CAHIERS ECO & MATHS

98.21

INVESTMENT CHOICE WITH POLLUTED CAPITAL

Johanna ETNER

Pierre-André JOUVET

CEME
INVESTMENT CHOICE WITH POLLUTED CAPITAL

Johanna Etner*

CEME, Paris 1 Panthéon-Sorbonne

Pierre-André Jouvet

GREQAM, Marseille et GREMARS, Université de Lille 3

<footnote>* Maison des Sciences Economiques - 106-112 Bd de l'Hôpital - 75647 Paris Cedex 13.
Tél. : 01 55 43 41 96 - E-mail : joetner@univ-paris1.fr.</footnote>
# INVESTMENT CHOICE WITH POLLUTED CAPITAL

**Johanna Etner**

CEME, Université Paris 1-Panthéon-Sorbonne

**Pierre-André Jouvet**

GREQAM, Marseille, et GREMARS, Université de Lille 3

## Résumé

Cet article développe un modèle à générations à deux secteurs dans lequel un secteur produit une externalité sur la qualité de l'environnement tandis que l'autre n'a aucun effet sur l'environnement. Nous supposons que la dégradation de la qualité de l'environnement résulte de de l'activité d'un secteur de production. Nous caractérisons alors la dynamique de ce système et établissons des conditions sous lesquelles il existe au moins un équilibre avec anticipations parfaites dans le cas d'une fonction de production Cobb-Douglas et d'une fonction d'utilité CES. Nous montrons que l'existence et la stabilité d'un équilibre stationnaire dépend des effets de substitution et de revenu créés par une augmentation de taux d'intérêt et du degré de pollution.

**Classification JEL :** D91, E22, O41, Q20

**Mots clés :** modèle à générations à deux secteurs, environnement, dynamique.

## Abstract

This paper develops a two-sector overlapping generations model in which one sector produces an externality on the environmental quality and the other has no effect. We assume that environmental quality degradation results from production activity of one sector. Then, we characterize the dynamical system globally and establish sufficient conditions for the global uniqueness of a perfect-foresight equilibrium path in the case of a Cobb-Douglas production function and a CES utility function. We show that the existence and the stability of the steady state depend on substitution and income effect and on the degree of pollution.

**JEL Classification:** D91, E22, O41, Q20

**Keywords:** two-sector overlapping generations model, environmental management, dynamic.
# 1 Introduction

The intertemporal allocation of natural resources has arguably been one of the most
debate issues in recent years. The recent conference of Kyoto underlines that human
society depends on a myriad of ways on the service provides by natural ecosystems
as it is shown by Chichilnisky G. and G. Heal[1993], and also by Daily [1997]. Tak-
ing account of production externalities problems and, more particularly pollution,
by political decisions conduces to questions on possibilities of economic growth and
optimization choices from governments (Chichilinsky [1993] and [1996]).

Nevertheless, we can observe that taking care of environmental variable is not the
only fact of governments strategic decisions. In fact, if consumers can take strategic
decisions, then, they take account of harmful effects from their behaviour on envi-
ronment and so on their welfare (Mourmouras [1991]). As also, investment decisions
integrate more and more externality effects of production on environment.

All of those strategic decisions that affect the environment both influence and re-
spond to macroeconomic variables, and industrial decisions as well as environmental
policy decisions have implications for economic growth and of course capital accu-
mulation.

Consequently, environmental modeling should focus on intertemporal issues and
externalities (Basking [1997]). Concerning intertemporal issues, there is a growing
interest in overlapping generations approach in the literature on environmental eco-
nomics (see for example John and Pecchenino [1994], John *et alii* [1995] or Marini
and Scaramozzinio [1995]). In fact, as argued by Solow [1986] the overlapping gen-
erations model seems to be a convenient framework for analysing intertemporal is-
sues related to environment degradation. Concern with environmental degradation,
standard economic analysis suggests that environment problem can be classified as
consumption and production externalities (Cropper and Oates [1992]).

Thus, studying environmental problems leads to analyses the economic dynamic
and so the conditions of existence and stability of equilibrium. It is interesting to
notice that one-sector growth model and one-sector overlapping generations model
have became the standard frameworks for analysis of economic dynamic phenom-
ena. Gale [1973] has demonstrated that, in one good nonmonetary pure-exchange
economy, with homogeneous two-period-lived agents within generation, if prefer-
ences satisfy gross substituability then the perfect-foresight equilibrium is uniquely
determined. Gale's results have been extended to a multi-commodity economy and
intertemporally separable preferences (Balasko and Shell [1981], Geanakopolos and Polemarchakis [1984]). In single dimensional dynamical system, that characterizes the one sector production economy, gross substitution ensures the global determinacy of perfect-foresight equilibrium (Galor and Ryder [1989]). But in multisector models, gross substitution in consumption is not sufficient to rule out indeterminacy (see Matsuyama [1991] and Galor [1992]). Therefore, gross substitution in consumption does not necessarily preclude indeterminacy in overlapping generations models with production.

