# Indeterminacy under input-specific externalities and implications for fiscal policy

Thomas Seegmuller

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Indeterminacy under Input-specific Externalities and Implications for Fiscal Policy

Thomas SEEGMULLER

2006.79

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

CENTRE NATIONAL
DE LA RECHERCHE
SCIENTIFIQUE

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

ISSN : 1624-0340
<figure><img src="image_2.png" /></figure>

Indeterminacy and Endogenous Fluctuations
under Input-Specific Externalities

Thomas SEEGMULLER

2006.79

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

CENTRE NATIONAL
DE LA RECHERCHE
SCIENTIFIQUE

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

ISSN : 1624-0340
Indeterminacy under Input-specific
Externalities and Implications for Fiscal
Policy

Thomas Seegmuller*

October 22, 2007

Abstract

In this paper, we introduce input-specific externalities in a dy-
namic general equilibrium model with heterogeneous households and
a finance constraint (Woodford (1986)). In contrast to existing papers,
average labor and capital have not a positive impact on the total pro-
ductivity of factors, but respectively on labor and capital efficiencies.
Focusing on not too low degrees of capital-labor substitution, we show
that indeterminacy requires not only a lower bound for the elasticity
of capital-labor substitution, but also an upper bound, although the
returns are increasing. As a direct implication, the well known wrong
slopes condition (labor demand steeper than labor supply) is neither
a necessary nor a sufficient condition for indeterminacy and larger in-
creasing returns promote sadlle-path stability when inputs are high
substitutes. Using this framework, we are also able to analyze the role
of variable tax rates on capital and labor income on the dynamics. In
contrast to existing results, we show that tax rates decreasing with
their tax base do not promote instability due to self-fulfilling expec-
tations when capital and labor are sufficiently high substitutes, but
rather have a stabilizing effect.

<footnote>*CNRS, Paris School of Economics, University Paris 1, Centre d'Economie de la Sorbonne, 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. I would like to thank Jean-Michel Grandmont, Teresa Lloyd-Braga, Leonor Modesto, Patrick Pintus and Alain Venditti for discussions on closely related issues. All remaining errors are mine. This paper benefits from the financial support of Agence Nationale de la Recherche (ANR-05-BLAN-0347-01).</footnote>
Indeterminacy and Endogenous Fluctuations under
Input-Specific Externalities

Thomas Seegmuller<fnref n="1" />

CNRS and Panthéon Sorbonne Economie,
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">¹I would like to thank Jean-Michel Grandmont, Teresa Lloyd-Braga, Leonor Modesto and Alain Venditti for discussions on closely related issues. All remaining errors are mine. This paper benefits from the financial support of ANR.</footnote>
JEL classification: C62, E32, H20.

Keywords: Indeterminacy, externalities, capital-labor substitution, fiscal policy.

# 1 Introduction

Considering dynamic general equilibrium one-sector models, a lot of papers have introduced productive externalities to analyze the role of increasing returns on local indeterminacy and the occurrence of endogenous cycles. In most of them, production, characterized by constant returns at the private level, benefits from externalities because the total productivity of factors increases with respect to average capital and labor ($Y = A(\bar{K}, \bar{L})F(K, L)$).<fnref n="1" /> The purpose of this paper is to go one step further by first analyzing the robustness of these results when productive externalities do no more affect the total productivity of factors, but rather have an influence on capital and labor efficiencies. In a second step, we will analyze the implications of such a specification of externalities on fiscal policy issues. More specifically, we will show that, in contrast to the existing literature, variable tax rates decreasing with their tax bases do not always promote fluctuations due to the volatility of expectations.

To become more precise, we introduce externalities as follows: we assume that average capital increases capital efficiency ($C(\bar{K})$) and average labor increases labor efficiency ($D(\bar{L})$). Therefore, average capital and labor do no more improve the aggregate production, but rather affect the capital-labor ratio, measured in efficient units ($Y = F(C(\bar{K}), K, D(\bar{L}), L)$).<fnref n="2" /> Obviously, we note that the existence of these input-specific externalities ensures that returns to scale are increasing at the social level.

We introduce such a production sector in a finance constrained economy with heterogeneous households as initially developed by Woodford (1986).<fnref n="3" />

<footnote n="1">¹Among others, one can refer to Barinci and Chéron (2001), Benhabib and Farmer (1994), Cazzavillan, Lloyd-Braga and Pintus (1998), Farmer and Guo (1994), Harrison and Weder (2002), Hintermaier (2003) or Pintus (2006). See Benhabib and Farmer (1999) for a survey.</footnote>

<footnote n="2">²In a previous version of this paper, we have considered a more general specification of externalities where average capital and labor increase both capital and labor efficiencies ($C(\bar{K}, \bar{L})$, $D(\bar{K}, \bar{L})$). It is shown that new results on indeterminacy emerge if the effect of average capital on labor efficiency and the effect of average labor on capital efficiency are not too large, providing a justification of the simpler specification we use in this work. For more details, see Seegmuller (2006).</footnote>

<footnote n="3">³In the last decade, this model has been extensively used in the literature. See, for instance, Barinci and Chéron (2001), Cazzavillan, Lloyd-Braga and Pintus (1998), Gokan</footnote>
Abstract

A lot of papers have analyzed the role of productive externalities on the
occurrence of indeterminacy and endogenous cycles, assuming that the total
productivity of factors increases with respect to average capital and labor.
In this paper, we extend such type of analysis introducing a more general
formulation of externalities, i.e. input-specific externalities. Indeed, we as-
sume that different externalities affect each input in the production function.
Considering a Woodford (1986) framework, we show that this generalized
form of externalities allows us to obtain new dynamic results concerning the
occurrence of local indeterminacy and endogenous cycles. Moreover, we ex-
hibit some configurations where the emergence of endogenous fluctuations
requires a weaker degree of increasing returns than in the usual case where
externalities are represented by the total productivity of factors.

*Keywords:* Indeterminacy, endogenous fluctuations, externalities, increasing returns, capital-labor substitution.

Résumé

A grand nombre de papiers ont analysé le rôle des externalités dans la
production sur l'indétermination et l'apparition de cycles endogènes, en sup-
posant que la productivité totale des facteurs augmente avec le capital et le
travail moyens. Dans ce papier, nous étendons ce type d'analyse en consid-
érant une forme plus générale d'externalités, c'est-à-dire spécifiques aux fac-
teurs de production. En effet, nous supposons que des externalités différentes
affectent chaque facteur dans la fonction de production. En considérant un
modèle à la Woodford (1986), nous montrons que cette généralisation des
externalités nous permet d'obtenir de nouveau résultats concernant la dy-
namique (indétermination et cycles endogènes). De plus, nous présentons
des configurations dans lesquelles l'émergence de fluctuations endogènes re-
quière des rendements d'échelle plus faiblement croissants que lorsque les
externalités apparaissent à travers la productivité totale des facteurs.

*Mots-clés:* Indétermination, fluctuations endogènes, externalités, rendements croissants, substitution capital-travail.

JEL classification: C62, E32.
Using this framework, we can easily analyze the stability properties of the steady state, i.e. the local indeterminacy and the occurrence of endogenous cycles, and compare our results with the case where productive externalities are introduced through the total productivity of factors, as analyzed by Cazzavillan, Lloyd-Braga, and Pintus (1998).

For sake of conciseness, we focus on cases where capital and labor are not weak substitutes.<fnref n="4" /> We show that indeterminacy and endogenous cycles require a not too weak effect of labor on labor efficiency ($D(L)$). If this necessary condition is satisfied, we prove that indeterminacy occurs if the capital-labor substitution is greater than a lower bound ($\sigma_H$). Note that this result is quite similar to the one obtained by Cazzavillan, Lloyd-Braga and Pintus (1998), even if the economic mechanism is different. However, in our framework, indeterminacy also requires a finite upper bound ($\sigma_T$) for the elasticity of capital-labor substitution: for all values of this elasticity larger than this upper bound, the steady state becomes a saddle.

As a direct implication, we show that the steady state can be locally determined when the wrong slopes condition is met on the labor market, i.e. the labor demand is steeper than the labor supply. Furthermore, since the upper bound value $\sigma_T$ is decreasing with the degree of increasing returns, we obtain another new conclusion. In contrast to the literature, larger increasing returns promote saddle-path stability for sufficiently high capital-labor substitutions.

To provide an economic intuition of our main result, i.e. indeterminacy does no more occur for a large enough elasticity of capital-labor substitution, we first underline that expectations can be self-fulfilling only if the real wage and real interest rate are sufficiently sensitive to capital and labor, respectively. When externalities affect the total productivity of factors (Cazzavillan, Lloyd-Braga and Pintus (1998)), this is possible as soon as externalities are not too weak. Indeed, in this case, externalities increase production and hence both the marginal productivities of capital and labor. On the contrary, in our framework, externalities mainly modify the capital-labor ratio. Since the elasticities of the marginal productivities of capital and labor with respect to the capital-labor ratio are inversely related to the degree of capital-labor substitution, the externalities do no more play a great role on dynamics if capital and labor are sufficiently high substitutes, explaining that the steady state becomes a saddle in such a case.

<footnote>(2006), Grandmont (2006), Grandmont, Pintus and de Vilder (1998), Lloyd-Braga and Modesto (2007), Lloyd-Braga, Modesto and Seegmuller (2006) and Pintus (2003, 2004).</footnote>

<footnote n="4">⁴Indeed, it is well known that in the Woodford (1986) model, endogenous fluctuations occur without externality for a weak elasticity of capital-labor substitution. See Grandmont, Pintus and de Vilder (1998).</footnote>
# 1 Introduction

Considering dynamic general equilibrium one-sector models, a lot of papers have introduced productive externalities to analyze the role of increasing returns on local indeterminacy and the occurrence of endogenous cycles.<fnref n="2" /> In most of them, production benefits from externalities because the total productivity of factors increases with respect to average capital and labor. In other words, these contributions depart from the perfectly competitive model by introducing a multiplicative term, that increases with respect to capital and labor, to the constant returns to scale production function ($Y = A(\bar{K}, \bar{L})F(K, L)$). Among others, one can refer to Barinci and Chéron (2001), Benhabib and Farmer (1994), Boldrin (1992), Cazzavillan (2001), Cazzavillan, Lloyd-Braga and Pintus (1998), Farmer and Guo (1994), Harrison and Weder (2002), Hintermaier (2003) or Pintus (2006).<fnref n="3" />

In this paper, we extend this type of analysis introducing a more general formulation of externalities in a one-sector model and we will study their implications on the dynamic stability properties and the occurrence of cycles. Instead of considering that externalities appear through the total productivity of factors, we introduce input-specific externalities. The production benefits from externalities because two different multiplicative terms, increasing with respect to average capital and labor, enter the production function, each one for a specific production factor, capital or labor ($Y = F(C(\bar{K}, \bar{L})K, D(\bar{K}, \bar{L})L)$). These two multiplicative terms, which represent externalities and ensure that returns to scale are increasing at the social level, can also be interpreted as capital and labor efficiencies. Evideently, when these two multiplicative terms are equal, we recover as a particular case externalities represented by the total productivity of factors ($C(\bar{K}, \bar{L}) = D(\bar{K}, \bar{L}) = A(\bar{K}, \bar{L})$).

We introduce such a production sector in a finance constrained economy with heterogeneous households as initially developed by Woodford (1986). With this framework, we can analyze the stability properties of the steady state, i.e. the local indeterminacy and the occurrence of endogenous cycles, using the geometrical method developed by Grandmont, Pintus, and de Vilder (1998). Moreover, we can easily compare our results with the case where productive externalities are introduced through the total productivity of factors as analyzed by Cazzavillan, Lloyd-Braga, and Pintus (1998).

Our main results concern the occurrence of indeterminacy and endoge-

<footnote n="2">²In two-sector models, externalities have also been introduced assuming constant returns to scale at the social level. For a seminal contribution, one can refer to Benhabib and Nishimura (1998).</footnote>

<footnote n="3">³For a survey, see Benhabib and Farmer (1999).</footnote>
Since the seminal contributions of Guo and Lansing (1998) and Schmitt-Grohé and Uribe (1997), several contributions have analyzed the (de-)stabilizing role of variable tax rates under a balanced budget. In one-sector models, one of the main results states that tax rates (at least on labor income) decreasing with their tax base have a destabilizing effect by promoting fluctuations due to the volatility of expectations.<fnref n="5" /> We reexamine this issue in our framework and obtain a less clear-cut conclusion. Indeed, variable tax rates decreasing with their tax base can be a source of determinacy if the capital-labor substitution is sufficiently large. The explanation of this result is quite simple: the introduction of tax rates decreasing with their tax base reduces the upper bound $\sigma_T$, promoting saddle-path stability.

This paper is organized as follows. In the next section, we present the model and define the intertemporal equilibrium. In Section 3, we establish the existence of a steady state. In Section 4, we analyze the occurrence of local indeterminacy and endogenous cycles, and provide an economic interpretation. In Section 5, we introduce variable tax rates through a balanced-budget and discuss dynamic implications of fiscal policy. Our main findings are summarized in Section 6.

## 2 The Model

We consider a competitive monetary economy with discrete time, $t = 1, 2, ..., +\infty$, and two types of infinitely lived households, workers and capitalists.<fnref n="6" /> Only workers supply labor, whereas both workers and capitalists consume the final good. Moreover, workers are more impatient than capitalists, i.e. they discount the future more than the latter. Following Woodford (1986), we assume that there is a financial market imperfection that prevent workers from borrowing against their wage earnings. Therefore, in a neighborhood of a monetary steady state, capitalists hold the whole capital stock and no money, and workers save their wage income in the form of money balances. In the production sector, firms produce the final good. The production benefits from productive externalities specific to each input.

<footnote n="5">⁵For instance, the reader can refer to Guo and Lansing (1998), Gokan (2006), Lloyd-Braga, Modesto and Seegmuller (2006), Pintus (2003) and Schmitt-Grohé and Uribe (1997).</footnote>

<footnote n="6">⁶This model with heterogeneous households has been introduced in the seminal contribution of Woodford (1986) and has been extensively used in the literature. See among others Barinci and Chéron (2001), Cazzavillan, Lloyd-Braga and Pintus (1998), Gokan (2006), Grandmont (2006), Grandmont, Pintus and de Vilder (1998), Lloyd-Braga and Modesto (2007), Lloyd-Braga, Modesto and Seegmuller (2006) and Pintus (2003, 2004).</footnote>
nous fluctuations when capital and labor are not weak substitutes. To fix
ideas, we call the direct effects of externalities the contributions of capital
(labor) to externalities that affect capital (labor) as an input, whereas we
call the crossing effects the contributions of capital (labor) to externalities
that affect labor (capital) as an input.

We show that, when the product of crossing effects are greater than the product of direct effects on externalities, we obtain results closely related to Cazzavillan, Lloyd-Braga, and Pintus (1998). Endogenous fluctuations can occur for all elasticities of capital-labor substitution greater than a lower bound and a sufficiently elastic labor supply. On the contrary, when the product of direct effects dominates the product of crossing effects, endogenous fluctuations can no more occur for a level of capital-labor substitution sufficiently high, in particular when the labor supply is highly elastic. We further exhibit some configurations where the greater the increasing returns are, the more relevant this conclusion is. Moreover, we prove that in contrast to several existing works, the steady state can be locally determinate under the wrong slopes on the labor market, i.e. the slope of labor demand is greater than the slope of labor supply.

