borrowing constraints, not only there exists a stationary state where impatient agents consume, but also persistent cycles arise (Becker (1980), Becker and Foias (1987, 1994), Sorger (1994)).

On the other side, many economists have analyzed the role of elastic labor supply on growth under the assumption of representative consumer. This literature succeeded in shedding a light on the interplay between the consumption-labor arbitrage and the mechanism of capital accumulation.<fnref n="2" />

Surprisingly, only few works have investigated the role of elastic labor supply on equilibrium transition and the long run when consumers are heterogeneous. In this respect, most papers focus on heterogeneity in wealth (Sorger (2000), Ghiglino and Sorger (2002), Garcia-Penalosa and Turnovsky (2006)).

In our paper, we are mainly interested in the effects of labor supply on the saddle-path stability when heterogeneity concerns not only consumers' endowments but also preferences. In this connection, we consider infinite-lived consumers with preferences additively separable in consumption and leisure, and over time. Heterogeneity is now threefold and turns on capital wealth, time preference and intra-temporal preferences. In addition, in line with Becker (1980), Becker and Foias (1987, 1994), Hernandez (1991) and Sorger (1994), we assume that consumers cannot borrow against their future labor income. This

<footnote n="1">¹Barro and Sala-i-Martin (1995) provide an introductory but representative survey of the literature.</footnote>

<footnote n="2">²The interested reader is referred among others to De Hek (1998) and Le Van and Vailakis (2004) for one-sector models; Bosi, Magris and Venditti (2005) for a two-sector model; Ladron-de-Guevara, Ortiguiera and Santos (1997) for endogenous growth; Nourry (2001) and Nourry and Venditti (2006) for OLG economies; Pintus (2006) and Garnier, Nishimura and Venditti (2006) for models with externalities.</footnote>
Abstract

In this paper we address the question of deterministic cycles in a Ram-
sey model with heterogeneous infinite-lived agents and borrowing constraints,
augmented to take into account the case of elastic labor supply. Under usual
restrictions, not only we show that the steady state is unique, but also we clarify
its stability properties through a local analysis. We find that, in many cases,
the introduction of elastic labor supply promotes convergence by widening the
range of parameters for saddle-path stability, and endogenous cycles can even-
tually disappear. These results are robustly illustrated by means of canonical
examples in which consumers have separable, KPR or homogeneous preferences.

Keywords: Saddle-path stability, endogenous cycles, heterogeneous agents, endogenous labor supply, borrowing constraint.

Résumé

Dans ce papier, nous nous intéressons à la question des cycles déterministes
dans le modèle de Ramsey avec agents hétérogènes et contrainte d'emprunt,
étendu au cas où l'offre de travail est élastique. Sous des restrictions habituelles,
non seulement nous montrons que l'état stationnaire est unique, mais analysons
aussi ses propriétés de stabilité à travers une analyse locale. Nous trouvons que,
dans la plupart des cas, l'introduction d'une offre de travail élastique favorise
la convergence, et les cycles endogènes peuvent même disparaître. Ces résultats
sont illustrés de manière robuste dans trois exemples dans lesquels les préférences
des consommateurs sont séparables, de type KPR, et homogènes.

Mots-clés: Stabilité de point-selle, cycles endogènes, agents hétérogènes, offre de travail endogène, contrainte d'emprunt.

JEL classification: C62, D30, E32.
borrowing constraint implicitly means that, in contrast to Le Van, Nguyen and Vailakis (2007), markets are incomplete.

Under these assumptions, we first derive sufficient conditions for the existence of an intertemporal equilibrium. We also show that, as in Becker (1980), Becker and Foias (1987, 1994), and Sorger (1994), the most patient household owns the whole capital stock at a steady state, whereas the others consume their per-period labor income.<fnref n="3" /> However, a new element emerges from the introduction of endogenous labor supply: there are two types of steady states, depending on the amount of leisure consumed by the most patient agent: one where he works and one where he supplies no labor.

We study the stability properties around the steady state where the most patient consumer not only owns the entire stock of wealth, as seen above, but also supplies no labor.<fnref n="4" /> Population splits in two classes: on the one side, a (patient) capitalist with no labor income, who spend his capital income to consume and invest; on the other side, (impatient) workers who neither get capital income nor invest.<fnref n="5" />

In order to study the convergence to the steady state, while underlining the role of heterogeneous preferences and elastic labor supply, we focus on their effects on the saddle-path stability and the occurrence of flip bifurcations. In economies with inelastic labor supply, the occurrence of endogenous cycles requires sufficiently weak capital-labor substitution and intertemporal substitution in consumption of the patient consumer.<fnref n="6" /> Introducing leisure in the utility functions, we find that the impatient agents' preferences play a crucial role on the stability properties of the steady state. More specifically, when the impatient agents' elasticity of intertemporal substitution becomes greater than one, the introduction of elastic labor supplies promotes stability, because the occurrence of cycles requires a weaker capital-labor substitution. However, the converse holds when their elasticity of intertemporal substitution in consumption is less than one. In addition, these results are reinforced by a larger elasticity of intertemporal substitution in leisure.

To understand why preferences play a role on the stability properties, we mind that, when each agent supplies inelastically one unit of labor, instability and endogenous cycles ensue from a negative response of the patient agent's (capital) income to a rise in the capital stock.<fnref n="7" /> Under elastic labor supply

<footnote n="3">³This result is in contrast with Sorger (2000) who obtains a continuum of long-run equilibria. Sorger's assumption of a common discount factor across the agents prevents a long-run concentration of wealth in the hands of the most patient individual. A result different from ours is also obtained by Le Van, Nguyen and Vailakis (2007) who show that without borrowing constraints consumption and leisure of impatient agents tend to zero in the long run.</footnote>

<footnote n="4">⁴While giving an interpretation of our results, briefly we will discuss also the local stability of the other steady state.</footnote>

<footnote n="5">⁵Woodford (1986) also studies a model with two types of households. However, he postulates the existence of two classes of consumers, workers and capitalists, within a monetary economy.</footnote>

<footnote n="6">⁶In Becker and Foias (1987, 1994) a similar result holds, but the patient agent supplies one unit of labor.</footnote>

<footnote n="7">⁷Notice that, in contrast to the model with inelastic labor supply (Becker and Foias (1994)), since we focus on the steady state where the patient consumer supplies no labor, his income</footnote>
# 1 Introduction

Growth theorists are usually confronted with the question of convergence of an economic system. In neoclassical model of capital accumulation, commonly addressed questions are whether economies converge to the same long-run equilibrium, how convergence takes place and whether is monotonic, how fundamentals affect the stability properties of equilibrium.<fnref n="2" />

The most influential growth model is undoubtedly Ramsey which is characterized, in its basic version, by a representative agent with exogenous labor supply, saddle-path stability, equilibrium uniqueness and optimality. In order to add a degree of realism, economic literature has not only introduced various kinds of market imperfections, but also considered elastic labor supply and agents' heterogeneity.

On the one side, a class of papers have been devoted in the last decade to shed a light on the interplay between the consumption-labor arbitrage and the mechanism of capital accumulation in models with a representative consumer. The reader is referred, among others, to De Hek (1998), a one-sector model with an infinite-lived agent; to Bosi, Magris and Venditti (2005), a two-sector model; to Ladron-de-Guevara, Ortiguiera and Santos (1997), an endogenous growth model; to Nourry (2001) and Nourry and Venditti (2006), overlapping generations economies; to, eventually, Pintus (2006) and Garnier, Nishimura and Venditti (2006), models with externalities.

On the other side, there are papers with heterogeneous consumers which focus on the convergence of capital accumulation and distribution in the long run. Unlike the case with a representative agent, consumers' heterogeneity promotes borrowing transactions. The existence of borrowing constraints plays a key role on the dynamics of capital accumulation. Becker (1980), Becker and Foias (1987, 1994) and Sorger (1994) study economies where agents discount future differently, while Hernandez (1991) consider heterogenous but equally patient consumers.

Surprisingly, few works have investigated the role of elastic labor supply on equilibrium transition and the long run when consumers are heterogeneous. In this respect, most papers focus on heterogeneity in wealth (Sorger (2000), Ghiglino and Sorger (2002), Garcia-Penalosa and Turnovsky (2006)).

In this paper, we are interested in the effects of endogenous labor supply on the saddle-path stability when consumers' heterogeneity concerns not only their endowments but also their preferences. In this connection, we consider infinite-lived consumers with preferences which are additively separable over time, but depend on consumption and leisure at each period. Heterogeneity is now threefold and turns on capital wealth, time preference and intra-temporal preferences. In addition, in line with Becker (1980), Becker and Foias (1987, 1994), Hernandez (1991) and Sorger (1994), we assume that consumers cannot borrow against their future labor income. This borrowing constraint implicitly

<footnote n="2">²Barro and Sala-i-Martin (1995) provide an introductory but representative survey of the literature.</footnote>
a second channel works: the capital stock affects the real wage which in turn affects the aggregate labor supply, the capital intensity and finally the return on capital. Since preferences underlie and determine the elasticity of labor supply with respect to the real wage, they have great influence on dynamics.

The rest of the paper is organized as follows. The model with heterogeneous consumers, elastic labor and borrowing constraints is presented in Section 2. In Section 3, we develop sufficient conditions for the existence of an intertemporal equilibrium. Steady states are analyzed in Section 4. Section 5 focuses on the stability properties and the occurrence of bifurcations. Section 6 concludes, while technical details are gathered in the Appendix.

## 2 Microeconomic behaviors

We address the saddle-path stability issue in a discrete time ($t = 0, 1, ..., \infty$) growth model with heterogeneous agents, endogenous labor supply and borrowing constraints. Consumers are differently endowed with capital and have different preferences. In this respect, we assume a twofold kind of heterogeneity in tastes: on the one hand, heterogeneous discounting; on the other hand, different instantaneous utilities in consumption and leisure across the households.

There is a finite number ($1+n$) of heterogeneous infinite-lived agents, who are progressively labeled by $i = 0, 1, ..., n$ according to their time preference, that is, to the (decreasing) ranking of their discount factors:

$$0 \le \beta_n \le \dots \le \beta_1 < \beta_0 < 1 \qquad (1)$$

We notice that a degree of heterogeneity between the most patient agent ($i = 0$) and the others is required at least.

At each period, the consumer $i$ is endowed with one unit of time that he shares between labor and leisure. We denote his consumption and labor supply at period $t$ with $c_{it}$ and $l_{it}$. Preferences are represented by a separable utility function

$$\sum_{t=0}^{\infty} \beta_t^i [u_i(c_{it}) + v_i(1 - l_{it})] \qquad (2)$$

and each consumer maximizes (2) with respect to a stream $(k_{it+1}, c_{it}, l_{it})_{t=0}^{\infty}$ under a sequence of budget constraints: $c_{it} + k_{it+1} - \Delta k_{it} \le r_t k_{it} + w_t l_{it}$, a sequence of borrowing constraints: $k_{it+1} \ge 0$, and a positive labor supply: $l_{it} \ge 0$. For each individual, the initial endowment of capital is given by $k_{i0} \ge 0$.

We choose the final good as the numeraire; $r$ and $w$ denote the real interest rate and the real wage, respectively, while $\delta = 1 - \Delta \in (0, 1)$ the depreciation rate of capital. The utility function satisfies the following assumption:

**Assumption 1** $u_i(c_i)$ and $v_i(1-l_i)$ are continuous functions defined on $[0, +\infty)$ and $[0, 1]$, and $C^2$ on $(0, +\infty)$ and $(0, 1)$, respectively; strictly increasing ($u'_i(c_i) > 0$, $v'_i(1-l_i) > 0$) and strictly concave ($u''_i(c_i) < 0$, $v''_i(1-l_i) < 0$)

<footnote>reduces to capital income.</footnote>
means that, in contrast to Le Van, Manh Hung and Vailakis (2006), markets
are incomplete.

Under these assumptions, we show that, as in Becker (1980), Becker and Foias (1987, 1994), and Sorger (1994), the most patient household owns the whole capital stock in a neighborhood of the steady state, whereas the others consume their per-period labor income. In the rest of the paper, since we are mainly interested in local dynamics around the steady state, we can focus directly on such equilibria with extreme distribution of capital.

Under usual hypotheses, we prove the existence of a unique steady state
in contrast with Sorger (2000) who obtains a continuum of long-run equilibria.
Sorger's assumption of a common discount factor across the agents prevents a
long-run concentration of wealth in the hands of the most patient individual.
Survival of many capital owners in the long run entails also a sensitivity of the
steady state to disaggregate initial conditions, namely the initial distribution of
wealth, accounting for a continuum of stationary equilibria.

When there are no impatient agents in the economy, our model gets closer to De Hek (1998) with a representative agent. However, normality assumptions on consumption and leisure prevents us to obtain multiple steady states in contrast with him, who obtains multiplicity under very unusual conditions.

In order to study the convergence to the (unique) steady state, while under-
lining the role of heterogeneous preferences and elastic labor supply, we focus on
their effects on the saddle-path stability and the occurrence of a flip bifurcation.
In economies with inelastic labor supply (Becker and Foias (1994)), the occur-
rence of endogenous cycles requires sufficiently weak capital-labor substitution.
Introducing leisure in the utility functions, we find that the impatient agent's
preferences play a role on the stability properties of the steady state.

We derive conditions for saddle-path stability and the occurrence of two-
period cycles under very general preferences in consumption and leisure. Then,
in order to provide a more explicit characterization and to check the robustness
of saddle-path stability, we focus on four particular cases.

First, the representative agent's case is nested in the general model and de-
serves some comparative comment with the existing literature. Second, hetero-
geneous preferences are considered, but separable in consumption and leisure,
in order to simplify and clarify conditions for cycles. Third, a KPR<fnref n="3" /> speci-
fication is taken into account to prove that an elastic labor supply promotes
stability. Eventually, the case of homogeneous preferences confirms how robust
saddle-path stability is.

In the first case (representative (patient) agent), we gets closer to De Hek
(1998). Under very usual restrictions, such as normality, we find a unique capital
path, which prevents the economy from deterministic fluctuations and, shortly,
we stress his lack of robustness.

In the other cases, heterogeneity is restored. When preferences are separa-
ble in consumption and leisure, the impatient agents’ intertemporal substitution
in consumption matters. More precisely, when their elasticity of intertemporal

<footnote n="3">³See King, Plosser and Rebelo (1988).</footnote>
0). In addition, the Inada conditions $\lim_{c_i \to 0} u'_i(c_i) = \lim_{l_i \to 1} v'_i(1-l_i) = +\infty$ are verified.

We denote the aggregate capital and labor with $k_t$ and $l_t$, and the capital intensity with $a_t \equiv k_t/l_t$. The representative firm produces the final good and maximizes the profits

$$y_t - r_t k_t - w_t l_t$$

with respect to the inputs $(k_t, l_t)$ under a technology constraint: $y_t \le F(k_t, l_t)$. The production function is also characterized:<fnref n="8" />

**Assumption 2** $F(k, l)$ is a continuous function defined on $[0, +\infty)^2$ and $C^2$ on $(0, +\infty)^2$, homogeneous of degree one, strictly increasing in each argument $(F_k(k, l) > 0, F_l(k, l) > 0)$ and strictly concave $(F_{kk}(k, l) < 0, F_{kl}(k, l) > F_{ll}(k, l) > F_{kl}(k, l)^2)$. In addition, $F(0, 0) = 0$ and the boundary (Inada) conditions $\lim_{a \to 0} f'(a) = +\infty$ and $\lim_{a \to +\infty} f'(a) < 1/\beta_0 - \Delta$ are satisfied, where $f(a) \equiv F(a, 1)$ denotes the product per worker.

In the next section, we define and prove the existence of an intertemporal equilibrium under sufficient conditions, before analyzing steady states and local dynamics.

# 3 Intertemporal equilibrium

A definition of equilibrium is now provided:

**Definition 1** An equilibrium of the economy $E = (F, (k_{i0}, \beta_i, u_i, v_i)_{i=0}^n)$ is an intertemporal sequence $(r_t, w_t, k_t, l_t, (k_{it}, l_{it}), (c_{it})_{i=0}^n)_{t=0}^\infty$ which satisfies the following conditions:

(D1) the sequence $(r_t, w_t)_{t=0}^\infty$ is strictly positive;

(D2) given $(r_t, w_t)_{t=0}^\infty$, $(k_t, l_t)$ solves the firm's program for $t = 0, \dots, \infty$;

(D3) given $(r_t, w_t)_{t=0}^\infty$, $(k_{it+1}, c_{it}, l_{it})_{t=0}^\infty$ solves the ith consumer's program for $i = 0, 1, \dots, n$;

(D4) the product market clears: $y_t = \sum_{i=0}^n (c_{it} + k_{it+1} - \Delta k_{it})$ for $t = 0, \dots, \infty$;

(D5) the capital market clears: $k_t = \sum_{i=0}^n k_{it}$ for $t = 0, \dots, \infty$;

(D6) the labor market clears: $l_t = \sum_{i=0}^n l_{it}$ for $t = 0, \dots, \infty$.