The current paper establishes conditions to rule out indeterminacy of equilibria in a multi-dimensional production economy with externality. More precisely, we develop a two-sector overlapping generations model in which one sector produces an externality on the environmental quality and the other has no effect. We assume that environmental quality degradation results from production activity of one sector. The very fact of producing pollutes: pollution is a "bad" joint product of the productive activity e.g. CO2, acid rain, noise, ozon layer depletion. In fact in this model, we suppose that pollution is then due to the using of one type of capital (we can think about utilization, for example, of oil or nuclear energy). Moreover, we assume that agents take into account the fact that invest in some firm increases pollution and so decreases their welfare. To ensure consistency in behaviour, we assume that individuals invest in different firms through the intermediation of homogeneous investors groups. Each individual behaves cooperatively towards the group to which he belongs, whereas those groups adopt a non-cooperative strategy when financing the firms capital accumulation. This assumption allows that all decisions regarding consumptions are made by individuals in a non-strategic, price taking way. We indeed assume that there is a continuum of individuals. Then, we characterize the dynamical system globally and establish sufficient conditions for the global uniqueness of a perfect-foresight equilibrium path in the case of a Cobb-Douglas production function and a CES utility function. We show that the existence and the stability of the steady state depend on substitution and income effect and on the degree of pollution. At last, we study effects of pollution on equilibrium.

This paper is organized as follows. In a second section, we develop our model. Then, we determine the existence of equilibrium with perfect foresight. In section 4, we examine the dynamic of capital accumulation. Then, we analyse sensibilities of steady state equilibrium to pollution (section 5). Section 6 contains a brief conclusion.
## 2 The model

We consider an infinite-horizon of two periods lives individuals. To ensure consistency in behaviour, we assume that individuals invest in capital accumulation through the intermediation of homogeneous investment groups. Each individual behaves cooperatively towards the group to which he belongs, whereas those groups adopt a non-cooperative strategy when financing the capital of firm. This assumption allows that all decisions regarding both savings and consumption are made by individuals in a non-strategic price taking way. We indeed assume that there is a constant population divided into N homogeneous groups of equal size normalized to 1 and consisting of a continuum of identical individuals.

### 2.1 Production sector and pollution

At each period $t$, production occurs according to a constant returns to scale production function using labor, $L_t$, and capital, $K_t$. Firms are perfectly competitive and capital is fully depreciates during the process of production. Labor is perfectly homogeneous with a perfect mobility. In this economy, there exist two kinds of capital, $K_n$ and $K_e$. Capital of type $K_n$ is used in firms of type $n$ and its using does not produce any effect on environment quality. On the contrary, using type of capital $K_e$ in firms of type $e$ induces harmful waste to environment (radioactivity, chemical pollution,...). Existence of two types of firms in the economy implies production of two types of consumption goods, $Y_n$ and $Y_e$, which we suppose perfectly substitutable. Firms of type $n$ and $e$ are then perfectly concurrentiable on goods and labour markets. We assume that these firms produce with a Cobb-Douglas technology with constant returns to scale. Production funtions of both type of firms are similar except a productivity parameter $A$. At period $t$, we respectively have for firms $e$ and $n$ with $\alpha \in ]0, 1[$:

$$Y_t^e = F(K_t^e, L_t^e) = (K_t^e)^\alpha (L_t^e)^{1-\alpha} \quad (1)$$

and

$$Y_t^n = G(K_t^n, L_t^n) = A(K_t^n)^\alpha (L_t^n)^{1-\alpha} \quad (2)$$

As firms are concurrential, production factors are paid to their marginal productivity. By assuming a perfect mobility of labour between firms, we obtain the
following equilibrium relations on production sector:

$$1 + r_t^e = R_t^e = F_K(K_t^e, L_t^e) = \alpha \left(\frac{L_t^e}{K_t^e}\right)^{1-\alpha} \quad (3)$$

$$1 + r_t^n = R_t^n = G_K(K_t^n, L_t^n) = \alpha A \left(\frac{L_t^n}{K_t^n}\right)^{1-\alpha} \quad (4)$$

and

$$w_t = F_L(K_t^e, L_t^e) = G_L(K_t^n, L_t^n) = (1-\alpha)\left(\frac{K_t^e}{L_t^e}\right)^\alpha = (1-\alpha)A\left(\frac{K_t^n}{L_t^n}\right)^\alpha \quad (5)$$

where $R_t^e$ and $R_t^n$ are respectively the return rates of capital in firms of type e and n, $w_t$ the real wage in the economy (Galor [1992]). From equations (3), (4) and (5), we deduce the following relations:

$$R_t^n = A^{\frac{1}{\alpha}} R_t^e \quad (6)$$

and

$$w_t = A(1-\alpha)\left(\frac{R_t^e}{\alpha}\right)^{-\frac{\alpha}{1-\alpha}} = (1-\alpha)\left(\frac{R_t^n}{\alpha}\right)^{-\frac{\alpha}{1-\alpha}} \quad (7)$$

Now let us turn to the stock of pollution. Its accumulation is proportional to the capital level used by firms of type e. The stock of pollution $P_t$ is proportional to the capital level used by firms of type e. At period t, it is given by a part $\gamma_t$ of the stock of capital $K_t^e$:

$$P_t = \gamma_t K_t^e \quad (8)$$

The level of environment quality, $S_t$, at period t is then defined by:

$$S_t = hS_{t-1} - P_t \quad (9)$$

where $h \ge 1$ is a natural improvement rate of environment. This rate is defined by analogy to the natural absorbtion rate of pollution which means a natural reduc- tion of pollution (see Pearce and Turner [1990]). This assumption means that there exists a natural improvement effect of environment when environment has been de- teriorated by polluted emissions. Moreover, if there no exists any source of pollution then, we consider that environmental stock is equal to its maximum level, $\bar{S}$.
## 2.2 Consumers

