Local indeterminacy under dynamic efficiency in a two-sector overlapping generations economy

(1)

HAL Id: halshs-00439240

https://halshs.archives-ouvertes.fr/halshs-00439240

Preprint submitted on 7 Dec 2009

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Local indeterminacy under dynamic eﬀiciency in a two-sector overlapping generations economy

Carine Nourry, Alain Venditti

To cite this version:

Carine Nourry, Alain Venditti. Local indeterminacy under dynamic eﬀiciency in a two-sector overlapping generations economy. 2009. �halshs-00439240�

(2)

GREQAM

Groupement de Recherche en Economie Quantitative d'Aix-Marseille - UMR-CNRS 6579 Ecole des Hautes Etudes en Sciences Sociales

Universités d'Aix-Marseille II et III

Document de Travail n°2009-27

LOCAL INDETERMINACY UNDER

DYNAMIC EFFICIENCY IN A TWO-SECTOR OVERLAPPING GENERATIONS ECONOMY

Carine NOURRY Alain VENDITTI

September 2009

(3)

Local indeterminacy under dynamic efficiency in a two-sector overlapping generations economy^∗

Carine NOURRY^†

Université de la Méditerranée and GREQAM

and

Alain VENDITTI^‡

CNRS - GREQAM and EDHEC

Abstract: We consider a two-sector two-periods overlapping generations model with inelastic labor, consumption in both periods of life and homothetic CES preferences.

Assuming gross substitutability and a capital intensive consumption good, we prove that when dynamic efficiency holds, local indeterminacy and sunspot fluctuations occur with low enough values for the sectoral elasticities of capital-labor substitution and we illustrate this finding within a standard example. This result shows that some fiscal policy rules can prevent the existence of business-cycle fluctuations in the economy by driving it to the optimal steady state as soon as it is announced, and thus shows that Reichlin’s [9] influential conclusion is a robust property in a two-sector OLG economy.

Keywords: Two-sector OLG model, dynamic efficiency, gross substitutability in consumption, local indeterminacy, stabilization policy

Journal of Economic Literature Classification Numbers: C62, E32, O41.

∗This work was supported by French National Research Agency Grant (ANR-08- BLAN-0245-01). We would like to thank G. Bloise, J.P. Drugeon, R. Farmer, C. Ghiglino, J.M. Grandmont and P. Pintus for useful comments and suggestions.

†E-mail: Carine.Nourry@univmed.fr

‡E-mail: Alain.Venditti@univmed.fr

(4)

1 Introduction

The existence of local indeterminacy and sunspot fluctuations under gross substitutability are well established facts whithin OLG economies. Consid- ering an aggregate model with endogenous labor and consumption in the second period of life only, Reichlin [9] has shown that Hopf cycles and thus local indeterminacy arise with a unique Pareto optimal steady state when the technology is Leontief. This conclusion has had a strong echo in the literature as it implies that the introduction of a public policy based on taxes and transfers could at the same time stabilize the economy and reach the Pareto optimal steady state along which all generations get an equal level of utility.

However, Cazzavillan and Pintus [3] have recently proved that the co- existence of local indeterminacy and dynamic efficiency is not robust to the consideration of any positive elasticity of capital-labor substitution. Indeed, apart from the very special case of Leontief technology, the steady state is always characterized by an over-accumulation of capital when local indeterminacy holds,¹ and policies targeting the steady state allocation generically leave room for welfare losses associated with productive inefficiency.

As initially shown in Galor [6], local indeterminacy also arises in two- sector OLG economies under gross substitutability. However, the Pareto optimality of the equilibrium has not been precisely discussed in such a framework. On the one hand, Reichlin [10] proves local indeterminacy of a dynamically inefficient steady state through the existence of periodic, quasi- periodic and chaotic dynamics under the assumption of Leontief technologies. On the other hand, Drugeon et al. [5] show that any dynamically efficient equilibrium path is locally determinate if the sectoral elasticities of capital-labor substitution are large enough.

