Median Voter Environmental Maintenance
Kirill Borissov∗, Thierry Bréchet†, Stéphane Lambrecht‡
July 6, 2009
Abstract
We assume that the population of infinitely-lived households of
the economy is split into two groups : one with a high discount factor
(the patient) and one with a low one (the impatient). The stock of
environmental quality is deteriorated through polluting emissions of
firms. The governmental policy consists in proposing households to
vote for a tax aimed at quality maintenance. We study the voting
equilibrium at steady states. We show that the resulting equilibrium
maintenance is the one of the median voter. We show that (i) an
increase in total factor productivity may produce effects described by
the Environmental Kuznets Curve, (ii) an increase in the patience of
impatient households may foster environmental quality if the median
voter is impatient and maintenance positive, finally (iii) a decrease
in inequality among the patient households leads to an increase in
environmental quality if the median voter is patient and maintenance
is positive.
Keywords: intertemporal choice and growth, discounting, government environmental policy, externalities, environmental taxes; voting equilibrium
JEL Classification: D90, Q58, H23, D72
∗
Saint-Petersburg Institute for Economics and Mathematics (RAS), 1 Tchaikovskogo
Str., Saint-Petersburg 191187, Russia, and European University in St. Petersburg, 3
Gagarinskaya Str., Saint-Petersburg 191187, Russia, email: [email protected]
†
Center for Operations Research and Econometrics (CORE), Université catholique
de Louvain, 34 voie du Roman pays, 1348 Louvain-la-Neuve, Belgium, email:
[email protected].
‡
Université Lille 1 Sciences et Technologies, Laboratoire EQUIPPE - Universités
de Lille, Cité scientifique, Bât. SH2, 59655 Villeneuve d’Ascq cedex, France, email:
[email protected], and Center for Operations Research and Econometrics
(CORE), Université catholique de Louvain, 34 voie du Roman pays, 1348 Louvain-laNeuve, Belgium.
1
Acknowledgements: part of this research was conducted during several
short visits of K. Borissov at the Université Lille 1 Sciences et Technologies,
laboratoire EQUIPPE - Universités de Lille, as an invited professor during
2007 and 2008. K. Borissov thanks the Russian Foundation for Basic Research for financial support (grant No. 08-06-00423).
2
1
Introduction
With the growing importance of the global warming and climate change
issues and the emphasis put on the general question of sustainable growth
and development, environmental policies and their financing have become a
major subject of discussion and concern in many developing or developed
countries.
Economists, and especially macro-economists, developed dynamic models based on the representative agent assumption to disentangle the nexus
between economic growth and pollution, or more generally environmental
quality (see among many others, John and Pecchenino (1994), Stokey (1998),
Jouvet et al. (2005)). For pollution problems, for instance, the emphasis is
put on the internalization of the pollution negative externality and on the
general efficieny gain for the economy as a whole.
Though it is striking to notice that the public debate about environmental
policies and their financing very often focus on the distributive aspect of the
policies, more precisely on the distribution of its weight among heterogenous
agents. To catch that dimension in their modeling tools, economists must use
macrodynamic models departing from the representative agent hypothesis.
Several ways of introducing heterogeneity are possible : e.g. heterogeneity
in wealth (Kempf and Rossignol (2007)) or in individual labor productivity
(Jouvet et al. (2004)), or in the exposition to damages.
In this paper we consider heterogeneity in the discount factor (for a survey of the literature on models of economic growth with consumers having
different discount factors, see, e.g. Becker (2006)). We assume that the population is exogenously divided into two groups: one with patient households
and the other with impatient households. Our consideration is restricted to
steady states only.
The questions we address are the following: (i) in a framework where