Comparing our formulation of externalities with the one usually adopted
through the total productivity of factors, we also exhibit some configurations
where the emergence of endogenous fluctuations requires weaker increasing
returns under the specification of externalities that we propose. In this sense,
the characterization of externalities developped in this paper promotes inde-
terminacy and endogenous cycles.

We finally remark that the Cobb-Douglas technology seems to be a par-
ticular case. Indeed, under a unit elasticity of capital-labor substitution, the
formulation of externalities that we present and the usual one are similar,
whereas we have seen that, in many cases, the dynamic stability properties
can be quite different.

This paper is organized as follows. In the next section, we present the
model and define the intertemporal equilibrium. In Section 3, we establish
the existence of a steady state. In Section 4, we analyze the occurrence
of local indeterminacy and endogenous cycles, and discuss our results. In
Section 5, we summarize our main findings, while some technical details can
be found in the Appendix.
## 2.1 Workers

The population size of these households is normalized to one. Each worker chooses his labor supply and his consumption maximizing the utility function:

$$ \sum_{t=1}^{\infty} \left[ \lambda^{t-1} U(C_t^w/B) - \lambda^t V(L_t) \right] \qquad (1) $$

where $C_t^w$ is the consumption in period $t$, $L_t$ the labor supply, $B > 0$ a scaling parameter and $\lambda \in (0, 1)$ the discount factor. The utility functions $U$ and $V$ satisfy the following assumption:<fnref n="7" />

**Assumption 1** The functions $U(x)$ and $V(L)$ are continuous for all $x \ge 0$ and $0 \le L \le \bar{L}$, where the labor endowment $\bar{L} > 1$ may be finite or infinite.<fnref n="8" /> They are $C^n$ for $x > 0$, $0 < L < \bar{L}$ and $n$ large enough, with $U'(x) > 0$, $U''(x) < 0$, $V'(L) > 0$ and $V''(L) > 0$. Moreover, $\lim_{L \to \bar{L}} V'(L) = +\infty$ and consumption and leisure are gross substitutes, i.e. $-xU''(x)/U'(x) < 1$.

In what follows, we respectively note $M_t^w$ and $K_t^w$ the money balances and the capital stock held by workers, $\delta \in (0, 1)$ the depreciation rate of capital, $r_t$ the nominal interest rate, $w_t$ the nominal wage and $P_t$ the price of the final good. Each worker maximizes his utility function (1) under the constraints:

$$ P_t (C_t^w + K_t^w - (1 - \delta)K_{t-1}^w) + M_t^w = M_{t-1}^w + r_t K_{t-1}^w + w_t L_t \quad (2) $$

$$ P_t (C_t^w + K_t^w - (1 - \delta)K_{t-1}^w) \le M_{t-1}^w + r_t K_{t-1}^w \quad (3) $$

Equation (2) represents the usual budget constraint, while (3) defines the liquidity constraint. The equilibria considered here are defined by:

$$ U'(C_t^w/B) > \lambda U'(C_{t+1}^w/B) [(1 - \delta) + r_{t+1}/P_{t+1}] \quad (4) $$

$$ (1 - \delta)P_{t+1} + r_{t+1} > P_t \qquad (5) $$

Workers always choose $K_t^w = 0$ and the financial constraint is binding, i.e. $P_t C_t^w = M_{t-1}^w$. Therefore, we deduce:

$$ u(C_{t+1}^w/B) = v(L_t) \qquad (6) $$

$$ P_{t+1} C_{t+1}^w = w_t L_t \qquad (7) $$

<footnote n="7">⁷For simplification, we note $x \equiv C^w/B$.</footnote>

<footnote n="8">⁸We assume that the labor endowment is strictly greater than 1, because as we will see in the next section, the labor supply will be normalized to 1 at the steady state.</footnote>
## 2 The Model

We consider a monetary economy with discrete time, $t = 1, 2, ..., \infty$, and two types of infinitely lived households, workers and capitalists.<fnref n="4" /> Only workers supply labor, whereas both workers and capitalists consume the final good. Moreover, workers are more impatient than capitalists, i.e. they discount the future more than the latter. Following Woodford (1986), we assume that there is a financial market imperfection that prevent workers from borrowing against their wage earnings. Therefore, in a neighborhood of a monetary steady state, capitalists hold the whole capital stock and no money, and workers save their wage income in the form of money balances. In the production sector, firms produce the final good. The production benefits from productive externalities specific for each input. As we will see, this type of externalities is more general than most of the formulations used in the literature and allows us to obtain new dynamic results.

### 2.1 Workers

Each worker chooses his labor supply and his consumption to maximize his utility function:

$$ \sum_{t=1}^{\infty} \left[ \lambda^{t-1} U(C_t^w / B) - \lambda^t V(L_t) \right] \quad (1) $$

where $C_t^w$ is the consumption in period $t$, $L_t$ the labor supply, $B > 0$ a scaling parameter and $\lambda \in (0, 1)$ the discount factor. The utility functions $U$ and $V$ are characterized by the following assumptions:

**Assumption 1** The functions $U(C^w/B)$ and $V(L)$ are continuous for all $C^w \ge 0$ and $0 \le L \le \bar{L}$, where the labor endowment $\bar{L} > 1$ may be finite or infinite.<fnref n="5" /> They are $C^n$ for $C^w > 0$, $0 < L < \bar{L}$ and $n$ large enough, with $U'(x) > 0$, $U''(x) < 0$, $V'(L) > 0$ and $V''(L) > 0$. Moreover, $\lim_{L \to \bar{L}} V'(L) = +\infty$ and consumption and leisure are gross substitutes, i.e., $-xU''(x)/U'(x) < 1$.

In what follows, we respectively note $M_t^w$ and $K_t^w$ the money balances and the capital stock held by workers, $\delta \in (0, 1)$ the depreciation rate of

<footnote n="4">⁴For more details on this model with heterogeneous households, see Woodford (1986). One can also refer to Barinci and Chéron (2001) or Grandmont, Pintus, and de Vilder (1998).</footnote>

<footnote n="5">⁵We assume that the labor endowment is strictly greater than 1, because as we will see in the next section, the labor supply will be normalized to 1 at the steady state.</footnote>
where $u(x) = xU'(x)$ and $v(L) = LV'(L)$. Under Assumption 1, there exists a function $\gamma \equiv u^{-1} \circ v$, such that $C_{t+1}^w/B = \gamma(L_t)$. Since consumption and leisure are gross substitutes, $\varepsilon_\gamma(L) \equiv \gamma'(L)L/\gamma(L) = [1+LV''(L)/V'(L)]/[1+xU''(x)/U'(x)] > 1$. Hence, the labor supply is increasing in the real wage, with an elasticity $1/(\varepsilon_\gamma(L) - 1) > 0$.

## 2.2 Capitalists

Capitalists behavior is represented by a representative agent who maximizes his lifetime utility function:

$$ \sum_{t=1}^{\infty} \beta^t \ln C_t^c \qquad (8) $$

where $\beta \in (\lambda, 1)$ is his discount factor and $C_t^c$ his consumption. At period $t$, the representative agent faces the following budget constraint:

$$ P_t (C_t^c + K_t^c - (1-\delta)K_{t-1}^c) + M_t^c = M_{t-1}^c + r_t K_{t-1}^c \qquad (9) $$

where $M_t^c$ is the money balances at period $t$ and $K_t^c$ the capital stock. Since we focus on equilibria satisfying $(1-\delta)P_{t+1} + r_{t+1} > P_t$, capitalists do not hold money ($M_t^c = 0$) because it has a lower return than capital. We obtain the optimal solution:

$$ C_t^c = (1 - \beta) R_t K_{t-1} \qquad (10) $$

$$ K_t = \beta R_t K_{t-1} \qquad (11) $$

where $R_t \equiv 1 - \delta + r_t/P_t$ is the real gross return on capital.<fnref n="9" />

## 2.3 Production Sector

A continuum of firms, of unit size, produce the final good using labor $L_t$ and capital $K_{t-1}$ with an internal constant returns to scale technology. However, production benefiting from externalities, returns to scale are increasing at the social level. In existing one-sector models, externalities are usually introduced through the total productivity of factors that increases with the average capital and labor.<fnref n="10" /> In this paper, we rather consider externalities specific to each input, assuming that efficiency of capital (labor) is increasing in the average capital (average labor).

<footnote n="9">⁹The superscript on $K_t^c$ is dropped because workers hold no capital.</footnote>

<footnote n="10">¹⁰See among others Barinci and Chéron (2001), Benhabib and Farmer (1994), Cazzavil- lan, Lloyd-Braga, and Pintus (1998), Farmer and Guo (1994), Harrison and Weder (2002), Hintermaier (2003), Pintus (2006). For a survey, one can refer to Benhabib and Farmer (1999).</footnote>
capital, $r_t$ the nominal interest rate, $w_t$ the nominal wage and $P_t$ the price of the final good. Each worker maximizes his utility function (1) under the constraints:

$$P_t (C_t^w + K_t^w - (1-\delta)K_{t-1}^w) + M_t^w = M_{t-1}^w + r_t K_{t-1}^w + w_t L_t \quad (2)$$

$$P_t (C_t^w + K_t^w - (1-\delta)K_{t-1}^w) \leq M_{t-1}^w + r_t K_{t-1}^w \quad (3)$$

Equation (2) represents the usual budget constraint, while (3) is the liquidity constraint. The equilibria considered here are defined by:

$$U'(C_t^w/B) > \lambda U'(C_{t+1}^w/B) [(1-\delta) + r_{t+1}/P_{t+1}] \quad (4)$$

$$(1-\delta)P_{t+1} + r_{t+1} > P_t \quad (5)$$

So, workers always choose $K_t^w = 0$ and the financial constraint is binding, i.e. $P_t C_t^w = M_{t-1}^w$. Hence, we obtain the two following relations:

$$u(C_{t+1}^w/B) = v(L_t) \quad (6)$$

$$P_{t+1}C_{t+1} = w_tL_t \quad (7)$$

where $u(x) = xU'(x)$ and $v(L) = LV'(L)$. Under Assumption 1, there exists a function $\gamma \equiv u^{-1} \circ v$, such that $C_{t+1}/B = \gamma(L_t)$. Since consumption and leisure are gross substitutes, $\varepsilon_\gamma(L) \equiv \gamma'(L)L/\gamma(L) = [1+LV''(L)/V'(L)]/[1+xU''(x)/U'(x)] > 1$, with $x = C/B$. We deduce that the labor supply increases with respect to the real wage, with an elasticity $1/(\varepsilon_\gamma(L) - 1) > 0$.

## 2.2 Capitalists

Capitalists behavior is represented by a representative agent who maximizes his lifetime utility function:

$$\sum_{t=1}^{\infty} \beta^t \ln C_t^c \quad (8)$$

where $\beta \in (\lambda, 1)$ is his discount factor and $C_t^c$ his consumption. At period $t$, the representative agent faces the following budget constraint:

$$P_t (C_t^c + K_t^c - (1-\delta)K_{t-1}^c) + M_t^c = M_{t-1}^c + r_t K_{t-1}^c \quad (9)$$
More specifically, $F(K_{t-1}, L_t)$ is a well defined strictly concave production function, homogeneous of degree one, increasing with each argument, and $f(a_t)$ the intensive production function satisfying the following assumption:

**Assumption 2** The intensive production function $f(a)$ is continuous for $a \ge 0$, positively valued and differentiable as many times as needed for $a > 0$, with $f'(a) > 0$ and $f''(a) < 0$.

At each period, the quantity of final good produced is given by:

$$ 
\begin{aligned}
Y_t &= AF(C(\bar{K}_{t-1})K_{t-1}, D(\bar{L}_t)L_t) \\
&= Af(a_t)D(\bar{L}_t)L_t,
\end{aligned} 
\quad (12) $$

where

$$ a_t = \frac{C(\bar{K}_{t-1})K_{t-1}}{D(\bar{L}_t)L_t} \quad (13) $$

is the capital-labor ratio measured in efficient units, $A > 0$ a scaling parameter, $\bar{K}_{t-1}$ average capital and $\bar{L}_t$ average labor. Furthermore, $C(\bar{K})$ represents externalities specific to capital and $D(\bar{L})$ externalities specific to labor. Note also that $C(\bar{K})K(D(\bar{L})L)$ can be interpreted as capital (labor) measured in efficient units. We further assume:

**Assumption 3** The functions $C(K)$ and $D(L)$ are continuous for all $K \ge 0$ and $L \ge 0$, positively valued and differentiable as many times as needed for $K > 0$ and $L > 0$. Moreover, we assume that $\varepsilon_{C,K}(K) = C'(K)K/C(K) \ge 0$ and $\varepsilon_{D,L}(L) = D'(L)L/D(L) \ge 0$.

Especially, this assumption states that the efficiency of capital $C(\bar{K})$ (labor $D(\bar{L})$) increases with average capital $\bar{K}$ (average labor $\bar{L}$).

Maximizing their profits, the producers take as given the level of externalities. We deduce the real interest rate $\rho_t$ and the real wage $\Omega_t$:

$$ \rho_t = C(\bar{K}_{t-1})A\rho(a_t) \quad (14) $$

$$ \Omega_t = D(\bar{L}_t)A\omega(a_t) \quad (15) $$

with

$$ \rho(a_t) = f'(a_t) \text{ and } \omega(a_t) = f(a_t) - a_t f'(a_t) $$

We notice that $\tilde{\rho}_t \equiv \rho_t/C(\bar{K}_{t-1})$ and $\tilde{\Omega}_t \equiv \Omega_t/D(\bar{L}_t)$ represent the marginal productivities of capital and labor measured in efficient units. Before determining the equilibrium, it is useful to define the following relationships. First, we note $s(a) \equiv \rho(a)a/f(a) \in (0,1)$ the capital share
where $M_t^c$ is the money balances at period $t$ and $K_t^c$ the capital stock. Since we focus on equilibria satisfying $(1 - \delta)P_{t+1} + r_{t+1} > P_t$, capitalists do not hold money ($M_t^c = 0$) because it has a lower return than capital. We deduce the following optimal solution:

$$C_t^c = (1 - \beta)R_t K_{t-1} \qquad (10)$$

$$K_t = \beta R_t K_{t-1} \qquad (11)$$

where $R_t = 1 - \delta + r_t / P_t$ is the real gross return on capital.<fnref n="6" />

## 2.3 Production Sector

All markets are perfectly competitive and a continuum of firms, of unit mass, produce the final good using labor and capital with an internal constant returns to scale technology. However, production benefiting from externalities, returns to scale are increasing at the social level. In most of the existing papers, externalities are introduced assuming that the total productivity of factors increases with respect to average capital and labor.<fnref n="7" /> Here, we introduce a more general formulation of externalities assuming that each input, labor and capital, is affected by a specific externality, increasing with respect to average capital and labor.

Consider that $F(K_{t-1}, L_t)$ is a well-defined strictly concave production function, homogeneous of degree one, increasing with each argument, $a_t \equiv K_{t-1}/L_t$ the capital-labor ratio and $f(x_t)$ the intensive production function that satisfies the following assumption:

**Assumption 2** The intensive production function $f(x)$ is continuous for $x \ge 0$, positively valued and differentiable as many times as needed for $x > 0$, with $f'(x) > 0$ and $f''(x) < 0$.