In the sequel, $R_t \equiv r_t + \Delta$ will denote the interest factor. The next proposition provides sufficient conditions to ensure the existence of such equilibrium.

<footnote n="8">⁸The following notations will hold across the paper: $F_{x_i}(x) \equiv \partial F(x)/\partial x_i$ and $F_{x_i x_j}(x) \equiv \partial^2 F(x)/\partial x_i \partial x_j$ with $i, j = 1, 2$ and $x \equiv (x_1, x_2) \equiv (k, l)$.</footnote>
substitution becomes greater than one, the introduction of elastic labor supplies
promotes stability by making conditions for cycles more demanding in terms of
technological parameters. However, the opposite conclusion holds when elastic-
ity of intertemporal substitution in consumption is less than one.

The class KPR of utility functions is used in growth literature to lighten restrictions on intra-temporal preferences in order to get positive growth rates. In this case, the patient agent's preferences matter, while the impatient agents' ones play no role on dynamics. More precisely, convergence to the steady state requires an elasticity of intertemporal substitution in consumption not too weak and a leisure utility not too concave, whatever the elasticity of capital-labor substitution.

Finally, when preferences are homogeneous, the patient agent's degree of
homogeneity becomes a key parameter. We prove that when it is sufficiently
close to one (including the CES and the Cobb-Douglas cases), we recover the
saddle-path stability, whatever shape the other fundamentals have.

To understand why preferences play a role on stability properties, we mind
that, when labor supply is inelastic, instability ensues from a negative response
of the patient agent's income to a rise in the capital stock. When labor supplies
become elastic, they also affect his income. In addition, the endogenous labor
supplies vary because an increase in the capital stock raises the real wage. Since
preferences underlie and determine the elasticities of labor supply with respect
to the real wage, unsurprisingly they have a great influence on dynamics.

The rest of the paper is organized as follows. The general model with het-
erogeneous consumers, elastic labor and borrowing constraints is presented in
Section 2. Section 3 is devoted to compute a two-dimensional system represent-
ing the equilibrium dynamics. Existence and uniqueness of the steady state are
proved in Section 4. Section 5 focuses on the stability properties and the occur-
rence of bifurcations under unspecified preferences. Sections 6 characterizes an
economy with representative consumer. Sections 7 to 9 revisit the general find-
ings in the case of separable, KPR or homogeneous preferences, respectively. In
Section 10, we provide an economic interpretation of the results, keeping in mind
the stability issue. Section 11 concludes, while technical details are gathered in
the Appendix.

2 A growth model with heterogeneous households

We address the saddle-path stability issue in a discrete time growth model with heterogeneous agents and borrowing constraints. Consumers are differently endowed with capital and have different preferences. In this respect, we assume a twofold kind of heterogeneity in tastes: on the one hand, heterogeneous discount factors; on the other hand, different instantaneous utilities in consumption and leisure across the households.
**Proposition 2** Let an economy E satisfying Assumptions 1 and 2, and a sequence $(k_t, l_t, (k_{it}, l_{it}, c_{it}))_{i=0}^n)_{t=0}^\infty$ verifying conditions (P1)-(P8) below for $t = 0, \dots, \infty$ and $i = 0, 1, \dots, n$.

(P1) $k_t > 0, l_t > 0, k_{it} \ge 0, 0 \le l_{it} < 1, c_{it} > 0;^9$

(P2) $u_i'(c_{it}) w_t \le v_i'(1 - l_{it})$, with equality when $l_{it} > 0$;

(P3) $u_i'(c_{it}) \ge \beta_i R_{t+1} u_i'(c_{it+1})$, with equality when $k_{it+1} > 0$;

(P4) $c_{it} + k_{it+1} - \Delta k_{it} = F_k(k_t, l_t) k_{it} + F_l(k_t, l_t) l_{it}$;

(P5) $k_t = \sum_{i=0}^{n} k_{it}$;

(P6) $l_t = \sum_{i=0}^{n} l_{it}$;

(P7) $\lim_{t \to +\infty} \beta_i^t u_i'(c_{it}) k_{it+1} = 0$.

Then, $(r_t, w_t, k_t, l_t, (k_{it}, l_{it}, c_{it}))_{i=0}^n)_{t=0}^\infty$ is an equilibrium of E with prices:

(P8) $r_t = F_k(k_t, l_t)$ and $w_t = F_l(k_t, l_t)$.

**Proof.** See the Appendix.

As we prove in the next section, at the steady state, capital concentrates in the hands of the most patient agent. However, different types of stationary solutions can exist depending on his labor supply.

# 4 Steady state

As in Becker (1980), heterogeneous discounting promotes wealth inequalities, but in our framework, labor supplies also play a significant role, as stated in the next proposition.

**Proposition 3** There exists a unique steady state defined by the following properties:

(Q1) $R = \Delta + r$ and $w$ are constant;

(Q2) $R = 1/\beta_0 < 1/\beta_1 \le \dots \le 1/\beta_n$;

(Q3) $k_0 > 0$ and $k_i = 0$ for $i \ge 1$;

(Q4) $u_0'(c_0) w \le v_0'(1 - l_0)$ and, for $i \ge 1$, $u_i'(c_i) w = v_i'(1 - l_i)$;

(Q5) $c_0 = (R - 1) k_0 + w l_0$ and $c_i = w l_i$ for $i \ge 1$;

(Q6) $k = k_0 > 0$;

<footnote n="9">⁹We notice that Inada conditions in Assumption 1 rule out equilibria with $c_{it} = 0$ or $l_{it} = 1$.</footnote>
## 2.1 Consumers

There is a finite number (1 + n) of heterogeneous infinite lived agents, who are identified by a progressive label i = 0, 1, ..., n according to their time preference, that is, to the (decreasing) ranking of their discount factors:

$$0 \le \beta_n \le \dots \le \beta_1 < \beta_0 < 1 \qquad (1)$$

We notice that a degree of heterogeneity between the more patient agent (i = 0) and the others is required at least.

At each period, the consumer i is endowed with one unit of time that he shares between labor and leisure. We denote his consumption and labor supply at period t with $c_{it}$ and $l_{it}$. Preferences are represented by an utility function in consumption and leisure which is separable over time:

$$\sum_{t=0}^{\infty} \beta_i^t u_i (c_{it}, 1 - l_{it}) \qquad (2)$$

Consumer i maximizes (2) under a sequence of budget constraints:

$$c_{it} + k_{it+1} - \Delta k_{it} \le r_t k_{it} + w_t l_{it} \qquad (3)$$

and a sequence of borrowing constraints:

$$k_{it} \ge 0 \qquad (4)$$

Notation is quite usual: r and w denote the interest rate and the wage, respectively, while $\Delta \equiv 1 - \delta$ with $\delta \in (0, 1)$ the depreciation rate of capital. The initial (heterogeneous) endowments $k_{i0} \ge 0$ are given, with at least one strictly positive.

**Assumption 1** The utility function $u_i(c_i, 1 - l_i)$ is defined on $[0, +\infty) \times [0, 1]$ for every $i = 0, \dots, n$; it is $C^2$ on $(0, +\infty) \times (0, 1)$, increasing in each argument and concave:<fnref n="4" />

$$u_{i11} < 0 < u_{i1} \qquad (5)$$

$$u_{i22} < 0 < u_{i2} \qquad (6)$$

$$u_{i12}u_{i21} \le u_{i11}u_{i22} \qquad (7)$$

and satisfies the boundary conditions $\lim_{c_i \to 0} u_{i1}(c_i, 1 - l_i) / u_{i2}(c_i, 1 - l_i) = +\infty$, $\lim_{l_i \to 1} u_{i1}(c_i, 1 - l_i) / u_{i2}(c_i, 1 - l_i) = 0$ on $(0, +\infty) \times (0, 1)$.

We also introduce a kind of normality between consumption and leisure.

**Assumption 2**

$$u_{i11}/u_{i1} < u_{i21}/u_{i2} \text{ and } u_{i22}/u_{i2} < u_{i12}/u_{i1} \qquad (8)$$

<footnote n="4">⁴In the following, we set $u_{ij} \equiv \partial u_i(x_1, x_2) / \partial x_j$ for $j = 1, 2$ and $u_{ijk} \equiv \partial u_{ij}(x_1, x_2) / \partial x_k$ for $k = 1, 2$.</footnote>
$$ (Q7) \quad l = \sum_{i=0}^{n} l_i. $$

**Proof.** See the Appendix.

We observe that this proposition does not exclude the existence of two types of steady state where, respectively, consumer $i=0$ supplies no labor ($l_0 = 0$) or a strictly positive amount ($0 < l_0 < 1$). Since the entire stock of capital is eventually held by one individual, on the one hand, the latter can renounce to work by consuming his capital income, while, on the other hand, the others are forced to work to consume.

In addition, the existence of borrowing constraints prevents the impatient agents from consuming more today and working hard tomorrow to refund the debt as in Le Van, Nguyen and Vailakis optimal growth model (2007), where impatient agents experience vanishing consumption bundles and leisure time in the long run.

We notice that, in contrast to Sorger (2000) who also introduces endogenous labor supply in an economy populated by heterogeneous agents, there is no longer room for a continuum of stationary solutions parametrized by the initial distribution of wealth. In Sorger's paper, agents start with heterogeneous endowments of capital, but share the same preferences and specifically the same discount factor. In our case, the most patient agent ends up to hold the whole capital stock and the stationary distribution of wealth is determined by his higher discount factor irrespective of any initial distribution.

The patient consumer's discount factor plays a great role in determining capital accumulation. To clarify notation, we will set from now on $\beta \equiv \beta_0$ and refer equivalently to the most patient agent as the "capitalist".

The steady state where the patient agent supplies no labor is undoubtedly of interest. Two social classes emerge: the capitalist who does not work but consume and invest the total amount of capital income, and $n$ workers who spend only their labor earnings without investing. In this respect, our model becomes close to Woodford (1986) who considers a monetary economy where impatient agents are subject to additional financial constraints and the existence of two classes of agents is postulated.

For the sake of conciseness but without significant loss of generality, we will focus on this steady state to characterize the local dynamics. Nevertheless, a short comment on the stability of the other steady state will be also provided with the interpretation of dynamic results.

# 5 Dynamics around the steady state without capitalist's labor supply

In this section, we study the stability properties and the occurrence of cycles in the neighborhood of the steady state where the capitalist supplies no labor. As seen above, we consider only equilibria where the most patient agent holds the
We observe that a more usual definition of normality is entailed by inequal-
ities (8) jointly with the consumption-leisure arbitrage ($u_{i2} = wu_{i1}$) as equilib-
rium condition in Proposition 1 below.

As intuition suggests and economic literature confirms,<fnref n="5" /> the patient agents
will hold the whole stock of capital near the steady state. In order to characterize
the equilibrium with high concentration of capital, we need a formal statement.

**Proposition 1** Near a steady state, the impatient agents (i = 1, ..., n) end up with no capital (k<sub>it</sub> = 0), while the patient one (i = 0) ends up to hold the entire amount of capital (k<sub>0t</sub> > 0).

On the one hand, a consumption-leisure arbitrage condition holds for every
agent including the patient one (i = 0, 1, ..., n):

$$
u_{i2}(c_{it}, 1 - l_{it}) = u_{i1}(c_{it}, 1 - l_{it}) w_t \quad (9)
$$

On the other hand, the patient capitalist (i = 0) and the impatient agents
(i ≥ 1) have different sequences of budget constraints:

$$
c_{0t} + k_{0t+1} = R_t k_{0t} + w_t l_{0t} \quad (10)
$$

$$
c_{it} = w_t l_{it}, \text{ for } i = 1, \dots, n \quad (11)
$$

and smooth consumption differently over time: the capitalist according to a Euler equation:

$$
\frac{u_{01}(c_{0t}, 1 - l_{0t})}{u_{01}(c_{0t+1}, 1 - l_{0t+1})} = \beta_0 R_{t+1} \quad (12)
$$

whereas the impatient consumers according to the following inequality:

$$
\frac{u_{i1}(w_t l_{it}, 1 - l_{it})}{u_{i1}(w_{t+1} l_{it+1}, 1 - l_{it+1})} > \beta_i R_{t+1} \quad (13)
$$

Individual transversality conditions are also satisfied.<fnref n="6" />

**Proof.** See the Appendix.

As equation (12) suggests, the patient consumer’s discount factor plays a
great role in determining capital accumulation. To lighten notation, we will set
from now on $\beta \equiv \beta_0$ and refer equivalently to the more patient agent as “the
capitalist”.

For the sake of conciseness, we introduce also the elasticities of marginal
utility through which slope and concavity of the utility functions will be char-
acterized:

$$
\begin{bmatrix} \epsilon_{i11} & \epsilon_{i12} \\ \epsilon_{i21} & \epsilon_{i22} \end{bmatrix} = \begin{bmatrix} \frac{u_{i11}c_i}{u_{i1}} & \frac{u_{i12}(1-l_i)}{u_{i1}} \\ \frac{u_{i21}c_i}{u_{i2}} & \frac{u_{i22}(1-l_i)}{u_{i2}} \end{bmatrix} \quad (15)
$$

$$
\lim_{t \to +\infty} \beta^t u_{01} (c_{0t}, 1 - l_{0t}) k_{t+1} = 0 \quad (14)
$$

<footnote n="5"><sup>5</sup>Among others, the reader is referred to the seminal work of Becker (1980).</footnote>

<footnote n="6"><sup>6</sup>Since $k_{it+1} \to 0$ for every $i \ge 1$, what actually matters is the capitalist's transversality condition:</footnote>
capital stock ($k_t = k_{0t}$) and impatient agents supply labor ($l_{0t} = 0$, $l_t = \sum_{i=1}^n l_{it}$, $i \ge 1$).

Some elasticities are introduced to simplify the analysis. The elasticities of intertemporal substitution in consumption and leisure evaluated at the steady state are respectively given by

$$ \mu_{1i} \equiv -u'_{i}(c_{i}) / [c_{i}u''_{i}(c_{i})] > 0 $$

$$ \mu_{2i} \equiv -v'_{i}(1 - l_{i}) / [(1 - l_{i}) v''_{i}(1 - l_{i})] > 0 $$

Moreover, the consumption-leisure arbitrage:

$$ u'_{i}(wl_{i})w = v'_{i}(1 - l_{i}) \qquad (3) $$

defines implicitly the labor supply $l_i(w)$ of each impatient agent $i \ge 1$, with elasticity:

$$ \xi_i \equiv \frac{l'_i(w)w}{l_i(w)} = \frac{\mu_{1i} - 1}{1 + \frac{\mu_{1i}}{\mu_{2i}} \frac{l_i}{1-l_i}} \qquad (4) $$

**Lemma 4** Under the Assumptions 1 and 2, the functions $l_i = l_i(w)$ are well defined.

**Proof.** See the Appendix.

It is important to note that we use a less common definition of labor supply. Indeed, labor supply is usually evaluated taking consumption or marginal utility of consumption as given.<fnref n="10" /> Then, the elasticity of labor supply simplifies to $\mu_{2i}(1 - l_i)/l_i$ and is always positive. On the contrary, our definition takes also into account the workers' budget constraint: $c_i = wl_i$. In this case, according to (4), labor supply is increasing in $w$ only if the intertemporal substitution in consumption is greater than one.

The weighted elasticity of labor supply $\xi = \sum_{i=1}^n \lambda_i \xi_i$, where weights are given by $\lambda_i \equiv l_i/l$, with $\sum_{i=1}^n \lambda_i = 1$, will play a noteworthy role in the sequel. This elasticity summarizes the workers' behavior and allows us to provide a specific contribution to the existing literature on heterogeneous discounting by analyzing the impact of endogenous labor supply on dynamics.

We will denote by $\sigma$ and $s$ the elasticity of capital-labor substitution and the capital share in total income both evaluated at the steady state.<fnref n="11" /> The labor market clearing condition $l = \sum_{i=1}^n l_i(w(k/l))$ implicitly defines an aggregate labor supply $l = l(k)$. We deduce the aggregate elasticity:<fnref n="12" />

$$ \varepsilon_{lk} \equiv \frac{l'(k)k}{l(k)} = \frac{\xi}{\xi + \sigma/s} $$

<footnote n="10">¹⁰When preferences are separable in consumption and leisure, these two definitions are equivalent.</footnote>

<footnote n="11">¹¹Setting $a \equiv k/l$ and $f(a) \equiv F(a, 1)$ (see Assumption 2), we obtain:</footnote>

<footnote>$$ s(a) \equiv af'(a)/f(a) \qquad (5) $$</footnote>

<footnote>$$ \sigma(a) \equiv [s(a) - 1]f'(a) / [af''(a)] \qquad (6) $$</footnote>

<footnote n="12">¹²In order to apply the Implicit Function Theorem, we require $\xi \neq -\sigma/s$.</footnote>
We notice that the concavity restrictions (5) and (6) (on the left-hand-side) and (7) now become:

$$ \varepsilon_{i11} < 0, \varepsilon_{i22} < 0 \text{ and } \varepsilon_{i11}\varepsilon_{i22} \geq \varepsilon_{i12}\varepsilon_{i21} \quad (16) $$

for $i = 0, 1, \dots, n$, respectively. Normality inequalities (8) (Assumption 2) also simplify:

$$ \varepsilon_{i11} < \varepsilon_{i21} \text{ and } \varepsilon_{i22} < \varepsilon_{i12} \quad (17) $$

for $i = 0, 1, \dots, n$.