the tax financing environmental maintenance is determined by vote, who
is the winner of the vote? Is it the median voter like in the simple static
unidimensional setting? And (ii) how is the voting equilibrium modified by
shocks on total factor productivity and shocks on the preferences (discount
factors) of patient and impatient households?
Section 2 present the main hypotheses of the model, defines the competitive equilibria and shows the existence of steady state equilibria for a
given policy. Section 3 endogenizes the voting procedure on environmental
maintenance, defines the intertemporal and steady state voting equilibria
3
and shows that the median voter result applies to our setting. Section 4 performs comparative statics exercises to determine under which circumstances
environmental quality is positively impacted by an increase in total factor
productivity, an increase in patience and a decrease in inequality. Section 5
concludes.
2
The model
The framework of analysis used in this paper is the one with infinitely-lived
consumers, supplying inelastically each one unit of labor and with a representative globally polluting firm.
2.1
Production and pollution
Output is determined by means of a neoclassical production function
F (Kt , Lt ) = Lf (kt ), where Kt is the demand for capital at time t, Lt is
the demand for labor, and kt = Kt /L is capital intensity, f (k) = F (k, 1) is
the production function in intensive form. Pollution (the emissions level) at
time t, Pt , is proportional to output:
Pt = λF (Kt , L) = λLf (kt ), λ > 0.
(1)
We denote by Qt an index of environmental quality at time t and by Mt
the maintenance of environmental quality. The dynamics of Qt is given by
Qt+1 = Ψ(Qt − Pt +
Mt
),
µ
(2)
where Ψ : R+ → R+ is a concave increasing function, µ > 0 is exogenously
given coefficient. Since ”marginal environmental productivity” of maintenance, ∂Qt+1 /∂Mt = Ψ0 (•)/µ, is negatively influenced by µ, we can interpret
1
as the environmental efficiency of maintenance. By Q̄ we denote a unique
µ
positive solution to the following equation: Ψ(Q) = Q, i.e. the stationary
value of environmental quality in the case with no pollution and no maintenance. For example, the following particular forms of Ψ(X) can be used:
Ψ(X) = X ν Q̄1−ν , with 0 < ν < 1, or Ψ(X) = νX +(1−ν)Q̄, with 0 < ν < 1.
Let Φ(·) = Ψ−1 (·). We can rewrite (2) as follows:
µΦ(Qt+1 ) = µ(Qt − Pt ) + Mt .
It should be noted that µΦ0 (Q) can be interpreted as the marginal cost of
quality improvement.
4
The representative firm maximize its profit πt under the constraint of
the technology F (Kt , Lt ) by choosing its preferred volumes of capital Kt and
labor Lt , considering the real wage and interest rates, wt and rt , as given.
The firm’s problem is summarized as follows:
max πt = F (Kt , Lt ) − (1 + rt )Kt − wt Lt ,
Kt ,Lt
(3)
and admits the following first-order conditions: FK0 (Kt , Lt ) = 1 + rt and
FL0 (Kt , Lt ) = wt , or in intensive terms: f 0 (kt ) = 1 + rt and f (kt ) − f 0 (kt )kt =
wt .
2.2
Consumers
Population consists of L consumers. Each consumer is endowed with one unit
of labor force. For simplicity, L is integer and odd. The objective function
of consumer i is
∞
X
βit [u(ct ) + v(Qt )],
t=0
where ct is his consumption at time t, βi is his discount factor. We assume
that u(c) and v(Q) satisfy the following conditions:
u0 (c) > 0, u00 (c) < 0, u0 (0) = ∞, v 0 (Q) > 0, v 00 (Q) < 0, v 0 (0) = ∞.
Each consumer i is patient (βi = β h ) or impatient (βi = β l ), 0 < β l < β h < 1.
We denote by Hh the set of patient consumers (with discount factor equal to
β h ) and by Hl the set of impatient consumers (those with β l ).
Each consumer pay a tax mt = Mt /L to finance the public provision of
environmental maintenance. Savings of each consumer at each time t, st ,
must be non-negative. Hence the budget constraints of a consumer at time
t are
ct + st + mt ≤ wt + (1 + rt )st−1 ,
ct ≥ 0, st ≥ 0,
where wt is the wage rate rt is the interest rate.
Suppose that consumer i is given his initial level of savings ŝi−1 , the initial
level of environmental quality Q̂0 , the stream of pollution (Pt )∞
t=0 and some
maintenance policy which is represented by a sequence m̄ = (m̄t )∞
t=0 . Then
5
the problem of this consumer is
P