Our main objective in this paper is to show that Reichlin’s [9] result is a robust property in a two-sector OLG economy. Assuming gross substitutability and a capital intensive consumption good sector, we prove indeed that local indeterminacy can occur under dynamic efficiency when interme- diary values for the elasticity of intertemporal substitution in consumption and low enough but positive sectoral elasticities of capital-labor substitution are considered. This conclusion shows that contrary to the aggregate framework, in two-sector OLG economies with capital-labor substitutability, some fiscal policy rules can eliminate business-cycle fluctuations in the economy

1See also Cazzavillan and Pintus [2] for an extension of this result to a model with positive first-period consumption.

(5)

by driving it to an optimal steady state as soon as it is announced.

2 The model

2.1 Production

The economy consists in one consumption goody₀ and one capital good y produced from capital and labor through standard constant returns to scale technologies:

y₀ =f⁰(k⁰, l⁰), y=f¹(k¹, l¹) with k⁰+k¹≤kand l⁰+l¹ ≤` (1) kbeing the total stock of capital and `the total amount of labor.

Assumption 1. Each production function fⁱ :R²+ → R+, i= 0,1, is C², increasing in each argument, concave, homogeneous of degree one and such that for any x >0, f₁ⁱ(0, x) =f₂ⁱ(x,0) = +∞, f₁ⁱ(+∞, x) =f₂ⁱ(x,+∞) = 0.

Asy≤f¹(k, `), Assumption 1 implies that there exists ¯k(`)>0 solution of k−f¹(k, `) = 0 such that f¹(k, `) > k when k < k(`), while¯ f¹(k, `) < k whenk >k(`). The set of admissible 3-uples (k, y, `) is thus defined as¯

K˜ =

(k, y, `)∈R³₊|0< `, 0≤k≤k(`),¯ 0≤y ≤f¹(k, `) (2) For any (k, y, `), profit maximization in the representative firm of each sector is equivalent to solving the following problem of optimal allocation of productive factors between the two sectors:

T(k, y, `) = max

k⁰,k¹,l⁰,l¹ f⁰(k⁰, l⁰) s.t. y≤f¹(k¹, l¹)

k⁰+k¹ ≤k l⁰+l¹≤` k⁰, k¹, l⁰, l¹≥0

(3)

The social production function T(k, y, `) gives the maximal output of the consumption good along interior temporary equilibria (k, y, `) ∈K. Under˜ Assumption 1,T(k, y, `) is homogeneous of degree one, concave andC² over K.˜ ² Denoting w the wage rate, r the gross rental rate of capital and p the price of investment good, all in terms of the price of the consumption good, we derive from the envelope theorem that

2See Benhabib and Nishimura [1].

2

(6)

r=T1(k, y, `), p=−T₂(k, y, `), w =T3(k, y, `) (4) The share of capital in total income is then given by

s(k, y, `) = _T_(k,y,`)+py^rk ∈(0,1) (5) 2.2 Consumption and savings

In each periodt,N_t agents are born, and they live for two periods. In their first period of life (when young), the agents are endowed with one unit of labor that they supply inelastically to firms. Their income results from the real wage and is allocated between current consumption and savings which are invested in the firms. In their second period of life (whenold), they are retired and their income resulting from the return on the savings is entirely consumed. The utility function of a representative agent, defined over his consumption bundle (c_t, when he is young, andd_t+1, when he is old), is

u(c_t, d_t+1) =h

c^1−1/γ_t +δ(d_t+1/B)^1−1/γiγ/(γ−1)

with δ ∈ (0,1) the discount factor, γ > 0 the elasticity of intertemporal substitution in consumption andB >0 a scaling parameter.

Each agent is assumed to have 1 + n > 0 children so that Nt+1 = (1 +n)N_t. Under perfect foresight, and considering the wage rate w_t and the gross rate of returnR_t+1 as given, a young agent maximizes his utility function over his life-cycle as follows:

ct,dmaxt+1,φt

u(ct, dt+1) s.t. wt=ct+φt

R_t+1φ_t=d_t+1 ct, dt+1, φt≥0

(6)

Solving the first order conditions gives:

c_t= _1+δγ(Rt+1^w^t/B)^γ−1 ≡α(R_t+1/B)w_t (7) with α(R/B) ∈ (0,1) the share of first period consumption over the wage income. Under gross substitutability, γ > 1 and the saving function φt = (1−α(R_t+1/B))w_t is increasing inR.

