Overlapping Generations Model with Endogenous

Labor Supply: General Formulation1,2

C. NOURRY3 AND A. VENDITTI4

Communicated by C. Deissenberg

1This paper is dedicated to the memory of Louis-Andre Gerard-Varet, Director of GREQAM from

1987 to 1999.

2We thank S. Catania, C. Deissenberg, J.-P. Drugeon, L.-A. Gerard-Varet, J.-M. Grandmont, A.

Kirman, T. Lloyd-Braga, P. Michel, P. Pintus, E. Thibault, B. Wigniolle and two anonymous referees for

helpful comments and suggestions. The paper benefited from comments received during a presentation

at the “Conference on Dynamic Equilibria, Expectations and Indeterminacies”, Paris, France, 1999.

3Professor, EPEE, Universite de la Mediterranee and GREQAM, Marseille, France

4Director of Research, CNRS - GREQAM, Marseille, France

Abstract: This paper develops a one-sector overlapping generations model with endoge-

nous labor supply and nonseparable preferences. It demonstrates that local indeterminacy

arises easily under gross substitutability as soon as there exist multiple steady states. We

show also that, depending on whether leisure and second period consumption are gross

substitutes, local indeterminacy holds for very different parameters configurations. If

gross substitutability is satisfied, the existence of multiple equilibrium paths requires the

share of capital in the total income to be strong enough with respect to the elasticity of

capital-labor substitution. On the other hand, if gross substitutability is violated, local

indeterminacy necessitates the share of capital in the total income to be weak enough

with respect to the elasticity of capital-labor substitution.

Key Words: Indeterminacy, endogenous cycles, overlapping generations, endogenous

labor supply.

2

1 Introduction

Since the seminal contribution of Cass et Shell (Ref. 1), a huge literature refers to the con-

cept of indeterminacy to analyze endogenous fluctuations and macroeconomic volatility.

From a general point of view, indeterminacy is associated with the existence of multiple

equilibria. Two types of concept may then be considered. Global indeterminacy is based

on the existence of multiple steady states and thus the existence of different equilibrium

paths converging to different long-run positions. Local indeterminacy means that there

exists a continuum of equilibrium path starting from the same initial condition, all of

which converging to the same steady state. Since the contribution of Woodford (Ref.

2), it is well known that local indeterminacy is a sufficient condition for the existence

of sunspot equilibria and stochastic fluctuations based upon extrinsic uncertainty. A

sunspot equilibrium is based on the fact that agents receive different allocations across

states while the fundamentals are identical. It follows that consumption and/or produc-

tion differ although the tastes of consumers and the production functions of firms are the

same. Local indeterminacy easily generates sunspot equilibria by randomizing the expec-

tations of agents over the continuum of equilibrium paths. Shocks on expectations then

provide a nice propagation mechanisms to explain part of the macroeconomic volatility.

In this paper we consider an extension of the Diamond (Ref. 3) overlapping generations

(OLG) model which is augmented to include endogenous labor supply. This model, which

1

is one of the main framework to analyze macroeconomic dynamics, is based on the co-

existence, at each date, of young agents who work, consume and save during a first period,

and old agents who are retired and consume on the basis of their saving returns during

a second period. The OLG model is then a counterpart to the Ramsey (Ref. 4) optimal

growth model in which agents are assumed to live over an infinite horizon.

In the recent period, the Ramsey model augmented to include endogenous labor supply

and market imperfections such that productive external effects has became a strandard

framework for the analysis of local indeterminacy and expectations-driven fluctuations

based on the existence of sunspot equilibria.5 A similar level of interest has not been

experienced by the OLG model with endogenous labor supply. Indeed, most of the recent

contributions are based on particular assumptions which either restrict preferences to

be additively separable or to exclude first period consumption, or restrict technologies to

Leontief or Cobb-Douglas formulations.6 Our goal is on the contrary to consider a general

OLG model and to provide a complete analysis of the dynamic properties of equilibria.

In a standard OLG model with inelastic labor, local indeterminacy of the steady

state is not possible. When endogenous labor is introduced in a model with separable

5External effects are feedbacks from the other agents in the economy who also face identical maximizing

problems. See Benhabib and Farmer (Ref. 5) for a survey.

6See Cazzavillan (Ref. 6), Cazzavillan and Pintus (Ref. 7), Lloyd-Braga (Ref. 8), Medio and Negroni

(Ref. 9) and Reichlin (Ref. 10).

2

preferences and Leontief technology, Reichlin (Ref. 10) shows on the contrary that local

indeterminacy may appear even under the gross substitute axiom.

Cazzavillan and Pintus (Ref. 7) consider an OLG model with consumption in both

periods and endogenous labor supply under assumptions of additively separable prefer-

ences and gross substitutability of consumptions and leisure. These restrictions allow the

authors to perform the dynamical analysis using a simple geometrical method recently

developed by Grandmont, Pintus and de Vilder (Ref. 11). They discuss the stability

properties of a steady state as function of the elasticity of capital-labor substitution and

the share of first period consumption in total income. On related topics, Nourry (Ref.

12) considers an overlapping generations model with endogenous labor and non separable

preferences but she still assumes gross substitutability between consumptions and leisure.

In this paper we assume general non separable preferences with consumption in both

periods and we do not a priori introduce any restriction on the substitutability properties.

Such a formulation implies that the geometrical methodology cannot in general be per-

formed. This is why we provide a formal stability analysis in a first part. The sufficient

conditions for global and local indeterminacy will be formulated with the elasticities of

the saving and labor supply functions. In a second part of the paper, we will however

consider a CES economy which will be analysed with the simple geometrical method.

The current paper establishes sufficient conditions for local and global indeterminacy of

perfect-foresight equilibria. The analysis demonstrates that gross substitutability in con-

3

sumptions and leisure only ensures local uniqueness of an equilibrium path if the steady

state is unique. When multiple steady states arise local indeterminacy is fully compatible

with a standard saving function which increases with the interest factor. When gross sub-

stitutability is violated, perfect-foresight equilibria are locally indeterminate if the labor

supply function is elastic enough with respect to the interest factor. We finally exhibit

a strong distinction depending on whether or not leisure and second period consumption

are gross substitutes: if gross substitutability holds, the existence of multiple equilibria

requires the share of capital in total income to be strong enough with respect to the

elasticity of capital-labor substitution. On the contrary, if gross substitutability does not

holds, local indeterminacy necessitates the share of capital to be weak enough with respect

to the elasticity of capital-labor substitution.