## 2.2 Firms

We denote the aggregate capital and labor with $k_t$ and $l_t$, respectively. The representative firm produces the final good $y_t = F(k_t, l_t) \equiv f(a_t)l_t$ with a constant returns to scale technology and a degree of capital intensity $a_t \equiv k_t/l_t$, in order to maximize the profit $\pi_t \equiv y_t - r_t k_t - w_t l_t$ at every period $t$. The intensive production function $f$ satisfies standard neoclassical requirements.

**Assumption 3** The intensive production $f(a)$ is a continuous function of the capital-labor ratio $a \geq 0$, positive-valued and differentiable, as many times as needed, for $a > 0$, with $f''(a) < 0 < f'(a)$ and $f(0) = 0$, $\lim_{a \to 0} f'(a) = +\infty$, $\lim_{a \to +\infty} f'(a) = 0$.

Firms' profit maximizing behavior entails that production takes place in period $t$ so that the capital-labor ratio $a_t$ equates the real rental rate of capital services $r_t$ with the marginal productivity of capital and the real wage $w_t$ with the marginal productivity of labor.

$$ r_t = f'(a_t) \equiv r(a_t) \quad (18) $$

$$ w_t = f(a_t) - f'(a_t) a_t \equiv w(a_t) \quad (19) $$

It is not unworthy to highlight the underlying relations existing between the elasticities of the interest rate with respect to capital and labor $r_k/k/r$, $r_l/l/r$, the analogous elasticities of the real wage $w_k/k/w$, $w_l/l/w$, the capital share in total income $s(a) \equiv f'(a)a/f(a) \in (0, 1)$ and the elasticity of capital-labor substitution: $\sigma(a) \equiv [af'(a)/f(a) - 1]f'(a)/[af''(a)] > 0$. Standard computations give $kr_k/r = -lr_l/r = -[1 - s(a)]/\sigma(a)$ and $kw_k/w = -lw_l/w = s(a)/\sigma(a)$.

## 2.3 Equilibrium

An intertemporal equilibrium is a sequence of prices (or, dually, of quantities) that clears the markets in every period. Equilibria in good and labor markets require:

$$ c_t + k_{t+1} - \Delta k_t = F(k_t, l_t) \quad \text{and} \quad l_t = \sum_{i=0}^{n} l_{it}, \quad (20) $$
Recall that the steady state where the capitalist's labor supply is zero ($l_0 = 0$) is defined by $r(k/l(k)) = 1/\beta - \Delta$ and $c_0 = (1/\beta - 1)k$. Around this steady state, dynamics are defined by a sequence $(c_{0t}, k_t)_{t=0}^{\infty}$, such that:

$$u'_{0}(c_{0t}) = \beta [\Delta + r(k_{t+1}/l(k_{t+1}))] u'_{0}(c_{0t+1}) \quad (7)$$

$$k_{t+1} = [\Delta + r(k_t/l(k_t))] k_t - c_{0t} \quad (8)$$

The issue of convergence to the steady state is now addressed through the analysis of a neighborhood of the steady state, where the intertemporal equilibrium defined in Proposition 2 precisely holds.

In order to characterize the stability properties of the steady state and the occurrence of local bifurcations, we proceed by linearizing the dynamic system (7)-(8) around the steady state $(k, c_0)$ defined in Proposition 3 (with $l_0 = 0$) and computing the Jacobian matrix $J$, evaluated at this steady state. Local dynamics are represented by a linear system $(dk_{t+1}/k, dc_{0t+1}/c_0)^T = J(dk_t/k, dc_{0t}/c_0)^T$. In the following, we exploit the fact that the trace $T$ and the determinant $D$ of $J$ are the sum and the product of the eigenvalues, respectively. As stressed by Grandmont, Pintus and de Vilder (1998), the stability properties of the system, that is, the location of the eigenvalues with respect to the unit circle, can be better characterized in the $(T, D)$-plane (see Figures 1-3).

More explicitly, we evaluate the characteristic polynomial $P(x) = x^2 - Tx + D$ at $-1$ and $1$. Along the line (AC), one eigenvalue is equal to $1$, i.e. $P(1) = 1 - T + D = 0$. Along the line (AB), one eigenvalue is equal to $-1$, i.e. $P(-1) = 1 + T + D = 0$. On the segment [BC], the two eigenvalues are complex and conjugate with unit modulus, i.e. $D = 1$ and $|T| < 2$. Therefore, inside the triangle ABC, the steady state is a sink, i.e. locally indeterminate ($D < 1$ and $|T| < 1 + D$). It is a saddle point if $(T, D)$ lies on the right sides of both (AB) and (AC) or on the left sides of both of them ($|1 + D| < |T|$). It is a source otherwise. A (local) bifurcation arises when an eigenvalue crosses the unit circle, that is, when the pair $(T, D)$ crosses one of the loci (AB), (AC) or [BC]. $(T, D)$ depends on the structural parameters. We choose a parameter of interest and we study how $(T, D)$ moves with it in the $(T, D)$-plane. More precisely, according to the changes in the bifurcation parameter, a transcritical bifurcation (generically) occurs when $(T, D)$ goes through (AC), a flip bifurcation (generically) arises when $(T, D)$ crosses (AB), whereas a Hopf bifurcation (generically) emerges when $(T, D)$ goes through the segment [BC].

Differentiating system (7)-(8) around the steady state $(k, c_0)$, we obtain:

$$\frac{dc_{0t+1}}{c_0} = -(1 - \beta\eta)\eta\mu_{10}\frac{dk_t}{k} + [1 + (1 - \beta)\eta\mu_{10}]\frac{dc_{0t}}{c_0}$$

$$\beta \frac{dk_{t+1}}{k} = (1 - \beta\eta) \frac{dk_t}{k} - (1 - \beta) \frac{dc_{0t}}{c_0}$$

where $\eta \equiv r(1 - \varepsilon_{lk})(1 - s)/\sigma$.

The determinant $D$ and the trace $T$ of the associated Jacobian matrix are,
where $c_t \equiv \sum_{i=0}^{n} c_{it}$ denotes the aggregate consumption, for $t = 0, 1, \dots, +\infty$. We observe that equation (20) is immediately obtained by aggregating budget constraints (10) and (11) across the individuals. Since the patient agent ends up to hold all the capital stock, we have also the following condition:

$$k_t = k_{0t} \qquad (21)$$

In order to find the equilibrium path prevailing around the steady state and characterized by a concentration of capital in the hands of a single agent (equation (21)), we derive the dynamic system and study the local dynamics around its stationary solution.

## 3 Dynamic system

Equation (9) is informative about the slope of the patient agent's labor supply $l_0$. We keep (9) with $i = 0$ and we solve (locally) for $l_0$:

$$l_{0t} = l_0(w_t, c_{0t}) \qquad (22)$$

Substituting (11) in (9), we transform the consumption-arbitrage condition of the $i$th impatient agent in an implicit equation relating his labor supply to the real wage:

$$u_{i2}(w_t l_{it}, 1 - l_{it}) = u_{i1}(w_t l_{it}, 1 - l_{it}) w_t \qquad (23)$$

or, locally,

$$l_{it} = l_{i}^{*}(w_{t}) \qquad (24)$$

We observe that the labor supply of an impatient agent just depends on the real wage, while the patient agent's labor supply depends also on his consumption demand. To ensure that labor supplies are correctly defined as functions, we use the assumptions made above on the shapes of utility and production functions.

**Lemma 2** If $\lim_{l_{0t} \to 0} u_{02}(c_{0t}, 1 - l_{0t})/u_{01}(c_{0t}, 1 - l_{0t}) < w_t$ and Assumptions 1-3 are satisfied, the functions $l_{it} = l_i^*(w_t)$ and $l_{0t} = l_0(c_{0t}, w_t)$ are well defined.

**Proof.** See the Appendix.

Since the purpose of the section is deriving a dynamic system, we need to reduce the number of variables, aggregating, for instance, the labor supplies. In this respect, putting together equations (19), (20), (22) and (24), we obtain:

$$l_t = l_0(w(k_t/l_t), c_{0t}) + \sum_{i=1}^{n} l_i^*(w(k_t/l_t)) \qquad (25)$$

that is an implicit equation relating the aggregate labor supply $l$ to the aggregate capital $k$ and the individual consumption $c_0$. More explicitly, but locally under the usual hypotheses of the implicit function theorem, we get:

$$l_t \equiv l(k_t, c_{0t}) \qquad (26)$$
respectively:

$$D(\sigma) = \frac{1}{\beta} - r \frac{1-s}{\sigma + s\xi}$$

$$T(\sigma) = \frac{1}{\beta} - r \frac{1-s}{\sigma + s\xi} \frac{1}{S} + 1$$

where

$$S \equiv [1 - (1 - \beta) \mu_{10}]^{-1} \notin [0, 1] \qquad (9)$$

In order to apply the geometrical method introduced by Grandmont, Pintus and de Vilder (1998), we need to choose carefully the parameter with respect to which the bifurcation analysis is led. An ideal choice should be a parameter significant in economic terms with a simple image, say linear, in the $(T, D)$-plane: for instance, an origin and a slope determines unambiguously a half-line. The elasticity of capital-labor substitution $\sigma$ meets both the criteria:<fnref n="13" /> the locus $\Sigma \equiv \{(T(\sigma), D(\sigma)) : \sigma \ge 0\}$ becomes either a (connected) segment or an (unconnected) half-line in the $(T, D)$-plane (see Figures 1-3) with origin $(\sigma = 0)$:

$$D(0) = \frac{1}{\beta} - r \frac{1-s}{s\xi}$$

$$T(0) = \frac{1}{\beta} - r \frac{1-s}{s\xi} \frac{1}{S} + 1$$

endpoint $(\sigma = +\infty)$:

$$D(+\infty) = 1/\beta > 1$$

$$T(+\infty) = 1 + 1/\beta = 1 + D(+\infty)$$

and slope $D'(\sigma)/T'(\sigma) = S$. From (9), we know that $S \notin [0, 1]$. Moreover, the endpoint $(T(+\infty), D(+\infty))$ lies on the line $(AC)$, above the point $C$.

The line including the locus $\Sigma$ always crosses the line $(AB)$,<fnref n="14" /> but, since $\Sigma$ is either a segment or a half-line, the intersection of $(AB)$ with $\Sigma$ could be empty. In order to characterize the shape of $\Sigma$ and provide clear-cut conditions of bifurcation, we need to understand the properties of trace and determinant in terms of $\sigma$.

Since $D'(\sigma) > 0$, when $\sigma$ increases from 0 to $+\infty$, the point $(T(\sigma), D(\sigma))$ moves always upward along $\Sigma$ from the origin $(T(0), D(0))$, to the endpoint $(T(+\infty), D(+\infty))$. If $\Sigma$ crosses the line $(AB)$, then a flip bifurcation (generically) occurs for a positive value of $\sigma$, say $\sigma_F$, corresponding to the intersection with $(AB)$ (equation $D(\sigma) = -T(\sigma) - 1$): $\sigma_F \equiv s(\varkappa - \xi)$ with

$$\varkappa \equiv \frac{1}{2} r \frac{1-s}{s} \frac{\beta}{1+\beta} \frac{1+S}{S} \qquad (10)$$

<footnote n="13">¹³In a Ramsey model with heterogeneous consumers and borrowing constraints, but inelastic labor supplies, Becker and Foias (1994) choose the same parameter to make the bifurcation analysis, while following a different (non-geometrical) approach.</footnote>

<footnote n="14">¹⁴Omit the case $S = -1$ generically.</footnote>
At the very core of dynamics is the capitalist's behavior: Replacing expressions (18), (19), (22) and (26) in his budget constraint (10) and in his Euler equation (12), eventually, we obtain the equilibrium dynamics:

$$ \frac{u_{01}(c_{0t}, 1 - l_0(w(k_t/l(k_t, c_{0t})), c_{0t}))}{u_{01}(c_{0t+1}, 1 - l_0(w(k_{t+1}/l(k_{t+1}, c_{0t+1})), c_{0t+1}))} = \beta \left[ \Delta + r \left( \frac{k_{t+1}}{l(k_{t+1}, c_{0t+1})} \right) \right] \quad (27) $$

$$ c_{0t} + k_{t+1} - \Delta k_t = r \left( \frac{k_t}{l(k_t, c_{0t})} \right) k_t + w \left( \frac{k_t}{l(k_t, c_{0t})} \right) l_0 \left( w \left( \frac{k_t}{l(k_t, c_{0t})} \right), c_{0t} \right) \quad (28) $$

where the initial condition $k_0 > 0$ is given<fnref n="7" /> and the transversality condition (14) holds.

Dynamic system (27)-(28) deserves some comments. It holds at every period, $t = 0, 1, \dots, +\infty$, and defines an intertemporal sequence $(c_{0t}, k_t)_{t \ge 0}$. Usually, the dynamic system of the Ramsey model is written in the representative agent's pair of variables $(k, c)$. The capital stock $k$ is a predetermined state variable, while the consumption expenditure $c$ jumps in order to make the equilibrium path positive and compatible with the transversality condition. In our model, the aggregate capital remains predetermined, but the aggregate consumption is replaced by the disaggregate capitalist's consumption.<fnref n="8" />

In the following, first, the uniqueness of the steady state will be stated; then, saddle-path stability and the existence of flip and period-doubling bifurcations will be characterized through the analysis of a linear approximation of system (27)-(28) around the steady state.

## 4 Steady state

A stationary state of dynamic system (27)-(28) is a solution $(k, c_0) = (k_t, c_{0t})$ of the following system for all $t = 0, 1, \dots, +\infty$:

$$ \frac{1}{\beta} - \Delta = r \left( \frac{k}{l(k, c_0)} \right) \quad (29) $$

$$ c_0 = \frac{1 - \beta}{\beta} k + w \left( \frac{k}{l(k, c_0)} \right) l_0 \left( w \left( \frac{k}{l(k, c_0)} \right), c_0 \right) $$

where the functions $l_0(w, c_0)$ and $l(k, c_0)$ are now defined at the steady state by (22) and (26), respectively. In the next proposition, we analyze the existence and the uniqueness of such stationary solution:

**Proposition 3** *Under Assumptions 1-3, there exists a unique steady state.*

<footnote n="7">⁷The initial aggregate capital stock is defined by $k_0 = \sum_{i=0}^{n} k_{i0}$. The equilibria considered in this section implicitly requires an initial distribution of capital close enough to that of Proposition 1.</footnote>

<footnote n="8">⁸Writing an equivalent system in the pair of variables $(k, c)$, where $c$ is the aggregate consumption, or in the pair $(k, l)$, will complicate needlessly the size of equations under study, whereas the associated eigenvalues are invariant to the choice of variables.</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 1: $S > 1$</figcaption></figure>

The necessary and sufficient conditions in order to obtain saddle-path stability or two-period cycles are summarized in the next proposition.

**Proposition 5** Let

$$ \sigma_F \equiv s(\varkappa - \xi) \qquad (11) $$

where $\varkappa$ is given by (10). Assuming that Assumption 1-2 hold, the stability properties of the steady state without capitalist's labor supply are generically the following.

(1) If the impatient agents' average elasticity of labor supply is greater than a threshold, that is, $\xi \ge \varkappa$, the steady state is a saddle, whatever the degree of capital-labor substitution $\sigma > 0$.

(2) If the impatient agents' average elasticity of labor supply is smaller than a threshold, that is, $\xi < \varkappa$, the steady state is a source when inputs are weak substitutes ($0 < \sigma < \sigma_F$), and it is a saddle when they are more substitutable ($\sigma > \sigma_F$). When $\sigma$ crosses $\sigma_F$, the system undergoes a flip bifurcation.

**Proof.** See the Appendix.
**Proof.** See the Appendix.

In contrast to Sorger (2000), there is no room for a continuum of stationary
solutions parametrized by the initial distribution of wealth. In Sorger's paper,
agents have heterogeneous initial capital endowments, but share the same pref-
erences and in particular the same discount factor. Since in our case the more
patient agent ends up to hold the whole capital stock, the stationary distribution
of wealth is determined by his heterogeneous discount factor and the existence
of borrowing constraints, and does not meet Sorger's conditions for multiplicity.