t
max(ct )+∞ ,(st )+∞ ,(Qt )+∞ ∞

t=0 βi [u(ct ) + v(Qt )],

t=0
t=0
t=0








subject to




P1 = 

µΦ(Qt+1 ) = µ(Qt − Pt ) + Lm̄t ,
ct + st + m̄t ≤ wt + (1 + rt )st−1 ,
s−1 = ŝi−1 , Q0 = Q̂0 ,
ct ≥ 0, st ≥ 0,



























t = 0, 1, . . . ,
t = 0, 1, . . . ,
t = 0, 1, . . . .














+∞
It should be noted that since m̄ = (m̄t )∞
t=0 is given, the sequence (Qt )t=0
is, in fact, predetermined by Q̂0 and m̄ and hence the utility consumer i
P
t
derives from environmental quality, ∞
t=0 βi v(Qt ), does not depend on his
choice here.
2.3
Competitive equilibrium paths and steady-state
equilibria
Let the sequence m̄ = (m̄t )∞
Let an initial state
t=0 be given.
i
L
i
{(ŝ−1 )i=1 , k̂0 , Q̂0 } also be given. Here ŝ−1 ≥ 0 are the initial savings of
consumers i = 1, . . . , L, k̂0 > 0 is the initial per capita stock of capital,
PL
i
i=1 ŝ−1 = Lk̂0 , Q̂0 > 0 is the initial value of environmental quality.
Definition 1. Competitive equilibrium path
i∗ L
∗
∗ ∞
Given m̄, the sequence {kt∗ , 1 + rt∗ , wt∗ , (si∗
t−1 , ct )i=1 , Pt , Qt }t=0 is called an
competitive equilibrium path starting from {(ŝi−1 )Li=1 , k̂0 , Q̂0 } if
1. capital and labor markets clear at the following prices: 1 + rt =
1 + rt∗ = f 0 (kt∗ ), wt = wt∗ = f (kt∗ ) − f 0 (kt∗ )kt∗ , t = 0, 1, . . . ;
∗ ∞
i∗
2. for each household i = 1, . . . , L the sequence (si∗
t−1 , ct , Qt )t=0 is a
solution to problem P1 at 1 + rt = 1 + rt∗ , wt = wt∗ , t = 0, 1, . . . ;
3.
PL
i=1
∗
si∗
t−1 = Lkt , t = 0, 1, . . . ;
4. Pt∗ = λLf (kt∗ ), t = 0, 1, . . . ;
5. µΦ(Q∗t+1 ) = µ(Q∗t − Pt∗ ) + Lm̄t , t = 0, 1, . . . .¤
6
Definition 2. Competitive steady state equilibrium path
Let an m̄ ≥ 0 be given and let m̄ = (m̄t )∞
t=0 , with m̄t = m̄, t = 0, 1, . . . .
∗
∗
∗
i∗ i∗ L
We call a tuple {k , 1 + r , w , (s , c )i=1 , P ∗ , Q∗ } a competitive steadyi∗ L
∗
∗ ∞
state equilibrium if the sequence {kt∗ , 1 + rt∗ , wt∗ , (si∗
t−1 , ct )i=1 , Pt , Qt }t=0
given for all t = 0, 1, . . . by
kt∗ = k ∗ , 1 + rt∗ = 1 + r∗ , wt∗ = w∗ ,
(4)
i∗ L
i∗ i∗ L
(si∗
t−1 , ct )i=1 = (s , c )i=1 ,
(5)
Pt∗ = P ∗ , Q∗t = Q∗ .
(6)
i
L
is an equilibrium path starting from the initial state {(ŝ−1 )i=1 , k̂0 , Q̂0 } =
{(si∗ )Li=1 , k ∗ , Q∗ }. ¤
Assume that m̄ is given and is not too large in the sense we clarify later.
Proposition 1. Existence of steady state equilibrium
The tuple {k ∗ , 1 + r∗ , w∗ , (si∗ , ci∗ )Li=1 , P ∗ , Q∗ } is a steady-state equilibrium
if and only if
1
βh =
, 1 + r∗ = f 0 (k ∗ ), w∗ = f (k ∗ ) − f 0 (k ∗ )k ∗ ,
(7)
1 + r∗
P ∗ = λLf (k ∗ ),
(8)
µΦ(Q∗ ) = µ(Q∗ − P ∗ ) + Lm̄,
(9)
si∗ = 0, i ∈ Hl ,
(10)
si∗ ≥ 0, i ∈ Hh ,
(11)
L
X
i=1
si∗ =
X
si∗ = Lk ∗ ;
(12)
i∈Hh
c∗ + s∗ + m̄ = w∗ + (1 + r∗ )s∗ .
Proof. See Appendix A.1. ¤
(13)
When we assumed that m̄ is not too large we meant that w∗ − m̄ > 0. In
a steady-state equilibrium, all impatient households find themselves at the
same position since they save nothing. As for the patient consumers, the
distribution of savings among them is arbitrary.
7
3
Voting equilibria
Since consumers in our model are heterogeneous it is reasonable to expect
that they will disagree over desired level of maintenance. One way to resolve
this disagreement is to choose it by majority voting. If the level of maintenance is to be determined by majority voting, which level will be chosen? If
policy choices were one-dimensional, we would refer to the median voter theorem, but in our intertemporal model this theorem cannot be applied directly
since, formally speaking, in such models policy choices are multi-dimensional.
However, if we constraint our consideration to steady states, median voter
approach to decision-making seems to be quite reasonable. At the same time,
we should not mistakenly think that mere consideration of steady states