2 Model with Endogenous Labor Supply

The economy is populated by finitely-lived agents. In each period t, Nt persons are born,

and they live for two periods. In their first period of life (when young), the agents are

working, supplying elastically to firms a portion l of one unit of labor: 0 ≤ l ≤ 1. Denoting

w the wage rate, their wage income is then equal to wl. They allocate this income between

current consumption ct and savings st which are invested in the firms. In their second

period of life (when old), they are retired. Their income, given by the return on savings

4

Rt+1st, is entirely consumed dt+1. The preferences of a representative agent born at time

t are thus defined over his consumption bundle and his leisure time (L = 1− l). They are

summarized by the utility function u(c,L, d).

(A1) u(c,L, d) is C2 with u1(c,L, d) > 0, u2(c,L, d) > 0, u3(c,L, d) > 0 and a negative

definite Hessian matrix over the interior of the set R+×[0, 1]×R+. Moreover, ∀c,L, d > 0,

u1(0,L, d) = u2(c, 0, d) = u3(c,L, 0) = ∞ and u2(c, 1, d) = 0.

Each agent is assumed to have 1+n children, with n ≥ 0. Under perfect foresight, and

considering the wage rate wt and the expected interest factor Rt+1 as given, he maximizes

his utility function over his life-cycle as follows:

maxct,Lt,dt+1,st

u(ct,Lt, dt+1) (1)

s.t. wt(1 − Lt) = ct + st (2)

Rt+1st = dt+1 (3)

0 ≤ Lt ≤ 1 (4)

Under (A1), the first order conditions:

−u1(wtlt − st, 1 − lt, stRt+1) + Rt+1u3(wtlt − st, 1 − lt, stRt+1) = 0 (5)

wtu1(wtlt − st, 1 − lt, stRt+1) − u2(wtlt − st, 1 − lt, stRt+1) = 0 (6)

give unique saving and labor supply functions, st = s(wt, Rt+1) and lt = l(wt,Rt+1), which

are differentiable for (w, R) ∈ R++ × R++, with values respectively in R+ and (0, 1). To

5

derive more information on these functions we introduce the following property.

Definition 2.1. Strong normality. Consider the following maximisation program

maxc,L,d

u(c,L, d)

s.t. w = c + wL + d/R

0 ≤ L ≤ 1

and the optimal demand functions c(w, R) and d(w, R). The consumptions c and d will

be called strongly normal goods if (∂c/∂w) ≥ 0 and (∂d/∂w) ≥ 0.7

We will use throughout the paper the following restriction:

(A2) Consumptions c and d are strongly normal goods.

Under (A2), the saving function is increasing with respect to the wage rate, i.e. sw(w, R) ≥

0. Notice that if leisure is also a strongly normal good, then Lw(w, R) ≥ 0 and the labour

supply is a decreasing function of the wage rate, i.e. lw(w, R) ≤ 0.

The production function of a representative firm, F (K, L), depends on the stock of

capital K and labor L, and is assumed to be concave and homogeneous of degree one.

7The standard definition of normality is based on the following program

maxc,L,d

u(c,L, d)

s.t. Ω = c + wL + d/R, 0 ≤ L ≤ 1

and implies (∂c/∂Ω) ≥ 0 and (∂d/∂Ω) ≥ 0. Besides this standard income effect, our definition includes

also a price effect which corresponds to the wage rate interpreted as the price of leisure. Strong normality

implies therefore that the income effect is greater than the price effect.

6

Under complete depreciation of capital, and denoting, for any L 6= 0, k = K/L the capital

stock per labor unit, we define the production function in intensive form as f(k) = F (k, 1).

(A3) f(k) is positively valued, C2, strictly increasing and strictly concave over R++.

Along a competitive equilibrium, the interest factor Rt and the wage rate wt satisfy:

Rt = f ′(kt) ≡ R(kt) wt = f(kt) − ktf′(kt) ≡ w(kt) (7)

Thus, the capital accumulation equation simply states as:

Kt+1 = Nts(wt, Rt+1) (8)

Since each young agent supplies lt unit of labor, the total labor in the economy during

period t is Lt = Ntlt, and equation (8) becomes:

(1 + n)kt+1l(w(kt+1), R(kt+2)) = s(w(kt), R(kt+1)) (9)

3 Steady State

The capital accumulation equation (9), together with the factor market equilibrium con-

ditions (7), (8) and the initial condition for the capital stock, fully characterize the in-

tertemporal equilibria with perfect foresight.

Definition 3.1. A steady state is a stationary capital-labor ratio k such that

(1 + n)kl(f(k) − kf ′(k), f ′(k)) = s(f(k) − kf ′(k), f ′(k)) (10)

7

The overlapping generations model may be characterized by either the non existence,

uniqueness or multiplicity of non-trivial steady states. Note that the proofs for all the

subsequent propositions are given in the Working Paper version of this document.8

Proposition 3.1. Under (A1)-(A3), there exists a trivial steady state k = 0 iff f(0) = 0.

The existence of a non-trivial steady state requires additional restrictions. Let us denote

φ(k) = (1 + n)k −s (f(k) − kf ′(k), f ′(k))

l (f (k) − kf ′(k), f ′(k))(11)

Proposition 3.2. Under (A1)-(A3),

(i) if lim supk→0 φ′(k) < 0, there exists a nontrivial steady state. Generically, the number

of nontrivial steady states is odd;

(ii) if lim infk→0 φ′(k) > 0, the number of non-trivial steady states is generically even.

We will analyse the local dynamics of equilibrium paths around a non-trivial steady state.

4 Analysis of the Local Dynamics

Consider εl,w(k), εl,R(k), εs,w(k), εs,R(k) the elasticities of the labor supply and saving

functions with respect to w and R evaluated at a non-trivial steady state k. We have,

for example, εl,w(k) = (w/l)(∂l(w, R)/∂w), where w and l are evaluated at the steady

state. (A2) implies that εs,w(k) > 0 while the sign of the other elasticities is a priori

8See (Ref. 13) available at http://www.vcharite.univ-mrs.fr/greqam/papers/paper fr.php?annee=2001.