Eventually, we notice that, setting $n = 0$ and $l_t = l_0(w_t, c_{0t})$, we get the
optimal growth model with representative consumer as particular case. In this
respect, Proposition 3 still applies. Our conventional assumptions entail results
which appear in open contrast with those obtained by De Hek (1998) who
analyzes an aggregate optimal growth model with leisure-depending preferences.
In order to reconcile the papers, we simply observe that his multiplicity of steady
states rest on some unusual assumptions, namely the lack of normality between
consumption and leisure.<fnref n="9" />

**5 On the saddle-path stability**

Convergence to the steady state is the core of the paper. This issue is ad-
dressed through a local analysis, i.e., through the analysis of a neighborhood of
the steady state, where the intertemporal equilibrium defined in Proposition 1
precisely holds.

In order to characterize the stability properties of the steady state and the
occurrence of local bifurcations, we proceed by linearizing the dynamic system
(27)-(28) around the steady state $(k, c_0)$ and computing the Jacobian matrix
$J$, evaluated at this steady state. Local dynamics are represented by a linear
system $(dk_{t+1}/k, dc_{0t+1}/c_0)^T = J (dk_t/k, dc_{0t}/c_0)^T$. In the following, we exploit
the fact that the trace $T$ and the determinant $D$ of $J$ are the sum and the product
of the eigenvalues, respectively. As emphasized by Grandmont, Pintus and de
Vilder (1998), the stability properties of the system, that is, the location of the
eigenvalues with respect to the unit circle, can be better characterized in the
$(T, D)$-plane (see Figures 1-3).

In concrete terms, we evaluate the characteristic polynomial $P(\alpha) \equiv \alpha^2 - T\alpha + D$ at $-1, 0, 1$. Along the line (AC), one eigenvalue is equal to 1, i.e., $P(1) = 1 - T + D = 0$. Along the line (AB), one eigenvalue is equal to $-1$, i.e., $P(-1) = 1 + T + D = 0$. On the segment [BC], the two eigenvalues are complex and conjugate with unit modulus, i.e., $D = 1$ and $|T| < 2$. Therefore, inside the triangle ABC, the steady state is a sink, i.e., locally indeterminate ($D < 1$ and $|T| < 1 + D$). It is a saddle point if $(T, D)$ lies on the right sides of both (AB) and (AC) or on the left sides of both of them ($|1 + D| < |T|$). It is a source otherwise. A (local) bifurcation arises when an eigenvalue crosses

<footnote n="9">⁹In Section 6, the unconventional equilibrium transitions found by De Hek (1998) are questioned on the same basis.</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 2: $S < -1$</figcaption></figure>

This proposition states that the steady state is always locally determinate. However, saddle-path stability and local convergence are not always ensured. When the substitution between capital and labor is sufficiently weak, the steady state can lose stability through the occurrence of cycles of period two.

In addition, as pointed out in Proposition 5, not only technology (namely, parameters $s$ and $\sigma$) and capitalist's preferences (see (9) and (10)) influence the dynamics, but also the workers' preferences ($i \ge 1$) summarized by $\xi$: the introduction of endogenous labor supply matters, that is the impatient agents' intratemporal arbitrage between consumption and leisure. In order to make this point clearer, let us first investigate the limit case where impatient consumers supply labor inelastically, and secondly the more general case with elastic labor supply.
the unit circle, that is, when the pair (T, D) crosses one of the loci (AB), (AC) or [BC]. (T, D) depends on the structural parameters. We choose a parameter of interest and we study how (T, D) moves with it in the (T, D)-plane. More explicitly, according to the changes in the bifurcation parameter, a transcritical bifurcation (generically) occurs when (T, D) goes through (AC), a flip bifurcation (generically) arises when (T, D) crosses (AB), whereas a Hopf bifurcation (generically) emerges when (T, D) goes through the segment [BC].

The expressions of T and D involve the elasticities of labor supply, which are a central link in the chain of effects accounting for the degree of stability. From (22) and (24), one defines these elasticities as $\xi_0 \equiv (\partial l_0 / \partial w)(w/l_0)$ for the capitalist, $\xi_i \equiv l_i^{*'}(w)w/l_i$ for the impatient agents. The average elasticity of labor supply $\xi \equiv \sum_{i=0}^n \lambda_i \xi_i$ is naturally defined, by weighting the individual elasticities with the individual labor shares in total labor supply $\lambda_i \equiv l_i/l_i$, $i = 0, 1, \dots, n$.<fnref n="10" /> Another average elasticity of labor supply of interest is $\tilde{\xi} \equiv \sum_{i=1}^n \lambda_i \xi_i / (1 - \lambda_0)$, which takes in account only the impatient agents' elasticities $\xi_i$'s and summarizes the role of their preferences on the dynamics. Clearly, $\xi = \lambda_0 \xi_0 + (1 - \lambda_0) \tilde{\xi}$. We also denote the elasticity of capital-labor substitution with $\sigma$ and the capital share in total income evaluated at the steady state with $s$. Eventually, we need to introduce a reduced parameter $\rho \equiv \beta r(1-s)/s = (1 - \beta \Delta)(1-s)/s$, before presenting the determinant and the trace:<fnref n="11" />

$$D(\sigma) = \frac{1}{\beta} - \frac{\rho}{\beta} \frac{1}{\tilde{\xi} - \tilde{\xi}^* + \frac{\sigma}{s} \frac{1}{1-\lambda_0}} \quad (30)$$

$$T(\sigma) = 1 + \frac{1}{\beta} - \frac{\rho}{\beta} \frac{1}{\tilde{\xi} - \tilde{\xi}^* + \frac{\sigma}{s} \frac{1}{1-\lambda_0}} \frac{1}{S} \quad (31)$$

where

$$\tilde{\xi}^* \equiv \frac{1 - l_0}{l - l_0} \frac{\varepsilon_{011} + \rho(\varepsilon_{011} - \varepsilon_{021})}{\varepsilon_{011}\varepsilon_{022} - \varepsilon_{012}\varepsilon_{021}} < 0 \quad (32)$$

$$S \equiv \left[ 1 - \frac{(1 - \beta + \rho)(\varepsilon_{021} - \varepsilon_{011})(1 - l_0) + (1 - \beta + \rho\lambda_0)(\varepsilon_{012} - \varepsilon_{022})l}{(\varepsilon_{011}\varepsilon_{022} - \varepsilon_{012}\varepsilon_{021})(l - l_0)} \right]^{-1} \quad (33)$$

In order to apply the geometrical method introduced by Grandmont, Pintus and de Vilder (1998), we need to choose carefully the parameter with respect to which the bifurcation analysis is leaded. An ideal choice is a significant parameters from an economic point of view, which has, in addition, a linear image in the (T, D)-plane: a half-line is unambiguously determined by the origin and the slope. Fortunately, the elasticity of capital-labor substitution $\sigma$ meets both the criteria:<fnref n="12" /> The locus $\Sigma \equiv \{(T(\sigma), D(\sigma)) : \sigma \ge 0\}$ is either a (connected)

<footnote n="10">¹⁰Notice also that $\sum_{i=0}^n \lambda_i = 1$.</footnote>

<footnote n="11">¹¹The detailed computations of J, T and D are relegated in the Appendix.</footnote>

<footnote n="12">¹²In a Ramsey model with heterogeneous consumers and borrowing constraints, but in-elastic labor supplies, Becker and Foias (1994) choose the same parameter to make the (flip) bifurcation analysis, while following a different (non-geometrical) approach.</footnote>
<figure><img src="image_6.png" /><figcaption>Figure 3: $-1 < S < 0$</figcaption></figure>

**5.1 Inelastic labor supply**

The *i*th agent's labor supply becomes inelastic when $\mu_{2i} = 0$. Assume now that all the impatient agents supply labor inelastically. This implies that also the average elasticity of labor supply is zero, $\xi = 0$, and that the bifurcation value becomes $\sigma_F = s\kappa$. Therefore, impatient consumers' preferences and, more precisely, the elasticity of intertemporal substitution in consumption, play no role on the stability properties of the steady state and the occurrence of cycles.

We further notice that the bifurcation value $\sigma_F$ is strictly positive if and only if the capitalist has a weak elasticity of intertemporal substitution in consumption, $\mu_{10} < 2/(1-\beta)$. In addition, $\sigma_F$ is decreasing in the capitalist's elasticity of intertemporal substitution in consumption $\mu_{10}$, generally meaning that a weaker elasticity of intertemporal substitution in consumption, by preventing the intertemporal arbitrage, promotes cycles.<fnref n="15" />

<footnote n="15">¹⁵The mechanism is deepened in a following interpretative section. See also Becker and Foias (1987, 1994) for a close interpretation. However in their papers, in contrast to our</footnote>
segment or an (unconnected) half-line in the (T, D)-plane (see Figures 1-3) with an origin ($\sigma = 0$):

$$D(0) = \frac{1}{\beta} - \frac{\rho}{\beta} \frac{1}{\tilde{\xi} - \tilde{\xi}^*} \qquad (34)$$

$$T(0) = 1 + \frac{1}{\beta} - \frac{\rho}{\beta} \frac{1}{\tilde{\xi} - \tilde{\xi}^*} \frac{1}{S} \qquad (35)$$

an endpoint ($\sigma = +\infty$):

$$(T(+\infty), D(+\infty)) = (1 + 1/\beta, 1/\beta) \qquad (36)$$

and a slope $D'(\sigma)/T'(\sigma) = S$. From (33), we note that $S \notin [0, 1]$. Moreover, the endpoint $(T_1(+\infty), D_1(+\infty))$ is on the line (AC), above the point C.

<figure><img src="image_5.png" /><figcaption>Figure 1: $S > 1$</figcaption></figure>

The locus $\Sigma$ is either a segment or a half-line. Even if the line including $\Sigma$ always crosses the line (AB),<fnref n="13" /> the intersection of (AB) with $\Sigma$ could be

<footnote n="13">¹³The case $S = -1$ is generically omitted.</footnote>
Now, let us come back to the case where the impatient consumers ($i \ge 1$) supply labor elastically.

## 5.2 Elastic labor supply

Without a great loss of generality, we assume that the impatient agents ($i \ge 1$) are identical, having the same preferences, the same elasticities $\mu_1 \equiv \mu_{1i}$, $\mu_2 \equiv \mu_{2i}$ and the same labor supply $l_i \equiv l/n$. Their individual (and average) elasticity of labor supply becomes:

$$ \xi = \frac{\mu_1 - 1}{1 + \frac{\mu_1}{\mu_2} \frac{l}{n-l}} \qquad (12) $$

Assume first that the consumption of impatient agents is sufficiently substitutable over time ($\mu_1 > 1$), i.e. labor supply turns out to be positive-sloped for all the agents ($\xi > 0$). A higher elasticity of intertemporal substitution in leisure $\mu_2$ (corresponding to a higher elasticity of labor supply $\xi$) lowers the critical elasticity of capital-labor substitution $\sigma_F$ (see equation (11)): the range of parameter compatible with the saddle-path stability ($\sigma_F, +\infty$) widens and instability and cycles become less likely.

Assume now that the impatient agents' intertemporal substitution in consumption is sufficiently weak ($\mu_1 < 1$), i.e. the labor supplies become negatively-sloped ($\xi < 0$). A higher elasticity of intertemporal substitution in leisure $\mu_2$ (corresponding to a higher elasticity of labor supply in absolute value $|\xi|$) raises the critical elasticity of capital-labor substitution $\sigma_F$ (equation (11)): the range of parameter compatible with the saddle-path stability ($\sigma_F, +\infty$) shrinks and instability and cycles become more likely, that is, compatible with a higher substitutability between capital and labor.

We eventually observe that the lower the elasticity of intertemporal substitution in consumption $\mu_1$ is, the greater the range $(0, \sigma_F)$ for instability. In the limit case where $\mu_1$ tends to 0, the bifurcation value $\sigma_F$ becomes equal to $s(1 + \kappa)$.

## 5.3 Interpretation

An economic interpretation of a bifurcation is a somewhat difficult task. In order to keep things as simple as possible, while preserving the main mechanism behind the emergence of cycles and the role of labor supply, it is not unworthy to compare the cases with elastic and inelastic labor supply.

The patient agent's budget constraint is the valuable information we need to start the analysis (see also Becker and Foias (1994)). In equilibrium, this constraint reduces to

$$ c_{0t} + k_{t+1} - \Delta k_t = r_t k_t \equiv I_t \qquad (13) $$

<footnote>assumption, the capitalist supplies inelastically one unit of labor.</footnote>
<figure><img src="image_6.png" /><figcaption>Figure 2: $S < -1$</figcaption></figure>

empty. In order to characterize the shape of $\Sigma$ and provide clear-cut bifurcation conditions, we need to detail the properties of trace and determinant as functions of $\sigma$.

Since $D'(\sigma) > 0$, when $\sigma$ increases from 0 to $+\infty$, the point $(T(\sigma), D(\sigma))$ moves always upward along $\Sigma$ from the origin $(T(0), D(0))$, to the endpoint $(T(+\infty), D(+\infty))$. If $\Sigma$ crosses the line (AB), then a flip bifurcation generically occurs for a positive value of $\sigma$, say $\sigma_F$, corresponding to the intersection and solving the equation $D(\sigma) = -T(\sigma) - 1$.

A quick look to the $(T, D)$-plane will convince the reader that few informations are required to locate $\Sigma$ and determine the stability properties of the steady state.

(1) Is the origin $(T(0), D(0))$ above or below the endpoint $(1 + 1/\beta, 1/\beta)$?

(2) Is the origin $(T(0), D(0))$ above or below the flip line (AB)?

(3) Is the slope steeper ($|S| > 1$) or flatter ($-1 < S < 0$)?
The economic intuition accounting for cycles is based on the (possibly) negative response of the capitalist's income $I_t$ to an increase in $k_t$, inducing a decrease of the future capital stock $k_{t+1}$.

Focus first on the simplest case: each impatient agent supplies an inelastic amount of labor, say, without loss of generality, one unit of time. We observe that heterogeneity still holds, because $l_{0t} = 0$ and $l_t = n$. The capitalist's income reduces to

$$I_t = f'(k_t/n) k_t \quad (14)$$

where $f(a) \equiv F(a, 1)$, and decreases in $k_t$, if and only if $f'(k_t/n) k_t$ does, which in turn requires a sufficiently weak capital-labor substitution

$$\sigma < 1 - s \quad (15)$$

A quick look to (13) will convince the reader that the future capital stock $k_{t+1}$ decreases if the consumption $c_{0t}$ is not too sensitive to changes in income $I_t$. That is why a weak capitalist's elasticity of intertemporal substitution in consumption contributes to promote cycles and instability.

Keeping in mind this basic mechanism, we can tackle the case of endogenous labor supply which makes slightly more complex the income response to a change in capital. A central question in our paper we cannot get round, is why, in some cases, endogenous labor promotes saddle-path stability while, in other cases, endogenous cycles and instability are more likely? Under elastic labor supply, the income $I_t$ becomes:

$$I_t \equiv I(k_t) = f'(k_t/l(k_t)) k_t \quad (16)$$

since $l_t = l(k_t)$. Then

$$I'(k) = f'(a) \left[ 1 - \frac{1 - s(a)}{\sigma(a) + s(a)\xi} \right]$$

where $s(a)$ and $\sigma(a)$ denote, respectively, the capital share in income and the elasticity of capital-labor substitution (see (5)-(6)). We notice that $I'(k) < 0$ if and only if

$$-s\xi < \sigma < 1 - s(1 + \xi) \quad (17)$$

(17) revisits inequality (15) for the occurrence of cycles, by encompassing the role of labor supply: when $\xi = 0$, labor supply is inelastic and we recover $\sigma < 1 - s$.

Assume now $\xi > 0$, which corresponds to an elasticity of intertemporal substitution in consumption higher than one for impatient consumers. Then, the higher bound $1 - s(1 + \xi)$ becomes smaller than $1 - s$, reducing the scope of capital-labor substitution for instability. Endogenous cycles can even be ruled out if $\xi$ is high enough, that is, when the elasticities of intertemporal substitution in consumption and leisure are sufficiently large (see (1) in Proposition 5).

On the contrary, when $\xi < 0$, i.e. the elasticity of intertemporal substitution in consumption for impatient consumers is less than one, the higher
<figure><img src="image_7.png" /><figcaption>Figure 3: $-1 < S < 0$</figcaption></figure>

We first find explicit conditions for points (1) and (2).

(1) The necessary and sufficient condition for the origin to stay above the endpoint is $D(0) > 1/\beta$, that is

$$ \tilde{\xi} < \tilde{\xi}^* \qquad (37) $$

In this case, $\Sigma$ is a half-line starting from $D = D(0) > 1/\beta$, going up to $D = +\infty$ and jumping to $D = -\infty$ for $\sigma = \sigma_\infty$, and going up to $D = 1/\beta$, with $\sigma_\infty \equiv s(1 - \lambda_0)(\tilde{\xi}^* - \tilde{\xi})$.