only makes policy choices one-dimensional. Fortunately, as we show in this
section, for some reasonable definition of voting equilibrium, in a voting
steady-state equilibrium the level of maintenance will be chosen just by the
median voter.
The optimal value of problem P1 for consumer i is a function of ŝ−1 , Q̂0
and m̄. We will denote this optimal value by Vi,0 (ŝ−1 , Q̂0 , m̄).
Definition 3. Preferred change in maintenance
If for some t, Vi,0 (ŝ−1 , Q̂0 , m̄) is differentiable in m̄t we say that consumer
,Q̂0 ,m̄)
> 0 and is in favor of
i is in favor of increasing m̄t if ∂Vi,0 (ŝ∂−1
m̄t
decreasing m̄t if m̄t > 0 and
∂Vi,0 (ŝ−1 ,Q̂0 ,m̄)
∂ m̄t
< 0. ¤
For an equilibrium path
i∗ L
∗
∗ ∞
E = {kt∗ , 1 + rt∗ , wt∗ , (si∗
t−1 , ct )i=1 , Pt , Qt }t=0
denote by Nt+ (E) the number of consumers who are in favor of increasing m̄t
and by Nt− (E) the number of consumers who are in favor of decreasing m̄t .
Definition 4. Intertemporal voting equilibrium
We say that an equilibrium path E is intertemporal voting equilibrium
path if, for all t = 0, 1, . . ., Nt+ (E) < L/2, Nt− (E) < L/2. ¤
8
Whether an equilibrium path is an intertemporal voting equilibrium path
or not crucially depends on the choice of (m̄t )∞
t=0 . We are not ready to
discuss the existence and properties of intertemporal voting equilibrium paths
starting from arbitrary initial states but we can describe steady state voting
equilibria.
Definition 5. Steady state voting equilibrium
Let an m̄ ≥ 0 be given and let m̄ = (m̄t )∞
t=0 with m̄t = m̄, t = 0, 1, . . .
∗
We call a steady-state equilibrium {k , 1 + r∗ , w∗ , (si∗ , ci∗ )Li=1 , P ∗ , Q∗ } a
steady state voting equilibrium if the corresponding equilibrium path is
an intertemporal voting equilibrium path. ¤
To answer the question of whether a steady state equilibrium is a steady
state voting equilibrium it is necessary to know which consumers are in favor
of increasing of m̄t = m̄ at each time t and which ones are in favor of its
decreasing.
We know that for each i the sequence (s̃it−1 , c̃it , Q̃t )∞
t=0 given by
s̃it−1 = si∗ , c̃it = ci∗ , Q̃t = Q∗ ,
(14)
is a solution to
max
+∞
∞
X
(ct )t=0 ,(Qt )+∞
t=0 t=0
βit [u(ct ) + v(Qt )],
(15)
µΦ(Qt+1 ) = µ(Qt − P ∗ ) + Lm̄t , t = 0, 1, . . . ,
ct + st + m̄t ≤ w∗ + (1 + r∗ )st−1 , t = 0, 1, . . . ,
(16)
(17)
si−1 = ŝi−1 , Q0 = Q̂0 ,
ct ≥ 0, st ≥ 0, Qt ≥ 0, t = 0, 1, . . .
(18)
(19)
at ŝi−1 = si∗ , Q̂0 = Q∗ .
Lemma 1. Differentiability of value function w.r.t.
nance and sign of derivative
mainte-
Let for some i the sequence (s̃it−1 , c̃it , Q̃t )∞
t=0 given by (14) be a solution to
problem (15)-(19) at a given m̄t = m̄ ≥ 0 and at ŝi−1 = si∗ , Q̂0 = Q∗ .
Then for all t = 0, 1, . . ., Vi,0 (si∗ , Q∗ , m̄) is differentiable in m̄t and
9
∂Vi,0 (si∗ , Q∗ , m̄)
> (<)0 ⇔ βi Lv 0 (Q∗ ) > (<)µu0 (c∗ )(Φ0 (Q∗ ) − βi ). (20)
∂ m̄t
Proof. See appendix A.2 ¤
The interpretation of lemma 1 runs as follows. Consider the first inequality of
equation (20) (>) at a given maintenance m̄t . In this case, out of a marginal
change in maintenance, the induced marginal utility of environmental quality,
i.e. the LHS of equation (20), is larger than the induced marginal utility of
consumption, i.e. the RHS of (20). This is likely to happen when the given
maintenance m̄t is low. This entails that the consumer is in favor of an
increase in maintenance. In the opposite case, the given maintenance m̄t
is likely to be large so that the induced marginal utility of consumption is
higher than the induced marginal utility of quality and the consumer is in
favor of decreasing maintenance.
To
check
whether
a
steady-state
equilibrium
{k ∗ , 1 +
∗
i∗ i∗ L
∗
∗
r , w , (s , c )i=1 , P , Q } is a voting steady-state equilibrium or not,
consider the following problem in which household i is free to choose mt :
∗
P