8

indeterminate. We also consider the share of capital in total income εf (k) = kf ′(k)/f (k)

and the index Af(k) = −kf ′′(k)/f ′(k). Denoting ς(k) the elasticity of capital-labor

substitution in the production function, it is easy to show that ς(k) = (1− εf (k))/Af (k).

We will assume that εl,R(k) 6= 0. Linearizing the difference equation (9) around k gives

the following characteristic polynomial in terms of elasticities:9

P(λ) = λ2 − λς + εf εl,w + (1 − εf )εs,R

(1 − εf )εl,R+

εf εs,w

(1 − εf)εl,R (12)

Definition 4.1. Let kt∞t=0 denote an equilibrium for an economy with initial condition

k0. We say that it is a locally indeterminate equilibrium if for every ε > 0 there exists

another sequence k′t

∞t=0, with 0 < |k′

1−k1| < ε and k′0 = k0, which is also an equilibrium.

If an equilibrium is not locally indeterminate, then we call it locally determinate. Actually,

k is locally indeterminate if and only if the local stable manifold is two-dimensional.

4.1 Case εl,R > 0.

We assume here that leisure and second period consumption are gross substitutes. In the

standard Diamond model, under the assumption εs,R ≥ 0, the following condition ensures

local stability of equilibrium paths:

ΛD(k) ≡ ς(k) + (1 − εf (k))εs,R(k) − εf(k)εs,w(k) > 0 (13)

9In the following we will omit the argument of elasticities when there is no ambiguity.

9

We may derive some conditions in the model with endogenous labor which can be inter-

preted as extensions of the above inequality.

Proposition 4.1. Let ΛL = ΛD + εfεl,w − (1 − εf )εl,R. Under (A1)-(A3) and εl,R > 0,

the following cases hold:

(C1) If ΛL + εf εs,w + (1 − εf)εl,R > 0, the steady state k is

(i) saddle-point stable iff ΛL > 0;

(ii) locally indeterminate iff ΛL < 0 and εf εs,w < (1 − εf )εl,R;

(iii) locally unstable iff ΛL < 0 and εfεs,w > (1 − εf )εl,R.

(C2) If ΛL + εf εs,w + (1 − εf)εl,R < 0, the steady state k is

(i) saddle-point stable iff ΛL + 2[εf εs,w + (1 − εf )εl,R] < 0;

(ii) locally indeterminate iff ΛL + 2[εf εs,w + (1 − εf )εl,R] > 0 and εf εs,w < (1 − εf)εl,R;

(iii) locally unstable iff ΛL + 2[εfεs,w + (1 − εf )εl,R] > 0 and εfεs,w > (1 − εf )εl,R.

Assume that there exist n non-trivial steady states, n ∈ N∗+, which are ordered as

k1 > k2 > ... > kn. We provide the following characterization:10

Proposition 4.2. Under (A1)-(A3), let εl,R(ki) > 0, i = 1, · · · , n. Then all steady states

with an odd index are saddle-point stable, whereas steady states with an even index

are locally indeterminate or locally unstable. Moreover, k = 0 is saddle-point stable if

lim infk→0 φ′(k) > 0, and locally indeterminate or locally unstable if lim supk→0 φ′(k) < 0.

10Assuming that ΛL + εfεs,w + (1 − εf)εl,R > 0, Nourry (Ref. 12) obtains a similar result.

10

Corollary 4.1. Under (A1)-(A3), let εl,R > 0. If there exists a unique k > 0, then it is

saddle-point stable while k = 0 is locally indeterminate or locally unstable.

Even if we impose gross substitutability, local indeterminacy of an interior steady state

is not ruled out. However, this result crucially depends on the existence of multiple non-

trivial steady states. If uniqueness holds, gross substitutability implies local determinacy.

Let us now consider that the elasticity of capital-labor substitution, ς , is constant.

Variations from the share of capital in total income εf could allow for the appearance of

bifurcations. Denoting the sum of characteristic roots as T (εf ), and the product as D(εf ),

T (εf ) =ς + εf εl,w + (1 − εf)εs,R

(1 − εf )εl,R, D(εf ) =

εfεs,w

(1 − εf )εl,R

we also introduce the following notations which will be convenient to state the results:

N (εf) = ς + εf εl,w + (1 − εf )εs,R

denotes the numerator of the trace,

P1(εf ) = (1 − εf )εl,R + εf εs,w −N (εf)

denotes the numerator of the characteristic polynomial when λ = 1, and

P−1(εf ) = (1 − εf )εl,R + εf εs,w + N (εf )

denotes the numerator of the characteristic polynomial when λ = −1.

11

Proposition 4.3. Under (A1)-(A3), let εl,R > 0 and the technology have a constant

elasticity of substitution ς ≥ 0. The following cases hold

(i) If N (εf ) > 0 for any εf ∈ (0, 1), limεf→0 P1(εf ) = εl,R − (ς + εs,R) < 0 and

limεf→1 P1(εf ) = εs,w − (ς + εl,w) > 0, there exists εTf ∈ (0, 1) such that the steady

state is saddle-point stable for any εf ∈ (0, εTf ) and locally indeterminate when εf belongs

to a right neighbourhood of εTf . An eigenvalue goes through one as εf crosses εT

f .11

(ii) If N (εf ) > 0 and P1(εf) > 0 for any εf ∈ (0, 1), limεf→0 D(εf) = 0 and

limεf→1 D(εf ) = +∞, there exists εHf ∈ (0, 1) such that the steady state is locally indeter-

minate for any εf ∈ (0, εHf ) and locally unstable when εf belongs to a right neighbourhood

of εHf . A Hopf bifurcation occurs at εH

f .

(iii) If N (εf ) < 0 for any εf ∈ (0, 1), limεf→0 P−1(εf ) = ς + εl,R + εs,R < 0 and

limεf→1 P−1(εf ) = ς + εs,w + εl,w > 0, there exists εFf ∈ (0, 1) such that the steady state

is saddle-point stable for any εf ∈ (0, εFf ) and locally indeterminate when εf belongs to a

right neighbourhood of εFf . A Flip bifurcation occurs at εF

f .

(iv) If N (εf) < 0 and P−1(εf ) > 0 for any εf ∈ (0, 1), limεf→0 D(εf ) = 0 and

limεf→1 D(εf ) = +∞, there exists εHf ∈ (0, 1) such that the steady state is locally indeter-

minate for any εf ∈ (0, εHf ) and locally unstable when εf belongs to a right neighbourhood

of εHf . A Hopf bifurcation occurs at εH

f .