2.3 Perfect-foresight competitive equilibrium

Total labor is given by the number Nt of young households, i.e., `t = Nt, and is increasing at raten, i.e.,`_t+1 = (1 +n)`_t. We also assume complete depreciation of capital within one period.

(7)

Definition 1. A sequence {k_t, yt, `t, ct, dt, rt, wt, pt}^∞_t=0, with (k0, `0) = (ˆk₀,`ˆ₀) given, is a perfect-foresight competitive equilibrium if:

i)c_t=α(R_t+1/B)w_t;

ii)`t(1−α(Rt+1/B))wt=ptyt; iii)y_t=k_t+1;

iv) `_t+1 = (1 +n)`_t;

v) `t[ct+dt/(1 +n)] =T(kt, yt, `t);

vi) (r_t, w_t, p_t) is given by (4);

vii) R_t+1=r_t+1/p_t.

Letκ=k/`and ¯κ be the solution ofκ−f¹(κ,1) = 0. The set of admissible paths given by (2) can be redefined as follows

K=

(κ_t, κ_t+1)∈R²+|0≤κ_t≤κ,¯ 0≤κ_t+1≤f¹(κ_t,1)/(1 +n) (8) A perfect-foresight competitive equilibrium then satisfies

(1 +n)κ_t+1+^T_T³^(κ^t^,(1+n)κ^t+1^,1)

2(κt,(1+n)κt+1,1)

h

1−α

−^T_T¹^(κ^t+1^,(1+n)κ^t+2^,1)

2(κt,(1+n)κt+1,1)B

i

= 0 (9) withα(R/B) given by (7), (κt, κt+1)∈ K and κ0 = ˆκ0= ˆk0/`ˆ0 given.

3 Steady state and dynamic efficiency

3.1 A normalized steady state

A steady state is defined asκ_t=κ^∗ for all t withκ^∗ solution of α

− ^T¹(κ,(1+n)κ,1) T2(κ,(1+n)κ,1)B

= 1 + (1 +n)^κT²(κ,(1+n)κ,1)

T3(κ,(1+n)κ,1) ≡Φ_κ∈(0,1) (10) We consider a family of economies parameterized byγ 6= 1. We follow the same procedure as in Drugeonet al. [5]: we use the scaling parameter B to ensure the existence of a normalized steady state (NSS) κ^∗ ∈ (0,κ) which¯ remains invariant as γ is varied. From (7) and (10) we get:

Proposition 1. Under Assumption 1, let γ 6= 1 and κ^∗ ∈ (0,¯κ). Then there exists a unique valueB(κ^∗)>0 as given by

B(κ^∗) =−^T_T¹^(κ^∗^,(1+n)κ^∗^,1)

2(κ^∗,(1+n)κ^∗,1)

_−(1+n)κ∗T2(κ^∗,(1+n)κ^∗,1)

δ^γ[T3(κ^∗,(1+n)κ^∗,1)+(1+n)κ^∗T2(κ^∗,(1+n)κ^∗,1)]

_1−γ¹

such thatκ^∗ is a steady state if and only ifB =B(κ^∗).

Proof: See Appendix 5.1.

In the rest of the paper we assume thatB =B(κ^∗) so thats=s(κ^∗, κ^∗,1) and α=α(R^∗/B(κ^∗)) = Φ_κ^∗ remain constant as γ is made to vary.

4

(8)

3.2 Dynamic efficiency

From Definition 1 and the homogeneity of T(k, y, `), considering that κ^∗T2/T3 = (T2/T1)(κ^∗T1/T3) = −s/R^∗(1−s), we derive the stationary gross rate of return along the NSS:

R^∗= _{(1−α)(1−s)}^(1+n)s (11)

Under-accumulation of capital is obtained if and only if R^∗ > 1 +n. As shown in Drugeon et al. [5] we have:

Proposition 2. Under Assumption 1, let γ 6= 1 and α = 1−s/(1−s).