At each period, individuals have a desutility from pollution which is perceived by investors groups. During their first period of life, each agent born in period $t$ supplies one inelastic unit of labour and receives the concurrential wage, $w_t$, which he consumes, $c_t$, and saves, $s_t$:

$$w_t = c_t + s_t \quad (10)$$

As consumption goods are supposed perfectly substitutable and consumers behave as price taker concerning their consumption, they do not make difference between goods produced by firms $e$ and firms $n$. At savings level, each investment group can choose to invest the sum of savings received in one type of firms or to share it between this two types. Thus, investment groups have to make a portfolio choice on two assets which returns are certain, but where one produces no externality on environment (investing in firms $n$) and the other induces an environment externality (investing in firms $e$):

$$s_t = a_t + e_t \quad (11)$$

In their second period of life, agents are retired and consume, $d_{t+1}$:

$$d_{t+1} = R_{t+1}^e e_t + R_{t+1}^n a_t \quad (12)$$

Agents' preferences are represented by a CES utility function:

$$U(c_t, S_t, d_{t+1}, S_{t+1}) = \frac{c_t^{1-\sigma}}{1-\sigma} + \frac{S_t^{1-\sigma}}{1-\sigma} + \delta \frac{d_{t+1}^{1-\sigma}}{1-\sigma} + \theta \frac{S_{t+1}^{1-\sigma}}{1-\sigma} \quad (13)$$

where $\sigma > 0$ is the elasticity of substitution, $\delta$ is the discount factor for consumption, and $\theta$, the one for environment. In fact, since pollution and consumption have not the same effect on welfare, it seems intuitive to assume that $\delta$ is different from $\theta$.

### 2.2.1 Groups' program

Let us now look at the behavior of investment groups. They consist of a continuum of identical members and each group chooses, in a cooperative way, investment rate in each firm such that it maximizes the utility of its members. All groups take the investment choice from the other groups as given.

At period $t$, groups decide their investment levels in firms $e$ and $n$. Therefore,
behavior of a group is characterized by the problem of a representative agent in this group, that is to maximize (13) subject to members budget constraints (10) (with (11)), (12) and to :

$$S_{t+1} = hS_t + \gamma_t e_t - \gamma_t \bar{E}_t \quad (14)$$

where $\bar{E}_t$ is the investment of all the other groups in firm e. Then, each group participating to capital accumulation of firms e is conscient that a part of its investment could contribute to environmental degradation.

We obtain the following first order conditions:

$$H_a \equiv -(w_t - a_t - e_t)^{-\sigma} + \delta R_{t+1}^n (R_{t+1}^e e_t + R_{t+1}^n a_t)^{-\sigma} \quad (15)$$

$$H_e \equiv -(w_t - a_t - e_t)^{-\sigma} + \delta R_{t+1}^e (R_{t+1}^e e_t + R_{t+1}^n a_t)^{-\sigma} - \theta(hS_t + \gamma_t e_t - \gamma_t \bar{E}_t)^{-\sigma} \quad (16)$$

We can notice that equations (15) and (16) gives a third relation:

$$\delta(R_{t+1}^n - R_{t+1}^e)(R_{t+1}^e e_t + R_{t+1}^n a_t)^{-\sigma} = -\theta(hS_t + \gamma_t e_t - \gamma_t \bar{E}_t)^{-\sigma} \quad (17)$$

which is verified only if $(R_{t+1}^n - R_{t+1}^e) < 0$. Thus, in order to verify the first order condition, it is necessary that return in firms n, which no pollute, is smaller than return in firms e, which pollute. In fact, if it were not the case, then, agents invest only in firms n, because of two effects, first, this type of firm would offer a larger return, and second, they do not produce any effect on environment quality.

Second order conditions are verified since Hessian is semi-defined negative, since $\sigma > 0$:

$$\frac{\partial H_a}{\partial a_t} < 0; \quad \frac{\partial H_e}{\partial e_t} < 0; \quad \frac{\partial H_a}{\partial a_t} \frac{\partial H_e}{\partial e_t} - \frac{\partial H_a}{\partial e_t} \frac{\partial H_e}{\partial a_t} > 0 \quad (18)$$

At optimum, we can exclude the case where $a_t$ or $e_t$ are negative. Moreover, we can never obtain $a_t = e_t = 0$. If $a_t$ or $e_t$ is negative, groups borrow to one type of firms in order to invest more in the other type. This attitude can come from two effects: first, a type of firms offers a better return (in our case, it is necessary firms e), second, one type (firms n) does not pollute and agents are very pollution averse. However, according to values of interest rate of firms n and e, and to values of utility parameters, groups could choose to borrow a level $a_t$ in firms n and invest $e_t$ in firms e.
### 2.2.2 Pollution sensibility

Before studying impact of a pollution sensibility on equilibrium, we are interested now in groups' reaction when environmental quality decreases. Then, we consider that $\gamma_t$ becomes larger. Static comparative result gives us:

$$ \operatorname{sign}(\frac{da_t}{d\gamma_t}) = \operatorname{sign}(\frac{\partial H_e}{\partial \gamma_t} \frac{\partial H_a}{\partial e_t}) > 0 \quad (19) $$

$$ \operatorname{sign}(\frac{de_t}{d\gamma_t}) = -\operatorname{sign}(\frac{\partial H_e}{\partial \gamma_t} \frac{\partial H_a}{\partial a_t}) < 0 \quad (20) $$

Consequently, we have the following result:

**Proposition 1** An increase in effect on environmental quality leads an investors group, who takes care of environment, to invest more in firms n, which no pollute, and less in firms e, which have a negative effect on environment although this last firms offer a better return.

## 3 Equilibrium

In this section, we study the effect of taking care of environment on equilibrium. We determine the equilibrium with rational expectations, then, we analyse the effect of a decrease in environmental quality.

### 3.1 Market conditions

In this economy with portfolio choice and Cobb-Douglas production function, at period t, equilibrium market conditions are the following.