11We cannot a prori distinguish between the transcritical, pitchfork or saddle-node bifurcations. In a

CES economy discussed in Section 5 we exhibit a transcritical bifurcation.

12

Interpretations: We have assumed that leisure and the second period consumption

are gross substitutes, i.e. εl,R > 0. If we also assume gross substitutability for consump-

tions of both periods then εs,R > 0. It follows that only cases (i) and (ii) in proposition

4.3 may appear. Note that leisure may be or not a strongly normal good but if strong

normality holds, i.e. εl,w < 0, εl,w cannot be too strong with respect to the elasticity of

capital-labor substitution ς . In case (i), the elasticities of the saving function needs to

be strong enough. In case (ii) on the contrary, the labour demand function needs to be

sufficiently elastic with respect to the interest factor. Note however that all these results

imply that there exist at least one other non-trivial steady state since if uniqueness holds

gross substitutability implies saddle-point stability.

When εs,R < 0 and if leisure is strongly normal, then cases (iii) and (iv) may appear.

However, εl,w and εs,R need to be strong enough in absolute value with respect to ς . More

precisely, we need to have in both cases εs,R < −ς − εl,R, but (iii) requires −ς > εl,w >

−ς − εs,w while (iv) requires εl,w < −ς − εs,w.

In cases (i) and (iii), local determinacy is ensured if the share of capital in total income

is weak enough with respect to the elasticity of capital-labor substitution. However, in

the others configurations, and even under gross substitutability, except uniqueness of the

steady state, there does not exist any simple condition to ensure local determinacy.

13

4.2 Case εl,R < 0.

If εl,R < 0 then leisure and second period consumption are not gross substitutes. Under

(A2), P(0) < 0 and the eigenvalues are real with opposite sign. We obtain:

Proposition 4.4. Under (A1)-(A3) and εl,R < 0, a steady state k is

(i) saddle-point stable iff ΛLΛL + 2[εf εs,w + (1 − εf)εl,R] > 0;

(ii) locally indeterminate iff ΛL > 0 > ΛL + 2[εf εs,w + (1 − εf )εl,R];

(iii) locally unstable iff ΛL + 2[εf εs,w + (1 − εf )εl,R] > 0 > ΛL.

As in the previous subsection, if there exist n steady states which are ordered as k1 >

k2 > ... > kn, then the following characterization holds:

Proposition 4.5. Under (A1)-(A3), let εl,R(ki) < 0, i = 1, · · · , n. Then the following

cases hold:

(i) If ΛL(ki) + εf (ki)εs,w(ki) + (1 − εf (ki))εl,R(ki) > 0, i = 1, · · · , n, all steady states

with an even index are locally unstable, whereas steady states with an odd index are

saddle-point stable or locally indeterminate. Moreover, the trivial steady state k = 0 is

locally unstable if lim supk→0 φ′(k) < 0, and saddle-point stable or locally indeterminate

if lim infk→0 φ′(k) > 0.

(ii) If ΛL(ki) + εf (ki)εs,w(ki) + (1 − εf (ki))εl,R(ki) < 0, i = 1, · · · , n, all steady states

with an odd index are locally indeterminate, while steady states with an even index

are saddle-point stable or locally unstable. Moreover, the trivial steady state k = 0 is

14

locally indeterminate if lim infk→0 φ′(k) > 0, and saddle-point stable or locally unstable if

lim supk→0 φ′(k) < 0.

Corollary 4.2. Under (A1)-(A3), let εl,R < 0. If there exits a unique non-trivial steady

state k > 0, it is saddle-point stable or locally indeterminate while the trivial steady state

k = 0 is locally unstable.

When the labor supply decreases with the interest factor, even if there exits a unique

non-trivial steady state, indeterminacy is not ruled out.

Considering again that the elasticity of capital-labor substitution ς is constant, we get:

Proposition 4.6. Under (A1)-(A3), let εl,R < 0 and the technology have a constant

given elasticity of substitution ς ≥ 0. Then the following cases hold

(i) If P−1(εf) < 0 for any εf ∈ (0, 1), limεf→0 P1(εf ) = εl,R − (ς + εs,R) < 0 and

limεf→1 P1(εf ) = εs,w − (ς + εl,w) > 0, th ere exists εTf ∈ (0, 1) such that the steady

state is locally indeterminate for any εf ∈ (0, εTf ) and saddle-point stable when εf belongs

to a right neighbourhood of εTf . An eigenvalue goes through one as εf crosses εT

f .

(ii) If P1(εf ) < 0 for any εf ∈ (0, 1), limεf→0 P−1(εf) = ς + εl,R + εs,R < 0 and

limεf→1 P−1(εf ) = ς + εs,w + εl,w > 0, there exists εFf ∈ (0, 1) such that the steady state

is locally indeterminate for any εf ∈ (0, εFf ) and saddle-point stable when εf belongs to a

right neighbourhood of εFf . A Flip bifurcation occurs at εT

f .

15

Interpretations: We have assumed that εl,R < 0. Case (i) requires that leisure is

strongly normal. If the labor supply is sufficiently elastic with respect to the wage rate

and the interest factor, indeterminacy is compatible with a saving function that increases

with R, i.e. with a gross substitutability between first and second period consumptions.

In case (ii), a steady state may be locally indeterminate even if leisure is not strongly

normal and the saving function increases with R. However the labor supply needs to be

strongly decreasing with respect to R.

Note that contrary to the configuration in Section 4.1, when leisure and second period

consumption are not gross substitutes, local determinacy of the steady state requires the

share of capital in total income to be strong enough with respect to the elasticity of

capital-labor substitution.

5 CES Economy

We consider an economy characterized by CES utility and production functions such that:

u(c,L, d) = (B/α) [σc−ρ + δL−ρ + βd−ρ]−α/ρ

f(k) = A [θk−γ + 1 − θ]−1/γ

with σ ≥ 0, δ, β > 0, θ ∈ (0, 1), α ≤ 1, ρ, γ > −1 and A, B > 0.12 The elasticity of

intertemporal substitution is given by η = 1/(1 + ρ) while the elasticity of capital-labor

12With B = (σ + δ + β)ρ/α, σ, δ and β may be interpreted as weighting parameters.