t
max(ct )+∞ ,(st )+∞ ,(mt )+∞ ,(Qt )+∞ ∞

t=0 βi [u(ct ) + v(Qt )],

t=0
t=0
t=0
t=0








subject to




P2 = 













µΦ(Qt+1 ) ≤ µ(Qt − P ∗ ) + Lmt ,
ct + st + mt ≤ w∗ + (1 + r∗ )st−1 ,
s−1 = ŝ−1 , Q0 = Q̂0 ,
ct ≥ 0, st ≥ 0, mt ≥ 0, Qt ≥ 0,















t = 0, 1, . . . ,
t = 0, 1, . . . ,
t = 0, 1, . . .














We say that (s̃, c̃, m̃, Q̃) ∈ R4+ determines a steady-state solution to this
problem if the sequence (s̃t−1 , c̃t , m̃t , Q̃t )∞
t=0 given by
s̃t−1 = s̃, c̃t = c̃, m̃t = m̃, Q̃t = Q̃
(21)
is its solution at ŝ−1 = s̃ and Q̂0 = Q̃.
Prior to formulating the following lemma, remind that β h (1 + r∗ ) = 1 and
hence that βi (1 + r∗ ) < 1, ∀i ∈ Hl , and βi (1 + r∗ ) = 1, ∀i ∈ Hh .
Lemma 2. Characterization of steady state voting equilibrium
The tuple (s̃, c̃, m̃, Q̃) ∈ R4+ determines a steady-state solution to P2 if
and only if
10
βi (1 + r∗ ) < 1 ⇒ s̃ = 0
(22)
βi Lv 0 (Q̃) ≤ µu0 (c̃)(Φ0 (Q̃) − βi ) (= if m̃ > 0)
(23)
c̃ = w∗ + r∗ s̃ − m̃
(24)
µ(Φ(Q̃) − Q̃ + P ∗ ) = Lm̃
(25)
Proof. See appendix A.3. ¤
To simplify the presentation we can get rid of m̃ by noticing that m̃ >
0 ⇔ c̃ < w∗ + r∗ s̃ and rewriting conditions (23)-(24) as follows:
µ ∗
µ
P ) + (Q̃ − Φ(Q̃)),
L
L
c̃ ≤ w∗ + r∗ s̃,
c̃ = (w∗ + r∗ s̃ −
βi Lv 0 (Q̃) ≤ µu0 (c̃)(Φ0 (Q̃) − βi ) (= if c̃ < w∗ + r∗ s̃).
(26)
(27)
(28)
Equation βi Lv 0 (Q) = µu0 (c)(Φ0 (Q) − βi ) implies an increasing dependence
of c on Q. As for equation c = (w∗ + r∗ s̃ − Lµ P ∗ ) + Lµ (Q − Φ(Q)), for any
given s̃, it specifies a dependence of c on Q which is simply decreasing or
first increasing and then decreasing (the increasing branch of this curve is
irrelevant for our purposes).
In what follows we assume that for each i = 1, . . . , L and for r∗ and w∗
given by (7) and for any s̃ ≥ 0 satisfying (22) there exist c̃ > 0 and Q̃ > 0
satisfying (26)-(28) i.e. there exists c̃ > 0, m̃ ≥ 0, Q̃ > 0 such that (s̃, c̃, m̃, Q̃)
determines a steady-state solution to problem P2 .
Now consider a steady-state equilibrium {(si∗ , ci∗ )Li=1 , k ∗ , Q∗ }, put all
households in ascending order of their savings, and take the median one,
im . The following theorem follows from Lemmas 1 and 2.
Theorem 1. Steady state voting equilibrium and median voter
A steady-state equilibrium {(si∗ , ci∗ )Li=1 , k ∗ , Q∗ } is a voting steady-state
equilibrium at some m̄ if and only if for i = im , the tuple (si∗ , ci∗ , m̄, Q∗ )
is a steady-state solution to problem P2 . ¤
It follows from this theorem that, in equilibrium, there are two possible
cases depending on whether cim ∗ = w∗ + r∗ sim ∗ (⇔ m̄ = 0) or cim ∗ < w∗ +
11
r∗ sim ∗ (⇔ m̄ > 0). They are illustrated by the left and right panel of Fig. 1,
on which we take sim ∗ as given. On these graphs the three curves C1 , C2 and
C3 are defined as follows:
C1 :
βim Lv 0 (Q) = µu0 (c)(Φ0 (Q) − βim )
∗
∗ im ∗
C2 :
c=w +r s
C3 :
c = (w∗ + r∗ sim ∗ −
(29)
(30)
µ
µ ∗
P ) + (Q − Φ(Q))
L
L
(31)
Let us describe more precisely these two regimes:
Regime 1 - Zero-maintenance The equilibrium point (Q∗ , cim ∗ ) is at the
intersection of the C2 curve and the C3 curve (see figure 1a) and, as far
as curve C1 is concerned, we have βim Lv 0 (Q∗ ) < µu0 (cim ∗ )(Φ0 (Q∗ )−βim ).
Regime 2 - Positive-maintenance The equilibrium point (Q∗ , cim ∗ ) is at
the intersection of the C1 curve with the C3 curve (see figure 1b) and,
as far as curve C2 is concerned, we have cim ∗ < w∗ + r∗ sim ∗ .
In combination with the above-mentioned two regimes (m̄ > 0 and m̄ =
0), two cases should be be distinguished:
Case 1 - Impatient median voter: βim = β l ;
Case 2 - Patient median voter: βim = β h .
In Case 1 savings of the median voter are determined uniquely, they are
nil: sim ∗ = 0. This makes comparative statics quite determinate. As for Case
2, where βim = β h , the savings of the median voter, sim ∗ , are not determined
2
uniquely. They can take any value in the interval [0, L+1
Lk ∗ ]. Therefore when
making a comparative statics exercise we should remember that a change in
a parameter will have an indeterminate effect on the savings of the median
voter. To circumvent this problem, when making all exercises except the last
one, we make the additional assumption that the ratio sim ∗ /k ∗ remains intact
when a parameter changes.
In both cases, the regime of equilibrium maintenance can be nil or positive.
12
C
C
C3
C1
C3
C1
E*
C2
C2
E*
Q*
Q
Q*
Q
Figure 1: Left: Zero-Maintenance Equilibrium (Regime 1) - Right: Positive