On labour market, where it assumed a perfect mobility:

$$ L_t^n + L_t^e = N \quad (21) $$

On neutral capital market:

$$ Na_{t-1} = K_t^n \quad (22) $$

On capital market with negative effect on environment:
$$Ne_{t-1} = K_t^e \qquad (23)$$

On consumption goods market:

$$Y_t = Y_t^n + Y_t^e = Nc_t + Nd_t + N(a_{t-1} + e_{t-1}) \qquad (24)$$

From these market conditions and optimal conditions of both of firms, we can determine the level of employment in both firms, capital returns and equilibrium wage. We obtain:

$$l_{t+1}^e = \frac{L_{t+1}^e}{N} = \frac{e_{t-1}}{e_{t-1} + a_{t-1}A^\frac{1}{\alpha}} \qquad (25)$$

$$l_{t+1}^n = \frac{L_{t+1}^n}{N} = \frac{a_{t-1}A^\frac{1}{\alpha}}{e_{t-1} + a_{t-1}A^\frac{1}{\alpha}} \qquad (26)$$

$$w_{t+1} = (1 - \alpha)[e_{t-1} + a_{t-1}A^\frac{1}{\alpha}]^\alpha \qquad (27)$$

$$R_{t+1}^e = \alpha[e_{t-1} + a_{t-1}A^\frac{1}{\alpha}]^{\alpha-1} \qquad (28)$$

$$R_{t+1}^n = \alpha A^\frac{1}{\alpha} [e_{t-1} + a_{t-1} A^\frac{1}{\alpha}]^{\alpha-1} \qquad (29)$$

Then, assumption of a larger capital return in firms polluting (firms e) than the one in firms with no effect on environment (firms n) is verified if and only if $A < 1$.

## 3.2 Equilibrium with perfect-foresight

Groups perfectly anticipate assets returns and choose the same portfolio (since agents are identical), then equilibrium with perfect expectations verified the following conditions.

$$H_a^o \equiv -((1 - \alpha)[e_{t-1} + a_{t-1}A^\frac{1}{\alpha}]^\alpha - a_t - e_t)^{-\sigma} + \delta\alpha^{1-\sigma}A^\frac{1}{\alpha}[e_t + a_tA^\frac{1}{\alpha}]^{-(1-\alpha)(1-\sigma)-\sigma} = 0 \qquad (30)$$

$$H_e^o \equiv -((1 - \alpha)[e_{t-1} + a_{t-1}A^\frac{1}{\alpha}]^\alpha - a_t - e_t)^{-\sigma} + \delta\alpha^{1-\sigma}[e_t + a_tA^\frac{1}{\alpha}]^{-(1-\alpha)(1-\sigma)-\sigma} - \theta\gamma_t(hS_t - N\gamma_t e_t)^{-\sigma} = 0 \qquad (31)$$

It is easy to verify that as $\sigma$ is positive, there exists a unique solution at equi-
librium given by equations (30) and (31). Moreover, a necessary and sufficient condition for having $e_t > 0$ at equilibrium is that $a_t > 0$, and conversely. As we showed before, it is not possible that these two levels are both nil, and negative by construction. Then, we have the existence proposition:

**Proposition 2** There exists a unique equilibrium with rational expectations ($a_t^*, e_t^*$) such as $a_t^* > 0$ and $e_t^* > 0$.

### 3.3 Pollution Sensibility

At equilibrium level, some spill-over effects appear when degradation by capital in firms e becomes more important. In fact, we obtain the following static result.

**Proposition 3** At equilibrium, an increase in the environmental degradation, $\gamma_t$, implies more capital of type n and less of type e.

**Proof:**

Form equations (30) and (31), we obtain:

$$ \mathrm{sign}(\frac{da_t^*}{d\gamma_t}) = \mathrm{sign}(\frac{\partial H_e^o}{\partial \gamma_t} \frac{\partial H_a^o}{\partial e_t}) > 0 \quad (32) $$

$$ \mathrm{sign}(\frac{de_t^*}{d\gamma_t}) = -\mathrm{sign}(\frac{\partial H_e^o}{\partial \gamma_t} \frac{\partial H_a^o}{\partial a_t}) < 0 \quad (33) $$

□

We can remark that at macroeconomic level, a degradation in effect on environment leads to less capital in firms e and more in firms n. We can distinguish different effects.

First, at individual level, agents want to protect themselves again a more harmful effect of investment in firms e. This attitude conduces to a more important level of capital in firms n and less in firms e.

Second, markets equilibria imply a decrease in capital return $R_{t+1}^n$, but an increase in $R_{t+1}^e$.

Third, agents through their groups anticipate these two effects which lead them to increase their level of investment in firms e and decrease in firms n. This last macroeconomic effect is opposite to the microeconomic one, but, not sufficient to reverse capital accumulation in both types of firms.
We have studied effects of a taking care of environment on investors decisions and capital accumulation in a static framework. Now, let us analyse these impacts on dynamic capital accumulation.

# 4 Dynamic of capital accumulation

In this section, we develop dynamic of capital e and n and we show that according to values of σ and γ, we obtain three possible configurations: no stationary solution, one stationary equilibrium or two stationary equilibria.