16

substitution is ς = 1/(1+γ). The share of capital in total income is εf = θ[θ+(1−θ)kγ]−1.

Contrary to the formulation used by Cazzavillan and Pintus (Ref. 7), the Arrow-Pratt

index for consumptions and labor are not independent. The case σ = 0 will correspond to

the Reichlin’s model (Ref. 10) in which the representative agent does not consume during

his first period of life. The saving and labor supply functions are:13

s(w, R) =w(βR)

11+ρ

wρ

1+ρ δ1

1+ρ R +[σ

11+ρ R + (βR)

11+ρ

] , l(w, R) σ1

1+ρ R+(βR)1

1+ρ

wρ

1+ρ δ1

1+ρ R+

[σ

11+ρ R+(βR)

11+ρ

]

with

w = w(k) = A(1 − θ) [θk−γ + 1 − θ]−(1+γ)/γ

R = R(k) = Aθk−1−γ [θk−γ + 1 − θ]−(1+γ)/γ

(14)

We study the dynamic behaviour of this economy using the geometric approach devel-

opped by Grandmont, Pintus and de Vilder (Ref. 11). We use the normalisation procedure

associated with the geometrical method which consists in finding some conditions on the

parameters A, δ and β such that one steady state k is always equal to one. The ob-

jective is to study geometrically the stability properties of this steady state in a simple

(Trace,Determinant) plan by varying the share of capital in total income and the elasticity

of capital-labor substitution.

Assuming that k = 1 is a steady state, let us first find some values for δ and β such

that the elasticities of the saving and labor supply functions when evaluated at k = 1

13While a CES utility function does not necessarily satisfy the Inada conditions in (A1), the optimiza-

tion program (4) provides interior solutions for consumptions, saving and labor supply.

17

are not affected when the share of capital in total income and the elasticity of capital

labor substitution are modified. Considering that w(1) = A(1 − θ) and R(1) = Aθ, some

tedious but straightforward computations show that if δ = [A(1 − θ)]−ρ and β = (Aθ)ρ,

the elasticities of the saving and labor supply functions when k = 1 are as follows:

εl,w = −ρ

(1 + ρ)(2 + σ1

1+ρ )εs,w = −

εl,w

ρ[1 + (1 + ρ)(1 + σ

11+ρ )] (15)

εs,R = εl,w(1 + σ1

1+ρ ) εl,R =εl,w

1 + σ1

1+ρ

(16)

Note that (A2) is satisfied since εs,w > 0 for any ρ > −1. Moreover εl,w, εl,R and εs,R

have the same sign given by the sign of −ρ. It follows that if ρ < 0, gross substitutability

between consumptions and leisure holds.

Equation (11) from Section 3 becomes

φ(k) = (1 + n)k −wR(βR)

11+ρ

σ1

1+ρ + β1

1+ρ R−ρ1+ρ

Considering the values of the wage rate and the interest factor given in (14), we have

w/R = (1 − θ)k1+γ/θ and a steady state is finally a solution of the following equation

φA(k) = k

1 + n −

(1−θ)kγ

θ

[βAθ[θ + (1 − θ)kγ ]−(1+γ)/γ

]1/(1+ρ)

σ1

1+ρ + β1

1+ρ (Aθ)−ρ/(1+ρ)[θ + (1 − θ)kγ]ρ(1+γ)/γ(1+ρ)

= 0

Using β = (Aθ)ρ, k = 1 is always a steady state, i.e. φA(1) = 0, if and only if

A = A∗ = (1 + n)1 + σ

11+ρ

1 − θ(17)

After substitution of A∗ into φA(k) we obtain:

18

φA∗(k) = (1 + n)k

1 − kγ(1 + σ

11+ρ )

σ1

1+ρ [θ + (1 − θ)kγ ]1+γ

γ(1+ρ) + [θ + (1 − θ)kγ]1+γ

γ

= 0

≡ (1 + n)kψ (k)

By definition we have ψ (1) = 0. Depending on γ the properties of ψ (k) are the following:

(i) If γ ≤ 0 then limk→0 ψ (k) = −∞, limk→+∞ ψ (k) = 1 and ψ ′(k) > 0.

(ii) If γ > 0 then ψ (0) = 1, limk→+∞ ψ (k) = 1 and there exists k > 0 such that

ψ ′(k) S 0 if and only if k S k. Note that k may be less or greater than one. Moreover

depending on the value of ρ + θ and σ, we have:

(a) If ρ + θ < 0 and σ ≥ [−θ(1 + ρ)/(ρ + θ)]1+ρ, ψ ′(1) < 0 ∀γ > 0;

(b) If ρ+ θ < 0 and σ < [−θ(1+ρ)/(ρ+ θ)]1+ρ, or ρ+ θ ≥ 0, there exists γ∗(θ) > 0

such that ψ ′(1) T 0 if and only if γ T γ∗(θ).14

In case (i) the function ψ (k) is monotone increasing so that k = 1 is the unique non-trivial

steady state. In case (ii), since k = 1 is always a steady state and ψ (0) = limk→+∞ ψ (k) =

1, there generically exists a second non-trivial steady state. Note also from proposition

3.1 that limk→0 φA(k) = 0 if and only if f(0) = 0, i.e. γ ≥ 0. In such a case only the trivial

steady state k = 0 also exists. We summarize the results into the following proposition

Proposition 5.1. Consider the CES economy with δ = [A∗(1 − θ)]−ρ, β = (A∗θ)ρ and

A∗ as defined in (17). There exists one trivial steady state k = 0 if and only if γ ≥ 0.

Moreover, the following cases hold:

14The exact value of γ∗(θ) is γ∗(θ) = (1 − θ)[1 + σ1/(1+ρ) + ρ

]/[σ1/(1+ρ)(ρ + θ) + θ(1 + ρ)].

19

(i) If γ ≤ 0, i.e. ς ≥ 1, k = 1 is the unique non-trivial steady state.