Maintenance Equilibrium (Regime 2)
4
Comparative statics
In this section we make the following
Assumption A:
w0 (k ∗ ) − m0 (k ∗ ) > 0,
where w(k) = f (k) − f 0 (k)k, m(k) = µλf (k).
To interpret this assumption, notice that a small increase in capital intensity k by, say, ∆k will result in an increase in pollution by ∆P ≈ Lf 0 (k)∆k.
To keep environmental quality intact it is necessary to increase maintenance by ∆M = µ∆P ≈ µλLf 0 (k)∆k and hence to increase tax rate by
∆m ≈ µλf 0 (k)∆k = m0 (k)∆k. At the same time wage rate also will increase by ∆w ≈ w0 (k)∆k. Thus Assumption A reads that the increase in tax
rate required for keeping environment intact after an increase in the capital
intensity is lower than the corresponding increase in wage rate.
If the production function is Cobb-Douglas, f (k) = k α , 0 < α < 1,
Assumption A can be rewritten in the following simpler form:
(1 − α) > µλ.
At a given value of α , this inequality fulfils if the proportion pollution/output
is not too high and the environmental efficiency of maintenance is not too
low.
13
Geometrically, Assumption A implies that the curve C3 will shift upwards
after an increase in the capital intensity. We will indicate which of our
conclusions are based on this assumption.
4.1
An increase in total factor productivity
In this subsection we assume an increase in total factor productivity by introducing a scale parameter a to the production function, which one becomes
aF (K, L) = Laf (k), where a represents the total factor productivity. The
impact of an increase in total factor productivity will depend on the regime
the economy follows in equilibrium.
Regime 1. Zero-maintenance Equilibrium
In this regime, a small increase in a will lead to an increase in k ∗ , w∗
and w∗ + r∗ sim ∗ . Hence, it will also increase the output level Lf (k ∗ ) and
pollution P ∗ but will not make maintenance positive. As a consequence,
the environmental quality Q∗ will decrease. Graphically (see Figure 2, left
panel), C2 will shift upwards due to the increase in w∗ + r∗ sim ∗ . C3 will also
shift upwards, but to a smaller extent, since both w∗ and P ∗ increase.
If the increase in a becomes too large, then the economy switches to
Regime 2, the Positive-maintenance Equilibrium.
Regime 2. Positive-maintenance Equilibrium
In that regime an increase in a will shift C3 upwards, as shown in Figure
2, right panel, and hence to an increase in Q∗ (this is not necessarily true if
Assumption A does not fulfil).
To sum up, if the economy starts under Regime 1, then an increase in at
from 0 to +∞ first leads to an decrease in the environmental quality Q∗ , and
then to a increase, as shown in Figure 2. If one consider that less developed
countries most likely correspond to Regime 1 and wealthy countries to Regime
2, then this conclusion means that technological progress first goes with a
decrease in environmental quality, and after some stage of development to an
increase in environmental quality. This conclusion provides a new rationale
for an Environmental Kuznets Curve to exist in the presence of voting (see
e.g. Stockey, 1998).
Let us now turn to two comparative statics related to households’ preferences.
14
C
C
C1
C3
C3
E*
C2
C1
C2
E*
Q*
Q
Q*
Q
Figure 2: An increase in total factor productivity in Regime 1 (Left) and
Regime 2 (Right)
4.2
Patient agents become more patient: an increase
in β h
Let us first consider an increase in β h , meaning that patient agents become
even more patient. The effects on the environmental quality will depend on
which regime the economy experiences.
Under Zero-Maintenance Equilibrium (Regime 1), a small increase in β h
leads to an increase in capital intensity k ∗ , wage rate w∗ , output Lf (k ∗ ) and
pollution P ∗ , but it cannot make maintenance positive. Hence Q∗ decreases
as β h increases under Regime 1. Graphically (see Fig. 2, left panel), C2 shifts
upwards due to the increase in w∗ ; C3 also shifts upwards, but to a smaller
extent (w∗ will increase but P ∗ will also increase).
If moreover the median voter is patient, Case 2, then, C1 shifts to the
right. As a consequence the economy may switch the economy to the Positivemaintenance regime (regime 2).
Under Positive-Maintenance Equilibrium (Regime 2, see Fig. 2b) an increase in β h will lead to an upward shift of C3 and, in Case 2, to a shift
of C1 to the right. Hence Q∗ will increase (this is not necessarily true if
Assumption A does not fulfil).