(2) The necessary and sufficient condition for the origin $(T(0), D(0))$ to lie above the line (AB) is $D(0) > -T(0) - 1$ or, equivalently,

$$ \varkappa / (\tilde{\xi} - \tilde{\xi}^*) < 1 \qquad (38) $$

where

$$ \varkappa \equiv \rho(1+S) / [2(1+\beta)S] \qquad (39) $$
bound $1 - s(1 + \xi)$ becomes greater than $1 - s$. This explains why instability and endogenous cycles occur under less demanding conditions on capital-labor substitution when the impatient consumers' intertemporal substitution in consumption is less than one.

Finally, notice how the capitalist's preferences matter: according to (13), as was the case under inelastic labor supply, endogenous cycles and instability occur more likely if $c_{0t}$ is little sensitive to variations in $k_t$. This requires a sufficiently weak capitalist's elasticity of intertemporal substitution in consumption, that is, a higher degree of concavity for consumption utility. In such a case, a decrease of $I_t$ will promote a decrease of $k_{t+1}$.

Before concluding, let us just move on a somewhat neglected but important point. One may care about the robustness of our findings in the case of a positive capitalist's labor supply. Actually, the stability analysis around the steady state where the capitalist supplies a positive amount of labor ($l_0 > 0$) changes only a little. In fact, applying the geometrical method gives the same qualitative pictures, the main difference concerning the level of critical values.

More precisely, at the steady state, $\sigma_F$ turns out to be smaller when capitalist supplies labor, which means that endogenous cycles and instability require a weaker substitution between capital and labor.<fnref n="16" /> In order to integrate the above economic interpretation, we simply observe that the capitalist's capital income $r_t k_t$ is now augmented by his labor earnings $w_t l_{0t}$. Since the real wage increases with the capital-labor ratio, following an increase of $k_t$, a decrease of $k_{t+1}$ requires a lower elasticity of capital-labor substitution.

# 6 Conclusion

We have addressed the question of saddle-path stability and dynamic convergence in an economy with heterogeneous consumers and elastic labor supply. In line with Becker (1980), Becker and Foias (1987, 1994), Hernandez (1991) and Sorger (1994), we have also supposed that agents face borrowing constraints, representing a kind of market incompleteness.

We first provide sufficient conditions for the existence of an intertemporal equilibrium. Focusing on the stationary solutions, we prove that, as was the case under inelastic labor supply, the capital stock is wholly owned by the most patient agent, whereas the others consume their labor income. However, two types of steady states may exist: one where the most patient agent supplies labor and one where he does not.

Without losing generality but gaining in conciseness, we characterize local dynamics around the second type of steady state and local dynamics around it to address saddle-path stability and convergence issues. In particular, we explain the role of preferences on the stability properties of the steady state and the occurrence of endogenous cycles. As was the case in economies with

<footnote n="16">¹⁶The reader is referred to Bosi and Seegmuller (2007) for more details. In this working paper, the stability properties of the steady state where the capitalist supplies labor are fully characterized.</footnote>
Using these two results, we discuss now the stability properties of the steady state and the occurrence of bifurcations considering two cases for the slope.

(3) We assume $|S| > 1$ and $-1 < S < 0$ in order.

(3.1) $|S| > 1$: $\varkappa$ is positive and three subcases matter (see Figures 1 and 2):

(3.1.1) If $\tilde{\xi} < \tilde{\xi}^*$, inequalities (37) and (38) are satisfied. $(T(0), D(0))$ is above the endpoint $(T(+\infty), D(+\infty))$ and the line (AB). $\Sigma$ is a jumping half-line that crosses the line (AB) at $\sigma = \sigma_F$. Therefore, the steady state is a source for $0 < \sigma < \sigma_F$ and a saddle for $\sigma > \sigma_F$.

(3.1.2) If $\tilde{\xi}^* < \tilde{\xi} < \tilde{\xi}^* + \varkappa$, inequalities (37) and (38) are no longer satisfied. The origin is below the endpoint and the line (AB). $\Sigma$ is a segment that crosses the line (AB). Therefore, the steady state is a source for $0 < \sigma < \sigma_F$ and a saddle for $\sigma > \sigma_F$.

(3.1.3) If $\tilde{\xi}^* + \varkappa < \tilde{\xi}$, inequality (37) is not satisfied, while (38) is. The origin is below the endpoint and above the line (AB). $\Sigma$ is a segment that does not cross the line (AB), which means that the steady state is a saddle whatever $\sigma$.

(3.2) $-1 < S < 0$: $\varkappa$ is negative and three subcases matter (see Figure 3):

(3.2.1) If $\tilde{\xi} < \tilde{\xi}^* + \varkappa$, inequalities (37) and (38) are satisfied. The origin is above the endpoint and the line (AB). $\Sigma$ is a jumping half-line that crosses the line (AB). Therefore, the steady state is a source for $0 < \sigma < \sigma_F$ and a saddle for $\sigma > \sigma_F$.

(3.2.2) If $\tilde{\xi}^* + \varkappa < \tilde{\xi} < \tilde{\xi}^*$, inequality (37) is satisfied, while (38) no longer holds. The origin is above the endpoint, but below the line (AB). $\Sigma$ is a jumping half-line that does not cross the line (AB). The steady state is a saddle whatever $\sigma$.

(3.2.3) If $\tilde{\xi}^* < \tilde{\xi}$, inequality (37) is no longer satisfied, while (38) is. The origin is below the endpoint and above the line (AB). $\Sigma$ is a segment that does not cross the line (AB). As in the previous subcase, the steady state is a saddle for all $\sigma$.

Summing up, we notice that, when $\tilde{\xi} < \tilde{\xi}^* + \varkappa$, the steady state is a source for $0 < \sigma < \sigma_F$ and a saddle for $\sigma > \sigma_F$, and the system undergoes a flip bifurcation at $\sigma = \sigma_F$ (subcases (3.1.1), (3.1.2), (3.2.1)). In contrast, when $\tilde{\xi}^* + \varkappa < \tilde{\xi}$, we get a saddle, whatever $\sigma$ (subcases (3.1.3), (3.2.2), (3.2.3)).

All these necessary and sufficient conditions are summarized in the next proposition.

**Proposition 4** Let

$$ \sigma_F \equiv s(1 - \lambda_0) (\tilde{\xi}^* + \varkappa - \tilde{\xi}) \quad (40) $$
inelastic labor supply, instability and endogenous cycles still require a weak intertemporal substitution in consumption for the most patient agent and a weak substitution between capital and labor. But, in addition, preferences of impatient consumers, say the interplay between intertemporal substitution and income effects, now matter: more explicitly, the range of inputs substitution for instability and cycles shrinks or widens, depending on whether or not the impatient consumers' elasticity of intertemporal substitution in consumption exceeds one.

We eventually notice that our paper addressed the question of convergence through a local analysis of the stability properties of the steady state. Future research should be concerned with a global analysis of stability as in the seminal work with inelastic labor supply by Becker and Foias (1987).

# 7 Appendix

**Proof of Proposition 2** Condition (D1) is ensured by (P8) and Assumption

2. To establish (D2), notice that for every alternative pair $(\tilde{k}_t, \tilde{l}_t)$, we have:

$$ 
\begin{aligned}
& F(k_t, l_t) - w_t l_t - r_t k_t - [F(\tilde{k}_t, \tilde{l}_t) - w_t \tilde{l}_t - r_t \tilde{k}_t] \\
&= F(k_t, l_t) - F(\tilde{k}_t, \tilde{l}_t) - r_t (k_t - \tilde{k}_t) - w_t (l_t - \tilde{l}_t) \\
&\geq F_k(k_t, l_t) (k_t - \tilde{k}_t) + F_l(k_t, l_t) (l_t - \tilde{l}_t) - r_t (k_t - \tilde{k}_t) - w_t (l_t - \tilde{l}_t) \\
&= 0
\end{aligned}
$$

Consider now a sequence $(\tilde{k}_{it}, \tilde{l}_{it}, \tilde{c}_{it})$ satisfying the constraints in the consumer's program and the initial condition. We have:

$$ 
\begin{aligned}
& \sum_{t=0}^{\infty} \beta_i^t \left[ (u_i(c_{it}) + v_i(1-l_{it})) - (u_i(\tilde{c}_{it}) + v_i(1-\tilde{l}_{it})) \right] \\
&= \sum_{t=0}^{\infty} \beta_i^t \left[ u_i(c_{it}) - u_i(\tilde{c}_{it}) + v_i(1-l_{it}) - v_i(1-\tilde{l}_{it}) \right] \\
&\geq \sum_{t=0}^{\infty} \beta_i^t \left[ u'_i(c_{it})(c_{it}-\tilde{c}_{it}) - v'_i(1-l_{it})(l_{it}-\tilde{l}_{it}) \right] \\
&= \sum_{t=0}^{\infty} \beta_i^t \left[ u'_i(c_{it}) (R_t(k_{it} - \tilde{k}_{it}) - (k_{it+1} - \tilde{k}_{it+1}) + w_t(l_{it} - \tilde{l}_{it})) \right. \\
& \qquad \left. - v'_i(1-l_{it})(l_{it}-\tilde{l}_{it}) \right]
\end{aligned}
$$
where $\tilde{\xi}^*$ and $\varkappa$ are given by (32) and (39), respectively. The following results generically hold.

If the impatient agents' average elasticity of labor supply is greater than a threshold, that is, $\tilde{\xi} \ge \tilde{\xi}^* + \varkappa$, the steady state is a saddle, whatever the degree of capital-labor substitution $\sigma > 0$.

If the impatient agents' average elasticity of labor supply is smaller than a threshold, that is, $\tilde{\xi} < \tilde{\xi}^* + \varkappa$, the steady state is a source when inputs are rather complementary ($0 < \sigma < \sigma_F$) and it is a saddle when they are rather substitutable ($\sigma > \sigma_F$). When $\sigma$ crosses $\sigma_F$, the system undergoes a flip bifurcation.

This proposition states that the steady state is always locally determinate. However, saddle-path stability and local convergence is not always ensured. When the substitution between capital and labor is sufficiently weak, the steady state can lose stability through the occurrence of a cycle of period two.

Proposition 4 is as general as somewhat cryptic. In order to make the role of preferences on (in)stability more explicit and provide an economic interpretation of bifurcation, we have to go beyond the generality of proposition and specify the fundamentals. Four canonical cases are retained.

At the first stage, we recover the particular case of a representative agent in order to compare our results with De Hek (1998).

At the second stage, we come back to heterogeneous consumers and we analyze three classes of preferences commonly employed in economic literature: separable, KPR<fnref n="14" /> and homogeneous utility functions in consumption and leisure.

# 6 Representative agent

In order to compare our model to the benchmark, we reduce the number of agents to one, the patient agent. In this case, we are also able to criticize De Hek (1998), who characterize transitional dynamics with representative consumer and elastic labor supply under somewhat unconventional assumptions.

Setting $n = 0$, we reduce agents' heterogeneity to the patient agent, now representative. Formally, let $\lambda_0 = 1$: the average elasticity of labor supply becomes $\xi = \xi_0$, while the locus $\Sigma$ becomes a horizontal segment defined by:

$$ \Sigma = \{(T, D) : D = 1/\beta, T(+\infty) \le T \le T(0)\} \quad (41) $$

where still $T(+\infty) = 1 + 1/\beta$, but now

$$ T(0) = 1 + \frac{1}{\beta} + \frac{\rho(1 - \beta + \rho)\varepsilon_{021} - \varepsilon_{011} + (\varepsilon_{012} - \varepsilon_{022})l_0/(1 - l_0)}{\rho(\varepsilon_{021} - \varepsilon_{011}) - \varepsilon_{011}} $$

$\Sigma$ lies entirely on the right-hand side of the endpoint $(1/\beta, 1 + 1/\beta)$ in the region of the saddle points. Indeed, it is enough to observe that the slope $S$ is zero (expression (33)), that $T'(\sigma) < 0$ (expression (31)) and that $T(0) >$

<footnote n="14">¹⁴Popularized by King, Plosser and Rebelo (1988), these preferences allow a positive growth rate in endogenous growth models.</footnote>
$$
\begin{align*}
&= \lim_{T \to +\infty} \left[ u'_i(c_{i0}) R_0(k_{i0} - \tilde{k}_{i0}) + \sum_{t=0}^{T-1} \beta_i^{t+1} u'_i(c_{it+1}) R_{t+1}(k_{it+1} - \tilde{k}_{it+1}) \right. \\
&\qquad \left. - \sum_{t=0}^{T-1} \beta_i^t u'_i(c_{it}) (k_{it+1} - \tilde{k}_{it+1}) - \beta_i^T u'_i(c_{iT}) (k_{iT+1} - \tilde{k}_{iT+1}) \right. \\
&\qquad \left. + \sum_{t=0}^{T} \beta_i^t (u'_i(c_{it}) w_t - v'_i(1-l_{it})) (l_{it} - \tilde{l}_{it}) \right] \\
&= \lim_{T \to +\infty} \left[ \sum_{t=0}^{T-1} \beta_i^t (\beta_i u'_i(c_{it+1}) R_{t+1} - u'_i(c_{it})) (k_{it+1} - \tilde{k}_{it+1}) \right. \\
&\qquad \left. - \beta_i^T u'_i(c_{iT}) (k_{iT+1} - \tilde{k}_{iT+1}) + \sum_{t=0}^{T} \beta_i^t (u'_i(c_{it}) w_t - v'_i(1-l_{it})) (l_{it} - \tilde{l}_{it}) \right] \\
&\ge \lim_{T \to +\infty} \left[ \sum_{t=0}^{T-1} \beta_i^t (\beta_i u'_i(c_{it+1}) R_{t+1} - u'_i(c_{it})) k_{it+1} - \beta_i^T u'_i(c_{iT}) k_{iT+1} \right. \\
&\qquad \left. + \sum_{t=0}^{T} \beta_i^t (u'_i(c_{it}) w_t - v'_i(1-l_{it})) l_{it} \right] = 0
\end{align*}
$$

This proves that condition (D3) holds. Finally, conditions (D4), (D5) and (D6)
are easily obtained using (P4), (P5) and (P6).

**Proof of Proposition 3** The proof is articulated in 4 steps.

(S1) For $i = 0$, (Q2)-(Q5) satisfy the optimality conditions in Proposition 2. Moreover, since $c_0$ is constant and $0 < \beta_0 < 1$, the transversality condition (P7): $\lim_{t \to +\infty} \beta_0^t u_0'(c_0)k_0 = 0$, holds.

(S2) For $i \ge 1$, consider the feasible sequence $(\tilde{k}_{it}, \tilde{l}_{it}, \tilde{c}_{it})$, starting from $\tilde{k}_{i0} = 0$. We now compare this path with the stationary solution $(c_i, l_i)$, such that $k_i = 0, 1 > l_i > 0$ and $c_i = w l_i$, and we show that the stationary solution is optimal. In details, we get:

$$
\begin{align*}
& \sum_{t=0}^{\infty} \beta_i^t \left[ u_i(c_i) + v_i(1-l_i) - (u_i(\tilde{c}_{it}) + v_i(1-\tilde{l}_{it})) \right] \\
&= \sum_{t=0}^{\infty} \beta_i^t \left[ u_i(wl_i) - u_i(\tilde{c}_{it}) + v_i(1-l_i) - v_i(1-\tilde{l}_{it}) \right] \\
&\ge \sum_{t=0}^{\infty} \beta_i^t \left[ u'_i(wl_i)(wl_i - \tilde{c}_{it}) - v'_i(1-l_i)(l_i - \tilde{l}_{it}) \right] \\
&= \sum_{t=0}^{\infty} \beta_i^t u'_i(wl_i) \left[ wl_i - \tilde{c}_{it} - w(l_i - \tilde{l}_{it}) \right]
\end{align*}
$$
$T(+\infty) = 1 + 1/\beta$, under the standard concavity and normality assumptions (16) and (17).

Let us summarize these findings as follows.

**Corollary 5** *Without heterogeneity, the steady state is always a saddle and the convergence to the steady state is monotonic.*

In contrast to De Hek (1998), we have proved that neither non-monotonic trajectories nor endogenous cycles occur in a one-sector optimal growth model with leisure in the utility function. The key assumption is that of normality (inequalities (17) entailing $T(0) > T(+\infty)$). As a matter of fact, De Hek not only introduces a production function which depends on past labor supply and is open to criticism; but also, more questionably, does not impose that consumption and leisure are normal goods at a macroeconomic level.

In the following, we come back to the model with heterogeneous agents and borrowing constraints. As seen above, to understand the role of preferences on saddle-path stability, we specify three classes of utility functions, widely used in economic models: the separable, the KPR and the homogeneous preferences.