(ii) If γ > 0, i.e. ς < 1, there exist two distinct non-trivial steady states, k = 1 and

k∗ > 0. Moreover:

(a) if ρ + θ < 0 and σ ≥ [−θ(1 + ρ)/(ρ + θ)]1+ρ, then k∗ > 1 ∀γ > 0;

(b) if ρ + θ < 0 and σ < [−θ(1 + ρ)/(ρ + θ)]1+ρ, or ρ + θ ≥ 0, then k∗ > 1 when

γ < γ∗(θ) and k∗ ∈ (0, 1) when γ > γ∗(θ). k = 1 and k∗ > 0 coincide when γ = γ∗(θ).

We now study the stability properties of k = 1. Let us introduce the notations:

T (σ, εf , ς) =ς + εf εl,w + (1 − εf )εs,R

(1 − εf )εl,R, D(σ, εf ) =

εf εs,w

(1 − εf )εl,R

for the trace T and the determinant D. Note that the following ratio of derivatives

∂D(σ, εf )/∂εf

∂T (σ, εf , ς)/∂εf≡ p(σ, ς) =

εs,w

εl,w + ς

does not depend on εf and corresponds to the slope of a line ∆ in the (T, D) plan which

gives a linear relationship between T and D. Substituting (15)-(16) into T , D and p gives

T (σ, εf , ς) =ς(1 + σ

11+ρ ) + εl,w(1 + σ

11+ρ )[εf + (1 − εf)(1 + σ

11+ρ )]

(1 − εf )εl,w

D(σ, εf) = −εf

ρ(1 − εf )(1 + σ

11+ρ )[1 + (1 + ρ)(1 + σ

11+ρ )]

p(σ, ς) = −εl,w[1 + (1 + ρ)(1 + σ1

1+ρ )]

ρ(εl,w + ς)

Note that D(σ, 0) = 0, T (σ, 0, ς) = (1 + σ1

1+ρ )[ς + εl,w(1 + σ1

1+ρ )]/εl,w. Moreover

limεf→1 D(σ, εf) = ±∞ if and only if ρ ≶ 0 and limεf→1 T (σ, εf , ς) = ±∞ depending

20

on the sign of ρ and the value of ς . Therefore varying εf from 0 to 1 describes a half-line

∆ defined by the equation

D = [1 + (1 + ρ)(1 + σ1

1+ρ )]

−

εl,w

ρ(εl,w + ς)T + (1 + σ

11+ρ )

ς + εl,w(1 + σ1

1+ρ )

ρ(εl,w + ς)

We divide the discussion into two subcases depending on the value of the elasticity of

intertemporal substitution η.

5.1 Case η > 1.

When ρ ∈ (−1, 0) and thus η > 1, we get εl,w, εl,R, εs,R > 0. We also have:

- ∀(σ, ς) ∈ R2+ p(σ, ς) > 0 with p(σ, 0) > 1, and limς→+∞ p(σ, ς) = 0.

- ∀(σ, ς) ∈ R2+ T (σ, 0, 0) = (1 + σ

11+ρ )2, limεf→1 T (σ, εf , ς) = +∞ and

limς→+∞ T (σ, 0, ς) = +∞.

- ∀σ ≥ 0 D(σ, 0) = 0 and limεf→1 D(σ, εf ) + ∞.

Note that T (0, 0, 0) = 1. In the (T, D) plan, a necessary condition for the existence of

local indeterminacy is T (σ, 0, 0) < 2, i.e. σ < σ(ρ) ≡ (√

2 − 1)1+ρ < 1. We may however

find a more accurate upper bound for σ. Since the slope is maximal when ς = 0, the

upper bound is defined as the solution of the following system:

T (σ, εf , 0) = 2 D(σ, εf ) = 1

From the second equation we derive

εHf =

−ρ

(1 + σ1

1+ρ )[1 + (1 + ρ)(1 + σ1

1+ρ )] − ρ(18)

21

After substitution into the first equation and denoting x = 1 +σ1

1+ρ , the upper bound for

σ is obtained from the following degree 3 polynomial

h(x) = (1 + ρ)x3 + x2 − 2(1 + ρ)x − 2 − ρ = 0 (19)

Since h(0) < 0 and h(√

2) > 0, there exists σ∗(ρ) ∈ (0, σ(ρ)) such that local indeterminacy

cannot hold as soon as σ ≥ σ∗(ρ). Note that this results does not necessarily implies that

indeterminacy is ruled out when first period consumption is too large. Considering indeed

that B = (σ + δ + β)ρ/α, we find that σ∗(ρ) gives a critical value for the discount factor

over the life-cycle. As it will be numerically illustrated below, local indeterminacy is

compatible with a representative agents which prefers present consumption.

Considering εHf as defined by (18), we may now find for any given ρ ∈ (−1, 0) and

σ ∈ [0, σ∗(ρ)), the value of ς , denoted ς, such that T (σ, εHf , ς) = 2, i.e such that the steady

state cannot be locally indeterminate for any ς ≥ ς :

ς = εl,w[1 + (1 + ρ)(1 + σ

11+ρ )][2 − (1 + σ

11+ρ )2] + ρ

(1 + σ1

1+ρ )[1 + (1 + ρ)(1 + σ1

1+ρ )] − ρ> 0 (20)

For any σ ∈ [0, σ∗(ρ)) we get p(σ, 1) = 1. Since ς is close to 0 when σ is close to σ∗(ρ), and

ς = −ρ/2 < 1 when σ = 0, for any σ ∈ [0, σ∗(ρ)), we have ς < 1. We may summarize the

results on indeterminacy by the following figure before stating a complete proposition.

Insert Figure 1

Proposition 5.2. Consider the bounds εHf and ς defined by (18) and (20), and assume

that η > 1, i.e. ρ ∈ (−1, 0). The following cases hold:

22

(C1) Let σ = 0.

(i) If ς = 0, the steady state is locally indeterminate for εf ∈ (0,−ρ/2), locally unstable

for εf > −ρ/2 and a Hopf bifurcation occurs at εHf = −ρ/2.

(ii) If ς ∈ (0,−ρ/2), there exists εTf ∈ (0, 1) such that the steady state is saddle-point

stable when εf ∈ (0, εTf ), locally indeterminate when εf ∈ (εT

f ,−ρ/2) and locally unstable

when εf > −ρ/2. A (reverse) transcritical bifurcation occurs at εTf , a Hopf bifurcation at

εHf = −ρ/2.

(iii) If ς = −ρ/2, the steady state is saddle-point stable when εf ∈ (0,−ρ/2), locally

unstable when εf > −ρ/2 and a co-dimension 2 bifurcation occurs at εHf = −ρ/2.