At each period, the quantity of final good produced is given by:

$$ 
\begin{aligned}
Y_t &= AF(C(\bar{K}_{t-1}, \bar{L}_t)K_{t-1}, D(\bar{K}_{t-1}, \bar{L}_t)L_t) \\
&= Af\left(\frac{C(\bar{K}_{t-1}, \bar{L}_t)}{D(\bar{K}_{t-1}, \bar{L}_t)} a_t\right) D(\bar{K}_{t-1}, \bar{L}_t)L_t
\end{aligned}
\qquad (12) $$

<footnote n="6">⁶The superscript on $K_t^c$ is dropped because workers hold no capital.</footnote>

<footnote n="7">⁷Such externalities have often been used in macroeconomic dynamics models. See among others Barinci and Chéron (2001), Benhabib and Farmer (1994), Boldrin (1992), Cazzavillan (2001), Cazzavillan, Lloyd-Braga, and Pintus (1998), Farmer and Guo (1994), Harrison and Weder (2002), Hintermaier (2003), Pintus (2006). For a survey, one can refer to Benhabib and Farmer (1999).</footnote>
in income. Moreover, the elasticity of capital-labor substitution in efficient units is defined by $\sigma(a) = dlna/dln(\tilde{\Omega}/\tilde{\rho}) \ge 0.$<fnref n="11" /> This implies that $1/\sigma(a) = d \ln \omega(a)/d \ln a - d \ln \rho(a)/d \ln a$. Since $\omega'(a) = -a\rho'(a)$, we deduce that:

$$ \varepsilon_{\omega}(a) \equiv \frac{\omega'(a)a}{\omega(a)} = \frac{s(a)}{\sigma(a)} \quad \text{and} \quad \varepsilon_{\rho}(a) \equiv \frac{\rho'(a)a}{\rho(a)} = -\frac{1-s(a)}{\sigma(a)} \quad (16) $$

Finally, note that the degree of returns to scale is determined by $1 + s(a)\varepsilon_{C,K}(\bar{K}) + (1-s(a))\varepsilon_{D,L}(\bar{L})$.<fnref n="12" />

## 2.4 Intertemporal Equilibrium

Equilibrium on labor market requires that $\bar{L}_t = L_t$ and:

$$ w_t/P_t = \Omega_t = D(L_t)A\omega(a_t) \equiv A\Omega(K_{t-1}, L_t) \quad (17) $$

with $a_t = C(K_{t-1})K_{t-1}/[D(L_t)L_t]$. Equilibrium on capital market is ensured by $\bar{K}_{t-1} = K_{t-1}$ and:

$$ r_t/P_t = \varrho_t = C(K_{t-1})A\rho(a_t) \equiv A\varrho(K_{t-1}, L_t) \quad (18) $$

Let $M > 0$ be the constant money supply. Since workers save their wage income in the form of money and capitalists do not hold money, the equilibrium condition on money market can be written:

$$ C_t^w = M/P_t = \Omega_t L_t \quad (19) $$

Finally, good market equilibrium is ensured by Walras law. Then, substituting (17) and (18) in (6) and (11), we obtain the two dynamic equations:

$$ A\Omega(K_t, L_{t+1})L_{t+1}/B = \gamma(L_t) \quad (20) $$

$$ K_t = \beta[1 - \delta + A\varrho(K_{t-1}, L_t)]K_{t-1} \quad (21) $$

and an intertemporal equilibrium is defined by:<fnref n="13" />

**Definition 1** Given $K_0 > 0$, an intertemporal equilibrium with perfect foresight is a sequence $(K_{t-1}, L_t) \in \mathbb{R}_{++}^2, t = 1, 2, ..., \infty$, such that (20) and (21) are satisfied.

<footnote n="11">¹¹A quite similar definition of the elasticity of capital-labor substitution is used by Pintus (2004).</footnote>

<footnote n="12">¹²Obviously, returns to scale are increasing when $s(a)\varepsilon_{C,K}(\bar{K}) + (1-s(a))\varepsilon_{D,L}(\bar{L}) > 0$.</footnote>

<footnote n="13">¹³Note that capital $K_{t-1}$ is the only one predetermined variable.</footnote>
where $A > 0$ is a scaling parameter, $\overline{K}_{t-1}$ average capital and $\overline{L}_t$ average labor. $C(\overline{K}, \overline{L})$ represents externalities specific to capital, whereas $D(\overline{K}, \overline{L})$ summarizes externalities specific to the second input, labor. Note that $C(\overline{K}, \overline{L})K$ (respectively $D(\overline{K}, \overline{L})L$) can also be interpreted as capital (respectively labor) measured in efficient units. Evidently, in the particular case where $C(\overline{K}, \overline{L}) = D(\overline{K}, \overline{L})$, we recover the usual form where externalities are represented by the total productivity of factors. We further assume:

**Assumption 3** The functions $C(K, L)$ and $D(K, L)$ are continuous for all $K, L \ge 0$, positively valued and differentiable as many times as needed for $K, L > 0$. Moreover, we assume that $\varepsilon_{C,K}(K, L)$, $\varepsilon_{C,L}(K, L)$, $\varepsilon_{D,K}(K, L)$ and $\varepsilon_{D,L}(K, L) \ge 0$, where we note $\varepsilon_{h,x}(K, L) \equiv \frac{\partial h(K,L)}{\partial x} \frac{x}{h(K,L)}$, with $h(K, L) = \{C(K, L), D(K, L)\}$ and $x = \{K, L\}$.

In particular, this assumption means that the contributions of capital and labor to the externalities $C(K, L)$ and $D(K, L)$ are always positive.

Maximizing their profits, the producers take as given the level of externalities. If $\varrho_t$ and $\Omega_t$ respectively denote the real interest rate and the real wage, we obtain:

$$ \varrho_t = C(\overline{K}_{t-1}, \overline{L}_t) A \rho(x_t) \quad (13) $$

$$ \Omega_t = D(\overline{K}_{t-1}, \overline{L}_t) A \omega(x_t) \quad (14) $$

with

$$ \rho(x_t) \equiv f'(x_t), \quad \omega(x_t) \equiv f(x_t) - x_t f'(x_t) \quad \text{and} \quad x_t \equiv \frac{C(\overline{K}_{t-1}, \overline{L}_t)}{D(\overline{K}_{t-1}, \overline{L}_t)} a_t $$

Remark that $\tilde{\varrho}_t \equiv \varrho_t/C(\overline{K}_{t-1}, \overline{L}_t)$ and $\tilde{\Omega}_t \equiv \Omega_t/D(\overline{K}_{t-1}, \overline{L}_t)$ represent the marginal productivities of capital and labor in efficient units.

Before determining the equilibrium, it is useful to define the following relationships. First, we note $s(x) \equiv \rho(x)x/f(x) \in (0, 1)$, with $x \equiv C(\overline{K}, \overline{L})a/D(\overline{K}, \overline{L})$, the capital share in income. Moreover, the elasticity of capital-labor substitution in efficient units is defined by $\sigma(x) = d\ln x/d\ln(\tilde{\Omega}/\tilde{\varrho}) \ge 0$.<fnref n="8" /> This implies that $1/\sigma(x) = d\ln\omega(x)/d\ln x - d\ln\rho(x)/d\ln x$. Since, $\omega'(x) = -x\rho'(x)$, we deduce that:

<footnote n="8">⁸A quite similar definition of the elasticity of capital-labor substitution is used by Pintus (2004).</footnote>
### 3 Existence of a Steady State

A steady state of the dynamic system (20)-(21) is a solution $(K, L) = (K_{t-1}, L_t)$ for all $t$, such that:

$$ \frac{\theta\Omega(K, L)}{B\beta\varrho(K, L)} = \frac{\gamma(L)}{L} \qquad (22) $$

$$ A\varrho(K, L) = \theta/\beta \qquad (23) $$

where $\theta \equiv 1 - \beta(1 - \delta) \in (0, 1)$.

Following Cazzavillan, Lloyd-Braga, and Pintus (1998), the existence of a steady state is established by choosing appropriately the two scaling parameters $A > 0$ and $B > 0$ so as to ensure that one stationary solution coincides with $(K, L) = (1, 1)$.<fnref n="14" /> From equation (23), we obtain a unique solution $A = A^*$, with:

$$ A^* = \frac{\theta}{\beta\varrho(1, 1)} > 0 \qquad (24) $$

Using (22), we get:

$$ u\left(\frac{\theta\Omega(1, 1)}{B\beta\varrho(1, 1)}\right) = v(1) \qquad (25) $$

From Assumption 1, $u$ is decreasing in $B$. Therefore, there exists a unique $B = B^* > 0$ satisfying (25) if and only if the following assumption is satisfied:

**Assumption 4** $\lim_{x \to 0} u(x) < v(1) < \lim_{x \to +\infty} u(x)$.

This result is summarized as follows:

**Proposition 1** Under Assumptions 1-4, $(K, L) = (1, 1)$ is a stationary solution of the dynamic system (20)-(21) if and only if $A^*$ and $B^*$ are the unique solutions of (24) and (25).

### 4 Local Dynamics

In this section, we analyze the stability properties of the steady state. More specifically, we focus on local indeterminacy and bifurcations to analyze the occurrence of fluctuations due to the volatility of expectations and endogenous cycles.

We first differentiate the dynamic system (20)-(21). If we note $\varepsilon_\gamma$ the elasticity of $\gamma(L)$, and $\varepsilon_{\varrho,K}$, $\varepsilon_{\varrho,L}$, $\varepsilon_{\Omega,K}$ and $\varepsilon_{\Omega,L}$ the elasticities of $\varrho(K, L)$ and

<footnote n="14">¹⁴For sake of conciseness, we do not analyze uniqueness or multiplicity of steady states.</footnote>
$$ \varepsilon_{\omega}(x) \equiv \frac{\omega'(x)x}{\omega(x)} = \frac{s(x)}{\sigma(x)} \quad \text{and} \quad \varepsilon_{\rho}(x) \equiv \frac{\rho'(x)x}{\rho(x)} = -\frac{1-s(x)}{\sigma(x)} \qquad (15) $$

Finally, note that the degree of returns to scale is determined by:

$$ 1 + s(x)[\varepsilon_{C,K}(\overline{K}, \overline{L}) + \varepsilon_{C,L}(\overline{K}, \overline{L})] + (1 - s(x)) [\varepsilon_{D,K}(\overline{K}, \overline{L}) + \varepsilon_{D,L}(\overline{K}, \overline{L})] $$

## 2.4 Intertemporal Equilibrium

Equilibrium on the labor market requires that $\overline{L}_t = L_t$ and $w_t/P_t = \Omega_t \equiv A\Omega(K_{t-1}, L_t)$, equilibrium on the capital market $\overline{K}_{t-1} = K_{t-1}$ and $r_t/P_t = \varrho_t \equiv A\varrho(K_{t-1}, L_t)$. Let $M > 0$ be the constant money supply. Since workers save their wage income in the form of money and capitalists do not hold money, the equilibrium condition on the money market can be written:

$$ C_t^w = M/P_t = \Omega_t L_t \qquad (16) $$

Finally, the good market equilibrium is ensured by Walras law. Then, using (6) and (11), we obtain the two following dynamic equations:

$$ (1/B)\Omega_{t+1}L_{t+1} = \gamma(L_t) \qquad (17) $$

$$ K_t = \beta R_t K_{t-1} \qquad (18) $$

where $R_t = 1 - \delta + \varrho_t$. Substituting expressions (13) and (14) into (17) and (18), we can define an intertemporal equilibrium:

**Definition 1** An intertemporal equilibrium with perfect foresight is a sequence $(K_{t-1}, L_t) \in \mathbb{R}_{++}^2$, $t = 1, 2, ..., \infty$, such that (17) and (18) are satisfied, where $\varrho_t$ and $\Omega_t$ are given by (13) and (14).

## 3 Existence of a Steady State

A steady state of the dynamic system (17)-(18) is a solution $(K, L) = (K_{t-1}, L_t)$ for all $t$, such that:

$$ \frac{\theta\Omega(K, L)}{B\beta\varrho(K, L)} = \frac{\gamma(L)}{L} \qquad (19) $$

$$ A\varrho(K, L) = \theta/\beta \qquad (20) $$
$\Omega(K, L)$ with respect to $K$ and $L$, evaluated at the steady state defined in Proposition 1, we get:

$$ \frac{dK_t}{K} = (\theta\varepsilon_{\varrho,K} + 1)\frac{dK_{t-1}}{K} + \theta\varepsilon_{\varrho,L}\frac{dL_t}{L} \quad (26) $$

$$ \frac{dL_{t+1}}{L} = -\frac{\varepsilon_{\Omega,K}(1 + \theta\varepsilon_{\varrho,K})}{1 + \varepsilon_{\Omega,L}} \frac{dK_{t-1}}{K} + \frac{\varepsilon_{\gamma} - \theta\varepsilon_{\Omega,K}\varepsilon_{\varrho,L}}{1 + \varepsilon_{\Omega,L}} \frac{dL_t}{L} \quad (27) $$

We also note $\varepsilon_{C,K} \equiv \varepsilon_{C,K}(1)$, $\varepsilon_{D,L} \equiv \varepsilon_{D,L}(1)$, $s \equiv s(C(1)/D(1))$ and $\sigma \equiv \sigma(C(1)/D(1))$. Using (16), (17) and (18), we obtain the following relationships:

$$ \begin{aligned} \varepsilon_{\varrho,K} &= \varepsilon_{C,K} - \frac{(1-s)(1+\varepsilon_{C,K})}{\sigma} \\ \varepsilon_{\varrho,L} &= \frac{(1-s)(1+\varepsilon_{D,L})}{\sigma} \\ \varepsilon_{\Omega,K} &= \frac{s(1+\varepsilon_{C,K})}{\sigma} \\ \varepsilon_{\Omega,L} &= \varepsilon_{D,L} - \frac{s(1+\varepsilon_{D,L})}{\sigma} \end{aligned} \quad (28) $$

Before determining the trace $T$ and the determinant $D$ of the associated Jacobian matrix, we clarify that, for simplification and briefness, we restrict our analysis to the case of an infinitely elastic labor supply,<fnref n="15" /> i.e.

**Assumption 5** $\varepsilon_{\gamma} = 1$.

Using this assumption and equations (26), (27), (28), we have:

$$ T = 1 + \frac{\sigma[1 + \theta\varepsilon_{C,K}(1 + \varepsilon_{D,L})] - \theta(1 + \varepsilon_{D,L})(1 - s + \varepsilon_{C,K})}{(\sigma - s)(1 + \varepsilon_{D,L})} \quad (29) $$

$$ D = \frac{\sigma(1 + \theta\varepsilon_{C,K}) - \theta(1 - s)(1 + \varepsilon_{C,K})}{(\sigma - s)(1 + \varepsilon_{D,L})} \quad (30) $$

Recall that when $1 - T + D > 0$, $1 + T + D > 0$ and $D < 1$, the steady state is a sink, i.e. locally indeterminate since one variable (the capital) is predetermined. When $1 - T + D > (<)0$ and $1 + T + D < (>0)$, the steady state is a saddle. Otherwise, it is a source. In these last two cases, the steady state is locally determinate. Moreover, when a parameter varies

<footnote n="15">¹⁵Note that this assumption is often used in the literature. The interested reader can however refer to a previous version of this paper (Seegmuller (2006)) where the analysis is led without assuming an infinitely elastic labor supply.</footnote>
where $\theta \equiv 1 - \beta(1 - \delta)$.