Then:

i) the NSS is characterized by an under-accumulation of capital if and only ifα≥α;

ii) an intertemporal competitive equilibrium converging towards the NSS is dynamically efficient ifα ∈(α,1)and dynamically inefficient ifα∈(0, α).

If the labor income is relatively larger than the capital income (s <1/2),α >

0 and young agents receive enough wage resources to provide a large amount of savings. But over-accumulation of capital can be avoided provided the share of first period consumption over the wage income is large enough, i.e.

the agent does not save too much.

4 Local indeterminacy under dynamic efficiency

Let us introduce the relative capital intensity difference across sectors b≡ k¹/l¹−k⁰/l⁰

l¹/y (12)

and the elasticity of the rental rate of capital

ε_rk=−T₁₁(κ^∗,(1 +n)κ^∗,1)κ^∗/T1(κ^∗,(1 +n)κ^∗,1) (13) evaluated at the NSS. From now on we consider a positive value forα= 1− s/(1−s), we assume gross substitutability, a capital intensive consumption good,³ and we consider dynamically efficient paths:

Assumption 2. s∈(0,1/2), γ >1, b <0 and α≥α.

We then prove that contrary to the aggregate OLG model, local indeterminacy arises under dynamic efficiency in a two-sector model with positive capital-labor substitution.

3Whenγ >1, this is a necessary condition for local indeterminacy (See Galor [6]).

(9)

Theorem 1. Under Assumptions 1-2, let¯b = −1/(1 +n) and b =−(1− α)/(1 +n)α. Then there exist ε_rk>0, γ >1 and ¯γ > γ such that the NSS is locally indeterminate if and only ifs∈(1/3,1/2),α ∈(α,1/2), b∈(b,¯b), εrk > ε_rk and γ ∈(γ,γ¯).

Theorem 1 is based on a large elasticity ε_rk. This restriction can be interpreted through the aggregate elasticity of capital-labor substitution Σ which is linked toε_rk as follows (see Drugeon [4]):

Σ = ^y_pyky⁰^+py

0 pyk⁰l⁰σ₀+y₀k¹l¹σ₁

, ε_rk= l⁰/y₀2 w(y0+py)

Σ (14)

with σ₀ and σ₁ the sectoral elasticities of input substitution. Local indeterminacy under dynamic efficiency then requires a low enough aggregate elasticity Σ, i.e. low enough but still positive sectoral elasticities. Note also that the consumption good sector has to be sufficiently capital intensive. If bis too close to 0, i.e. if the model is close to the aggregate formulation, we get the same conclusion as Cazzavillan and Pintus [2, 3]: local indeterminacy under dynamic efficiency can never arise with any positive elasticity of capital-labor substitution.

Remark 1: Note that ¯γ is generically a flip bifurcation value giving rise to period-two cycles which are locally indeterminate (or unstable) in a right (or left) neighborhood of ¯γ, whileγis generically a transcritical bifurcation value leading to the existence of a second steady state which is locally unstable (saddle-point stable) in a right (left) neighborhood of γ (see Figure 1 in Appendix 5.2).

Through the NSS, the conditions on α, s, b and ε_rk in Theorem 1 are based on joint restrictions of the technologies. We need therefore to show that they can be satisfied with standard production functions. Consider the following CES formulations:

f⁰(k⁰, l⁰) =

χ(k⁰)^−ς+ (1−χ)(l⁰)^−ς−¹_ς

f¹(k¹, l¹) =

1 2

k¹ η

−ρ

+^(l₂¹⁾^−ρ −¹

ρ (15)

withχ∈(0,1),ς, ρ >−1 andη >0. The sectoral elasticities of capital-labor substitution are given by σ0 = 1/(1 +ς) and σ1 = 1/(1 +ρ). We assume for simplicity a constant population, i.e. n = 0. Note that when σ₁ = 0 the investment good technology becomes Leontief as limρ→+∞f¹(k¹, l¹) =