# 7 Separable utilities

Assume the utility functions of all consumers to be separable in consumption and leisure, i.e., take the form $u_i(c_{it}, 1 - l_{it}) \equiv v_i(c_{it}) + w_i(1 - l_{it})$, with $v'_i(c_i) > 0$, $w'_i(1 - l_i) > 0$, $v''_i(c_i) \le 0$, $w''_i(1 - l_i) \le 0$. Still applying the results of Proposition 4, we will able to capture the role of concavity in consumption and leisure on convergence.

Separability and concavity of the utility functions of all consumers $i = 0, 1, ..., n$ entail $\varepsilon_{i12} = \varepsilon_{i21} = 0$ and $\varepsilon_{i11} < 0, \varepsilon_{i22} < 0$, respectively. Then, the critical values for the impatient agents' average elasticity of labor supply and the elasticity of capital-labor substitution, involved in Proposition 4, simplify:

$$ \tilde{\xi}^* \equiv -\frac{1+\rho}{1-\lambda_0}\lambda_0\xi_0 \quad (42) $$

$$ \tilde{\xi}^* + \varkappa = \frac{1}{1-\lambda_0}\left[\frac{\sigma_I}{s} - \lambda_0\xi_0\left(1+\frac{\rho}{2}\frac{3+\beta+\rho}{1+\beta}\right)\right] $$

$$ \sigma_F \equiv \sigma_I - s\left[(1 - \lambda_0)\tilde{\xi} + \lambda_0\xi_0\left(1 + \frac{\rho}{2}\frac{3 + \beta + \rho}{1 + \beta}\right)\right] \quad (43) $$

where $\xi_0 = -(1 - l_0) / (l_0\varepsilon_{022})$ is now the capitalist's (partial) elasticity of labor supply and:

$$ \sigma_I \equiv \frac{s\rho}{1+\beta}\left(1 - \lambda_0 + \frac{1-\beta+\rho\lambda_0}{2\varepsilon_{011}}\right) \quad (44) $$

It is not unworthy to focus on the limit case with inelastic labor supply, already investigated by Becker and Foias (1987, 1994), in order to make ideas clearer about the role of endogenous labor supplies on the stability properties.
$$
\begin{align*}
&= u'_{i}(wl_{i}) \sum_{t=0}^{\infty} \beta_{i}^{t} (wl_{it} - \tilde{c}_{it}) \\
&= u'_{i}(wl_{i}) \sum_{t=0}^{\infty} \beta_{i}^{t} (\tilde{k}_{it+1} - \tilde{k}_{it}/\beta_{0}) \\
&= u'_{i}(wl_{i}) \lim_{T \to +\infty} \left[ \beta_{i}^{T} \tilde{k}_{iT+1} + (1/\beta_{i} - 1/\beta_{0}) \sum_{t=1}^{T} \beta_{i}^{t} \tilde{k}_{it} - \tilde{k}_{i0}/\beta_{0} \right] \\
&\geq -u'_{i}(wl_{i}) \tilde{k}_{i0}/\beta_{0} = 0
\end{align*}
$$

because $\beta_0 > \beta_i$ and $\tilde{k}_{it} \ge 0$ for all $i \ge 1$ and $t \ge 0$. Notice that $l_i = 0$ is not
optimal for $i \ge 1$. Indeed, since $c_i = wl_i$ and $\lim_{c_i \to 0} u'_i(c_i) = +\infty$, a small
increase of $l_i$ would imply an infinite increase of welfare.

(S3) Under Assumption 2, there is a unique finite and strictly positive value of a such that $R = f'(a) + \Delta = 1/\beta_0$.

(S4) We conclude this proof by showing that $R = 1/\beta_0$ (with $k = k_0$ and $k_i = 0$ for all $i \ge 1$) is the only stationary solution:

If $R > 1/\beta_0$, then it is optimal for the consumer $i = 0$ to increase capital.
However, this cannot be a stationary solution because of decreasing returns.

If $R < 1/\beta_0 < 1/\beta_1 \le \dots \le 1/\beta_n$, then it is optimal for each household to decumulate to zero in a finite time. Therefore, $a_t \to 0$ and $f'(a_t) \to +\infty$, violating stationarity.

**Proof of Lemma 4** We prove the existence of the function $l_i(w)$ for $i \ge 1$. We need to solve the implicit equation:

$$
\varphi_i(w, l_i) = w \tag{18}
$$

where $\varphi_i(w, l_i) \equiv v'_i(1-l_i)/u'_i(wl_i)$. We notice that, under Assumption 2, $\partial\varphi_i/\partial l_i = -(u'_i v''_i + u''_i v'_i w)/u'^2_i > 0$. Then $\varphi_i$ is a continuous and strictly increasing function of $l_i$. Given $w$, we have, under Assumption 1, $\lim_{l_i \to 0^+} \varphi_i(w, l_i) = 0$ and $\lim_{l_i \to 1^-} \varphi_i(w, l_i) = +\infty$. Then $\varphi_i$ crosses $w$ once and only once. In other terms, the solution of equation (18) exists and is unique, ensuring that $l_i(w)$ is a well-defined function. ■

**Proof of Proposition 5** Looking at the $(T, D)$-plane will convince the reader that few informations are required to locate $\Sigma$ and determine the stability properties of the steady state.

(1) Is the origin $(T(0), D(0))$ above or below the endpoint $(1+1/\beta, 1/\beta)$?

(2) Is the origin $(T(0), D(0))$ above or below the flip line (AB)?

(3) Is the slope steeper ($|S| > 1$) or flatter ($-1 < S < 0$)?
## 7.1 Inelastic labor supply

When agents supply labor inelastically, the leisure elasticity $\varepsilon_{i22}$ falls to $-\infty$ and, according to formulas (42), (70) and (72), $\xi_i = \tilde{\xi} = \tilde{\xi}^* = 0$ for every $i \ge 0$. By Proposition 4, the occurrence of a flip bifurcation needs $\tilde{\xi} < \tilde{\xi}^* + \kappa$, that is, now, a positive critical elasticity $\sigma_F = \sigma_I$. In the light of equation (44), the patient consumer is required to have a sufficiently weak intertemporal substitution in consumption, i.e., $-1/\varepsilon_{011} < 2(1 - \lambda_0)/(1 - \beta + \rho\lambda_0)$.

According to expression (44), the bifurcation value $\sigma_I$ decreases in the elasticity of intertemporal substitution in consumption $-1/\varepsilon_{011} > 0$. Since $\sigma_I < s\rho/(1+\beta)$ and $\rho = (1-\beta\Delta)(1-s)/s$, we get $\sigma_I < 1-s$. Consequently, in a model with inelastic labor supply, the existence of cycles and instability requires the capital income ($rk$) to decrease when capital increases. A weak elasticity of intertemporal substitution in consumption $-1/\varepsilon_{011} > 0$ prevents the intertemporal arbitrage (see Becker and Foias (1987, 1994)).

We further notice that, in contrast to the case with endogenous labor supply, the preferences of impatient consumers play no role on the stability properties of the steady state and the occurrence of cycles.

## 7.2 Elastic labor supply

The opposite case with elastic labor supply is now considered. Without a significant loss of generality, we assume that all the impatient agents ($i = 1, \dots, n$) share the same instantaneous utility function and, in consequence, the same labor supply $l_i \equiv m$ as well as the same elasticities $\varepsilon_{i11} \equiv \varepsilon_{11}$, $\varepsilon_{i22} \equiv \varepsilon_{22}$. According to (72) (see the Appendix), their individual (and average) elasticity of labor supply becomes:

$$ \tilde{\xi} = - \frac{1 + \varepsilon_{11}}{\varepsilon_{11} + \varepsilon_{22}m/(1-m)} \quad (45) $$

while the capitalist's elasticity of labor supply with respect to the real wage is defined as above by $\xi_0 = -(1-l_0)/(l_0\varepsilon_{022}) > 0$. In the following, we consider two polar cases.

Assume first that the impatient agents' consumption is sufficiently substitutable over time ($-1/\varepsilon_{11} > 1$). Hence, labor supply turns out to be positively sloped for all the agents. We notice that the more elastic the labor supplies (higher $\xi_0$ and $\tilde{\xi}$),<fnref n="15" /> the weaker the bifurcation elasticity of capital-labor substitution $\sigma_F$ (see (43)).<fnref n="16" /> This means that the range of parameter compatible with the saddle-path stability ($\sigma_F, +\infty$) widens with the elasticity of labor supply and restrictions for instability and cycles become more demanding than under an exogenous labor supply.

If the impatient agents' intertemporal substitution in consumption is sufficiently weak ($-1/\varepsilon_{11} < 1$), then, according to (45), $\tilde{\xi}$ becomes negative (while

<footnote n="15">¹⁵Notice that the less concave the disutilities of labor, the more elastic the labor supplies.</footnote>

<footnote n="16">¹⁶In any case, $\sigma_F < \sigma_I$ because the term into brackets on the RHS of (43) is negative.</footnote>
Let us find explicit conditions for points (1) and (2).

(1) The necessary and sufficient condition for the origin to stay above the endpoint is $D(0) > 1/\beta$, that is

$$ \xi < 0 \qquad (19) $$

In this case, the half-line $\Sigma$ starts from $D = D(0) > 1/\beta$, goes up to $D = +\infty$ and jumps to $D = -\infty$ for $\sigma = \sigma_{\infty} \equiv -s\xi$, and, eventually, reaches $D = 1/\beta$.

(2) The necessary and sufficient condition for the origin $(T(0), D(0))$ to lie above the line $(AB)$ is $D(0) > -T(0) - 1$ or, equivalently,

$$ \varkappa/\xi < 1 \qquad (20) $$

Combining these results and distinguishing two cases for the slope, we can discuss the stability properties of the steady state and the occurrence of bifurcations.

(3) First, assume $|S| > 1$; then, $-1 < S < 0$.

(3.1) $|S| > 1$: $\varkappa$ is positive and three subcases arise (see Figures 1 and 2):

(3.1.1) If $\xi < 0$, inequalities (19) and (20) are satisfied. $(T(0), D(0))$ is above the endpoint $(T(+\infty), D(+\infty))$ and the line $(AB)$. $\Sigma$ is a jumping half-line that crosses the line $(AB)$ at $\sigma = \sigma_F$. Therefore, the steady state is a source for $0 < \sigma < \sigma_F$ and a saddle for $\sigma > \sigma_F$.

(3.1.2) If $0 < \xi < \varkappa$, inequalities (19) and (20) are no longer satisfied. The origin is below the endpoint and the line $(AB)$. $\Sigma$ is a segment that crosses the line $(AB)$. Therefore, the steady state is a source for $0 < \sigma < \sigma_F$ and a saddle for $\sigma > \sigma_F$.

(3.1.3) If $\varkappa < \xi$, inequality (19) is not verified, but (20) is. The origin is below the endpoint and above the line $(AB)$. $\Sigma$ is a segment that does not cross the line $(AB)$, which means that the steady state is a saddle whatever $\sigma$.

(3.2) $-1 < S < 0$: $\varkappa$ is negative and three subcases hold (see Figure 3):

(3.2.1) If $\xi < \varkappa$, inequalities (19) and (20) are satisfied. The origin is above the endpoint and the line $(AB)$. $\Sigma$ is a jumping half-line that crosses the line $(AB)$. Therefore, the steady state is a source for $0 < \sigma < \sigma_F$ and a saddle for $\sigma > \sigma_F$.

(3.2.2) If $\varkappa < \xi < 0$, inequality (19) is verified, while (20) no longer holds. The origin is above the endpoint, but below the line $(AB)$. $\Sigma$ is a jumping half-line that does not cross the line $(AB)$. The steady state is a saddle whatever $\sigma$.
$\xi_0$ still remains positive) and the critical elasticity $\sigma_F$ can be greater than $\sigma_I$. By direct inspection of equation (43), this requires:

$$ \tilde{\xi} < - \frac{\lambda_0 \xi_0}{1 - \lambda_0} \left( 1 + \frac{\rho}{2} \frac{3 + \beta + \rho}{1 + \beta} \right) $$

We deduce that $\sigma_F > \sigma_I$ if the patient agent's labor supply is not too elastic and the negative aggregate elasticity $\tilde{\xi}$ not too close to zero, that is, the impatient agents' elasticity of intertemporal substitution in consumption $(-1/\varepsilon_{11})$ remains sufficiently weak. In this case, the introduction of a consumption-leisure arbitrage promotes instability and cycles, because the range of parameters for saddle-path stability $(\sigma_F, +\infty)$ shrinks and the flip bifurcation occurs under a higher substitutability between capital and labor.

Now, let us provide two examples based on non-separable utility functions. Two popular specifications in macrodynamic literature are retained.

# 8 KPR preferences

In this section, we consider preferences à la King, Plosser and Rebelo (1988). They have the advantage of lightening restrictions on the structural parameters in order to obtain positive growth rates in models with capital accumulation. Agent i's utility function is specified as follows:

$$ u_i(c_i, 1 - l_i) \equiv [c_i v_i (1 - l_i)]^{1+\varepsilon_i} / (1 + \varepsilon_i) \quad \text{if } \varepsilon_i \le 0, \varepsilon_i \ne -1 \quad (46) $$

$$ u_i(c_i, 1 - l_i) \equiv \ln c_i + \ln v_i (1 - l_i) \quad \text{if } \varepsilon_i = -1 \quad (47) $$

The parameter $-1/\varepsilon_i$ is still interpretable as elasticity of intertemporal substitution in consumption. In addition, we need to define (first and second order) elasticities of leisure utility: $\varepsilon_{v_i} \equiv v'_i (1 - l_i) (1 - l_i) / v_i (1 - l_i)$ and $\varepsilon_{v'_i} \equiv v''_i (1 - l_i) (1 - l_i) / v'_i (1 - l_i)$. The KPR functional form (46) implies:

$$ \begin{bmatrix} \varepsilon_{i11} & \varepsilon_{i12} \\ \varepsilon_{i21} & \varepsilon_{i22} \end{bmatrix} = \begin{bmatrix} \varepsilon_i & (1 + \varepsilon_i) \varepsilon_{v_i} \\ 1 + \varepsilon_i & \varepsilon_{v'_i} + \varepsilon_i \varepsilon_{v_i} \end{bmatrix} \quad (48) $$

Usual assumptions on preferences are maintained.

**Assumption 4 (KPR)** $v_i (1 - l_i)$ is a continuous function on $[0, 1]$, differentiable as many times as needed, increasing and concave on $(0, 1)$, that is, $\varepsilon_{v'_i} \le 0 \le \varepsilon_{v_i}$. We further assume:

$$ (1 + 2\varepsilon_i) \varepsilon_{v_i} - \varepsilon_i \varepsilon_{v'_i} \le 0 \quad (49) $$

Assumption 4 deserves some comments. First, $\varepsilon_{v'_i} \le 0 \le \varepsilon_{v_i}$ and $\varepsilon_i \le 0$ imply $\varepsilon_{i11} \le 0$, $\varepsilon_{i22} \le 0$, while (49) is equivalent to $\varepsilon_{i11}\varepsilon_{i22} \ge \varepsilon_{i12}\varepsilon_{i21}$. So, concavity conditions (16) are met. Second, normality conditions (17) ensue from $\varepsilon_{v'_i} \le \varepsilon_{v_i}$.

Let us now define $Q \equiv (1 - l_0) (1 - \beta + \rho) + l (1 - \beta + \rho\lambda_0) (\varepsilon_{v_0} - \varepsilon_{v'_0})$. Applying Proposition 4 to KPR preferences, we make the bifurcation through which cycles arise more explicit.
(3.2.3) If $0 < \xi$, inequality (19) is no longer satisfied, while (20) is. The origin is below the endpoint and above the line (AB). $\Sigma$ is a segment that does not cross the line (AB). As in the previous subcase, the steady state is a saddle for all $\sigma$.

Summing up, we notice that, when $\xi < \kappa$, the steady state is a source for $0 < \sigma < \sigma_F$ and a saddle for $\sigma > \sigma_F$, and the system undergoes a flip bifurcation at $\sigma = \sigma_F$ (subcases (3.1.1), (3.1.2), (3.2.1)). In contrast, when $\kappa < \xi$, we get a saddle, whatever $\sigma$ (subcases (3.1.3), (3.2.2), (3.2.3)). ■

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**Corollary 6** The system undergoes a flip bifurcation at $\sigma = \sigma_F > 0$, if and only if both the inequalities are satisfied:

$$2\varepsilon_{v_0} - \varepsilon_{v'_0} > (1 - l_0)(1 + \beta)/[\rho(l - l_0)] \quad (50)$$

$$-\frac{1}{\varepsilon_0} < \frac{(l - l_0)(2\varepsilon_{v_0} - \varepsilon_{v'_0}) - (1 - l_0)(1 + \beta)/\rho}{(l - l_0)\varepsilon_{v_0} + (1 - l_0)(1 + \beta) + Q/2} \quad (51)$$

Otherwise saddle-path stability prevails whatever $\sigma > 0$.