Following Cazzavillan, Lloyd-Braga, and Pintus (1998), the existence of a steady state is established by choosing appropriately the two scaling parameters $A > 0$ and $B > 0$ so as to ensure that one steady state coincides with $(K, L) = (1, 1)$. From equation (20), we obtain a unique solution:

$$A = \frac{\theta}{\beta\varrho(1, 1)} > 0 \qquad (21)$$

Using relation (19), we get:

$$u\left(\frac{\theta\Omega(1, 1)}{B\beta\varrho(1, 1)}\right) = v(1) \qquad (22)$$

From Assumption 1, $u$ is decreasing in $B$. Therefore, there exists a unique $B > 0$ satisfying this last equation if and only if $\lim_{x \to 0} u(x) < v(1) < \lim_{x \to +\infty} u(x)$.<fnref n="9" />

**Proposition 1** Under $\lim_{x \to 0} u(x) < v(1) < \lim_{x \to +\infty} u(x)$ and Assumptions 1-3, $(K, L) = (1, 1)$ is a stationary solution of the dynamic system (17)-(18) if and only if A and B are the unique solutions of (21) and (22).

# 4 Local Dynamics and Bifurcation Analysis

In this section, we analyze the local stability of the steady state and the occurrence of bifurcations. We will see that the key parameters of this analysis will be the elasticities of labor supply, of capital-labor substitution, and of externalities with respect to capital and labor. To ease the presentation of the results, we use the geometrical method developed by Grandmont, Pintus, and de Vilder (1998), which applies to discrete time nonlinear two-dimensional dynamic systems. Therefore, we first differentiate the dynamic system (17)-(18). If we note $\varepsilon_\gamma$ the elasticity of $\gamma(L)$, and $\varepsilon_{\varrho,K}$, $\varepsilon_{\varrho,L}$, $\varepsilon_{\Omega,K}$ and $\varepsilon_{\Omega,L}$ the elasticities of $\varrho(K, L)$ and $\Omega(K, L)$ with respect to $K$ and $L$, evaluated at the steady state defined in Proposition 1, we get:

$$\frac{dK_t}{K} = (\theta\varepsilon_{\varrho,K} + 1)\frac{dK_{t-1}}{K} + \theta\varepsilon_{\varrho,L}\frac{dL_t}{L} \qquad (23)$$

$$\frac{dL_{t+1}}{L} = -\frac{\varepsilon_{\Omega,K}(1 + \theta\varepsilon_{\varrho,K})}{1 + \varepsilon_{\Omega,L}}\frac{dK_{t-1}}{K} + \frac{\varepsilon_{\gamma} - \theta\varepsilon_{\Omega,K}\varepsilon_{\varrho,L}}{1 + \varepsilon_{\Omega,L}}\frac{dL_t}{L} \qquad (24)$$

<footnote n="9">⁹In order to be as short as possible, we do not analyze uniqueness or multiplicity of steady states.</footnote>
continuously, a flip bifurcation generically occurs when $1+T+D=0$, a transcritical bifurcation generically occurs when $1-T+D=0$ and a Hopf bifurcation generically occurs when $1-T+D>0$, $1+T+D>0$ and $D=1$.<fnref n="16" />

Before beginning the analysis, we further assume:

**Assumption 6**

$$ (i) \quad s > \theta(1-s)(1+\varepsilon_{C,K})(2+\varepsilon_{D,L})/(2+\theta\varepsilon_{C,K}); $$

$$ (ii) \quad \sigma > s. $$

Note that, in this model, the length of period is small, implying values of the depreciation rate of capital $\delta$ close to 0 and of the discount factor $\beta$ close to 1. Therefore, $\theta$ is small and close to 0, providing a justification of inequality (i).<fnref n="17" /> Assumption 6 (ii) imposes that we do not focus on sufficiently weak degrees of capital-labor substitution. In addition, it is already well known that, in the Woodford (1986) framework, local indeterminacy and endogenous cycles can occur without externalities only if $\sigma < s$ (see Grandmont, Pintus and de Vilder (1998)).

To determine the local stability properties of the steady state, we first compute $1+T+D$. Using (29) and (30), we obtain:

$$ 1+T+D = \frac{2(2+\varepsilon_{D,L})(2+\theta\varepsilon_{C,K})(\sigma-\sigma_F)}{(\sigma-s)(1+\varepsilon_{D,L})} \quad (31) $$

with

$$ \sigma_F = \frac{(1+\varepsilon_{D,L})[2s+\theta(1-s+\varepsilon_{C,K})] + \theta(1-s)(1+\varepsilon_{C,K})}{2(2+\varepsilon_{D,L})(2+\theta\varepsilon_{C,K})} \quad (32) $$

Using Assumption 6 (i), we can prove that $\sigma_F < s$. Therefore, $1+T+D > 0$ for all $\sigma > s$. We compute now $1-T+D$:

$$ 1-T+D = \frac{\theta\varepsilon_{C,K}\varepsilon_{D,L}(\sigma_T-\sigma)}{(\sigma-s)(1+\varepsilon_{D,L})} \quad (33) $$

with

$$ \sigma_T = 1 + \frac{1-s}{\varepsilon_{C,K}} + \frac{s}{\varepsilon_{D,L}} \quad (34) $$

<footnote n="16">¹⁶See Azariadis (1993) and Grandmont, Pintus and de Vilder (1998) for more details.</footnote>

<footnote n="17">¹⁷We remark that without externalities, inequality (i) becomes $s > \theta(1-s)$, which is an assumption introduced in the perfectly competitive economy analyzed by Grandmont, Pintus and de Vilder (1998).</footnote>
Using (15) and noting $\varepsilon_{h,x} = \varepsilon_{h,x}(1, 1)$, with $h = \{C, D\}$ and $x = \{K, L\}$, $s \equiv s(1)$ and $\sigma \equiv \sigma(1)$, we obtain the following elasticities evaluated at the steady state:

$$ \varepsilon_{\varrho,K} = \varepsilon_{C,K} - \frac{(1-s)(\varepsilon_{C,K} - \varepsilon_{D,K} + 1)}{\sigma} \quad (25) $$

$$ \varepsilon_{\varrho,L} = \varepsilon_{C,L} - \frac{(1-s)(\varepsilon_{C,L} - \varepsilon_{D,L} - 1)}{\sigma} \quad (26) $$

$$ \varepsilon_{\Omega,K} = \varepsilon_{D,K} + \frac{s(\varepsilon_{C,K} - \varepsilon_{D,K} + 1)}{\sigma} \quad (27) $$

$$ \varepsilon_{\Omega,L} = \varepsilon_{D,L} + \frac{s(\varepsilon_{C,L} - \varepsilon_{D,L} - 1)}{\sigma} \quad (28) $$

Using these expressions, the trace $T$ and the determinant $D$ of the associated Jacobian matrix, which represent respectively the sum and the product of the two eigenvalues, i.e. the roots of the characteristic polynomial $Q(\lambda) \equiv \lambda^2 - T\lambda + D = 0$, can be written:

$$ T = \frac{\sigma}{\sigma(1 + \varepsilon_{D,L}) - s(1 + \varepsilon_{D,L} - \varepsilon_{C,L})} (\varepsilon_{\gamma} - 1) + T_1(\sigma), \quad \text{with} \\ T_1(\sigma) = 1 + \frac{\sigma[1 + \theta(\varepsilon_{C,K}(1 + \varepsilon_{D,L}) - \varepsilon_{D,K}\varepsilon_{C,L})]}{\sigma(1 + \varepsilon_{D,L}) - s(1 + \varepsilon_{D,L} - \varepsilon_{C,L})} \quad (29) \\ - \frac{\theta[\varepsilon_{C,K} + s\varepsilon_{C,L} + (1 - s)(1 + \varepsilon_{D,L}) + \varepsilon_{D,L}\varepsilon_{C,K} - \varepsilon_{D,K}\varepsilon_{C,L}]}{\sigma(1 + \varepsilon_{D,L}) - s(1 + \varepsilon_{D,L} - \varepsilon_{C,L})} $$

$$ D = \varepsilon_{\gamma} D_{1}(\sigma), \quad \text{with} \quad D_{1}(\sigma) = \frac{\sigma(1 + \theta\varepsilon_{C,K}) - \theta(1 - s)(1 + \varepsilon_{C,K} - \varepsilon_{D,K})}{\sigma(1 + \varepsilon_{D,L}) - s(1 + \varepsilon_{D,L} - \varepsilon_{C,L})} \quad (30) $$

Following Grandmont, Pintus, and de Vilder (1998), we study the local stability (indeterminacy) of the steady state and the occurrence of bifurcations by analyzing the trace $T$ and the determinant $D$ in the plane (see Figures 1-4). On the line (AC), one eigenvalue is equal to 1. On the line (AB), one eigenvalue is equal to -1. On the segment [BC], the two eigenvalues are complex conjugates and have an unit modulus. Furthermore, if $(T, D)$ is inside (ABC), the steady state is a sink, i.e. locally indeterminate since one variable is predetermined. When $1 - T + D > (<)0$ and $1 + T + D < (>0)$, the steady state is a saddle. Otherwise, it is a source. In the last two cases, the steady state is locally determinate.
We deduce that $1-T+D > 0$ for $s < \sigma < \sigma_T$, $1-T+D = 0$ for $\sigma = \sigma_T$ and $1-T+D < 0$ for $\sigma > \sigma_T$. Note that in the particular case where $\varepsilon_{C,K} = 0$ and/or $\varepsilon_{D,L} = 0$, $\sigma_T$ is not defined and $1-T+D > 0$ for all $\sigma > s$.

Using Assumption 6 (i), we remark that $D$ is a decreasing function of $\sigma$ (see equation (30)), from $+\infty$ when $\sigma \to s$ to $(1 + \theta\varepsilon_{C,K})/(1 + \varepsilon_{D,L})$ when $\sigma \to +\infty$. Therefore, the inequality

$$ \varepsilon_{D,L} > \theta\varepsilon_{C,K} \qquad (35) $$

is a necessary condition to have $D < 1$. Indeed, when (35) is satisfied, $D > 1$ for $s < \sigma < \sigma_H$, $D = 1$ for $\sigma = \sigma_H$ and $D < 1$ for $\sigma > \sigma_H$, with:

$$ \sigma_H \equiv \frac{s(1 + \varepsilon_{D,L}) - \theta(1 - s)(1 + \varepsilon_{C,K})}{\varepsilon_{D,L} - \theta\varepsilon_{C,K}} \qquad (36) $$

Otherwise, $D > 1$ for all $\sigma > s$. We finally remark that $\sigma_T > \sigma_H$ if and only if:

$$ \frac{\varepsilon_{D,L}^2}{1 + \varepsilon_{D,L}} > \theta \frac{s}{1 - s} \frac{\varepsilon_{C,K}^2}{1 + \varepsilon_{C,K}} \qquad (37) $$

Using all these results, we deduce the stability properties of the steady state:

**Proposition 2** Under Assumptions 1-6, the following generically holds.

(i) If $\varepsilon_{D,L} \le \theta\varepsilon_{C,K}$ or $\varepsilon_{D,L} > \theta\varepsilon_{C,K}$ and $\frac{\varepsilon_{D,L}^2}{1+\varepsilon_{D,L}} < \theta\frac{s}{1-s}\frac{\varepsilon_{C,K}^2}{1+\varepsilon_{C,K}}$, the steady state is a source for $s < \sigma < \sigma_T$ and is a saddle for $\sigma > \sigma_T$.

(ii) If $\varepsilon_{D,L} > \theta\varepsilon_{C,K}$ and $\frac{\varepsilon_{D,L}^2}{1+\varepsilon_{D,L}} > \theta\frac{s}{1-s}\frac{\varepsilon_{C,K}^2}{1+\varepsilon_{C,K}}$, the steady state is a source for $s < \sigma < \sigma_H$, is a sink for $\sigma_H < \sigma < \sigma_T$ and is a saddle for $\sigma > \sigma_T$.

A transcritical bifurcation generically occurs when $\sigma$ crosses $\sigma_T$ and a Hopf bifurcation generically occurs when $\sigma$ crosses $\sigma_H$.

This proposition states first that endogenous fluctuations can occur only if inequalities (35) and (37) are satisfied. This means that the response of labor efficiency ($D(L)$) to an increase of labor has to be large enough with respect to the response of capital efficiency ($C(K)$) to an increase of capital. In such a case, indeterminacy requires elasticities of capital-labor substitution higher than a lower bound ($\sigma_H$). This result can be related to Cazzavillan, Lloyd-Braga and Pintus (1998) who analyze, in a Woodford (1986) model,
Suppose now that $T$ and $D$ change when a parameter, called the bifurcation parameter, varies continuously. When $(T, D)$ crosses the line (AC), a transcritical or a pitchfork bifurcation generically occurs.<fnref n="10" /> When $(T, D)$ crosses the line (AB), one gets a flip bifurcation, i.e. a cycle of period 2 appears around the steady state. When $(T, D)$ crosses the segment $[BC]$, one gets a Hopf bifurcation, i.e. an invariant closed curve appears around the steady state.<fnref n="11" /> Moreover, sunspot equilibria can appear around a steady state if it is locally indeterminate. They can also occur in the neighborhood of a cycle of period two if it is locally stable and in a neighborhood of an invariant closed curve if the Hopf bifurcation is supercritical.<fnref n="12" />

In order to keep the analysis as simple as possible, we assume in what follows:

**Assumption 4** (i) $\theta$ is sufficiently small, such that the following inequalities are satisfied:

* $s > \theta(1-s);$

* $s(1 + \varepsilon_{D,L} - \varepsilon_{C,L})(1 + \theta\varepsilon_{C,K}) > \theta(1 + \varepsilon_{D,L})[\varepsilon_{C,K} + s\varepsilon_{C,L} + (1-s)(1 + \varepsilon_{D,L})] + \theta(\varepsilon_{C,K}\varepsilon_{D,L} - \varepsilon_{D,K}\varepsilon_{C,L})[(1-s)(1 + \varepsilon_{D,L}) + s\varepsilon_{C,L}];$

* $s(1 + \varepsilon_{D,L} - \varepsilon_{C,L}) > \theta[(1 + \varepsilon_{D,L})(1 - s + \varepsilon_{C,K}) + \varepsilon_{C,L}(s - \varepsilon_{D,K})];$

(ii) $\varepsilon_{h,x} < 1-s$, with $h = \{C, D\}$ and $x = \{K, L\}$.

Note that, in this type of model, the length of period is small. Then, the assumption that $\theta$ is small is not very restrictive, because $\delta$ is usually close to 0 and $\beta$ close to 1. The second part of Assumption 4 means that externalities cannot be too strong. This last remark is in accordance with empirical studies establishing that if returns to scale are increasing, they are not too strong.

We choose $\varepsilon_\gamma \in (1, +\infty)$ as the bifurcation parameter. It is closely related to the elasticity of labor supply $1/(\varepsilon_\gamma - 1)$. From equations (29) and (30), we notice that $(T, D)$ describes a half-line $\Delta$ when $\varepsilon_\gamma$ varies in $(1, +\infty)$. This half-line $\Delta$ starts from $(T_1(\sigma), D_1(\sigma))$ when $\varepsilon_\gamma$ tends to 1 and has a slope $S(\sigma)$ equal to:

<footnote n="10">¹⁰When one eigenvalue crosses the value 1, a saddle, a transcritical or a pitchfork bifurcation generically occurs. However, since there exists at least one steady state (see Proposition 1), it excludes saddle bifurcations.</footnote>

<footnote n="11">¹¹For further information about local bifurcation theory, see for example Grandmont (1988) and Wiggins (1990).</footnote>

<footnote n="12">¹²For more details, see Guesnerie and Woodford (1992) and Grandmont, Pintus, and de Vilder (1998).</footnote>
the more usual case where externalities affect the total productivity of factors.<fnref n="18" /> They show that, when capital and labor are not too weak substitutes, indeterminacy and endogenous cycles require a contribution of labor to externalities sufficiently large with respect to the contribution of capital to these externalities. Even if these two results seem to be connected, they have different economic meanings. Indeed, in Cazzavillan, Lloyd-Braga and Pintus (1998), an increase of average labor (capital) increases production through the total productivity, whereas in our paper, this modifies labor (capital) efficiency and therefore capital-labor ratio.