## 4.1 Dynamic

In order to study capital dynamic, we explain it with capital return in firms e. Remind that there is a closed relation between the two capital returns (equation (6)) and we can rewrite the concurrential wage as a function of capital returns (equation (7)). Now, using the two first order conditions (10) and (12) and optimal firms conditions, more particularly, relation (28), we can express, at period t, the level of total investment in firms n and e as functions of capital return in firms e, $R_{t+1}^e$:

$$e_t^* = -\frac{1}{N\gamma_t}\left(\frac{\delta(1 - A^\frac{1}{\alpha})}{\alpha^{\frac{\sigma}{1-\alpha}}\theta\gamma_t}\right)^{-\frac{1}{\sigma}}(R_{t+1}^e)^{-\frac{1-\alpha+\alpha\sigma}{(1-\alpha)\sigma}} + \frac{hS_t}{N\gamma_t} \quad (34)$$

$$a_t^* = w_t + \left( \frac{1}{N\gamma_t} \left( \frac{\delta(1 - A^\frac{1}{\alpha})}{\alpha^{\frac{\sigma}{1-\alpha}}\theta\gamma_t} \right)^{-\frac{1}{\sigma}} - \frac{(\delta A^\frac{1}{\alpha})^{-\frac{1}{\sigma}}}{\alpha^{\frac{1}{\alpha_1}}} \right) (R_{t+1}^e)^{-\frac{1-\alpha+\alpha\sigma}{(1-\alpha)\sigma}} - \frac{hS_t}{N\gamma_t} \quad (35)$$

By using these expressions and by expressing $w_t$ and $S_t$ as functions of $R_{t+1}^e$, we obtain the dynamic of our economy in capital return of type of firms e:

$$- \lambda_{1t} (R_{t+1}^e)^{-\frac{1-\alpha+\alpha\sigma}{(1-\alpha)\sigma}} - \lambda_{2t} (R_{t+1}^e)^{-\frac{1}{1-\alpha}} + \lambda_{3t} (R_{t+1}^e)^{-\frac{\alpha}{1-\alpha}} + \lambda_{4t} (R_{t+1}^e)^{-\frac{1-\alpha+\alpha\sigma}{(1-\alpha)\sigma}} = 0 \quad (36)$$

where

$$\lambda_{1t} = \left(\frac{\delta}{\theta\gamma_t}\right)^{-\frac{1}{\sigma}} \frac{(1 - A^\frac{1}{\alpha})^{1-\frac{1}{\sigma}}}{\alpha^{-\frac{1}{1-\alpha}} N \gamma_t} + \frac{\delta^{-\frac{1}{\sigma}} (A^\frac{1}{\alpha})^{1-\frac{1}{\sigma}}}{\alpha^{\frac{1}{1-\alpha}}} > 0 \quad (37)$$

$$\lambda_{2t} = \frac{1}{\alpha^{-\frac{1}{1-\alpha}}} > 0 \quad (38)$$
$$ \lambda_{3t} = \frac{(1 - \alpha)A^{\frac{1}{\alpha}}}{\alpha^{-\frac{\alpha}{1-\alpha}}} > 0 \qquad (39) $$

$$ \lambda_{4t} = h\left(\frac{\delta}{\theta\gamma_{t-1}}\right)^{-\frac{1}{\sigma}} \frac{(1 - A^{\frac{1}{\alpha}})^{1-\frac{1}{\sigma}}}{\alpha^{-\frac{1}{1-\alpha}}N\gamma_t} > 0 \qquad (40) $$

Since we consider complementary factors (capital and labor), an increase in capital stock implies an increase in the competitive wage and so an increase in the total amount of savings, therefore we easily obtain the following characteristic of dynamic:

**Proposition 4** *The dynamic of this economy is increasing:*

$$ \frac{dR_{t+1}^e}{dR_t^e} > 0 \text{ and then } \frac{da_t}{da_{t-1}} > 0, \frac{de_t}{de_{t-1}} > 0. $$

## 4.2 Existence and stability of stationary equilibrium

From equation (36), stationary equilibrium, if it exists, verifies the following expression:

$$ f(R) = \Lambda R^{-\frac{1-\alpha+\alpha\sigma}{(1-\alpha)\sigma}} - \lambda_2 R^{-\frac{1}{1-\alpha}} + \lambda_3 R^{-\frac{\alpha}{1-\alpha}} = 0 \qquad (41) $$

where

$$ \Lambda = (h-1)\left(\frac{\delta}{\theta\gamma_t}\right)^{-\frac{1}{\sigma}} \frac{(1-A^{\frac{1}{\alpha}})^{1-\frac{1}{\sigma}}}{\alpha^{-\frac{1}{1-\alpha}}N\gamma_t} - \frac{\delta^{\frac{1}{\sigma}}(A^{\frac{1}{\alpha}})^{1-\frac{1}{\sigma}}}{\alpha^{\frac{1}{1-\alpha}}} \qquad (42) $$

Then, the study of stationary equilibrium depends on the sign of the expression (41) but also on values of elasticity of substitution $\sigma$. Consequently, we will analyse this problem by distinguishing three cases, $\sigma = 1$ (case of logarithmic function), $\sigma > 1$ (the income effect is greater than the substitution effect) and $\sigma < 1$.

### 4.2.1 Case $\sigma = 1$

When $\sigma$ is equal to one, that is when an increase in interest rate does not influence the level of investment, we can show that there can be no stationary equilibrium or a unique one:

**Proposition 5** *In the case of $\sigma = 1$,*

if $N < (h-1)\frac{\theta}{1+\delta}$, then there is no stationary equilibrium;

if $N > (h-1)\frac{\theta}{1+\delta}$, then there exists a unique stationary equilibrium $\tilde{R}$ locally stable.
**Proof:**

The stationary equilibrium must verify:

$$f(R) = (\Lambda - \lambda_2)R^{-\frac{1}{1-\alpha}} + \lambda_3 R^{-\frac{\alpha}{1-\alpha}} = 0$$

where function has the following properties:

$$\lim_{R \to 0} f(R) = +\infty$$

if $\Lambda - \lambda_2 < 0$,

$$\lim_{R \to 0} f(R) = -\infty$$

else, and

$$\lim_{R \to \infty} f(R) = 0.$$

We can show that if $\Lambda - \lambda_2 < 0$, then $f'(R) < 0$ and there is no solution to equation $f(R) = 0$. But, in the opposite case, we can easily determine the unique solution to $f(R) = 0$, which is $\tilde{R} = \frac{\lambda_2 - \Lambda}{\lambda_3}$. Moreover, this point is inferior to $\tilde{R}$, with $\tilde{R}$ defined as $f'(\tilde{R}) = 0$. As $f'(\tilde{R}) > 0$ which is equivalent to $\frac{dR_{t+1}^e}{dR_t^e}|_{\tilde{R}^*} < 1$, then $\tilde{R}$ is locally stable.