15
4.3
Impatient agents become less impatient: an increase in β l
Now, let us consider an increase in β l , which means that impatient agents
become less impatient. The effect on Q∗ will depend on whether the median
consumer is impatient or patient, what we referred to as Case 1 and Case 2,
respectively.
In the case where the median voter is impatient (Case 1, βim = β l ), then
the two regimes must be considered.
• under Zero-Maintenance Equilibrium (Regime 1), a small increase in
β l does not change k ∗ , w∗ , Lf (k ∗ ) or P ∗ . It neither changes Q∗ . This
case results in a shift of C1 to the right, as illustrated in Fig. 3. Still,
if the increase in β l becomes large enough, then the economy switches
to Regime 2;
• under Positive-Maintenance Equilibrium (Regime 2), a small increase
in β l does not change k ∗ , w∗ , Lf (k ∗ ) or P ∗ , but it does increase Q∗ , as
illustrated in Fig. 3.
In the case where the median voter is patient (Case 2, βim = β h ), then it
is clear that changing β l has no effect on Q∗ .
4.4
An increase in sim ∗ other things equal
In this last subsection we carry out a comparative statics exercise relevant
only in Case 2, where the median voter is patient and, consequently, his
savings can be positive.
Here we do not make the assumption that sim ∗ /k ∗ remains unchanged.
On the contrary we assume that an increase in sim ∗ is not due to an increase
in capital intensity k ∗ and, therefore, that it only reflects a change in the
distribution of savings among the patient consumers and hence a change in
income distribution (more precisely, an increase in the median income relative
to the mean).
16
C
C1Regime 1
C3
C1Regime 2
C2
Q
Figure 3: Impatient agents become less impatient
• under Zero-Maintenance Equilibrium (Regime 1), a small increase in
sim ∗ , other things equal, will shift C2 and C3 upwards by the same
magnitude. Hence, consumption of the median voter cim ∗ will increase,
but the environmental quality Q∗ will remain unchanged. A larger
increase in sim ∗ may lead the economy to Regime 2.
• under Positive-Maintenance Equilibrium (Regime 2), a small increase
in sim ∗ , other things equal, will shift C3 upwards, while letting C1 untouched. Hence the environmental quality Q∗ will increase.
Following the politico-economic literature about income inequality (see
e.g. Meltzer and Richard (1981)), an income distribution is called more
equal, the higher the median income is relative to the mean (this is only reasonable in the case where the median income does not exceed the mean). For
our model this implies that in developed economies, where m̄ > 0, lower inequality has a positive effect on environmental quality, whereas in less developed economies, where m̄ = 0, inequality itself does not effect environmental
quality.
5
Conclusion
In this paper, we assumed that the population is exogenously divided into
two groups: one with patient households and the other with impatient house17
holds. The environmental maintenance is voted by the households. We look
for steady state voting equilibria and find that the median voter theorem
applies.
We show that (i) an increase in total factor productivity may produce
effects described by the Environmental Kuznets Curve, (ii) an increase in
the patience of impatient households may improve the environmental quality
if the median voter is impatient and maintenance positive, finally (iii) if the
median income is lower than the mean, a decrease in inequality can lead to
an increase in the environmental quality if the median voter is patient and
maintenance is positive.
It is of interest to compare our model with a representative agent model.
Such a model can be identified with the particular case of our model where
all consumers are patient and savings are distributed evenly across agents.
In the case of impatient median voter, the level of environmental quality
predicted by our model is lower than predicted by the representative agent
model. The same is true if the median voter is patient, but the median
income is lower that the mean. Only in the case where the median income
is higher than the mean our model predicts a higher level of environmental
quality. However, this case should not be regarded as typical.
One observation concerning possible policy implications of our model can
be made. In the case where a majority of agents are impatient, the median