Another difference between the two papers concerns the range of elasticities of capital-labor substitution such that indeterminacy emerges. In contrast to Cazzavillan, Lloyd-Braga and Pintus (1998), local indeterminacy requires a capital-labor substitution less than an upper bound ($\sigma_T$). Hence, in our paper, a high enough capital-labor substitution ($\sigma > \sigma_T$) does not promote indeterminacy, but is rather a source of saddle-path stability. Since the upper bound $\sigma_T$ decreases with $\varepsilon_{C,K}$ and $\varepsilon_{D,L}$ (see equation (34)), that is, with the degree of increasing returns, determinacy emerges more easily for larger degree of increasing returns, if capital and labor are sufficiently high substitutes. This result is also new since most of the existing papers rather show that higher increasing returns promote fluctuations due to self-fulfilling expectations.

It is also interesting to connect our analysis to Benhabib and Farmer (1994). As it is well known, these authors have shown, considering an infinitely lived agent model, that local indeterminacy requires wrong slopes on the labor market, i.e. a slope of labor demand greater than the slope of labor supply. Since this seminal contribution, several authors have enriched the model to show that fluctuations can occur without the wrong slopes.<fnref n="19" /> In our model, the slope of labor supply $\varepsilon_\gamma - 1$ is equal to 0 (Assumption 5), whereas the slope of labor demand is given by $\varepsilon_{\Omega,L} = \varepsilon_{D,L} - s(1 + \varepsilon_{D,L})/\sigma$ (see (28)). Therefore, we get the wrong slopes if:

$$ \sigma > \frac{s(1 + \varepsilon_{D,L})}{\varepsilon_{D,L}} \equiv \tilde{\sigma} \qquad (38) $$

where $\sigma_H < \tilde{\sigma} < \sigma_T$. Since $\varepsilon_{\Omega,L}$ is an increasing function of $\sigma$, when $\sigma_H < \sigma < \tilde{\sigma}$, indeterminacy occurs with a negatively slopped labor demand, i.e. without the wrong slopes.<fnref n="20" /> On the contrary, when $\tilde{\sigma} < \sigma < \sigma_T$, there

<footnote n="18">¹⁸In their model, the production is given by $Y_t = A(\bar{K}_{t-1}, \bar{L}_t)F(K_{t-1}, L_t)$, where the total productivity of factors $A(\bar{K}_{t-1}, \bar{L}_t)$ is increasing in the average capital $\bar{K}_{t-1}$ and labor $\bar{L}_t$.</footnote>

<footnote n="19">¹⁹See Benhabib and Farmer (1999) for a survey.</footnote>

<footnote n="20">²⁰See Barinci and Chéron (2001) for a related result under a Cobb-Douglas technology.</footnote>
$$S(\sigma) = 1 + \theta\varepsilon_{C,K} - \theta(1-s)\frac{1 + \varepsilon_{C,K} - \varepsilon_{D,K}}{\sigma} \quad (31)$$

Consequently, studying local indeterminacy and the occurrence of endogenous cycles only requires to analyze $(T_1(\sigma), D_1(\sigma))$, the slope $S(\sigma)$ and how they change when the elasticity of capital-labor substitution $\sigma$ varies.

Under Assumptions 3-4, we first notice that the slope of the half-line $\Delta$ increases with respect to $\sigma$, from $-\infty$ when $\sigma$ tends to 0, to $1 + \theta\varepsilon_{C,K} \ge 1$ when $\sigma$ tends to $+\infty$. In particular, $S(\sigma_{F_1}) = -1$ with $\sigma_{F_1} \equiv \theta(1-s)(1 + \varepsilon_{C,K} - \varepsilon_{D,K})/(2 + \theta\varepsilon_{C,K})$, $S(\sigma_0) = 0$ with $\sigma_0 \equiv \theta(1-s)(1 + \varepsilon_{C,K} - \varepsilon_{D,K})/(1 + \theta\varepsilon_{C,K})$, and $S(\sigma_T) = 1$ with $\sigma_T \equiv (1-s)(1 + \varepsilon_{C,K} - \varepsilon_{D,K})/\varepsilon_{C,K}$.<fnref n="13" />

Secondly, when $\sigma$ varies, $(T_1(\sigma), D_1(\sigma))$ describes a line $\Delta_1$ whose the slope $S_1$ is defined by:

$$S_1 = \frac{D'_1(\sigma)}{T'_1(\sigma)} = \frac{N}{M}, \quad \text{with} \\ N \equiv \theta(1 + \varepsilon_{D,L})(1 - s)(1 + \varepsilon_{C,K} - \varepsilon_{D,K}) - s(1 + \varepsilon_{D,L} - \varepsilon_{C,L})(1 + \theta\varepsilon_{C,K}) \\ M \equiv \theta(1 + \varepsilon_{D,L})[\varepsilon_{C,K} + s\varepsilon_{C,L} + (1 - s)(1 + \varepsilon_{D,L})] + \theta(\varepsilon_{C,K}\varepsilon_{D,L} \\ - \varepsilon_{D,K}\varepsilon_{C,L})[(1 - s)(1 + \varepsilon_{D,L}) + s\varepsilon_{C,L}] - s(1 + \varepsilon_{D,L} - \varepsilon_{C,L})(1 + \theta\varepsilon_{C,K}) \quad (32)$$

Using Assumptions 3-4, we deduce that $D_1(\sigma)$ and $T_1(\sigma)$ are both decreasing, and $S_1 \ge 1$.<fnref n="14" /> On $\Delta_1$, the two extreme points are $(T_1(0), D_1(0))$ and $(T_1(+\infty), D_1(+\infty))$. The first one is determined by:

$$T_1(0) = 1 + \theta \frac{\varepsilon_{C,K} + s\varepsilon_{C,L} + (1-s)(1+\varepsilon_{D,L}) + \varepsilon_{D,L}\varepsilon_{C,K} - \varepsilon_{D,K}\varepsilon_{C,L}}{s(1+\varepsilon_{D,L}-\varepsilon_{C,L})}, \\ D_1(0) = \frac{\theta(1-s)(1+\varepsilon_{C,K}-\varepsilon_{D,K})}{s(1+\varepsilon_{D,L}-\varepsilon_{C,L})} > 0, \text{ and } 1 - T_1(0) + D_1(0) = \\ - \theta \frac{s(\varepsilon_{C,K} + \varepsilon_{C,L}) + \varepsilon_{D,K}(1-s-\varepsilon_{C,L}) + \varepsilon_{D,L}(1-s+\varepsilon_{C,K})}{s(1+\varepsilon_{D,L}-\varepsilon_{C,L})} \le 0 \quad (33)$$

Note that $(T_1(0), D_1(0))$ is on the right side of $(AC)$, above the horizontal axis. $(T_1(+\infty), D_1(+\infty))$ is defined as follows:

<footnote n="13">¹³Evidently, the existence of $\sigma_T > 0$ requires $\varepsilon_{C,K} > 0$.</footnote>

<footnote n="14">¹⁴Indeed, Assumption 3 ensures that $M < 0$. Then, one can prove that $S_1 \ge 1$ is equivalent to $(1 - s)(1 + \varepsilon_{D,L})[\varepsilon_{D,L}(1 + \varepsilon_{C,K}) + \varepsilon_{D,K}(1 - \varepsilon_{C,L})] + (1 + \varepsilon_{D,L})s\varepsilon_{C,K} + s\varepsilon_{C,L}[1 + \varepsilon_{D,L}(1 + \varepsilon_{C,K}) - \varepsilon_{D,K}\varepsilon_{C,L}] \ge 0$, which is always satisfied.</footnote>
is indeterminacy under the wrong slopes condition. More interestingly, the steady state is a saddle for all $\sigma > \sigma_T$, i.e. when the condition on the wrong slopes is ensured. Hence, the wrong slopes condition is neither a necessary nor a sufficient condition for indeterminacy in our framework.

To give now an economic interpretation of our results, we recall that the dynamics are defined by:

$$C_{t+1}^w / B = \gamma(L_t) \qquad (39)$$

$$K_t = \beta(1 - \delta + \varrho_t)K_{t-1} \qquad (40)$$

where $C_{t+1}^w = (P_t/P_{t+1})\Omega_t L_t = \Omega_{t+1} L_{t+1}$. To provide an intuition on the occurrence of fluctuations due to self-fulfilling expectations, assume that workers expect a decrease of the inflation factor $(P_{t+1}/P_t)$, i.e. an increase of future consumption. Since consumption and leisure are gross substitutes, they increase their labor supply $L_t$ (see (39)), which has a positive effect on the real interest rate $\varrho_t$. Therefore, capital accumulation $K_t$ increases (see (40)), which implies a larger real wage $\Omega_{t+1}$.

To summarize, following a decrease of inflation, future consumption will effectively increase if two effects are large enough:

(i) the positive impact of labor on the real interest rate;

(ii) the positive impact of capital on the real wage.

Under perfect competition (no externality), these two effects require a sufficiently weak capital-labor substitution (see (28)).<fnref n="21" /> When production benefits from externalities through the total productivity of factors (Cazzavillan, Lloyd-Braga and Pintus (1998)), expectations can be self-fulfilling under a high degree of capital-labor substitution, because a larger level of labor (capital) increases the productivity, i.e. the aggregate production, and therefore the real wage and the real interest rate.

In our framework, the mechanism is quite different because now externalities affect the capital-labor ratio $a = [C(K)K]/[D(L)L]$ measured in efficient units. Hence, a higher level of labor implies a sufficiently large increase of the real interest rate $\varrho = C(K)A\rho(a)$, because labor efficiency $D(L)$, increasing in labor, reinforces the decrease of the capital-labor ratio $a$. In a quite similar way, a higher level of capital has a sufficiently positive impact on the real wage $\Omega = D(L)A\omega(a)$, because it raises capital efficiency $C(K)$, which reinforces the increase of the capital-labor ratio $a$. However, in contrast to Cazzavillan, Lloyd-Braga and Pintus (1998), indeterminacy does no more occur if the elasticity of capital-labor substitution is sufficiently large because

<footnote n="21">²¹See Grandmont, Pintus and de Vilder (1998) and Woodford (1986) for more details.</footnote>
$$ T_1(+\infty) = 1 + \frac{1 + \theta(\varepsilon_{C,K}(1 + \varepsilon_{D,L}) - \varepsilon_{D,K}\varepsilon_{C,L})}{1 + \varepsilon_{D,L}}, \quad D_1(+\infty) = \frac{1 + \theta\varepsilon_{C,K}}{1 + \varepsilon_{D,L}} > 0 \\ \text{and } 1 - T_1(+\infty) + D_1(+\infty) = \theta \frac{\varepsilon_{D,K}\varepsilon_{C,L} - \varepsilon_{C,K}\varepsilon_{D,L}}{1 + \varepsilon_{D,L}} \quad (34) $$

$(T_1(\sigma), D_1(\sigma))$ decreases from $(T_1(0), D_1(0))$ to $(-\infty, -\infty)$ when $\sigma = \sigma_\infty \equiv s(1+\varepsilon_{D,L}-\varepsilon_{C,L})/(1+\varepsilon_{D,L})$, and from $(+\infty, +\infty)$ to $(T_1(+\infty), D_1(+\infty))$ when $\sigma$ is greater than $\sigma_\infty$ and increases to $+\infty$. Therefore, $D_1(+\infty) < 1$ is a necessary condition to have indeterminacy when capital and labor are not too weak substitutes. In the rest of the paper, we restrict our attention to this more interesting case, assuming:

**Assumption 5** $\varepsilon_{D,L} > \theta\varepsilon_{C,K}$.

Furthermore, we can notice that the position of $(T_1(+\infty), D_1(+\infty))$ with respect to the line (AC) depends on the value of $\varepsilon_{D,K}\varepsilon_{C,L} - \varepsilon_{C,K}\varepsilon_{D,L}$. We have two configurations:

(i) if $\varepsilon_{D,K}\varepsilon_{C,L} \ge \varepsilon_{C,K}\varepsilon_{D,L}$, $(T_1(+\infty), D_1(+\infty))$ is above or on (AC) (see Figure 1);

(ii) if $\varepsilon_{D,K}\varepsilon_{C,L} < \varepsilon_{C,K}\varepsilon_{D,L}$, $(T_1(+\infty), D_1(+\infty))$ is below (AC) (see Figure 2).

Under Assumption 5, we define $\sigma_{H_2}$ such that $D_1(\sigma_{H_2}) = 1$. We obtain:

$$ \sigma_{H_2} \equiv \frac{s(1 + \varepsilon_{D,L} - \varepsilon_{C,L}) - \theta(1 - s)(1 + \varepsilon_{C,K} - \varepsilon_{D,K})}{\varepsilon_{D,L} - \theta\varepsilon_{C,K}} \quad (35) $$

If $\varepsilon_{D,K}\varepsilon_{C,L} \ge \varepsilon_{C,K}\varepsilon_{D,L}$, $(T_1(\sigma), D_1(\sigma))$ crosses the segment $[BC]$ for $\sigma = \sigma_{H_2}$ and is inside (ABC) for all $\sigma \ge \sigma_{H_2}$. If $\varepsilon_{D,K}\varepsilon_{C,L} < \varepsilon_{C,K}\varepsilon_{D,L}$, $(T_1(\sigma), D_1(\sigma))$ can cross the segment $[BC]$ for $\sigma = \sigma_{H_2}$, can be inside (ABC) for $\sigma \in [\sigma_{H_2}, \sigma_{I_2}]$ and is below the (AC) line for all $\sigma > \sigma_{I_2}$,<fnref n="15" /> where $\sigma_{I_2}$ is defined by $1 - T_1(\sigma_{I_2}) + D_1(\sigma_{I_2}) = 0$, i.e.

$$ \sigma_{I_2} \equiv 1 + \frac{s(\varepsilon_{C,K} + \varepsilon_{C,L}) + (1 - s)(\varepsilon_{D,K} + \varepsilon_{D,L})}{\varepsilon_{C,K}\varepsilon_{D,L} - \varepsilon_{D,K}\varepsilon_{C,L}} \quad (36) $$

<footnote n="15">¹⁵We will see later that, in the case studied in this paper, $\Delta_1$ is on the left of point C and goes inside (ABC).</footnote>
the two effects just described are dampened. Indeed, when $\sigma$ is high enough, a decrease (increase) of the capital-labor ratio $a$ measured in efficient units only induces a small variation of the real interest rate (wage) (see (28)).

# 5 Implications for Fiscal Policy

Since the seminal contributions of Guo and Lansing (1998) and Schmitt-Grohé and Uribe (1997), several papers have analyzed the (de-)stabilizing role of fiscal policy rules under a balanced budget. In this section, we contribute to this debate by introducing variable tax rates on labor and capital incomes in the previous model. While existing results show that, in one-sector models, tax rates decreasing with their tax base always have a destabilizing effect,<fnref n="22" /> we will show that less clear-cut conclusions are obtained in our framework.

