ZHANG ARTICOLO FORMATTED 2
3
DYNAMIC INTERACTIONS AMONG
GROWTH, ENVIRONMENTAL CHANGE,
HABIT FORMATION, AND PREFERENCE
CHANGE
WEI-BIN ZHANG
Ritsumeikan Asia Pacific University, Japan
Received: March 27, 2013 Accepted: July 30, 2013 Online Published: October 10, 2013
Abstract
The purpose of this study is to construct an economic growth model with environmental
change and preference formation. The paper is focused on dynamic interactions among
capital accumulation, environmental change, habit formation, preference change, and
division of labor in perfectly competitive markets with environmental taxes on production,
wealth income, wage income and consumption. The model integrated the dynamic economic
mechanisms in the neoclassical growth theory, the environmental dynamics in traditional
models of environmental economics, and the literature of economic growth with habit
formation and within a comprehensive framework. It is showed that the motion of the
economic system is given by three nonlinear autonomous differential equations. We simulate
the time-invariant system. The simulation demonstrates some dynamic interactions among the
economic variables which can be predicted neither by the neoclassical growth theory nor by
the traditional economic models of environmental change. For instance, if the past
consumption has weaker impact on the current consumption, although the long-term
equilibrium of the dynamic system will not be affected, the transitional paths are shifted as
follows: initially the transitional path of the stock habit becomes lower than its original path;
the consumption level falls initially in association with falling in the propensity to consume;
the exogenous disturbance causes the propensity to consume to fall and the propensity to save
to rise; the national wealth and capital inputs of the two sectors are augmented; the labor
force is shifted initially from the industrial sector to the environmental sector, but
subsequently the direction is opposite before the labor distribution comes to its original
equilibrium point; the wage rate is enhanced in association with falling in the rate of interest;
the level of pollution falls initially, but rise subsequently; the output levels of the two sectors
and the total tax income are enhanced before they come back to their original paths.
Keywords: Habit formation; preference change; environmental tax; pollution; economic
growth.
Acknowledgements: The author is grateful for the financial support from the Grants-in-Aid
for Scientific Research (C), Project No. 25380246, Japan Society for the Promotion of
Science.
4
1. Introduction
Economic growth has close relationships with environmental changes. As far as neoclassical
growth theory is concerned, early formal modeling on tradeoffs between consumption and
pollution were found in the seminal papers by Plourde (1972) and Forster (1973). Interactions
between environmental change and economic growth have received increasingly attention in the
literature of economic growth and development (see, for instance, Pearson, 1994; Grossman,
1995; Copeland and Taylor, 2004; Stern 2004; Brock and Taylor, 2006; Dasgupta, et al.
2006). In the literature of economic development and environmental change many researches are
conducted to confirm or question the environmental Kuznets curve. The environmental Kuznets
curve refers to the phenomenon that per capita income and environmental quality follow an
inverted U-curve. A recent survey on the literature of the curve is given by Kijima et al.
(2010). In fact, in the increasing literature of empirical studies on relations between growth and
environmental quality rather than the suggested environmental Kuznets curve one finds different
relations such as inverted U-shaped relationship, a U-shaped relationship, a monotonically
increasing or monotonically decreasing relationship (Dinda, 2004; Managi, 2007; Tsurumi and
Managi, 2010). The various relations between economic development and environmental quality
the inability of economic theories for properly explaining these observed phenomena implies the
necessity that more comprehensive theories are needed. The purpose of this study is to introduce
habit formation and preference change into the literature of economic growth and environmental
change. As far as I am aware, there is no formal neoclassical growth model based on micro
economic foundation which deals with economic growth, environmental change, habit
formation, and preference change in an integrated framework.
People behave under influences of their habits. These habits are formed over years. People
may change their habits with different speeds. Habits are parts of preferences. Different people
have different preferences. The preference is also changeable over time for the same person. This
study tries to integrate habit formation, preference change and economic growth within a
comprehensive framework. Preferences are changeable and many factors may attribute to these
changes. For instance, Becker and Mulligan (1997) found that expenditures in health and
education have positive impacts saving (see also, Fuchs, 1982; Shoda et al., 1990; Olsen, 1993;
Kirby et al. 2002; Chao et al., 2009). Many empirical studies identify relations between
preference changes and other changes in social and economic conditions (e.g., Horioka, 1990;
Sheldon, 1997, 1998). In the literature of economic growth with preference change, economists
analyze preference change mainly by introducing time preference change in the Ramsey growth
model. A main approach to modeling relations between growth and preference change is the so-
called endogenous time preference. The formal modeling in continuous time formation was
initiated with Uzawa’s seminal paper (Uzawa, 1968). Following Uzawa, Lucas and Stokey
(1984) and Epstein (1987) establish relations between time preference and consumptions. Becker
and Barro (1988) build a time preference change model in which the parent’s generational
discount rate is connected to their fertility. There are other studies on the implications of
endogenous time preference for the macroeconomy (see, e.g., Epstein and Hynes, 1983;
Obstfeld, 1990; Shin and Epstein, 1993; Palivos et al., 1997; Drugeon, 1996, 2000; Stern, 2006;
Meng, 2006; Dioikitopoulos and Kalyvitis, 2010). The idea of analyzing change in impatience in
this study is influenced by the literature of time preference. We introduce changes in impatience
in an alternative utility proposed by Zhang (1993). Except the literature on time preference
change, this study is also influenced by the so-called habit formation or habit persistence model.