voter saves nothing. He/she spends his/her wage on consumption and maintenance. Thus, the higher the wage rate, the higher the level of maintenance
supported by the median voter. It follows that any macroeconomic policy
having an impact on income distribution will effect the level of maintenance
supported by the median voter. For example, Mankiw (2000) shows that, in
a model with heterogeneous consumers, government debt has no impact on
the steady-state capital stock and national income, but it does effect income
distribution. Namely, an increase in government debt will decrease the posttax wage rate. It follows that an increase in government debt will decrease
the quality of environment.
6
References
• Becker, R. A. (1980) ”On the long-run steady-state in a simple dynamic model of equilibrium with heterogeneous households” Quarterly
Journal of Economics, Vol. 94, pp. 375-383.
• Becker, R. A. (2006) ”Equilibrium dynamics with many agents”, in:
18
Dana R.-A., Le Van, C., Mitra, T. and Nishimura, K. (Eds.) Handbook
of Optimal Growth 1. Discrete Time. Springer, 2006.
• John, A. and Pecchenino R. (1994), “An overlapping generations model
of growth and the environment”, The Economic Journal, vol. 104, pp.
1393-1410.
• Jouvet, P.-A., Ph. Michel and P. Pestieau (2004) “Public and private environmental spending. A political economy approach.”, CORE
Discussion Paper 2004/68.
• Jouvet, P.-A., Ph. Michel and G. Rotillon (2005) , “Optimal growth
with pollution: How to use pollution permits?”, Journal of Economic
Dynamics and Control,vol. 29 (9) pp. 1597-1609.
• Kempf, H. and S. Rossignol (2007), “Is inequality harmful for the environment in a growing economy?”, Politics and Economics, vol. 19(1),
pp. 53-71
• Mankiw, N.G., (2000) “The savers-spenders theory of fiscal policy.”
American Economic Review 90, 120-125.
• McKenzie, L. (1986) ”Optimal economic growth, turnpike theorems
and comparative dynamics”, in Arrow K.J. and Intriligator M. D. (eds.)
Handbook of Mathematical Economics, Vol. 3, pp.1281-1355.
• Meltzer, A.H., Richard, S.F. (1981) ”A rational theory of the size of
government”, Journal of Political Economy 89, 914-927.
• Stokey, N. L. (1998), “Are there limits to growth?”, International Economic Review, 39: 1-31.
A
A.1
Appendices
Proof of Proposition 1
It is sufficient to notice that since in a steady-state equilibrium we have
µΦ(Q∗ ) = µ(Q∗ − P ∗ ) + Lm̄,
and for each i, the sequence
(s̃it−1 , c̃it )∞
t=0
19
given by
s̃it−1 = si∗ , c̃it = ci∗ ,
is a solution to
max
∞
X
βit u(ct ),
t=0
∗
ct + st ≤ (w − m̄) + (1 + r∗ )st−1 ,
si−1 = si∗ ,
ct ≥ 0, st ≥ 0,
and to refer to Becker (1980). ¤
A.2
Proof of Lemma 1
Define the value functions Vi,t by
Vi,t (st−1 , Qt , m̄t ) = max{
∞
X
βiτ (u(ct+τ ) + v(Qt+τ ))
τ =0
| µΦ(Qt+τ +1 ) ≤ µ(Qt+τ − P ∗ ) + Lm̄t+τ ,
ct+τ + st+τ + m̄t+τ ≤ w∗ + (1 + r∗ )st+τ −1 ,
Qt+τ +1 ≥ 0, ct+τ ≥ 0, st+τ ≥ 0, τ = 0, 1, . . .}.
We have:
Vi,t (si∗ , Q∗ , m̄t ) = u(ci∗ ) + v(Q∗ ) + βi Vi,t+1 (si∗ , Q∗ , m̄t+1 ), t = 0, 1, . . . ,
where m̄t = (m̄t , m̄t+1 , . . .) = (m̄, m̄, . . .) and m̄t+1 = (m̄t+1 , m̄t+2 , . . .) =
(m̄, m̄, . . .).
It is clear that
∂Vi,t (si∗ , Q∗ , m̄t )
∂Λi,t (Q∗ , m̄t ) ∂Γi,t (si∗ , m̄t )
=
+
,
∂ m̄t
∂ m̄t
∂ m̄t
where the functions Λi,t and Γi,t are defined as follows:
Λi,t (Qt , m̄t ) = max{
∞
X
βiτ v(Qt+τ )
τ =0
| µΦ(Qt+τ +1 ) ≤ µ(Qt+τ − P ∗ ) + Lm̄t+τ , Qt+τ +1 ≥ 0, τ = 0, 1, . . .},
20
Γi,t (st−1 , m̄t ) = max{
∞
X
βiτ u(ct+τ )
τ =0
| ct+τ + st+τ + m̄t+τ ≤ w∗ + (1 + r∗ )st+τ −1 ,
ct+τ ≥ 0, st+τ ≥ 0, τ = 0, 1, . . .}.
It is not difficult to check that
∂Λi,t (Q∗ , m̄t )
Lv 0 (Q∗ )
= βi
∂ m̄t
µ(Φ0 (Q∗ ) − βi )
and
∂Γi,t (si∗ , m̄t )
= −u0 (c∗ ).
∂ m̄t
Therefore,
∂Vi,t (si∗ , Q∗ , m̄)
Lv 0 (Q∗ )
∂Vi,0 (si∗ , Q∗ , m̄)
= βit
= βit+1
− βit u0 (c∗ ),
0
∗
∂ m̄t
∂ m̄t
µ(Φ (Q ) − βi )
which implies (20). ¤
A.3
Proof of Lemma 2
Using a traditional argument (see e.g. McKenzie (1986)) we can prove that
a sequence (s̃t−1 , c̃t , m̃t , Q̃t )∞
t=0 given by (21) is a solution to problem P2 if
and only if there exist q and p such that for
pt = βi pt−1 = ... = βit p,
qt+1 = βi qt = ... = βit+1 q.
the following relationships hold:
βit u0 (c̃t ) = pt ,
βit v 0 (Q̃t ) + qt+1 µ − qt µΦ0 (Q̃t ) = 0,
(1 + r∗ )pt ≤ pt−1 (= if s̃t−1 > 0),
qt+1 L − pt ≥ 0 (= if m̃t > 0),
qt+1 Q̃t + pt s̃t−1 →t→∞ 0,
or, equivalently,
u0 (c̃) = p,
v 0 (Q̃) = µq(Φ0 (Q̃) − βi ),
21
1
(= if s̃ > 0),
1 + r∗
βi Lq − p ≥ 0 (= if m̃ > 0).
βi ≤
The existence of such q and p is equivalent to conditions (22)-(23). ¤
22