To be as short as possible we do not present the model with taxation in details but only focus on the main differences regarding the model developed in Section 2. We assume that there exists now a government that levies taxes on labor income $\Omega_t L_t$ and capital income $\varrho_t K_{t-1}$, at the rate $\tau_t^L$ and $\tau_t^K$ respectively. These two tax rates that depend on the tax base are defined by the government as follows:<fnref n="23" />

$$ \tau_t^L = 1 - z_L \left( \frac{\Omega(K_{t-1}, L_t)L_t}{\Omega(1,1)} \right)^{\xi_L} \equiv \tau^L(K_{t-1}, L_t) \quad (41) $$

$$ \tau_t^K = 1 - z_K \left( \frac{\varrho(K_{t-1}, L_t)K_{t-1}}{\varrho(1,1)} \right)^{\xi_K} \equiv \tau^K(K_{t-1}, L_t) \quad (42) $$

with $z_i \in (0,1)$ for $i = K, L$. Note that the term into the brackets corresponds to the tax base (labor and capital income respectively), divided by its value at the steady state. Therefore, $1 - z_i \in (0,1)$ determines the level of the tax rate at the steady state and when the tax rate does not depend on the tax base ($\xi_i = 0$). Finally, the tax rate $\tau_t^i$ is decreasing (increasing) in the tax base if $\xi_i > 0$ ($\xi_i < 0$).

The government determines the level of public spending $G_t$ according to the balanced-budget rule:<fnref n="24" />

$$ G_t = \tau_t^L \Omega_t L_t + \tau_t^K \varrho_t K_{t-1} \quad (43) $$

<footnote n="22">²²See among others Guo and Lansing (1998), Gokan (2006), Lloyd-Braga, Modesto and Seegmuller (2006), Pintus (2003) and Schmitt-Grohé and Uribe (1997).</footnote>

<footnote n="23">²³In their paper, Guo and Lansing (1998) use quite similar specifications of the tax rates.</footnote>

<footnote n="24">²⁴Note that $G_t$ neither enters the production function nor the households preferences.</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 1: Representation of $\Delta_1$ when $\varepsilon_{D,K}\varepsilon_{C,L} \ge \varepsilon_{C,K}\varepsilon_{D,L}$</figcaption></figure>

Using these notations and geometrical results, we first study the local stability of the steady state and the occurrence of bifurcations for a weak substitution between capital and labor ($\sigma < \sigma_\infty$).

Since the locus $(T_1(\sigma), D_1(\sigma))$ decreases along the line $\Delta_1$ and $S(\sigma)$ increases with respect to $\sigma$, $\Delta$ makes a counterclockwise rotation around $\Delta_1$. In particular when $\sigma = 0$, the half-line $\Delta$ is vertical, crossing (AC) ($\varepsilon_\gamma = \varepsilon_{\gamma_T}$) between the horizontal axis and the point C, and the segment $[BC]$ ($\varepsilon_\gamma = \varepsilon_{\gamma_H}$).<sup>16</sup> This first case applies for all $\sigma \le \sigma_{F_1}$. In what follows, note that $\sigma_{H_1}$ is such that the half-line $\Delta$ goes through the point B, $\sigma_{I_1}$ is such that $\Delta$ goes through A and $\sigma_{F_2}$ is such that the line $\Delta_1$ crosses (AB).<sup>17</sup> When $\sigma$ belongs to ($\sigma_{F_1}, \sigma_{H_1}$), $\Delta$ does not only cross (AC) and $[BC]$, but also the line (AB) ($\varepsilon_\gamma = \varepsilon_{\gamma_F}$) above B. When $\sigma$ becomes greater than $\sigma_{H_1}$ and is smaller than $\sigma_{I_1}$, the half-line $\Delta$ only crosses (AC), and (AB) between

<footnote n="16"><sup>16</sup>Note that under Assumption 4, $T_1(0)$ is strictly smaller than 2.</footnote>

<footnote n="17"><sup>17</sup>Unfortunately, we cannot explicitly determine $\sigma_{H_1}$ and $\sigma_{I_1}$, while $\sigma_{F_2}$ is given in the Appendix.</footnote>
Following Gokan (2006), Lloyd-Braga, Modesto and Seegmuller (2006), Pintus (2003) and Schmitt-Grohé and Uribe (1997), we assume that households know the tax rates but take them as given when they determine their optimal behaviors. We deduce that all the analysis of Section 2 applies, except that the real wage $\Omega_t = A\Omega(K_{t-1}, L_t)$ and the real interest rate $\varrho_t = A\varrho(K_{t-1}, L_t)$ have to be replaced by the after tax real wage $\bar{\Omega}_t$ and real interest rate $\bar{\varrho}_t$. Using equations (41) and (42), we get:

$$ \begin{aligned} \bar{\Omega}_t &= (1 - \tau_t^L)\Omega_t = z_L \left( \frac{\Omega(K_{t-1}, L_t)L_t}{\Omega(1, 1)} \right)^{\xi_L} A\Omega(K_{t-1}, L_t) \quad (44) \\ &\equiv A\bar{\Omega}(K_{t-1}, L_t) \end{aligned} $$

$$ \begin{aligned} \bar{\varrho}_t &= (1 - \tau_t^K)\varrho_t = z_K \left( \frac{\varrho(K_{t-1}, L_t)K_{t-1}}{\varrho(1, 1)} \right)^{\xi_K} A\varrho(K_{t-1}, L_t) \quad (45) \\ &\equiv A\bar{\varrho}(K_{t-1}, L_t) \end{aligned} $$

Substituting these two expressions into (20) and (21), the definition of the intertemporal equilibrium and the existence of a steady state can be conducted in a similar way. Focusing now on the dynamic implications of the fiscal policy, we observe that the introduction of the two variable tax rates only modify the elasticities of the (after tax) real wage and real interest rate. Defining $\varepsilon_{\bar{\varrho},K}$, $\varepsilon_{\bar{\varrho},L}$, $\varepsilon_{\bar{\Omega},K}$ and $\varepsilon_{\bar{\Omega},L}$ the elasticities of $\bar{\varrho}(K, L)$ and $\bar{\Omega}(K, L)$ with respect to K and L, evaluated at the steady state $(K, L) = (1, 1)$, and using (28), (44) and (45), we get:

$$ \begin{aligned} \varepsilon_{\bar{\varrho},K} &= \bar{\varepsilon}_{C,K} - \frac{(1-s)(1+\bar{\varepsilon}_{C,K})}{\sigma} \\ \varepsilon_{\bar{\varrho},L} &= \frac{(1-s)(1+\bar{\varepsilon}_{D,L})}{\sigma} \\ \varepsilon_{\bar{\Omega},K} &= \frac{s(1+\bar{\varepsilon}_{C,K})}{\sigma} \\ \varepsilon_{\bar{\Omega},L} &= \bar{\varepsilon}_{D,L} - \frac{s(1+\bar{\varepsilon}_{D,L})}{\sigma} \end{aligned} \quad (46) $$

where $\bar{\varepsilon}_{C,K}$ and $\bar{\varepsilon}_{D,L}$ are defined by:

$$ \begin{aligned} \bar{\varepsilon}_{C,K} &= \xi_K(1 + \varepsilon_{C,K}) + \varepsilon_{C,K} \\ \bar{\varepsilon}_{D,L} &= \xi_L(1 + \varepsilon_{D,L}) + \varepsilon_{D,L} \end{aligned} \quad (47) $$

We notice that substituting $\bar{\varepsilon}_{C,K}$ and $\bar{\varepsilon}_{D,L}$ by $\varepsilon_{C,K}$ and $\varepsilon_{D,L}$ respectively, the elasticities $\varepsilon_{\bar{\varrho},K}$, $\varepsilon_{\bar{\varrho},L}$, $\varepsilon_{\bar{\Omega},K}$ and $\varepsilon_{\bar{\Omega},L}$ are identical to $\varepsilon_{\varrho,K}$, $\varepsilon_{\varrho,L}$, $\varepsilon_{\Omega,K}$ and
<figure><img src="image_5.png" /><figcaption>Figure 2: Representation of $\Delta_1$ when $\varepsilon_{D,K}\varepsilon_{C,L} < \varepsilon_{C,K}\varepsilon_{D,L}$</figcaption></figure>

A and B. When $\sigma$ becomes greater than $\sigma_{I_1}$, the half-line $\Delta$ goes below the point A. We obtain two different cases: When $\sigma_{I_1} < \sigma < \sigma_{F_2}$, $\Delta$ crosses (AB) and (AC) and when $\sigma_{F_2} \le \sigma < \sigma_\infty$, $\Delta$ only crosses (AC).

These results which are common to all the configurations can be summarized as follows (see also Figure 3):

**Proposition 2** (Local dynamics for a weak capital-labor substitution) Assuming that there is a steady state (Proposition 1) and that Assumptions 1-4 are satisfied, the following generically holds.<fnref n="18" />

(i) When $0 < \sigma \le \sigma_{F_1}$, the steady state is a saddle for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_T}$, undergoes a transcritical or a pitchfork bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_T}$, is a sink for $\varepsilon_{\gamma_T} < \varepsilon_\gamma < \varepsilon_{\gamma_H}$, undergoes a Hopf bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_H}$, is a source for $\varepsilon_\gamma > \varepsilon_{\gamma_H}$.

<footnote n="18">¹⁸The values $\varepsilon_{\gamma_T}$, $\varepsilon_{\gamma_H}$ and $\varepsilon_F$ are given in the Appendix.</footnote>
$\varepsilon_{\Omega,L}$. Therefore, the analysis of Section 4 applies and the dynamic stability properties of the steady state are still given by Proposition 2, substituting $\varepsilon_{C,K}$ and $\varepsilon_{D,L}$ by $\bar{\varepsilon}_{C,K}$ and $\bar{\varepsilon}_{D,L}$. However, we need to impose $\bar{\varepsilon}_{C,K} \ge 0$ and $\bar{\varepsilon}_{D,L} \ge 0$, i.e.

**Assumption 7** $\xi_K \ge -\varepsilon_{C,K}/(1 + \varepsilon_{C,K})$ and $\xi_L \ge -\varepsilon_{D,L}/(1 + \varepsilon_{D,L})$.

On the one hand, note that when the tax rates are decreasing in their tax base, i.e. $\xi_K > 0$ and $\xi_L > 0$, this assumption is satisfied. In this case, we further note that $\bar{\varepsilon}_{C,K} > \varepsilon_{C,K}$ and $\bar{\varepsilon}_{D,L} > \varepsilon_{D,L}$. On the other hand, when $\xi_K = \xi_L = 0$, $\varepsilon_{\bar{\theta},K}$, $\varepsilon_{\bar{\theta},L}$, $\varepsilon_{\bar{\Omega},K}$ and $\varepsilon_{\bar{\Omega},L}$ are equal to $\varepsilon_{\theta,K}$, $\varepsilon_{\theta,L}$, $\varepsilon_{\Omega,K}$ and $\varepsilon_{\Omega,L}$, respectively. Hence, the fiscal policy has no influence on local dynamics when the tax rates are constant.

In what follows, focusing on tax rates decreasing with their tax base ($\xi_K > 0$ and $\xi_L > 0$), we analyze the implications of fiscal policy on indeterminacy. Applying the results of Proposition 2, we notice that indeterminacy requires a sufficiently large $\bar{\varepsilon}_{D,L}$ regarding the value of $\bar{\varepsilon}_{C,K}$. This is ensured by a not too flat tax rate on labor income and a not too decreasing tax rate on capital income. Otherwise, indeterminacy is ruled out. Considering now that this condition is satisfied, Proposition 2 states that indeterminacy occurs for $\sigma_H < \sigma < \sigma_T$, with:

$$ \sigma_H \equiv \frac{s(1 + \bar{\varepsilon}_{D,L}) - \theta(1 - s)(1 + \bar{\varepsilon}_{C,K})}{\bar{\varepsilon}_{D,L} - \theta\bar{\varepsilon}_{C,K}} \quad (48) $$

$$ \sigma_T \equiv 1 + \frac{1 - s}{\bar{\varepsilon}_{C,K}} + \frac{s}{\bar{\varepsilon}_{D,L}} \quad (49) $$

Obviously, we remark that the higher bound $\sigma_T$ is decreasing in $\bar{\varepsilon}_{C,K}$ and $\bar{\varepsilon}_{D,L}$, i.e. in $\xi_K$ and $\xi_L$. Moreover, we can prove that under not restrictive conditions, $\sigma_H$ is decreasing in $\bar{\varepsilon}_{D,L}$ (or $\xi_L$) and increasing in $\bar{\varepsilon}_{C,K}$ (or $\xi_K$).$^{25}$

We deduce that a more variable (decreasing) tax rate on capital income ($\xi_K$ higher) promotes stability by increasing $\sigma_H$ and reducing $\sigma_T$. A more variable (decreasing) tax rate on labor income ($\xi_L$ higher) implies a decrease of both $\sigma_H$ and $\sigma_T$. Therefore, under more decreasing tax rates, indeterminacy occurs less easily for high elasticities of capital-labor substitution ($\sigma$ close to $\sigma_T$). Tax rates characterized by larger values of $\xi_K$ and $\xi_L$ stabilize the economy by promoting saddle-path stability. As we have already underlined, this result is in contrast with the existing literature introducing distortionary taxes and balanced-budget rules in one-sector models.

<footnote>$^{25}$Using equation (48), we obtain $\partial\sigma_H/\partial\bar{\varepsilon}_{D,L} < 0$ if $\theta(1 - 2s)\bar{\varepsilon}_{C,K} < s - \theta(1 - s)$ and $\partial\sigma_H/\partial\bar{\varepsilon}_{C,K} > 0$ if $\theta(1 - 2s)\bar{\varepsilon}_{D,L} < s - \theta(1 - s)$. These two inequalities are satisfied if $\bar{\varepsilon}_{C,K}$ and $\bar{\varepsilon}_{D,L}$ are not excessively large and $\theta$ is small.</footnote>
(ii) When $\sigma_{F_1} < \sigma < \sigma_{H_1}$, the steady state is a saddle for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_T}$, undergoes a transcritical or a pitchfork bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_T}$, is a sink for $\varepsilon_{\gamma_T} < \varepsilon_\gamma < \varepsilon_{\gamma_H}$, undergoes a Hopf bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_H}$, is a source for $\varepsilon_{\gamma_H} < \varepsilon_\gamma < \varepsilon_{\gamma_F}$, undergoes a flip bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_F}$, is a saddle for $\varepsilon_\gamma > \varepsilon_{\gamma_F}$.

(iii) When $\sigma_{H_1} < \sigma < \sigma_{I_1}$, the steady state is a saddle for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_T}$, undergoes a transcritical or a pitchfork bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_T}$, is a sink for $\varepsilon_{\gamma_T} < \varepsilon_\gamma < \varepsilon_{\gamma_F}$, undergoes a flip bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_F}$, is a saddle for $\varepsilon_\gamma > \varepsilon_{\gamma_F}$.

(iv) When $\sigma_{I_1} < \sigma < \sigma_{F_2}$, the steady state is a saddle for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_F}$, undergoes a flip bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_F}$, is a source for $\varepsilon_{\gamma_F} < \varepsilon_\gamma < \varepsilon_{\gamma_T}$, undergoes a transcritical or a pitchfork bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_T}$, is a saddle for $\varepsilon_\gamma > \varepsilon_{\gamma_T}$.

(v) When $\sigma_{F_2} \le \sigma < \sigma_\infty$, the steady state is a source for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_T}$, undergoes a transcritical or a pitchfork bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_T}$, is a saddle for $\varepsilon_\gamma > \varepsilon_{\gamma_T}$.