The model was initially proposed in formal economic analysis by Duesenberry (1949). Becker
(1992) explains the role of habit in affecting human behavior as follows: “the habit acquired as a
child or young adult generally continue to influence behavior even when the environment
5
changes radically. For instance, Indian adults who migrate to the United States often eat the same
type of cuisine they had in India, and continue to wear the same type clothing.” Habit formation
is also applied to different fields of economic analysis (for instance, Pollak, 1970; Mehra and
Prescott, 1985; Sundaresan, 1989; Constantinides, 1990; Campbell and Cochrane, 1999; de la
Croix, 1996; Boldrin et al., 2001; Christiano et al. 2005; Ravn et al., 2006; Huang, 2012). It
should be noted that since the research by Abel (1990), ‘catching up with the Joneses’ is often
used exchangeable with external habit formation.
The purpose of this paper is to study economic growth with environmental change and
preference change on the basis of the Solow one-sector growth model, Zhang’s approach to
household behavior, the neoclassical growth models with environmental change, the literature of
time preference and the literature of habit formation. The model in this paper is an extension of
Zhang’s two models on environmental change and habit formation. The interdependence
between savings and dynamics of environment is mainly based on Zhang (2011), while the habit
formation and preference change are based on Zhang (2012). Section 2 introduces the basic
model with wealth accumulation, environmental dynamics, habit formation and preference
change. Section 3 studies dynamic properties of the model and simulates the model, identifying
the existence of a unique equilibrium and checking the stability conditions. Section 4 conducts
comparative dynamic analysis with regard to some parameters. Section 5 concludes the study.
The appendix proves the analytical results in Section 3.
2. The Basic Model
The production side of the economy consists of one industrial sector and one environmental
sector. The industrial sector is similar to the standard one-sector growth model (see Burmeister
and Dobell 1970; Barro and Sala-i-Martin, 1995). The economy has only one (durable) good and
one pollutant in the economy under consideration. In the literature of environmental economics,
there are different kinds of environmental variables (e.g., Moslener and Requate, 2007; Repetto,
1987; Leighter, 1999; and Nordhaus, 2000). Capital of the economy is owned by the
households who distribute their incomes to consume the commodity and to save. Exchanges take
place in perfectly competitive markets. The population N is fixed and homogenous. The labor
force is fully employed by the two sectors. The commodity is selected to serve as numeraire
(whose price is normalized to 1), with all the other prices being measured relative to its price.
The industrial sector
Economic productivities are affected by pollution through different channels. For instance,
pollution may directly affect production technology or the productivity of any input (Grimaud,
1999; Chao and Peck, 2000; Gradus and Smulders, 1996; Ono, 2002) for the impact on the
productivity of any input. We assume that production is to combine labor force, ( ),tN i and
physical capital, ( ).tK i We add environmental impact to the conventional production function.
The production function is specified as follows
( ) ( )( ) ( ) ( ) ,1,0,,, =+>Γ= iiiiiiiiii AtNtKtEAtF ii βαβαβα (1)
where ( )tFi is the output level of the industrial sector at time ,t ( )EiΓ is a function of the
environmental quality measured by the level of pollution, ( ),tE iA is the total productivity, and
iα and iβ are respectively the output elasticities of capital and labor. The environmental impact
on the productivity ( )EiΓ is specified as follows
6
( )( ) ( ) .0, ≤=Γ ibi btEtE i
In perfectly competitive markets are competitive, labor and capital earn their marginal
products. The environmental quality is not decided by any individuals firm. Let ( )tr and ( )tw
represent respectively the rate of interest and wage rate as follows
( ) ( ) ( )
( )
( ) ( ) ( )
( )
,
1
,
1
tN
tF
tw
tK
tF
tr
i
iii
i
iii
k
τβτα
δ
−
=
−
=+ (2)
where kδ is the fixed depreciation rate of physical capital and iτ is the fixed tax rate,
.10 <<< iτ
Consumer behaviors
The representative household decides how much to consume and how much to save. This applies
the approach to behavior of the household proposed by Zhang (1993). Per capita wealth is
denoted by ( ).tk We have ( ) ( ) ,/ NtKtk = where is the total capital stock. The per capita
disposable current income which is the sum of the interest payment ( ) ( )tktr and the wage
payment ( )tw after taxation is given by
( ) ( ) ( ) ( ) ( ) ( ),11 twtktrty wk ττ −+−=
where kτ and wτ are respectively the tax rates on the interest payment and wage income. The
per capita disposable income is
( ) ( ) ( ).ˆ tktyty += (3)
The disposable income is distributed between saving and consumption. The representative
household spends the total available budget on saving, ( ),ts and the commodity, ( ).tc The
budget constraint is
( ) ( ) ( ) ( ),ˆ1 tytstcc =++ τ (4)
where cτ is the tax rate on the consumption. In this study we neglect the possibility that
consumers explicitly take care of environment. For modern economies, consumers tend to make
efforts in improving environment, for instance, by preferring to environment-friendly goods. As
observed by Selden and Song (1995), when society has a lower level of pollution, the
representative agent may not care much about environment and spends his resource on
consumption; however, as the environmental quality lowers and the agent earns more, the agent
may spend more resources on environmental improvement.