**Proof.** See the Appendix.

We observe that, when preferences are KPR, the elasticities of labor supply of impatient consumers ($i \ge 1$) become zero, which implies in turn $\tilde{\xi} = 0$. Therefore, preferences of impatient agents play no role on local dynamics, as it was the case under inelastic labor supply.

As seen above, a sufficiently low intertemporal substitution in patient agent's consumption ($-1/\varepsilon_0$) is needed to recover instability and the occurrence of endogenous cycles. This result is also in accordance with what happens under exogenous labor supply. However, by direct inspection of (51), we notice that a higher degree of concavity of $v_0$ (patient agent's utility of leisure) also promotes macroeconomic instability. Indeed, taken $\varepsilon_{v_0}$ as given, the left-hand side of (50) and the right-hand side of (51) increase with the degree of concavity $-\varepsilon_{v'_0}$, making both the inequalities and instability more likely. Conversely, since the elasticity of labor supply with respect to the real wage $\xi_0 = (1 - l_0) / [l_0(\varepsilon_{v_0} - \varepsilon_{v'_0})]$ decreases with $-\varepsilon_{v'_0}$, a more elastic labor supply of the patient agent promotes saddle-path stability and dynamic convergence.

# 9 Homogeneous utilities

One of the most popular classes of non-separable preferences is represented by homogeneous utility functions. In contrast to most existing papers, we don't reduce the analysis to homogeneity of degree one, but we allow the utility functions to have a degree less or equal to one.

The degrees of homogeneity and concavity are closely related.<fnref n="17" /> The former plays a key role on the stability properties, as the latter did in the KPR case. That's why taking into account a degree less than one matters.

**Assumption 5 (homogeneity)** The utility function $u_i(c_{it}, 1 - l_{it})$ is homogeneous of degree $\nu_i \le 1$, for every $i = 0, \dots, n$.

Restriction $\nu_i \le 1$ is needed to ensure the concavity and meet the second order conditions for intertemporal maximization. We also notice that Cobb-Douglas and CES utility functions satisfy Assumption 5, being homogeneous of degree one.

<footnote n="17">¹⁷The higher the degree of homogeneity, the lower the degree of concavity.</footnote>
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As in the KPR case, the patient agent matters more than the others in the stability analysis. We will show, precisely, that the steady state becomes a saddle point as soon as the patient agent's degree $\nu_0$ becomes close to one, while, in contrast, the occurrence of cycles requires low degrees of homogeneity. In this connection, we need an additional notation.

$$s_i \equiv u_{i1}c_i/u_i \in (0, 1) \qquad (52)$$

will denote the agent $i$'s consumption share in total utility and $\varepsilon_i \equiv c_i u_{i1}/u_{i1} < 0$ the elasticity of marginal utility with respect to consumption, while $-1/\varepsilon_i$ will represent, as usual, the elasticity of intertemporal substitution in consumption.

Under Assumption 5, the homogeneity property and normality between consumption and leisure entail, respectively:<fnref n="18" />

$$s_i < \nu_i \le 1 \quad \text{and} \quad \varepsilon_i \nu_i + (1 - \nu_i) s_i < 0 \qquad (53)$$

for every $i = 0, \dots, n$. Under properties (53), Proposition 4 is explicitly revisited.

**Proposition 7** *Proposition 4 still holds, where now:*

$$\tilde{\xi} = - \sum_{i=1}^{n} \frac{l_i (1 - l_i)}{l - l_0} \left( 1 + \frac{\nu_i - s_i}{\varepsilon_i \nu_i + (1 - \nu_i) s_i} \right) \qquad (54)$$

$$\tilde{\xi}^* = - \frac{1 - l_0}{(1 - \nu_0)(l - l_0)} \left[ \rho + \varepsilon_0 \frac{\nu_0 - s_0}{\varepsilon_0 \nu_0 + (1 - \nu_0) s_0} \right] < 0 \qquad (55)$$

$$\tilde{\xi}^* + \varkappa = \tilde{\xi}^* + \frac{\rho}{1 + \beta} \left[ 1 - \frac{\rho + (1 - \beta)(1 + l - l_0)}{2(1 - \nu_0)(l - l_0)} \right] \qquad (56)$$

**Proof.** See the Appendix.

Proposition 7 gives necessary and sufficient conditions for saddle-path stability, instability and deterministic cycles. In order to appreciate the economic intuition behind, we provide sufficient conditions for saddle-path stability in terms of the most significant parameter, the degree of homogeneity.

**Corollary 8** *Under Assumption 5, if the degree of homogeneity of the patient agent is sufficiently high, that is,*

$$\nu_0 > 1 - \frac{\rho + (1 - \beta)(1 + l - l_0)}{2(l - l_0)} \qquad (57)$$

while the impatient agents' elasticities of intertemporal substitution in consumption are sufficiently large, that is,

$$-1/\varepsilon_i > 1/(1 - s_i) \qquad (58)$$

for every $i = 1, \dots, n$, then the steady state is a saddle point (there is no room for cycles).

<footnote n="18">¹⁸See the proof of Proposition 7 in the Appendix</footnote>
**Proof.** See the Appendix.

Condition (58) is somewhat close to those we got above (see for instance, the separable case with elastic labor supply). Condition (57), which involves only the patient consumer's degree of homogeneity, is more puzzling and deserves great regard.

The next corollary stresses the prominence of (57), by asserting that restrictions (58) become superfluous to get saddle-path stability, when $v_0$ lies in a sufficiently small neighborhood of one.

**Corollary 9** *Under Assumption 5, when the capitalist's degree of homogeneity $v_0$ is sufficiently close to one the steady state is saddle-point stable and cycles are definitely ruled out.*

*In particular, when the preferences of the patient agent are homogeneous of degree one, saddle-path stability prevails.*

**Proof.** See the Appendix.

Corollaries (8)-(9) undoubtedly show how the degree of homogeneity in the patient agent's utility matters to determine the stability properties of the economy. Homogeneous utilities of degree one, so popular in economic literature, have a very specific impact on dynamics and must be questioned on the ground of robustness. In this model, they imply dynamic convergence whatever the other parameter values, whereas stability is no longer ensured as soon as the degree of homogeneity lowers.

Eventually, we observe that lower the degree of homogeneity, higher the degree of concavity, more likely instability and cycles. We recover some results already found in the separable case (namely, under elastic labor supply) and in the KPR case. In both the cases the impact of concavity through the elasticity of labor supply emerges as dominant feature.

# 10 Interpretation

An economic interpretation of a bifurcation is a somewhat difficult task. In order to simplify this task, while preserving the main mechanism behind the emergence of cycles and the role of labor supply, it is not unworthy to compare the cases with elastic and inelastic labor supply.

The patient agent's budget constraint is the valuable information we need to start the analysis (see also Becker and Foias (1994)). In equilibrium, the constraints reduces to

$$c_{0t} + k_{t+1} - \Delta k_t = r_t k_t + w_t l_{0t} \equiv I_t$$

The economic intuition is based on the (possibly) negative response of the income $I_t$ to an increase in $k_t$.
As announced, we first assume an inelastic labor supply: for simplicity, each agent supplies one unit of time. Therefore, $l_{0t} = 1$ and $l_t = 1 + n$. Using (18) and (19), we obtain:

$$I_t = f\left(\frac{k_t}{1+n}\right) + f'\left(\frac{k_t}{1+n}\right) \frac{nk_t}{1+n} \quad (59)$$

We notice immediately that, in the case of a representative agent ($n = 0$), $I_t$ is increasing in $k_t$. On the contrary, when consumers are heterogeneous ($n > 0$), $I_t$ can decrease in $k_t$, if $f'(k_t/(1+n))k_t$ lowers enough, which in turn requires a sufficiently weak capital-labor substitution, i.e., $\sigma < 1 - s$. In this case, the future capital stock $k_{t+1}$ decreases if the consumption $c_{0t}$ is not too sensitive to changes in income $I_t$. This explains why a sufficiently weak elasticity of intertemporal substitution in consumption is necessary for the occurrence of cycles and instability and why concavity promotes cycles too.

Keeping in mind the basic mechanism, we can tackle the case of endogenous labor supply which makes more complex the income response to a change in capital.

The crucial question of the paper, we cannot get round, is why, in many cases, endogenous labor promotes saddle-path stability? Under elastic labor supply, the income $I_t$ becomes:

$$I_t = f(a_t) l_{0t} + f'(a_t) a_t \sum_{i=1}^{n} l_{it} \quad (60)$$

with $l_t = \sum_{i=0}^{n} l_{it}$ and $a_t = k_t/l_t$. All the labor supplies $l_{it}$ ($i = 0, 1, ..., n$) depend on the real wage which is increasing in $k_t$. Therefore, an increase in $k_t$ generates two new effects: on the one hand, an impact on $I_t$ through $a_t$, that is, through $f(a_t)$ and $f'(a_t)$ $a_t$; on the other hand, an impact through the weights $l_{0t}$ and $\sum_{i=1}^{n} l_{it}$.

In order to evaluate the first effect, we observe that $l_t$ is an implicit function of $k_t$ and $c_{0t}$ (equation (26)), and the capital-labor ratio is as well: $a_t = k_t/l_t(k_t, c_{0t}) \equiv a(k_t, c_{0t})$. Then, using (73) (see the Appendix), we obtain:

$$\frac{\partial a/a}{\partial k/k} = \frac{\sigma(a)/s(a)}{\sigma(a)/s(a) + \lambda_0 \xi_0 + (1 - \lambda_0) \tilde{\xi}}$$

As soon as $\lambda_0\xi_0 + (1 - \lambda_0)\tilde{\xi} > 0$, this elasticity is less than one,<fnref n="19" /> which means that, following an initial increase in $k_t$, the endogenous labor response reduces the increase of the capital-labor ratio and possibly dampens oscillations along non-monotonic capital paths.

In order to evaluate the second effect (on the weights $l_{0t}$ and $\sum_{i=1}^{n} l_{it}$), we consider the $a_t$'s appearing directly in equation (60) as fixed. Under this restriction, the derivative of (60) with respect to $k_t$ is given by:

$$k \left[ f(a) \lambda_0 \xi_0 + a f'(a) (1 - \lambda_0) \tilde{\xi} \right] \frac{s(a)}{\sigma(a)} \frac{\partial a}{\partial k} \quad (61)$$

<footnote n="19">¹⁹We notice that, when labor supply is inelastic, the capital-labor ratio is equal to $k/(1+n)$.</footnote>
The second effect also promotes stability if the income $I_t$ increases with $k_t$ through the weights $l_{0t}$ and $\sum_{i=1}^n l_{it}$. So, oscillations and endogenous cycles are dampened if (61) is positive, which depends on the sign of the term into the brackets and the sign of $\partial a/\partial k$. Both of them are positive if $\tilde{\xi}$ is not too negative:

$$ \tilde{\xi} > -\min \left\{ \frac{\lambda_0 \xi_0}{s(a)(1-\lambda_0)}, \frac{\lambda_0 \xi_0 + \sigma(a)/s(a)}{1-\lambda_0} \right\} $$

The addition of both the effects can explain why, when the slope of the impatient agents' labor supply is not too negative or the degree of concavity of leisure utility is sufficiently weak, endogenous labor supply promotes saddle-path stability. One immediately sees that the same conclusion holds for the patient agent, through $\xi_0$.

Finally, let us notice that, as was the case under inelastic labor supply, endogenous cycles and instability occur more likely if $c_{0t}$ is little sensitive to variations in $k_t$.<fnref n="20" /> This requires a sufficiently weak elasticity of intertemporal substitution in consumption, a high degree of concavity of consumption utility or a degree of homogeneity sufficiently less than one.

# 11 Conclusion

We have addressed the question of saddle-path stability and dynamic convergence in an economy with heterogeneous consumers and elastic labor supply. In line with Becker (1980), Becker and Foias (1987, 1994), Hernandez (1991) and Sorger (1994), we have also supposed that agents face borrowing constraints, representing a kind of market incompleteness.

As in the case of inelastic labor supply, a state with a positive amount of capital in the hands of some impatient household cannot be stationary. This means that in a neighborhood of a steady state, the capital stock is wholly owned by the most patient agent, whereas the others consume their labor income. Focusing on such equilibria, we first establish the existence and uniqueness of the steady state.

At a second stage, we analyze the local dynamics to address the saddle-path stability and convergence issues. In particular, we focus on the role of preferences on the stability properties of the steady state and the occurrence of endogenous cycles. As was the case in economies with inelastic labor supply, instability and endogenous cycles require sufficiently weak capital-labor substitution. However, now, the taste for leisure and the impatient agents' behavior have a great influence on dynamics. We first derive conditions for saddle-path stability and the occurrence of flip bifurcation under general preferences in consumption and leisure. Then, we apply our results to particular classes of preferences widely employed in macrodynamics.

In order to fix ideas, it is not unworthy to recover the representative agent as a limit case. We find that the steady state is always a saddle point, in contrast

<footnote n="20">²⁰We observe that, under elastic labor supply, $c_{0t}$ also affects $l_{0t}$ and, consequently, $l_t$ and $a_t$.</footnote>
to De Hek (1998) who obtains persistent fluctuations. However, his analysis is somewhat misleading, because founded on questionable assumptions such as the lack of normality in consumption and leisure.

Coming back to the model with heterogeneous consumers, we make explicit the role of preferences on dynamics, by presenting three classes of utility functions commonly used in economic literature.

When preferences are separable in consumption and leisure, the introduction of elastic labor supply promotes saddle-path stability if the impatient agents easily smooth consumption over time. If conversely, their intertemporal substitution in consumption is too weak, there is room for instability and deterministic cycles.

When preferences are KPR (King, Plosser and Rebelo (1988)), the impatient agents' utility functions cease from having any influence on dynamics, while only the patient agent's tastes matter. An elasticity of intertemporal substitution in consumption which is not too weak and a leisure utility which is not too concave are needed to ensure convergence, whatever the technological parameters.

Eventually, when preferences are homogeneous, the degree of homogeneity of the patient agent's utility naturally emerges as crucial parameter. In particular, we prove a somewhat intriguing result: when the degree is close to one, saddle-path stability prevails.

# 12 Appendix

**Proof of Proposition 1.** The ith agent maximizes the intertemporal utility function (2) under the sequences of budget constraints (3) and borrowing constraints (4). The first order conditions reduce to the Euler equations:

$$ \frac{u_{i1}(c_{it}, 1 - l_{it})}{u_{i1}(c_{it+1}, 1 - l_{it+1})} \geq \beta_i R_{t+1}, \text{ with equality if } k_{it+1} > 0 \quad (62) $$

the consumption-labor arbitrages: $u_{i2}(c_{it}, 1 - l_{it}) = u_{i1}(c_{it}, 1 - l_{it}) w_t$, the budget constraints (3) (now binding), and, eventually, the individual transversality conditions: $\lim_{t \to +\infty} \beta_i^t u_{i1}(c_{it}, 1 - l_{it}) k_{it+1} = 0$.

As seen above, heterogeneity in time preference leads the more patient agent to hold the entire capital stock near a stationary solution. More precisely, equation (62) becomes $R \leq 1/\beta_i$ in a neighborhood of the steady state. But, since $\beta_0 > \beta_i$ for $i > 0$, the equilibrium interest factor will prevail, will be the cheapest rental factor:

$$ R = \min\{1/\beta_i\}_{i=0}^n = 1/\beta_0 \quad (63) $$

(assumption (1)), and $1/\beta_i > R$ at the stationary equilibrium for $i \geq 1$. Around the steady state, $u_{i1}(c_{it}, 1 - l_{it})/u_{i1}(c_{it+1}, 1 - l_{it+1}) > \beta_i R_{t+1}$, for $i \geq 1$, according to equations (62) and (63). Firms borrow the cheapest capital from the individual $i = 0$, while individuals $i \geq 1$ end up to hold no capital: $k_{0t} > 0$ and $k_{it} = 0$ for $i \geq 1$.
The capitalist's equation (62) ($i=0$) will hold with equality, jointly with the consumption-leisure arbitrage (9), the budget constraint (10) and the transversality condition.

The impatient agents' budget constraints (3) simplify to (11), because now $k_{it} = 0$ (for $i \ge 1$). The consumption-leisure arbitrages (9) still hold, where $c_{it}$ is given by (11). Eventually, the transversality conditions are satisfied.