This proposition establishes that endogenous fluctuations can occur for sufficiently weak elasticities of capital-labor substitution. Indeed, local indeterminacy can emerge for $0 < \sigma < \sigma_{I_1}$ and deterministic cycles can appear through the occurrence of a Hopf bifurcation for $0 < \sigma < \sigma_{H_1}$ and through the occurrence of a flip bifurcation for $\sigma_{F_1} < \sigma < \sigma_{F_2}$. One can remark that such results are not so different than those obtained under perfect competition and constant returns to scale by Grandmont, Pintus, and de Vilder (1998). The main difference concerns the occurrence of a transcritical or pitchfork bifurcation, which is not possible in their framework.

We now analyze what happens for $\sigma > \sigma_\infty$. As we have already noticed, when $\sigma$ increases from $\sigma_\infty$ to $+\infty$, $D_1(\sigma)$ decreases from $+\infty$ to $D_1(+\infty) \in (0, 1)$ along the line $\Delta_1$. Two different configurations can arise depending on the value of $\varepsilon_{C,K}\varepsilon_{D,L} - \varepsilon_{D,K}\varepsilon_{C,L}$.

To begin, assume that the first configuration applies, i.e. $\varepsilon_{D,K}\varepsilon_{C,L} \ge \varepsilon_{C,K}\varepsilon_{D,L}$, which means that $(T_1(\sigma), D_1(\sigma))$ is inside (ABC) for all $\sigma > \sigma_{H_2}$. In order to simplify the presentation, we restrict our attention to the following case:

$$ \textbf{Assumption 6} \quad \frac{(1-s)\varepsilon_{D,L}}{1+\varepsilon_{D,L}-\varepsilon_{C,L}} > \frac{s\varepsilon_{C,K}}{1+\varepsilon_{C,K}-\varepsilon_{D,K}}. $$

This means that $\sigma_T > \sigma_{H_2}$. We deduce that when $\sigma_\infty < \sigma \le \sigma_{H_2}$, the half-line $\Delta$ only crosses (AC). When $\sigma_{H_2} < \sigma < \sigma_T$, $\Delta$ crosses [BC]
# 6 Conclusion

In this paper, we introduce input-specific externalities in a model with in-
finitely lived agents and a finance constraint as developed in the seminal con-
tribution of Woodford (1986). Instead of assuming, as usually, that average
labor and capital increase the total productivity of factors, we consider that
average labor (capital) raises labor (capital) efficiency. Focusing on not too
weak elasticities of capital-labor substitution, we show that indeterminacy
does not only require a lower bound for the capital-labor substitution, but
also a finite higher bound. As a direct implication of this result, the wrong
slopes condition (slope of labor demand greater than slope of labor supply)
does not ensure an indeterminate steady state. Moreover, larger increas-
ing returns improve the scope of capital-labor substitution for saddle-path
stability. Introducing variable tax rates on capital and labor income in our
framework, we reexamine the role of a balanced-budget fiscal policy rule on
endogenous fluctuations. In contrast to existing results in one-sector models,
tax rates decreasing with their tax base do not always promote instability
coming from the volatility of expectations.

# References

[1] Azariadis, C. (1993): *Intertemporal Macroeconomics*. Cambridge MA and Oxford: Blackwell.

[2] Barinci, J.-P., and A. Chéron (2001): “Sunspot and the Business Cycle in a Finance Constrained Model”, *Journal of Economic Theory*, 97, 30-49.

[3] Benhabib, J., and R. E. A. Farmer (1994): “Indeterminacy and Increasing Returns”, *Journal of Economic Theory*, 63, 19-41.

[4] Benhabib, J., and R. E. A. Farmer (1999): “Indeterminacy and Sunspots in Macroeconomics”, in *Handbook of Macroeconomics*, ed. by J. B. Taylor, and M. Woodford, pp. 387-448, Amsterdam. North-Holland.

[5] Cazzavillan, G., T. Lloyd-Braga, and P. Pintus (1998): “Multiple Steady States and Endogenous Fluctuations with Increasing Returns to Scale in Production”, *Journal of Economic Theory*, 80, 60-107.

[6] Farmer, R. E. A., and J.-T. Guo (1994): “Real Business Cycles and the Animal Spirits Hypothesis”, *Journal of Economic Theory*, 63, 42-72.
<figure><img src="image_6.png" /><figcaption>Figure 3: Representation of Δ when $\epsilon_{D,K}\epsilon_{C,L} \ge \epsilon_{C,K}\epsilon_{D,L}$</figcaption></figure>

and (AC) above C. Finally, when $\sigma \ge \sigma_T$, the half-line Δ only crosses the segment [BC].<fnref n="19" />

Consider now that the second configuration applies, i.e. $\epsilon_{D,K}\epsilon_{C,L} < \epsilon_{C,K}\epsilon_{D,L}$. Recall that now, $(T_1(\sigma), D_1(\sigma))$ crosses the line (AC) for $\sigma = \sigma_{I_2}$ and is below (AC) for $\sigma > \sigma_{I_2}$. We further note that, under Assumption 6, we have $\sigma_{I_2} > \sigma_T > \sigma_{H_2}$. This last inequality implies that the line $\Delta_1$ crosses (ABC) and is on the left side of C. Then, Δ only crosses (AC) for $\sigma_\infty < \sigma \le \sigma_{H_2}$, crosses the segment [BC] and the line (AC) above C for $\sigma_{H_2} < \sigma < \sigma_T$ and only crosses [BC] for $\sigma_T < \sigma \le \sigma_{I_2}$. For greater values of the elasticity of capital-labor substitution ($\sigma > \sigma_{I_2}$), there exist some differences between the two following cases<fnref n="20" />:

(i) if $1 + \epsilon_{D,L} \le (1 + \theta\epsilon_{C,K})\epsilon_{C,L}\epsilon_{D,K}/(\theta\epsilon_{C,K}^2)$, Δ crosses (AC) below C and

<footnote n="19">¹⁹Indeed, as it is shown in the Appendix, in the configuration where $\epsilon_{D,K}\epsilon_{C,L} \ge \epsilon_{C,K}\epsilon_{D,L}$, Δ always crosses [BC] when $\sigma > \sigma_{H_2}$.</footnote>

<footnote n="20">²⁰See the Appendix for more details.</footnote>
[7] Gokan, Y. (2006): "Dynamic Effects of Government Expenditure in a Finance Constrained Economy", *Journal of Economic Theory*, 127, 323-333.

[8] Grandmont, J.-M. (2006): "Negishi-Solow Efficiency Wages, Unemployment Insurance and Dynamic Deterministic Indeterminacy", University Ca' Foscari of Venice, Dept. of Economics Research Paper Series No. 60-06.

[9] Grandmont, J.-M., P. Pintus, and R. de Vilder (1998): "Capital-labour Substitution and Competitive Nonlinear Endogenous Business Cycles", *Journal of Economic Theory*, 80, 14-59.

[10] Guo, J.T. and K. Lansing (1998), "Indeterminacy and Stabilization Policy", *Journal of Economic Theory*, 82, 481-490.

[11] Harrison, S. G., and M. Weder (2002): "Tracing Externalities as Sources of Indeterminacy", *Journal of Economic Dynamics and Control*, 26, 851-867.

[12] Hintermaier, T. (2003): "On the Minimum Degree of Returns to Scale in Sunspot Models of the Business Cycle", *Journal of Economic Theory*, 110, 400-409.

[13] Lloyd-Braga, T., and L. Modesto (2007): "Indeterminacy in a Finance Constrained Unionized Economy", *Journal of Mathematical Economics*, 43, 347-364.

[14] Lloyd-Braga, T., L. Modesto and T. Seegmuller (2006), "Tax Rate Variability and Public Spending as Sources of Indeterminacy", CEPR, DP5796, http://www.cepr.org/pubs/dps/DP5796.asp.

[15] Pintus, P. (2003): "Aggregate Instability in the Fixed-Cost Approach to Public Spending", mimeo, Aix-Marseille.

[16] Pintus, P. (2004): "Expectations-Driven Fluctuations when Factor Utilization Is Variable", *Macroeconomic Dynamics*, 8, 3-26.

[17] Pintus, P. (2006): "Indeterminacy with almost Constant Returns to Scale: Capital-labor Substitution Matters", *Economic Theory*, 28, 633-649.

[18] Schmitt-Grohé, S., and M. Uribe (1997): "Balanced-Budget Rules, Distortionary Taxes, and Aggregate Instability", *Journal of Political Economy*, 105, 976-1000.
<figure><img src="image_7.png" /><figcaption>Figure 4: Representation of Δ when $\varepsilon_{D,K}\varepsilon_{C,L} < \varepsilon_{C,K}\varepsilon_{D,L}$</figcaption></figure>

[BC] for all $\sigma > \sigma_{I_2}$;

(ii) if $1 + \varepsilon_{D,L} > (1 + \theta\varepsilon_{C,K})\varepsilon_{C,L}\varepsilon_{D,K}/(\theta\varepsilon_{C,K}^2)$, $\Delta$ crosses (AC) below C and [BC] for $\sigma_{I_2} < \sigma < \sigma_{I_3}$ and only crosses (AC) above C for $\sigma > \sigma_{I_3}$.

All these results can be summarized as follows (see also Figures 3 and 4):

**Proposition 3** (Local dynamics when capital-labor substitution is not too weak) Assuming that there is a steady state (Proposition 1) and that Assumptions 1-6 are satisfied, the following generically holds.<fnref n="21" />

1. $\varepsilon_{D,K}\varepsilon_{C,L} \ge \varepsilon_{C,K}\varepsilon_{D,L}$:

(i) When $\sigma_\infty < \sigma \le \sigma_{H_2}$, the steady state is a source for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_T}$, undergoes a transcritical or a pitchfork bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_T}$, is a saddle for $\varepsilon_\gamma > \varepsilon_{\gamma_T}$.

<footnote n="21">²¹The values $\varepsilon_{\gamma_T}$, $\varepsilon_{\gamma_H}$ and $\varepsilon_{\gamma_F}$ and some technical details are given in the Appendix.</footnote>
[19] Seegmuller, T. (2006), "Indeterminacy and Endogenous Fluctuations under Input-specific Externalities", Cahiers de la MSE 2006.79, Université Paris 1.

[20] Woodford, M. (1986): "Stationary Sunspot Equilibria in a Finance Constrained Economy", *Journal of Economic Theory*, 40, 128-137.
(ii) When $\sigma_{H_2} < \sigma < \sigma_T$, the steady state is a sink for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_H}$, undergoes a Hopf bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_H}$, is a source for $\varepsilon_{\gamma_H} < \varepsilon_\gamma < \varepsilon_{\gamma_T}$, undergoes a transcritical or a pitchfork bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_T}$, is a saddle for $\varepsilon_\gamma > \varepsilon_{\gamma_T}$.

(iii) When $\sigma \ge \sigma_T$, the steady state is a sink for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_H}$, undergoes a Hopf bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_H}$, is a source for $\varepsilon_\gamma > \varepsilon_{\gamma_H}$.

2. $\varepsilon_{D,K}\varepsilon_{C,L} < \varepsilon_{C,K}\varepsilon_{D,L}$:

(a) $1 + \varepsilon_{D,L} \le (1 + \theta\varepsilon_{C,K})\varepsilon_{C,L}\varepsilon_{D,K}/(\theta\varepsilon_{C,K}^2)$:

(i) When $\sigma_\infty < \sigma \le \sigma_{H_2}$, the steady state is a source for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_T}$, undergoes a transcritical or a pitchfork bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_T}$, is a saddle for $\varepsilon_\gamma > \varepsilon_{\gamma_T}$.

(ii) When $\sigma_{H_2} < \sigma < \sigma_T$, the steady state is a sink for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_H}$, undergoes a Hopf bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_H}$, is a source for $\varepsilon_{\gamma_H} < \varepsilon_\gamma < \varepsilon_{\gamma_T}$, undergoes a transcritical or a pitchfork bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_T}$, is a saddle for $\varepsilon_\gamma > \varepsilon_{\gamma_T}$.

(iii) When $\sigma_T \le \sigma \le \sigma_{I_2}$, the steady state is a sink for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_H}$, undergoes a Hopf bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_H}$, is a source for $\varepsilon_\gamma > \varepsilon_{\gamma_H}$.

(iv) When $\sigma > \sigma_{I_2}$, the steady state is a saddle for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_T}$, undergoes a transcritical or a pitchfork bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_T}$, is a sink for $\varepsilon_{\gamma_T} < \varepsilon_\gamma < \varepsilon_{\gamma_H}$, undergoes a Hopf bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_H}$, is a source for $\varepsilon_\gamma > \varepsilon_{\gamma_H}$.

(b) $1 + \varepsilon_{D,L} > (1 + \theta\varepsilon_{C,K})\varepsilon_{C,L}\varepsilon_{D,K}/(\theta\varepsilon_{C,K}^2)$:

(i) When $\sigma_\infty < \sigma \le \sigma_{H_2}$, the steady state is a source for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_T}$, undergoes a transcritical or a pitchfork bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_T}$, is a saddle for $\varepsilon_\gamma > \varepsilon_{\gamma_T}$.

(ii) When $\sigma_{H_2} < \sigma < \sigma_T$, the steady state is a sink for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_H}$, undergoes a Hopf bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_H}$, is a source for $\varepsilon_{\gamma_H} < \varepsilon_\gamma < \varepsilon_{\gamma_T}$, undergoes a transcritical or a pitchfork bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_T}$, is a saddle for $\varepsilon_\gamma > \varepsilon_{\gamma_T}$.

(iii) When $\sigma_T \le \sigma \le \sigma_{I_2}$, the steady state is a sink for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_H}$, undergoes a Hopf bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_H}$, is a source for $\varepsilon_\gamma > \varepsilon_{\gamma_H}$.

(iv) When $\sigma_{I_2} < \sigma < \sigma_{I_3}$, the steady state is a saddle for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_T}$, undergoes a transcritical or a pitchfork bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_T}$, is a sink for $\varepsilon_{\gamma_T} < \varepsilon_\gamma < \varepsilon_{\gamma_H}$, undergoes a Hopf bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_H}$, is a source for $\varepsilon_\gamma > \varepsilon_{\gamma_H}$.
(v) When $\sigma > \sigma_{I_3}$, the steady state is a saddle for $1 < \varepsilon_\gamma < \varepsilon_{\gamma_T}$, undergoes a transcritical or a pitchfork bifurcation for $\varepsilon_\gamma = \varepsilon_{\gamma_T}$, is a source for $\varepsilon_\gamma > \varepsilon_{\gamma_T}$.

This proposition summarizes the conditions for the emergence of endoge-
nous fluctuations under a not too weak capital-labor substitution ($\sigma > \sigma_\infty$).
In what follows, to ease the discussion, we call $\varepsilon_{C,K}$ and $\varepsilon_{D,L}$ the direct effects
of externalities and $\varepsilon_{D,K}$ and $\varepsilon_{C,L}$ the crossing effects. Indeed, the first ones
correspond to the contribution of capital (labor) on externalities representing
capital (labor) efficiency in the production function. On the contrary, the
last ones correspond to the contribution of capital (labor) on externalities
representing labor (capital) efficiency.