<figure><img src="image_1.png" /></figure>

Consequently, for $\sigma = 1$, we can have two possible configurations as it is shown on figure 1. In fact, for $\sigma = 1$ interest rates have no influence on the level of savings and agents are unsensible to variation of pollution rate $\gamma$, then the existence of those two configurations is directly linked to the well known subscription effect since groups' strategy is Nash non cooperative (see for more details on this point Bergstron et al. [1986] or Broadway et al. [1989]).

### 4.2.2 Case $\sigma > 1$

In this case, we can show that there always exists a unique stationary equilibrium, denoted by $\tilde{R}^*$:

**Proposition 6** In the case of $\sigma > 1$, there exists a unique stationary equilibrium $\tilde{R}^*$ locally stable.

**Proof:**

The stationary equilibrium verifies equation (41). The function $f(.)$ has the following
properties:

$$ \lim_{R \to 0} f(R) = +\infty $$

and

$$ \lim_{R \to +\infty} f(R) = 0^+ $$

The first derivative of $f(\cdot)$ is written by:

$$ f'(R) = \frac{R^{-\frac{1}{1-\alpha}-1}}{(1-\alpha)\sigma} k(R) $$

where

$$ k(R) = -(1 - \alpha + \alpha\sigma)\Lambda R^{1-\frac{1}{\sigma}} + \lambda_3\sigma - \alpha\sigma\lambda_4 R $$

And, we can show that there exists a unique point $\bar{R}$, such that $k(\bar{R}) = 0$, that is $f'(R) = 0$.

Consequently, function $f(\cdot)$ becomes nil only on one point inferior to $\bar{R}$ which we denote by $\tilde{R}^*$. Moreover, this stationary equilibrium is locally stable if $\frac{dR_t^e}{dR_t^e}|_{\tilde{R}^*} < 1$, which is equivalent to $f'(\tilde{R}^*) > 0$, that equivalent to $\tilde{R}^* < \bar{R}$. This last expression is always true by construction.

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

For $\sigma > 1$, the income effect is greater than the substitution effect, in other words agents prefer to decrease their savings level in response to an increase in interest rate. Therefore, if we first consider an initial capital level in firms $e$ too high relative to the equilibrium level, then the level of interest rate in those firms is too weak. Consequently, agents will prefer to invest in firms $n$. But, an increase in investment in favor to firms $n$ will diminish capital return in those firms and increase the firms $e$ one. This adjustment will be continue until the two capital returns will be equal (by taking account of compensation of externality from firms $e$). If at the end of this capital level adjustment in both type of firms is too weak, which induces a decrease in global portfolio return, then, under the assumption $\sigma > 1$, agents will increase their savings level until stationary equilibrium will be realized.

The effect of pollution desutility reinforces capital adjustment in firms $e$. Thus, this adjustment becomes faster than firms $n$ one.

Consequently, in the case $\sigma > 1$ we have only one possible configuration (see figure 2).
### 4.2.3 Case $\sigma < 1$

According to values of $\gamma$, we can obtain none stationary equilibrium, one locally stable or two, one locally stable and the other not. Let us define $\hat{\gamma}$ and $\hat{\hat{\gamma}}$, respectively the values of $\gamma$ such as $\Lambda = 0$ and the second derivative of function $f(\cdot)$ is nil:

$$ \hat{\gamma} = (h-1) \frac{\theta^{\frac{1}{\sigma}}}{N} (A^{-\frac{1}{\alpha}} - 1) \quad (43) $$

$$ \hat{\hat{\gamma}} = \left( \frac{\beta_2 + \beta_3}{\beta_1} \right)^{\frac{\sigma}{1-\sigma}} \quad (44) $$

where

$$ \beta_1 = (h-1) \left(\frac{\delta}{\theta}\right)^{-\frac{1}{\sigma}} \frac{(1-A^{\frac{1}{\alpha}})^{1-\frac{1}{\sigma}}}{\alpha^{-\frac{1}{1-\alpha}} N} \quad (45) $$

$$ \beta_2 = \frac{\delta - \frac{1}{\sigma}}{\alpha^{-\frac{1}{1-\alpha}}} (A^{\frac{1}{\alpha}})^{1-\frac{1}{\sigma}} \quad (46) $$

$$ \beta_3 = \left( \frac{\lambda_3 \sigma (1-\sigma)}{(2-\sigma) \alpha \lambda_4 \sigma^2} \right)^{\frac{1}{\sigma}} \frac{\alpha \sigma^2 \lambda_4}{(1-\alpha + \alpha \sigma)(1-\sigma)} \quad (47) $$

We thus have the following proposition,

**Proposition 7** In the case of $\sigma < 1$,

if $\gamma \ge \hat{\gamma}$, then there exists a unique stationary equilibrium locally stable denoted $\tilde{R}$; if $\gamma < \hat{\gamma}$, then

- if $\gamma < \hat{\hat{\gamma}}$, then there exist two stationary equilibria, $\hat{R}$ and $\tilde{R}$ with $\hat{R} < \tilde{R}$ and $\hat{R}$ is unstable whereas $\tilde{R}$ is locally stable;

- if $\gamma > \hat{\hat{\gamma}}$, then there is no stationary equilibrium.