6

(10)

min{k¹/η, l¹}. Assume in a first step thatσ1 = 0. Asy=k¹/η =l¹, we get T⁰(k, y, `) =

χ(k−ηy)^−ς+ (1−χ)(`−y)^−ς−1/ς

Following Proposition 1, we consider a NSSκ=κ^∗∈(0,1) and we compute α⁰= ^{(1−χ)(1−η)}_{(1−χ)(1−η)}^1+ς^κ^∗ς^(1−κ1+ς^∗^{)−ηχ(1−κ}κ^∗ς ^∗⁾^1+ς, s⁰ = _{(1−χ)(1−η)}^χ(1−κ^∗⁾^1+ς1+ςκ^∗ς

α⁰ = ^{(1−χ)(1−η)}_{(1−χ)(1−η)}^1+ς^κ^∗ς^−χ(1−κ1+ςκ^∗ς ^∗⁾^1+ς, b⁰ = ^η−κ_1−κ^∗∗ andε⁰_rk= (1+ς)(1−χ)κ^∗ς (1−η)h

χ“

1−κ∗ κ∗(1−η)

”ς

+1−χi

together with the bounds ¯b=−1,

b⁰ = −_{(1−χ)(1−η)}^{(1−χ)(1−η)}_1+ς^1+ς_κ_∗ς^κ_(1−κ^∗1+ς^+ηχ(1−κ_∗_{)−ηχ(1−κ}^∗⁾^1+ς_∗₎_1+ς, ε⁰_rk = ^(κ_(2κ^∗^{−η)(1−κ}∗−1−η)(1−η)ηχ^∗^)(1−χ)

χ(1+η)+(1−χ)“_κ∗(1−η) 1−κ∗

”1+ς

χ

“ 1−κ∗ κ∗(1−η)

”ς

+1−χ

In a first step we derive that ifκ^∗ ∈(1/2,1),χ∈(χ⁰₁,χ¯⁰) with χ⁰₁ = _(1−κ∗^(1−η))^1+ς+(1−η)^ς^κ^∗1+ς^ςκ^∗1+ς, χ¯⁰ = _(1−κ∗^(1−η))^1+ς+(1−η)^1+ς^κ^∗ς^1+ςκ^∗ς

andη <η¯⁰ = min{1−κ^∗,2κ^∗−1}, thens⁰ ∈(1/3,1/2),α⁰∈(α⁰,1/2) and b⁰∈(b⁰,−1). In a second step, we derive that if ς > ς⁰ with

ς⁰ = ^κ

∗[^{1−η(1−η)−κ}^∗2]

(2κ^∗−1−η)η

then there existsχ⁰₂ ∈(0,χ) given by¯ χ⁰

2 = ^(κ

∗−η)(1−κ^∗)

“κ∗(1−η) 1−κ∗

”1+ς

ς(2κ^∗−1−η)η−κ^∗[1+η²−κ^∗(1+η)]+(κ^∗−η)(1−κ^∗)“_κ∗(1−η) 1−κ∗

”1+ς

such that ε_rk > ε⁰_rk when χ > χ⁰₂. Denoting χ⁰ = max{χ⁰

1, χ⁰₂}, we have then proved:

Lemma 1. Let the production functions be given by (15) with σ₁ = 0. If κ^∗ ∈ (1/2,1), η < η¯⁰, ς > ς⁰ and χ ∈ (χ⁰,χ¯⁰), there exist γ⁰ > 1 and

¯

γ⁰ > γ⁰ such that the NSS is locally indeterminate for any γ ∈(γ⁰,¯γ⁰).

Assume now that σ₁ >0. The social production function derived as the value function of program (3) is parameterized by σ₁, namely T^σ¹(k, y, `).