Recall first that endogenous fluctuations can occur only if Assumption 5 is
satisfied. This means that the contribution of labor to externalities $D(K, L)$
has not to be too weak with respect to the contribution of capital to external-
ities $C(K, L)$. In other words, only direct effects have a role on this necessary
condition for indeterminacy. This first comment can be related to the paper
by Cazzavillan, Lloyd-Braga, and Pintus (1998) who analyze, in a Woodford
(1986) model, the usual case where externalities are represented by the total
productivity of factors, i.e. $C(K, L) = D(K, L)$. They establish that the
emergence of indeterminacy and cycles for a high elasticity of capital-labor
substitution requires that the contribution of labor to externalities has to
be sufficiently high with respect to the contribution of capital. The more
general formulation of externalities used in this paper allows to clarify that
the necessary condition for indeterminacy only requires a not too weak direct
effect of labor on externalities that affect labor efficiency. Moreover, Proposi-
tion 3 establishes that, due to the more general formulation of externalitites,
two configurations emerge depending on the level of the direct effects with
respect to the crossing effects on externalities.

If the product of crossing effects are greater than the product of direct
effects ($\varepsilon_{D,K}\varepsilon_{C,L} \ge \varepsilon_{C,K}\varepsilon_{D,L}$), indeterminacy and endogenous cycles can ap-
pear for all elasticities of capital-labor substitution greater than $\sigma_{H_2}$. This
result is quite similar to the one obtained by Cazzavillan, Lloyd-Braga, and
Pintus (1998) where the product of crossing effects are, by assumption, equal
to the product of direct effects ($\varepsilon_{D,K}\varepsilon_{C,L} = \varepsilon_{C,K}\varepsilon_{D,L}$). In other words, we
provide here a generalization of their result to the case where the product of
crossing effects are not only equal but also greater than the product of direct
effects of externalities.

In this case, it is also interesting to relate our analysis to Benhabib and
Farmer (1994). As it is well-known, these authors have shown, considering an
infinitely lived agent model, that local indeterminacy requires wrong slopes
on the labor market, i.e. the slope of labor demand must be greater than the slope of labor supply.<fnref n="22" /> In our framework, when $ε_{D,K}ε_{C,L} ≥ ε_{C,K}ε_{D,L}$ and capital-labor is not weak, the occurrence of indeterminacy requires that:

$$ε_{γ} < ε_{γH} = \frac{1 + ε_{D,L} - s(1 + ε_{D,L} - ε_{C,L})/σ}{1 + θε_{C,K}(1 - σ_{T}/σ)} \quad (37)$$

Using equation (28), we remark that $ε_{D,L} - s(1 + ε_{D,L} - ε_{C,L})/σ$ represents the slope of labor demand, whereas $ε_{γ} - 1$ is the slope of labor supply. Therefore, when $σ > σ_{T}$, local indeterminacy requires the wrong slopes, whereas when $σ ∈ (σ_{H₂}, σ_{T})$, indeterminacy can occur when the slope of labor demand is smaller than the slope of labor supply. This result already obtained by Barinci and Chéron (2001) is extended here to the case of a non unitary elasticity of capital-labor substitution and a more general formulation of externalities.

Consider now that the product of crossing effects are weaker than the product of direct effects ($ε_{D,K}ε_{C,L} < ε_{C,K}ε_{D,L}$). As in the previous case, endogenous fluctuations can occur for a sufficiently elastic labor supply as soon as $σ$ becomes greater than $σ_{H₂}$. However, for high enough elasticities of capital-labor substitution ($σ > σ_{I₂}$), indeterminacy and endogenous cycles can no more occur, particularly under a highly elastic labor supply is. This means that, in this configuration and in contrast to a lot of existing results,<fnref n="23" /> endogenous fluctuations do no more emerge when the real wage is sufficiently increasing in labor, production factors are substitutes and the elasticity of labor supply is high enough. Furthermore, consider the case where the crossing effects are equal to zero, i.e. $ε_{D,K} = ε_{C,L} = 0$. Therefore, $σ_{I₂} = 1 + s/ε_{D,L} + (1 - s)/ε_{C,K}$ decreases with the level of externalities. This means that the greater the increasing returns are, the more relevant our last conclusion is. Stronger increasing returns do not always promote local indeterminacy.

Note also that the steady state can be locally determinate under the wrong slopes. To illustrate this last remark, consider that $ε_{D,L} > θε_{C,K} > 0$ and $ε_{D,K} = ε_{C,L} = 0$. In the limit case where $ε_{γ}$ tends to 1 and $σ$ to $+∞$, the labor supply has a slope equal to 0 and the labor demand to $ε_{D,L}$. In this case, the steady state is a saddle even if the positive slope of labor demand is higher than the slope of labor supply.

We will now exhibit some simple configurations where, in comparison with the usual type of externalities (see Cazzavillan, Lloyd-Braga, and Pin-

<footnote n="22">²²Note that this result can be relaxed when one does not restrict the analysis to an unit elasticity of factor substitution. See Pintus (2006).</footnote>

<footnote n="23">²³See Benhabib and Farmer (1999) for a survey.</footnote>
tus (1998)), our formulation of externalities can provide results promoting indeterminacy and endogenous cycles. In what follows, consider the case where $\varepsilon_{D,K} = \varepsilon_{C,K} = 0$. Indeterminacy requires $\sigma > \sigma_{H_2}$ and $\varepsilon_{\gamma} < \varepsilon_{\gamma_H}$. In the case where externalities enter the total productivity of factors, we have $\varepsilon_{C,L} = \varepsilon_{D,L} \equiv \varepsilon_{E,L}$, where $\varepsilon_{E,L}$ represents the level of increasing returns. Using (35) and (42), we obtain $\sigma_{H_2} = \frac{s-\theta(1-s)}{\varepsilon_{E,L}}$ and $\varepsilon_{\gamma_H} = \frac{\sigma(1+\varepsilon_{E,L})-s}{\sigma-\theta(1-s)}$. Consider now the model developed in this paper assuming further $\varepsilon_{C,L} = 0$. Then, $(1-s)\varepsilon_{D,L}$ represents the level of increasing returns and we have $\sigma_{H_2} = \frac{s(1+\varepsilon_{D,L})-\theta(1-s)}{\varepsilon_{D,L}}$ and $\varepsilon_{\gamma_H} = \frac{(\sigma-s)(1+\varepsilon_{D,L})}{\sigma-\theta(1-s)}$.

Equalizing first the two expressions of $\sigma_{H_2}$, we get $\varepsilon_{E,L} = \frac{(s-\theta(1-s))\varepsilon_{D,L}}{s(1+\varepsilon_{D,L})-\theta(1-s)}$ which is greater than $(1-s)\varepsilon_{D,L}$ for all $\varepsilon_{D,L} < \frac{s-\theta(1-s)}{1-s}$. This means that, under a not too strong degree of externalities, one needs a smaller level of increasing returns to obtain the same range of elasticities of capital-labor substitution compatible with indeterminacy. In other words, for the same level of increasing returns, the range of $\sigma$ compatible with indeterminacy is greater in the model developed in this paper.

Equalizing now the two expressions of $\varepsilon_{\gamma_H}$, we obtain $\varepsilon_{E,L} = \varepsilon_{D,L}(1-s/\sigma)$, which is greater than $(1-s)\varepsilon_{D,L}$ for $\sigma > 1$ and smaller otherwise. Therefore, when $\sigma > 1$, elasticities of labor supply compatible with indeterminacy require a smaller level of increasing returns in the model developed in this paper or, in other terms, for the same level of increasing returns, less elastic labor supply are compatible with indeterminacy. Evidently, we obtain the opposite conclusion for $\sigma < 1$.

To summarize this last discussion, in some configurations, endogenous fluctuations can occur when capital and labor are substitutes under less restrictive conditions in our framework than in the model where externalities appear through the total productivity of factors, because, taken as given $\sigma$ and $\varepsilon_{\gamma}$, indeterminacy requires weaker increasing returns.

Before providing concluding remarks, let us note the following. In this paper, we have shown that, considering a more general formulation of externalities, local indeterminacy and endogenous fluctuations can occur under different conditions than in the more usual case studied by Cazzavillan, Lloyd-Braga, and Pintus (1998) where externalities are represented by the total productivity of factors. However, in the Cobb-Douglas case ($\sigma = 1$), the two formulations of externalities are equivalent. Therefore, we can argue that the Cobb-Douglas technology is very specific. To prove this last remark, note that when $\sigma = 1$, the production is given by:

$$Y = A(C(K, L)K)^s (D(K, L)L)^{1-s} = AC(K, L)^s D(K, L)^{1-s} K^s L^{1-s} \quad (38)$$
Therefore, the level of externalities represented by $C(K, L)^s D(K, L)^{1-s}$ plays exactly the same role than externalities that should appear through the total productivity of factors. Using our notations, the contributions of labor and capital to these total externalities are respectively defined by $s\varepsilon_{C,L} + (1-s)\varepsilon_{D,L}$ and $s\varepsilon_{C,K} + (1-s)\varepsilon_{D,K}$.

# 5 Conclusion

In this paper, we introduce input-specific externalities in a Woodford (1986) model to analyze their role on the stability properties of the steady state. This specification of externalities is a generalization of the usual type of productive externalities characterized by a total productivity of factors that increases with respect to average capital and labor.

In this framework, there exist configurations of parameters such that, in contrast to previous results, endogenous fluctuations cannot occur for a sufficiently high capital-labor substitution, particularly when the labor supply is highly elastic. We also stress that, in some cases, this result is more relevant under a stronger level of increasing returns. Another direct implication of this result is that the steady state can be locally determinate under the wrong slopes on the labor market.

Comparing our conclusions with the case where productive externalities enter the total productivity of factors, we exhibit some configurations where, in our framework, indeterminacy and endogenous cycles can occur under a weaker degree of increasing returns when capital and labor are substitutes.

Finally, we also conclude that the Cobb-Douglas technology appears to be quite specific. Indeed, under a unit elasticity of capital-labor substitution, our form of externalities and the usual one are equivalent, whereas we have seen that, when we consider a more general production function, dynamic properties can be quite different.

# Appendix

## Hopf bifurcation for $\sigma > \sigma_{H_2}$ and existence of $\sigma_{I_3}$

Assuming $\sigma > \sigma_{H_2}$, we establish the conditions such that $T < 2$ when $D = 1$. $D = 1$ is equivalent to $\varepsilon_\gamma = 1/D_1(\sigma)$. Substituting this expression into $T < 2$, we obtain:

$$a\sigma^2 + b\sigma + c > 0 \qquad (39)$$
with

$$
\begin{align}
a &= \theta\varepsilon_{C,K}(\varepsilon_{D,L} - \theta\varepsilon_{C,K}) + \theta(1 + \theta\varepsilon_{C,K})(\varepsilon_{D,K}\varepsilon_{C,L} - \varepsilon_{D,L}\varepsilon_{C,K}) \nonumber \\
b &= \theta\varepsilon_{D,K}[(1 - s)(1 + \varepsilon_{D,L}) - \varepsilon_{C,L}] + \theta\varepsilon_{C,L}[\theta(1 - s)\varepsilon_{D,K}^2 \nonumber \\
  &\quad + (1 + \varepsilon_{C,K})(s - \theta(1 - s)\varepsilon_{D,K})] + \theta^2\varepsilon_{C,K}[(1 + \varepsilon_{D,L})(1 - s)(1 - \varepsilon_{D,K}) \nonumber \\
  &\quad + s\varepsilon_{C,L} + (1 + \varepsilon_{D,L})(1 - s + (2 - s)\varepsilon_{C,K}) - \varepsilon_{D,K}\varepsilon_{C,L}] \tag{40} \\
c &= \theta(1 - s)(1 + \varepsilon_{C,K} - \varepsilon_{D,K})[s(1 + \varepsilon_{D,L} - \varepsilon_{C,L}) \nonumber \\
  &\quad - \theta((1 + \varepsilon_{D,L})(1 - s + \varepsilon_{C,K}) + \varepsilon_{C,L}(s - \varepsilon_{D,K}))] \nonumber
\end{align}
$$

Under Assumption 4, we deduce that $b > 0$ and $c > 0$. Then, we have
two cases:

(i) if $1 + \varepsilon_{D,L} \le (1 + \theta\varepsilon_{C,K})\varepsilon_{C,L}\varepsilon_{D,K}/(\theta\varepsilon_{C,K}^2)$ (i.e. $a \ge 0$), the inequality (39) is satisfied for all $\sigma > 0$;

(ii) if $1 + \varepsilon_{D,L} < (1 + \theta\varepsilon_{C,K})\varepsilon_{C,L}\varepsilon_{D,K}/(\theta\varepsilon_{C,K}^2)$ (i.e. $a < 0$), there exists a unique $\sigma_{I_3} > 0$ such that the inequality (39) is satisfied for all $\sigma < \sigma_{I_3}$.

This means that in case (i), $\Delta$ always crosses $[BC]$ when $\sigma > \sigma_{H_2}$. In particular, this arises when $\varepsilon_{D,K}\varepsilon_{C,L} \ge \varepsilon_{D,L}\varepsilon_{C,K}$. In case (ii), $\Delta$ crosses $[BC]$ for $\sigma < \sigma_{I_3}$, but does not cross any more $[BC]$ when $\sigma > \sigma_{I_3}$. Note that, by continuity, $\sigma_{I_3}$ is strictly greater than $\sigma_{I_2}$.

The value of $\sigma_{F_2}$

$\sigma_{F_2}$ is defined by $1 + T_1(\sigma) + D_1(\sigma) = 0$. After some computations, we obtain:

$$
\sigma_{F_2} = \frac{2s(1 + \varepsilon_{D,L} - \varepsilon_{C,L}) + \theta[(1 - s)(2 + \varepsilon_{D,L} - \varepsilon_{D,K}) + (2 - s)\varepsilon_{C,K}]}{2(2 + \varepsilon_{D,L}) + \theta[\varepsilon_{C,K}(2 + \varepsilon_{D,L}) - \varepsilon_{D,K}\varepsilon_{C,L}]} + \frac{s\varepsilon_{C,L} + \varepsilon_{D,L}\varepsilon_{C,K} - \varepsilon_{D,K}\varepsilon_{C,L}}{2(2 + \varepsilon_{D,L}) + \theta[\varepsilon_{C,K}(2 + \varepsilon_{D,L}) - \varepsilon_{D,K}\varepsilon_{C,L}]} \quad (41)
$$

The value of $\varepsilon_{\gamma H}$

$\varepsilon_{\gamma H}$ is defined by $D = 1$. We obtain:

$$
\varepsilon_{\gamma H} \equiv \frac{1}{D_1(\sigma)} = \frac{\sigma(1 + \varepsilon_{D,L}) - s(1 + \varepsilon_{D,L} - \varepsilon_{C,L})}{\sigma(1 + \theta\varepsilon_{C,K}) - \theta(1 - s)(1 + \varepsilon_{C,K} - \varepsilon_{D,K})} \quad (42)
$$
The value of $\varepsilon_{\gamma_F}$

$\varepsilon_{\gamma_F}$ is defined by $1 + T + D = 0$. After some computations, we obtain:

$$ \varepsilon_{\gamma_F} = 1 + \frac{2(2 + \varepsilon_{D,L}) + \theta[\varepsilon_{C,K}(2 + \varepsilon_{D,L}) - \varepsilon_{D,K}\varepsilon_{C,L}]}{2 + \theta\varepsilon_{C,K}} \frac{\sigma_{F_2} - \sigma}{\sigma - \sigma_{F_1}} \quad (43) $$

The value of $\varepsilon_{\gamma_T}$

$\varepsilon_{\gamma_T}$ is defined by $1 - T + D = 0$. After some computations, we obtain:

$$ \varepsilon_{\gamma_T} = 1 + \frac{(\varepsilon_{D,L}\varepsilon_{C,K} - \varepsilon_{D,K}\varepsilon_{C,L})(\sigma - \sigma_{I_2})}{\varepsilon_{C,K}(\sigma - \sigma_T)} \quad (44) $$

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