**Proof:**

The stationary equilibrium verifies equation (41). The function $f(\cdot)$ has the following properties:

$$ \lim_{R \to +\infty} f(R) = 0^+. $$

and if $\gamma < \hat{\gamma}$ then $\Lambda > 0$ and

$$ \lim_{R \to 0} f(R) = +\infty $$

if $\gamma \ge \hat{\gamma}$, then $\Lambda \le 0$ and

$$ \lim_{R \to 0} f(R) = +\infty $$

and
The first derivative of $f(.)$ is written by:

$$f'(R) = \frac{R^{-\frac{1}{1-\alpha}-1}}{(1-\alpha)\sigma}k(R)$$

where

$$k(R) = -(1 - \alpha + \alpha\sigma)\Lambda R^{1-\frac{1}{\sigma}} + \lambda_3\sigma - \alpha\sigma\lambda_4 R.$$

If $\Lambda \le 0$, we can show that there exists a unique point, $\bar{R}$, such that $k(\bar{R}) = 0$, that is $f'(\bar{R}) = 0$. Consequently, function $f(.)$ becomes nil only on one point inferior to $\bar{R}$ which we denote by $\tilde{R}$. Moreover, this stationary equilibrium is locally stable since $f'(\tilde{R}) > 0$.

If $\Lambda > 0$, then, there exists a value of $R$, denoted by $\bar{R}$ such that $k(\bar{R}) = 0$. We can easily verify that $R < \bar{R}$ implies $k' < 0$, and we have the following properties:

$$\lim_{R \to 0} k(R) = +\infty$$

and

$$\lim_{R \to +\infty} k(R) = -\infty$$

Consequently, if $k(\bar{R}) \le 0$ (which equivalent to $\gamma > \hat{\gamma}$), there is no stationary equilibrium since $f'(R)$ is always negative, but in the opposite case, there are two stationary equilibria, $\hat{R}$ and $\tilde{R}$. In fact, as $k(\bar{R}) \ge 0$, there are two values $R_1$ and $R_2$ such $k(R_1) = k(R_2) = 0$, with $R_1 < \bar{R} < R_2$. So, as $f'(R_1) = f'(R_2) = 0$, we have two stationary equilibria such as $\hat{R} < R_1 < \tilde{R} < R_2$. As above, we can show that $\hat{R}$ is locally stable, but $\hat{R}$ is not.

□

The assumption of $\sigma < 1$ means that substitution effect is upper than income effect. In this case, since an increase in interest rate implies an increase in savings, the existence of equilibrium depends only on adjustment relative speed of capital levels in both type of firms. In other terms, it depends on externality level produced by firms e. Consequently, for $\sigma < 1$, we obtain three possible configurations (see figure 3).
# 5 Impact of polluted emissions on stationary equilibrium

Now, we analyse the impact of an increase in $\gamma$ on stationary equilibrium. In the case of $\sigma = 1$, $\gamma$ has no influence on equilibrium. Consequently, in this section, we study only two cases.

## 5.1 $\sigma > 1$

In this case, since stationary capital is an increasing function of $\tilde{R}$ and as $\tilde{R}$ is an increasing function of $\gamma$, we can easily show the following result:

**Proposition 8** In the case of $\sigma > 1$, an increase in $\gamma$ leads to an increase in stationary global capital.

An increase in $\gamma$ has several effects on capital.

First, a microeconomic effect is that an increase in $\gamma$ conduces to an increase in level of investment in firms $n$ and a decrease in investment in firms $e$, because agents are pollution averse. Thus, interest rate $R^n$ decreases whereas $R^e$ increases. We can then show that the global return $\frac{aR^n + eR^e}{a+e}$ decreases. As $\sigma$ is superior to one, savings is a decreasing function of global return, and thus, savings increases. However, a second effect appears. As interest rate $R^e$ increases at equilibrium, the concurrential wage decreases, then, this last effect discourages savings. Finally, stationary global capital increases (for an illustration, see figure 4).

## 5.2 $\sigma < 1$

In this case, we immediately deduce the following result:

**Proposition 9** In the case of $\sigma < 1$,

if $\gamma \ge \hat{\gamma}$, then an increase in $\gamma$ leads to a decrease in stable stationary global capital;

if $\gamma < \hat{\gamma}$ and if $\gamma < \tilde{\gamma}$, an increase in $\gamma$ leads to a decrease in stable stationary global capital and an increase in the unstable one.

An increase in $\gamma$ has two effects on capital.

Consider first a stable stationary equilibrium. An increase in $\gamma$ conduces to an increase in level of investment in firms $n$ and a decrease in investment in firms $e$. Thus, interest rate $R^n$ decreases whereas $R^e$ increases. We can then show that the
global return decreases. As $\sigma$ is inferior to one, savings is an increasing function of global return, and thus, savings decreases. Moreover, the second effect discourages savings since the concurrential wage decreases. Therefore stable stationary global capital decreases (see figure 5).

Now, let us study the case of a unstable equilibrium. In this case, we can show that the global return increases. Consequently, savings increases too. Moreover, second effect encourages savings since the concurrential wage now increases. Thus, the unstable stationary global capital increases (see figure 6).