(iv) If ς ∈ (ς, 1), there exists εTf ∈ (0, 1) such that the steady state is saddle-point

stable for εf ∈ (0, εTf ), locally unstable for εf > εT

f and a transcritical bifurcation occurs

at εTf .

(v) If ς ≥ 1, the steady state is saddle-point stable for all εf ∈ (0, 1).

(C2) Let σ ∈ (0, σ∗(ρ)) with σ∗(ρ) defined by (19).

(i) If ς ∈ [0, ς), there exists εTf ∈ (0, 1) such that the steady state is saddle-point stable

when εf ∈ (0, εTf ), locally indeterminate when εf ∈ (εT

f , εHf ) and locally unstable when

εf > εHf . A (reverse) transcritical bifurcation occurs at εT

f , a Hopf bifurcation at εHf .

(ii) If ς = ς , the steady state is saddle-point stable for εf ∈ (0, εHf ), locally unstable

for εf > εHf and a co-dimension 2 bifurcation occurs at εH

f .

(iii) If ς ∈ (ς , 1), there exists εTf ∈ (0, 1) such that the steady state is saddle-point

23

stable when εf ∈ (0, εTf ), locally unstable for εf > εT

f and a transcritical bifurcation occurs

at εTf .

(iv) If ς ≥ 1, the steady state is saddle-point stable for all εf ∈ (0, 1).

(C3) Let σ ≥ σ∗(ρ).

(i) If ς ∈ [0, 1), there exists εTf ∈ (0, 1) such that the steady state is saddle-point stable

when εf ∈ (0, εTf ), locally unstable for εf > εT

f and a transcritical bifurcation occurs at εTf .

(ii) If ς ≥ 1, the steady state is saddle-point stable for all εf ∈ (0, 1).

This proposition shows in particular how a Cobb-Douglas technology is specific since local

indeterminacy and cycles cannot occur when ς = 1.

Codimension 2 Global Bifurcation: In cases (C1)-(iii) and (C2)-(ii), a co-

dimension 2 Bogdanov-Takens bifurcation occurs. There exist two eigenvalues equal to 1

when the pair (ς, εf ) is equal to (−ρ/2, εHf ), with σ = 0, or (ς , εH

f ), with σ ∈ (0, σ∗(ρ)).

This strong resonance implies a global bifurcation. Its study requires different changes

of variables first to embed the non linear map into an approximated flow, and second

to use the Bogdanov-Takens bifurcation analysis (see Kuznetsov (Ref. 14)). Under non

degeneracy conditions, the approximated continuous time system will exhibit a saddle-

connection characterized by a homoclinic orbit and a limit cycle. But within the original

discrete time system, the coincidence of the stable and unstable manifolds of the saddle

is generically replaced by a transversal intersection, giving rise to a homoclinic structure

24

and complicated (“chaotic”) dynamics.

Comparisons with Related Papers: It is well-known from Reichlin (Ref. 10),

Lloyd-Braga (Ref. 8) and Cazzavillan and Pintus (Ref. 7) that under an additive sepa-

rable utility function, a Hopf bifurcation is only possible if the elasticity of capital-labor

substitution is lower than the share of capital in total income.15 In our CES economy, we

find a similar result: as soon as σ ∈ (0, σ∗(ρ)), the critical value for the inputs elasticity

of substitution above which local indeterminacy and Hopf bifurcation no longer exist is

less than the Hopf bifurcation value for the share of capital in total income, i.e. εHf > ς .

Cazzavillan and Pintus (Ref. 7) also prove that local indeterminacy only occurs when

consumption in the first period of life is a small fraction of wage income. Proposition 5.2

exhibits a result close to this conclusion. Local indeterminacy and Hopf bifurcation do

not occur anymore as soon as σ is higher than σ∗(ρ) < 1. However, as mentioned above,

this result is compatible with a discount factor characterizing a preference for present

consumption. Such a conclusion may be axplained by the fact that, contrary to the

Cazzavillan-Pintus formulation,16 the elasticity of intertemporal substitution in consump-

15Lloyd-Braga (Ref. 8) however proves that a Hopf bifurcation occurs while the elasticity of substitution

is higher than the share of capital when imperfect competition and increasing returns to scale internal to

the firm are considered. The same result also holds when increasing returns are generated by externalities

(See Cazzavillan (Ref. 6)).

16Their additive separable formulation does not allow to define the elasticity of intertemporal substi-

25

tion associated with non-separable CES preferences may be large enough to compensate

an increase in first period consumption by a decrease in second period consumption.

Notice also that we may find a Hopf bifurcation value εHf such that the share of capital

in total output εf is equal to 1/3. Consider ρ = −0.74 so that η ≈ 3.85, σ∗(−0.74) ≈ 0.66

and the value of σ derived from solving εHf = 1/3 is σ ≈ 0.6. In this case, we derive

B = (3.9)ρ/α and σ/(3.9) ≈ 0.438, δ/(3.9) ≈ 0.265, β/(3.9) ≈ 0.297. The implicit discount

factor is 0.6. The upper bound for ς above which local indeterminacy is precluded is finally

ς ≈ 0.24. It follows that for any ς ∈ [0, 0.24) the steady state is locally indeterminate

when εf < 1/3 and, if the Hopf bifurcation is supercritical, indeterminate quasi-periodic

cycles exist when εf is in a right neighbourhood of 1/3.

5.2 Case η < 1.

When ρ > 0 and thus η < 1, we get εl,w, εl,R, εs,R < 0, and the following results hold:

- There exists ς∞ = −εl,w = ρ/[(1 + ρ)(2 + σ1/(1+ρ)] < 1 such that p(σ, ς∞) = ∞

∀σ ≥ 0. Therefore p(σ, ς) < 0 when ς ∈ [0, ς∞) and p(σ, ς) > 0 when ς > ς∞. Moreover

∀σ ≥ 0 p(σ, 0) < −1 and limς→+∞ p(σ, ς) = 0.

- T (σ, 0, 0) = (1 + σ1

1+ρ )2, T (σ, 0, ς∞) = σ1

1+ρ (1 + σ1

1+ rho ) and limς→+∞ T (σ, 0, ς) =

−∞. Moreover, limσ→+∞ T (σ, 0, ς) = +∞ and limεf→1 T (σ, εf , ς) = +∞ if ς ∈ [0, ς∞),

limεf→1 T (σ, εf , ς) = −∞ if ς > ς∞.

tution in consumption.