Considering the same NSS κ = κ^∗ ∈ (0,1), we can similarly compute α^σ¹, s^σ¹,α^σ¹,b^σ¹,ε^σ_rk¹,b^σ¹ and ε^σ_rk¹. We have the following property:

Lemma 2. The social production function T^σ¹(k, y, `) is continuous in (k, y, `, σ1), and as σ1 → 0, converges to T⁰(k, y, `) uniformly in (k, y, `), i.e. for any > 0, there is ξ > 0 such that if 0 < σ₁ < ξ, |T^σ¹(k, y, `)− T⁰(k, y, `)|< for any (k, y, `)∈K.˜

(11)

It follows from these Lemmas that lim_σ₁→0(α^σ¹, s^σ¹, α^σ¹, b^σ¹, ε^σ_rk¹, b^σ¹, ε^σ_rk¹) = (α⁰, s⁰, α⁰, b⁰, ε⁰_rk, b⁰, ε⁰_rk). We have then proved:

Proposition 3. Let the production functions be given by (15). If κ^∗ ∈ (1/2,1), there isσ¯1>0such that for all σ1 ∈[0,σ¯1)there existη¯^σ¹ ∈(0,1), ς^σ¹ > 0, χ^σ¹ ∈ (0,1), χ¯^σ¹ ∈ (0,1), γ^σ¹ >1 and γ¯^σ¹ > γ^σ¹ such that when η < η¯^σ¹, ς > ς^σ¹ and χ ∈ (χ^σ¹,χ¯^σ¹), the NSS is locally indeterminate for anyγ ∈(γ^σ¹,¯γ^σ¹).

Theorem 1 and Proposition 3 show that in a two-sector OLG economy, the existence of local indeterminacy under dynamic efficiency is compatible with positive capital-labor substitution in each sector. Following Reichlin [9], our conclusion suggests that the Pareto optimal steady state can be considered as a target of the policymaker. Indeed the introduction of a public policy based on taxes and transfers could at the same time rule out business-cycle fluctuations in the economy and drive the equilibrium to the optimal steady state which provides an equal level of utility to all generations.

Let us prove now along the line of Reichlin [9] that such a policy exists under the assumption that agents and public authority do not make forecasting mistakes. Assume that the public authority buys goods, levies taxes and makes transfers. Let g_t be the flow of consumption goods which is bought, τ_t^y <(>)0 the taxes (transfers) on the income of the young and τ_tô<(>)0 the taxes (transfers) on the income of the old. We assume a bal- anced budget rule, i.e. g_t+τ_t^y+τ_tô= 0. The agent’s first and second period budget constraints becomewt+τ_t^y =ct+φtandRt+1φt+τ_t+1ô =dt+1. From utility maximization, we get the optimal saving

φ_t= ^B(δR^t+1^/B)

γ(wt+τ_t^y)−τ_t+1^o

Rt+1+B(δRt+1/B)^γ (16)

and a perfect-foresight competitive equilibrium satisfies (1 +n)κ_t+1p_t= ^B(δR^t+1^/B)

γ(wt+τ_t^y)−τ_t+1^o

Rt+1+B(δRt+1/B)^γ (17)

Now consider the expression of the normalization constant as given in Propo- sition 1 evaluated at some point κ ∈ (0,κ), namely¯ B = B(κ) such that:

B(κ) = _p(κ)^r(κ) _(1+n)κp(κ)

δ^γ[w(κ)−(1+n)κp(κ)]

_1−γ¹

with r(κ) = T₁(κ,(1 + n)κ,1), p(κ) = −T₂(κ,(1 +n)κ,1) and w(κ) = T₃(κ,(1 +n)κ,1). Let

8

(12)

ˆ

τ_t+1^o = (1 +n) n

κp(κ) [R(κ) +B(κ)(δR(κ)/B(κ))^γ] Rt+1

R(κ)

γ

− κt+1pt[Rt+1+B(κ)(δRt+1/B(κ))^γ] o

ˆ

τ_t^y = w(κ)−wt

(18)

withR(κ) =r(κ)/p(κ). Pluggingτ_t^y = ˆτ_t^y, τ_t+1ô = ˆτ_t+1ô and τ_t+1^y =τ_t+2ô = 0 into (17) gives

(1 +n)κp(κ) = (δR(κ)/B(κ))^γB(κ)w(κ)

R(κ)+(δR(κ)/B(κ))^γ (19)

As shown in Proposition 1, there exists a solutionκ=κ^∗ which corresponds to the normalized stationary Pareto optimal perfect-foresight competitive equilibrium. It follows that if agents believe in this announced policy rule, they will expect the optimal NSS allocation to hold in the future. This expectation in turn drives the sytem to the NSS and keep it there forever.