## 5.3 Consequences of agents attitude on stationary equilibrium

As in previous subsections, we show the possibilities of existence of a stationary equilibrium, now, we can exhibit the consequences of agents attitude depicted by values of $\sigma$ on equilibrium.

We can show that if $\gamma$ is relatively high, stationary capital stock is a decreasing function of $\sigma$ (see figures 8 and 9). According to previous propositions, more the effect of income created by an increase in interest rate becomes important more capital stock in our economy decreases because of pollution effect (agents want a high compensation between the two firms).

On the contrary, if $\gamma$ is relatively weak, stationary capital stock is an increasing function of $\sigma$ (see figure 7).

# 6 Conclusion

The introduction of taking care of environmental quality in investors decisions con- duces to the possibility of several configurations: none stationary equilibrium, a unique which is stable or two equilibria, according to agents attitudes and to values of polluted emission.

When the substitution effect created by an increase in global return is exactly equal to the income effect, then there is a unique stationary equilibrium if polluted emis- sion is relatively high and none else. In the case of substitution effect dominates income effect, we can have the three possible configurations. And, in the last case,
it always exists a unique stable stationary equilibrium.

The existence of stationary equilibrium depends on substitution and income effect (see also Galor [1992]). But, in our case, it depends on the level of externality degree on firms environment. For exemple, when substitution effect is upper than income effect, the more pollution rate is important the less existence possibilities of a stationary equilibrium are.

In a second time, we analysed the effect of an increase in polluted emission on capital stock. We showed that this increase leads to an increase in stationary global capital if substitution effect is upper than income effect and to a decrease in the opposite case. Consequently, according to agents attitude, a decrease in polluted emission could reduce capital stock.

At last, introducing environmental dimension in public or private decisions implies more and more to take account of uncertainties linked to environment since existence possibilities of equilibrium depend on possible effects of polluted emission.

## References

Balasko Y. and K. Shell [1981], “The overlapping generations model: the case of log-linear utility functions”, *Journal of Economic Theory*, **24**, 143-152.

Baskin Y., [1997], *The works of Nature's*, Island Press, Washington D.C.

Bergstrom B., L. Blume and H. Varian [1986]: “On the private provision of public goods”, *Journal of Public Economics*, **29**, 25-49.

Boadway R., P. Pestieau and D. Wildasin [1989]: “Tax-transfert policies and the voluntary provision of public goods”, *Journal of Public Economics*, **39**, 157-176.

Chichilnisky G., [1993], *Biodiversity and the pharmaceutical industry*, Columbia Business School.

Chichilnisky G. et G. Heal, [1993], “Global environmental risks”, *Journal of Economic Perspectives*, **7**, 65-86.

Chichilnisky G., [1996], “An axiomatic approach to sustainable development?”, *Social Choice and Welfare*, **13**, 231-257.

Cropper M.L. and W.E. Oates, [1992], “Environmental economics: a survey”, *Journal of Economic Literature*, **30**, 657-740.
Daily G., [1997], *Nature's service: societal dependence on natural ecosystems*, Island Press, Washington D.C.

Gale D., [1973], "Pure exchange equilibrium of dynamic economic models", *Journal of Economic Theory*, **6**, 12-36.

Galor O., [1992], "A two-sector overlapping-generations model: a global characterization of the dynamical system", *Econometrica*, **60**, 1351-1386.

Geanakopolos J.D. and H.P. Polemarchakis [1984], "Intertemporally separable overlapping generations models", *Journal of Economic Theory*, **34**, 207-215.

Galor O. and H.E. Ryder [1989], "Existence, uniqueness, and stability of equilibrium in an overlapping generations model with productive capital", *Journal of Economic Theory*, **49**, 360-375.

John A., R. Pecchenino, D. Schimmelpfennig and S. Schreft [1995]: "Short lived agents and the long lived environment", *Journal of Public Economics*, **58**, 127-141.

Marini G. and P. Scaramazzino [1995]: "Overlapping generations and environmental control", *Journal of Environmental Economics and Management*, **29**, 64-77.

Matsuyama K., [1991], "Increase returns, industrialization and inderterminancy of equilibrium", *Quaterly Journal of Economics*, **106**, 615-650.

Mourmouras A., [1991], "Competitive equilibria and sustainable growth in a life-cycle model with natural resources", *Scand. Journal of Economics*, **93**, 585-591.

Pearce D.W. and R.K. Turner, [1990], *Economics of natural resources and the environment*, Harvester W., G.B..

Solow R. M. [1986]: "On the intergenerational allocation of natural resources", *Scandinavian Journal of Economics*, **88**, 141-149.
$\sigma = 1$

<figure><img src="image_3.png" /><figcaption>FIGURE 1</figcaption></figure>

$\sigma > 1$

<figure><img src="image_4.png" /><figcaption>FIGURE 2</figcaption></figure>

$\sigma < 1$

<figure><img src="image_5.png" /><figcaption>FIGURE 3</figcaption></figure>
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>FIGURE 4</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>FIGURE 5</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIGURE 6</figcaption></figure>
$$ \gamma < \frac{\zeta}{\theta} (1 - A^{1/\gamma}) $$

<figure><img src="image_10.png" /><figcaption>FIGURE 7</figcaption></figure>

$$ \gamma > \frac{1 - A^{1/\gamma}}{A^{1/\gamma}\theta} $$

<figure><img src="image_11.png" /><figcaption>FIGURE 8</figcaption></figure>

$$ \frac{1 - A^{1/\gamma}}{A^{1/\gamma}} \theta > \gamma > \frac{\zeta}{\theta} (1 - A^{1/\gamma}) $$

<figure><img src="image_12.png" /><figcaption>FIGURE 9</figcaption></figure>