26

- ∀σ ≥ 0 D(σ, 0) = 0 and limεf→1 D(σ, εf ) −∞.

For any σ ≥ 0, we may find the value of ς , denoted ς , such that the line ∆ goes through

the point (T, D) = (0,−1) by solving T (σ, εf , 0) = 0 and D(σ, εf ) = −1. From D we get

εFTf =

ρ

(1 + σ1

1+ρ )[1 + (1 + ρ)(1 + σ1

1+ρ )] + ρ(21)

After substitution into T we find

ς = −εl,w(1 + σ

11+ρ )2[1 + (1 + ρ)(1 + σ

11+ρ )] + ρ

(1 + σ1

1+ρ )[1 + (1 + ρ)(1 + σ1

1+ρ )] + ρ> 0 (22)

We may also compute the value of ς , denoted ςT , such that T (σ, 0, ς) = 1.

ςT =ρ[(1 + σ

11+ρ )2 − 1]

(1 + ρ)(1 + σ1

1+ρ )(2 + σ1

1+ρ )< 1 (23)

It is easy to show that ςT < ς and ς∞ < ς. We need finally to compute the value of ς ,

denoted ςF , such that T (σ, 0, ς) = −1:

ςF =ρ[1 + (1 + σ

11+ρ )2]

(1 + ρ)(1 + σ1

1+ρ )(2 + σ1

1+ρ )< 1 (24)

We summarize all the results by the following figure and proposition.

Insert Figure 2

Proposition 5.3. Consider the bounds εFTf , ς , ςT , ςF defined by (21), (22), (23), (24),

and assume that η < 1, i.e. ρ > 0. The following cases hold:

(C1) Let σ = 0.

27

(i) If ς = 0, there exists εTf ∈ (0, 1) such that the steady state is saddle-point stable

when εf ∈ (0, εTf ), locally unstable when εf > εT

f and a transcritical bifurcation occurs at

εTf .

(ii) If ς ∈ (0, ρ/[2(1 + ρ)]), there exist εTf ∈ (0, 1) and εF

f ∈ (0, 1) such that the steady

state is locally indeterminate when εf ∈ (0, εTf ), saddle-point stable when εf ∈ (εT

f , εFf )

and locally unstable when εf > εFf . A transcritical bifurcation occurs at εT

f , a Flip at εFf .

(iii) If ς = ρ/[2(1 + ρ)], the steady state is locally indeterminate for εf ∈ (0, εF Tf ),

locally unstable for εf > εFTf and a co-dimension 2 bifurcation occurs at εFT

f .

(iv) If ς ∈ (ρ/[2(1+ρ)], ςF ), there exist εFf ∈ (0, 1) and εT

f ∈ (0, 1) such that the steady

state is locally indeterminate when εf ∈ (0, εFf ), saddle-point stable when εf ∈ (εF

f , εTf )

and locally unstable when εf > εTf . A Flip bifurcation occurs at εF

f , a transcritical at εTf .

(v) If ς ∈ (ςF , 1), there exists εTf ∈ (0, 1) such that the steady state is saddle-point

stable when εf ∈ (0, εTf ), locally unstable when εf > εT

f and a transcritical bifurcation

occurs at εTf .

(vi) If ς ≥ 1, the steady state is saddle-point stable for any εf ∈ (0, 1).

(C2) Let σ > 0.

(i) If ς ∈ [0, ςT ], there exists εFf ∈ (0, 1) such that the steady state is saddle-point

stable when εf ∈ (0, εFf ), locally unstable when εf > εF

f and a Flip bifurcation occurs at

εFf .

(ii) If ς ∈ (ςT , ς), there exist εTf ∈ (0, 1) and εF

f ∈ (0, 1) such that the steady state is

28

locally indeterminate when εf ∈ (0, εTf ), saddle-point stable when εf ∈ (εT

f , εFf ) and locally

unstable when εf > εFf . A transcritical bifurcation occurs at εT

f , a Flip at εFf .

(iii) If ς = ς, the steady state is locally indeterminate for εf ∈ (0, εFTf ), locally unstable

for εf > εFTf and a co-dimension 2 bifurcation occurs at εF T

f .

(iv) If ς ∈ (ς , ςF ), there exist εFf ∈ (0, 1) and εT

f ∈ (0,1) such that the steady state is

locally indeterminate when εf ∈ (0, εFf ), saddle-point stable when εf ∈ (εF

f , εTf ) and locally

unstable when εf > εTf . A Flip bifurcation occurs at εF

f , a transcritical at εTf .

(v) If ς ∈ [ςF , 1), there exists εTf ∈ (0, 1) such that the steady state is saddle-point

stable when εf ∈ (0, εTf ), locally unstable when εf > εT

f and a trnascritical bifurcation

occurs at εTf .

(vi) If ς ≥ 1, the steady state is saddle-point stable for any εf ∈ (0, 1).

This proposition confirms the singularity of Cobb-Douglas technologies since local inde-

terminacy and cycles cannot occur when ς = 1. We have also proved:

Corollary 5.1. In a CES economy, if the elasticity of capital-labor substitution is greater

than or equal to 1, the steady state is saddle-point stable.

This corollary therefore implies that in a CES economy, uniqueness of the steady state

implies uniqueness of the equilibrium path even if gross substitutability does not hold.

29

6 Concluding Comments

This paper develops a one-sector overlapping generations model with endogenous labor

supply. It exhibits a stong interaction between global and local indeterminacy by showing

that local indeterminacy arises under gross substitutability as soon as there exist multiple

steady states. It also shows that depending on whether or not leisure and second period

consumption are gross substitutes, local indeterminacy holds for very different parameters

configurations. If gross substitutability is satisfied, the existence of multiple equilibrium

paths requires the share of capital in total income to be strong enough with respect to

the elasticity of capital-labor substitution. On the contrary, if gross substitutability is

violated, local indeterminacy necessitates the share of capital in total income to be weak

enough with respect to the elasticity of capital-labor substitution.

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32

List of Figures:

Figure 1. Indeterminacy and bifurcations when σ ∈ (0, σ∗).

33

Figure 2. Indeterminacy and bifurcations when σ > 0.

34