Note however that contrary to Riechlin [9], when local indeterminacy holds, the NSS is not the unique stationary solution of equation (19). Indeed, as mentioned in Remark 1, a second steady state generically exists through a transcritical bifuration occurring at γ and is locally unstable for γ > γ.

It follows that the stabilization policy (18) has to be parameterized by the right NSS in order to jump on the optimal stationary allocation.

5 Appendix

5.1 Proof of Proposition 1

Consider the setKas defined by (8) and the expression ofα(R/B) as given by (7). κ^∗∈(0,κ) is a solution of (10) if¯

1 1+δ^γ

“

−_T^T¹⁽^κ^∗^,(1+n)κ^∗^,1)

2(κ∗,(1+n)κ∗,1)B

”γ−1 = 1 + (1 +n)^κ^∗_T^T²^(κ^∗^,(1+n)κ^∗^,1)

3(κ^∗,(1+n)κ^∗,1) ∈(0,1) (20) Then there exists a unique positive value ofB solution of (20) given by

B(κ^∗) =−^T_T¹^(κ^∗^,(1+n)κ^∗^,1)

2(κ^∗,(1+n)κ^∗,1)

_−(1+n)κ∗T2(κ^∗,(1+n)κ^∗,1)

[T3(κ^∗,(1+n)κ^∗,1)+(1+n)κ^∗T2(κ^∗,(1+n)κ^∗,1)]δ^γ

_1−γ¹

and κ^∗ is a steady state if and only if B =B(κ^∗).

5.2 Proof of Theorem 1 From (7), we get

α⁰(R/B) = (1−γ)α(R/B)(1−α(R/B))B/R (21)

(13)

Under Assumptions 1 and 2, we get from the first order conditions of program (3)

T₁₂=−T₁₁b <0, T₂₂=T₁₁b² <0, T₃₁=−T₁₁a >0, T₃₂=T₁₁ab >0 (22) with a ≡ k⁰/l⁰ > 0, b as defined by (12) and T11 < 0. Considering (13) together with (22),T₁κ^∗/T₃ =s/(1−s), −T₁/T₂ =R^∗ =s/(1−α)(1−s) and the fact that homogeneity ofT(k, y, `) impliesa= [1−(1 +n)b]κ^∗, total differenciation of (9) using (4), (5) and (21) evaluated at the NSS gives the characteristic polynomialP(λ) =λ²−λT(γ) +D(γ) with

D(γ) = s[(1+n)bα(γ−1)+1−α+α(1+n)b]

(1+n)b(1−α)(1−s)α(γ−1) , T(γ) = _(1+n)bε¹

rkα(γ−1) +^1+D(1+n)_(1+n)b²^b² WhenB=B(κ^∗), the NSS,α,sandεrkremain constant for anyγ 6= 1. We then study the variations ofT(γ) and D(γ) in the (T,D) plane as γ varies within (1,+∞). SolvingT(γ) and D(γ) with respect toα(γ−1) yields to

D= ∆(T) = ^ε^rks[1−α+α(1+n)b]T

(1−α)(1−s)+ε_rks(1+n)b[1−α+α(1+n)b]

− ^ε^rks[1−α+α(1+n)b]−s(1+n)b

(1+n)b[(1−α)(1−s)+εrks(1+n)b(1−α+α(1+n)b)]

and allows to use the methodology introduced by Grandmontet al. [7]. As γ spans the interval (1,+∞), T(γ) and D(γ) vary linearly along the line

∆(T). The fundamental properties of ∆(T) depend on its extremities. The starting point of the pair (T(γ),D(γ)) is obtained whenγ = +∞:

D(+∞) =D_∞= _{(1−α)(1−s)}^s , T(+∞) =T_∞= (1−α)(1−s)+(1+n)²b²s

(1+n)b(1−α)(1−s) (23) while the end point is obtained when γ converges to 1 from above. Let

¯b=−1/(1 +n),b=−(1−α)/(1 +n)α and ˆb=−(1−α)(1−s)/(1 +n)s.