Inequalities (13) must hold at every period in order to ensure the capitalistic distribution as an intertemporal equilibrium. ■

**Proof of Lemma 2.** First, we prove that $l_0(w, c_0)$ is well defined from:

$$ \varphi_0(l_0, c_0) = w \qquad (64) $$

where $\varphi_0(l_0, c_0) \equiv u_{02}(c_0, 1-l_0)/u_{01}(c_0, 1-l_0)$. We observe that under Assumption 1, $\lim_{l_0 \to 1^-} \varphi_0(l_0, c_0) = +\infty$. $\varphi_0$ crosses at least once $w$ because, by hypothesis, $\lim_{l_0 \to 0} u_{02}(c_0, 1-l_0)/u_{01}(c_0, 1-l_0) < w$. Moreover, under Assumption 2, $\partial\varphi_0(l_0, c_0)/\partial l_0 = -(u_{022}-u_{012}u_{02}/u_{01})/u_{01} > 0$. Then, $\varphi_0$ crosses $w$ only once and the solution of (64) is unique.

Second, we prove the existence of the function $l_i(w)$ for $i \ge 1$. We need to solve the implicit equation:

$$ \varphi_i(l_i) = w \qquad (65) $$

where, as above, $\varphi_i(l_i) \equiv u_{i2}(wl_i, 1-l_i)/u_{i1}(wl_i, 1-l_i)$. Given $w$, we have, under Assumption 1, $\lim_{l_i \to 0^+} \varphi_i(l_i) = 0$ and $\lim_{l_i \to 1^-} \varphi_i(l_i) = +\infty$. By continuity, at least a solution of equation (65) exists. In addition, since $\varphi'_i(l_i) = [u_{i12}u_{i2}/u_{i1} - u_{i22} + w(u_{i21} - u_{i11}u_{i2}/u_{i1})]/u_{i1} > 0$ under Assumption 2, the function $\varphi_i$ crosses $w$ only once. The solution $l_i(w)$ of equation (65) is thus unique. ■

**Proof of Proposition 3.** As a preliminary result, we observe that, under Assumption 1, the labor supply of each consumer is always smaller than one, i.e., $l_i < 1$ for all $i \ge 0$. Now, notice that a steady state is defined by:

$$ \begin{aligned} r(k/l) &= 1/\beta - \Delta \\ l &= l_0(w(k/l), c_0) + \sum_{i=1}^{n} l_i(w(k/l)) \\ c_0 &= (r(k/l) - \delta)k + w(k/l)l_0(w(k/l), c_0) \end{aligned} $$

Under Assumption 3, there exists a unique solution $k/l \equiv \alpha$ to the first equation, where $\alpha > 0$ is a constant. The wage $w(k/l) = w(\alpha)$ is constant too and the functions $l_i(w)$ as well (denote them $\bar{l}_i$). Finally, the agent 0's labor supply is defined by:

$$ u_{02}((1/\beta - 1)k + wl_0, 1 - l_0) = wu_{01}((1/\beta - 1)k + wl_0, 1 - l_0) \qquad (66) $$
and determines a continuous function $l_0 = \hat{l}_0(k)$, with $k = \alpha l$. Since the sum of continuous functions is continuous,

$$ \lambda(l) = \hat{l}_0(\alpha l) + \sum_{i=1}^{n} \bar{l}_i \qquad (67) $$

is continuous. A steady state is a solution to the equation $l = \lambda(l)$, where $l$ belongs to $(0, 1+n)$. Using our preliminary result, we deduce that $\lambda(0) < 1+n$. Moreover, we have:

$$ \lambda'(l) = -\alpha \frac{1-\beta}{\beta} \left( w + \frac{c_0}{1-l_0} \frac{\varepsilon_{022} - \varepsilon_{012}}{\varepsilon_{011} - \varepsilon_{021}} \right)^{-1} < 0, $$

according to (17). Thus, there is a unique solution $l^* = \lambda(l^*)$, which is a steady state and determines the other variables, namely $k^*$ and $l_0^*$. ■

**Local Dynamics.** In order to get local information on dynamics, we linearize the dynamic system (27)-(28) around the steady state. Equations (27) and (28) become, respectively:

$$ \begin{aligned} & -\frac{u_{012}(1-l_0)}{u_{01}} \frac{l_0}{1-l_0} \frac{l_{0w}w}{l_0} \left( \frac{w_k k}{w} + \frac{w_{l1} l_k k}{w l} \right) \frac{dk_t}{k} \\ & + \left[ \frac{u_{011}c_0}{u_{01}} - \frac{u_{012}(1-l_0)}{u_{01}} \frac{l_0}{1-l_0} \left( \frac{l_{0c_0}c_0}{l_0} + \frac{l_{0w}w}{l_0} \frac{w_{l1} l_{c_0} c_0}{w l} \right) \right] \frac{dc_{0t}}{c_0} \\ & + \left[ \frac{u_{012}(1-l_0)}{u_{01}} \frac{l_0}{1-l_0} \frac{l_{0w}w}{l_0} \left( \frac{w_k k}{w} + \frac{w_{l1} l_k k}{w l} \right) - \beta r \left( \frac{r_k k}{r} + \frac{r_{l1} l_k k}{r l} \right) \right] \frac{dk_{t+1}}{k} \\ & - \left[ \frac{u_{011}c_0}{u_{01}} - \frac{u_{012}(1-l_0)}{u_{01}} \frac{l_0}{1-l_0} \left( \frac{l_{0c_0}c_0}{l_0} + \frac{l_{0w}w}{l_0} \frac{w_{l1} l_{c_0} c_0}{w l} \right) + \beta r \frac{r_{l1} l_{c_0} c_0}{r l} \right] \frac{dc_{0t+1}}{c_0} \\ & = 0 \end{aligned} \qquad (68) $$

and

$$ \begin{aligned} & \left( \left[ \Delta + \left( 1 + \frac{r_k k}{r} + \frac{r_{l1} l_k k}{r l} \right) r \right] k + w l_0 \left( \frac{w_k k}{w} + \frac{w_{l1} l_k k}{w l} \right) \left( 1 + \frac{l_{0w}w}{l_0} \right) \right) \frac{dk_t}{k} \\ & + \left( \frac{r_{l1} l_{c_0} c_0}{r l} r k + w l_0 \left[ \frac{l_{0c_0}c_0}{l_0} + \frac{w_{l1} l_{c_0} c_0}{w l} \left( 1 + \frac{l_{0w}w}{l_0} \right) \right] - c_0 \right) \frac{dc_{0t}}{c_0} \\ & - k \frac{dk_{t+1}}{k} = 0 \end{aligned} \qquad (69) $$

To achieve the reduction in elasticities, we need those of labor supply. We know that $l_0 = l_0(w, c_0)$, $l_i^* = l_i(w)$ and $l = l(k, c_0)$ are respectively given by (9), (23) and (25). Total differentiating (9), we get:

$$ \varepsilon_{w0} \equiv \frac{l_{0w}w}{l_0} = \frac{1-l_0}{l_0} \frac{1}{\varepsilon_{012} - \varepsilon_{022}} \equiv \xi_0 > 0 \qquad (70) $$

$$ \varepsilon_{c0} \equiv \frac{l_{0c_0}c_0}{l_0} = \frac{1-l_0}{l_0} \frac{\varepsilon_{021} - \varepsilon_{011}}{\varepsilon_{022} - \varepsilon_{012}} < 0 \qquad (71) $$
Totally differentiating (23), we obtain:

$$ \xi_i \equiv \frac{l_i^{*\prime}(w)w}{l_i} = \frac{1 + \varepsilon_{i11} - \varepsilon_{i21}}{\varepsilon_{i21} - \varepsilon_{i11} + (\varepsilon_{i12} - \varepsilon_{i22})l_i / (1 - l_i)} \quad (72) $$

for $i = 1, \dots, n$, while, totally differentiating (25), we find:

$$ \frac{l_k k}{l} = \frac{\lambda_0 \varepsilon_{w0} + \sum_{i=1}^{n} \lambda_i \xi_i}{\lambda_0 \varepsilon_{w0} + \sum_{i=1}^{n} \lambda_i \xi_i + \sigma/s} \quad (73) $$

$$ \frac{l_{c0} c_0}{l} = \frac{\lambda_0 \varepsilon_{c0} \sigma / s}{\lambda_0 \varepsilon_{w0} + \sum_{i=1}^{n} \lambda_i \xi_i + \sigma / s} \quad (74) $$

where $\lambda_i$ is the individual labor share in total labor supply. Substituting the elasticities (70), (71), (73) and (74) in equations (68) and (69), we get a reduced system:

$$ \begin{aligned} & -\frac{(1 + \rho x)}{k} \frac{dk_{t+1}}{dt} + \left[ \lambda_0 \varepsilon_{c0} \left(1 + \rho x\right) - z \frac{\varepsilon_{011}\varepsilon_{022} - \varepsilon_{012}\varepsilon_{021}}{\varepsilon_{012}} \right] \frac{dc_{0t+1}}{c_0} \\ & = -\frac{dk_t}{k} + \left[ \lambda_0 \varepsilon_{c0} - z \frac{\varepsilon_{011}\varepsilon_{022} - \varepsilon_{012}\varepsilon_{021}}{\varepsilon_{012}} \right] \frac{dc_{0t}}{c_0} \\ & \quad -\beta \frac{dk_{t+1}}{k} = \left[ \rho \frac{1 - \lambda_0 - \lambda_0 \varepsilon_{w0}}{z - 1} \right] \frac{dk_t}{k} \\ & \quad + \left[ \rho \lambda_0 \left( 1 - \varepsilon_{c0} \left[ 1 + \frac{1 - \lambda_0 - \lambda_0 \varepsilon_{w0}}{z} \right] \right) + 1 - \beta \right] \frac{dc_{0t}}{c_0} \end{aligned} $$

where

$$ x \equiv 1 - \varepsilon_{022} / \varepsilon_{012} \quad (75) $$

$$ z \equiv \sigma/s + \xi \quad (76) $$

Afterwards, we evaluate the Jacobian matrix $J$ of system (27)-(28) at the steady state:

$$ \begin{bmatrix} -(1 + \rho x) & \lambda_0 \varepsilon_{c0} (1 + \rho x) - z \frac{\varepsilon_{011}\varepsilon_{022} - \varepsilon_{012}\varepsilon_{021}}{\varepsilon_{012}} \\ -\beta & 0 \end{bmatrix}^{-1} \begin{bmatrix} -1 & \lambda_0 \varepsilon_{c0} - z \frac{\varepsilon_{011}\varepsilon_{022} - \varepsilon_{012}\varepsilon_{021}}{\varepsilon_{012}} \\ \rho \frac{1 - \lambda_0 - \lambda_0 \varepsilon_{w0}}{z - 1} & \rho \lambda_0 \left(1 - \varepsilon_{c0} \left[1 + \frac{1 - \lambda_0 - \lambda_0 \varepsilon_{w0}}{z}\right] + 1 - \beta\right) \end{bmatrix} $$

We deduce the trace $T$ and the determinant $D$ of the Jacobian matrix using (75) and (76).

**Proof of Corollary 6.** We apply the results obtained in Proposition 4. Using (48), the critical values $\tilde{\xi}^*$ and $\varkappa$ are explicitly computed:

$$ \tilde{\xi}^* = \frac{1 - l_0}{l - l_0} \frac{\rho - \varepsilon_0}{(1 + 2\varepsilon_0)\varepsilon_{v_0} - \varepsilon_0\varepsilon_{v_0'}} (< 0) \quad (77) $$

$$ \varkappa = \frac{\rho}{1 + \beta} \left( 1 + \frac{Q}{2(l - l_0) \left[ (1 + 2\varepsilon_0)\varepsilon_{v_0} - \varepsilon_0\varepsilon_{v_0'} \right]} \right) \quad (78) $$
Since $\xi_i = 0$ for all $i \ge 1$ (expression (72)), we have also $\tilde{\xi} = 0$. As highlighted in Proposition 4, a flip bifurcation generically arises if and only if $\sigma_F = s(1 - \lambda_0)(\tilde{\xi}^* + \varkappa) > 0$, which requires $\tilde{\xi}^* + \varkappa > 0$. Using (49), (77) and (78), this last inequality holds if and only if (50) and (51) are satisfied. ■

**Proof of Proposition 7.** Since the utility function $u_i$ is homogeneous of degree $\nu_i$, $u_i(\mu c_i, \mu(1-l_i)) = \mu^{\nu_i}u_i(c_i, 1-l_i)$ for any positive real number $\mu$. Homogeneity implies some well-known properties: $s_{i1} + s_{i2} = \nu_i$, $\varepsilon_{i11} + \varepsilon_{i12} = \nu_i - 1 = \varepsilon_{i21} + \varepsilon_{i22}$ and

$$ \frac{s_{i1}}{s_{i2}} = \frac{\nu_i - 1 - \varepsilon_{i22}}{\nu_i - 1 - \varepsilon_{i11}} \qquad (79) $$

for $i = 0, \dots, n$, where $s_{i1} \equiv u_{i1}c_i/u_i$ and $s_{i2} \equiv u_{i2}(1-l_i)/u_i$ are the consumption and leisure shares in total utility and $\varepsilon_{ij1}$ and $\varepsilon_{ij2}$ the elasticities of marginal utilities (definition (15)).$^{21}$

In our analysis, the key variables are the consumption share in total utility $s_{i1}$ and the elasticity of intertemporal substitution in consumption $(-1/\varepsilon_{i11})$. Simplifying notation as follows: $\varepsilon_i \equiv \varepsilon_{i11}$ and $s_i \equiv s_{i1}$, we get $s_{i2} = \nu_i - s_i$ and:

$$ \varepsilon_{i12} = \nu_i - 1 - \varepsilon_i \qquad (80) $$

$$ \varepsilon_{i21} = \frac{s_i}{\nu_i - s_i} (\nu_i - 1 - \varepsilon_i) \qquad (81) $$

$$ \varepsilon_{i22} = \nu_i - 1 - \frac{s_i}{\nu_i - s_i} (\nu_i - 1 - \varepsilon_i) \qquad (82) $$

The first inequality in (53) ensues from $s_{i1}, s_{i2} \in (0, \nu_i)$. From (80), (81) and (82), we compute other expressions of interest appearing in $\tilde{\xi}^*$, $\tilde{\xi}^* + \varkappa$ and $\tilde{\xi} = \sum_{i=1}^n \lambda_i \xi_i / (1 - \lambda_0)$, that is in equations (32), (33) and (72), respectively:

$$ \varepsilon_{i11} - \varepsilon_{i21} = \left[ \frac{\varepsilon_i \nu_i + (1 - \nu_i) s_i}{\nu_i - s_i} \right] / (\nu_i - s_i) = \varepsilon_{i22} - \varepsilon_{i12} \qquad (83) $$

$$ \varepsilon_{i11}\varepsilon_{i22} - \varepsilon_{i12}\varepsilon_{i21} = -(1 - \nu_i)(\varepsilon_{i11} - \varepsilon_{i21}) \qquad (84) $$

Normality conditions (16) and (17) immediately entail the second inequality in (53). Substituting (80)-(84) into (32), (33) and (72), we eventually obtain (54), (55) and (56). ■

**Proof of Corollary 8.** Saddle-path stability requires $\sigma > \sigma_F$. Then $\sigma_F < 0$ or, equivalently $\tilde{\xi} \ge \tilde{\xi}^* + \varkappa$, implies convergence. Replacing (54), (55) and (56)

<footnote>$^{21}$These properties are derived using the Euler's law ($u_{i1}c_i + u_{i2}(1-l_i) = \nu u_i$) and the fact that the marginal utilities are homogeneous of degree $\nu_i - 1$. Equation (79) is obtained differentiating the Euler's law w.r.t. to $c_i$ and $1 - l_i$ and noticing that the cross derivatives are equal.</footnote>
in $\tilde{\xi} \ge \tilde{\xi}^* + \varkappa$, we get

$$ \sum_{i=1}^{n} l_i (1 - l_i) \left( 1 + \frac{\nu_i - s_i}{\varepsilon_i \nu_i + (1 - \nu_i) s_i} \right) \\ \le \frac{1 - l_0}{1 - \nu_0} \left( \rho + \varepsilon_0 \frac{\nu_0 - s_0}{\varepsilon_0 \nu_0 + (1 - \nu_0) s_0} \right) \\ - \frac{\rho}{1 + \beta} \left( l - l_0 - \frac{\rho + (1 - \beta) (1 + l - l_0)}{2 (1 - \nu_0)} \right) \quad (85) $$

A sufficient condition for the left-hand side in inequality (85) to be negative is $1 + (\nu_i - s_i) / [\varepsilon_i \nu_i + (1 - \nu_i) s_i] < 0$ for every $i$, that is (58) under (53). A sufficient condition for the right-hand side to be positive is $l - l_0 - [\rho + (1 - \beta) (1 + l - l_0)] / [2 (1 - \nu_0)] < 0$, that is (57) under (53). ■

**Proof of Corollary 9.** We notice that, under (53), as $\nu_0$ tends to $1^-$, the left-hand side of inequality (85) in the Proof of Corollary 8 gets a finite value, while the right-hand side goes to $+\infty$. Then inequality (85) asymptotically holds. ■

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