As lim_γ→1⁺D(γ) = lim_γ→1⁺T(γ) = ∞, ∆(T) is a half-line starting from (T∞,D∞) and pointing downwards or upwards depending on the sign of

D⁰(γ) = b(1−α)(1−s)(γ−1)^s(b−b) ² (24) Under Assumptions 1-2, we get from (23) thatD_∞=R^∗/(1 +n)>1. Local indeterminacy then requires that D⁰(γ) > 0, i.e. b ∈ (b₁,0) as shown by (24). But in this case,D= 1 impliesT <−2. Indeed, solving D= 1 gives

α(1−γ) = b(1−s)(α−α)^αs(b−b) (25)

which can be satisfied if and only ifb∈(b,0). Moreover, substitutingD= 1 into the expression of T allows to get

T + 2 = _(1+n)bε¹

rkα(γ−1) +^[1+(1+n)b]_(1+n)b ² <0 (26) Therefore, whenb∈(b,0), the only possibility to get local indeterminacy is that the ∆-half-line crosses the interior of segment [AC] as follows

10

(14)

A

B C

6

-

@

T D

HH HH

HH HHH γ=+∞

¯

γ γ

∆

Figure 1: Local indeterminacy with dynamic efficiency.

This requires that 1+T_∞+D_∞= (¯b−b)(b−ˆb)/bˆb <0 ⇔ b∈(−∞,¯b)∪(ˆb,0), and thatD=−1 impliesT >0. SolvingD=−1 gives

α(1−γ) = b[1−α(1−s)]^αs(b−b) (27) which holds if and only ifb∈(b,0). SubstitutingD=−1 into T gives

T = ^ε^rkαs(b−b)(1+n)²(¯b²−b²)−b[1−α(1−s)]

(1+n)bεrkαs(b−b) (28)

Noting that b < ¯b ⇔ α < 1/2 and α < 1/2 ⇔ s > 1/3, we derive from (28) that ∆(T) = −1 implies T > 0 if and only if s ∈ (1/3,1/2), α∈(α,1/2),b∈(b,¯b) and εrk > ε_rk with

ε_rk= b[1−α(1−s)]

(1+n)²(b²₀−b²)αs(b−b1) >0 (29) The result follows.

5.3 Proof of Lemma 2

We use the same kind of argument as Nishimura and Yano ([8], Lemma 1, p.

229). Define the setE(k, y, `, σ1) ={(k⁰, l⁰)≥0|y≤f¹(k−k⁰, `−l⁰)}. Then T^σ¹(k, y, `) = maxf⁰(k⁰, l⁰) s.t. (k⁰, l⁰) ∈E(k, y, `, σ1). SinceE(k, y, `, σ1) is lower-semi-continuous in (k, y, `, σ₁), T^σ¹(k, y, `) is continuous. It follows also that the functions (k⁰_σ₁(k, y, `), l⁰_σ₁(k, y, `)) = argmaxf⁰(k⁰, l⁰) s.t.

(k⁰, l⁰) ∈ E(k, y, `, σ1) are continuous. Moreover, as limρ→+∞f¹(k¹, l¹) = min{k¹/η, l¹}, we can find a sub-sequence σ_1i such that when σ_1i → 0, (k⁰_σ₁(k, y, `), l⁰_σ₁(k, y, `)) → (k−ηy, `−y) for any (k, y, `) ∈ K. Therefore,˜ the uniform convergence follows and limσ1→0T^σ¹(k, y, `) =T⁰(k, y, `).