The household decides the two variables, ( )ts and ( ).tc This study specifies the utility
function as follows
( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ,0,,, 000000 >= − χλξχλξ tttEtstctU ttt
7
where ( )t0ξ is the utility elasticity of the commodity, called the propensity to consume, ( )t0λ is
the utility elasticity of saving, called the propensity to own wealth, and 0χ is the elasticity of
environmental quality. This type of utility function was initially proposed by Zhang (1993). As
Balcao (2001) and Nakada (2004), we assume that utility is negatively to pollution, which is a
side product of the production process. According to Munro (2009: 43), “environmental
economics has been slow to incorporate the full nature of the household into its analytical
structures. … [A]n accurate understanding household behavior is vital for environmental
economics.” In our approach,
For the representative household, )(tw and )(tr are given in markets. Maximizing )(tU
subject to (4) yields
( ) ( ) ( ) ( ) ( ) ( ),ˆ,ˆ tyttstyttc λξ == (5)
where
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .
1
,,
1 00
0
0
tt
ttt
tt
t
c λξ
ρλρλ
τ
ξρ
ξ
+
≡≡
+
≡
We call ( )tξ and ( )tλ respectively the relative propensities to consume and to save. It is the
values of the relative propensities, not the propensities, which matter in determining the
expenditure allocation.
Dynamics of wealth accumulation
According to the definition of ( ),ts the change in the household’s wealth is given by
( ) ( ) ( ).tktstk −=& (6)
The equation simply states that the change in wealth is equal to saving minus dissaving.
The demand and supply balance
The output of the industrial sector equals the sum of the level of consumption, the
depreciation of capital stock and the net savings. Hence we have
( ) ( ) ( ) ( ) ( ),tFtKtKtStC ik =+−+ δ (7)
where ( ) ( ) NtctC = is the total consumption, and ( ) ( ) ( )tKtKtS kδ+− is the sum of the net
saving and depreciation, where ( ) ( ) .NtstS ≡
Full employment of production factors
We use N te ( ) and K te ( ) to respectively stand for the labor force and capital stocks employed by
the environmental sector. As full employment of labor and capital is assumed, we have
( ) ( ) ( ),tKtKtK ei =+ ( ) ( ) .NtNtN ei =+ (8)
8
Environmental change
We now describe dynamics of the stock of pollutants ( ).tE Both production and consumption
pollute environment. The dynamics of the stock of pollutants is specified as follows
( ) ( ) ( ) ( ) ( ),0 tEtQtCtFtE ecif θθθ −−+=& (9)
where q qf c, , and q0 are positive parameters and
( ) ( ) ( ) ,0,,,)( >Γ= eeeeeeee AtNtKEAtQ ee βαβα (10)
where ,eA ,eα and eβ are positive parameters, and )0()( ≥Γ Ee is a function of .E As in
Gutiérrez (2008), the emission of pollutants during production processes is linearly positively
proportional to the output level. This is reflected by Ffθ in (10). As in John and Pecchenino
(1994), John et al. (1995), and Prieur (2009), in consuming one unit of the good the quantity cθ
is left as waste. We consider cθ is related to the technology and environmental sense of
consumers. The term E0θ is the rate that the nature purifies environment, where 0θ is called the
rate of natural purification. We use the term, ,ee ee NK
βα in Qe to reflect that that the purification
rate of environment is positively related to capital and labor inputs. The function )(EeΓ means
that the purification efficiency is related to the stock of pollutants. For simplicity, we specify eΓ
as follows ( ) ,ebee EE θ=Γ where eθ and eb are parameters.
The behavior of the environmental sector
In this study we consider that the environmental sector is financially supported by the
government. The sector decides the number of labor force and the level of capital employed. The
government’s tax revenue consists of the tax incomes on the industrial sector, consumption,
wage income and wealth income. Hence, the government’s income is given by
( ) ( ) ( ) ( ) ( ) ( ).tKtrtwNtCtFtY kwciie ττττ +++= (11)
As in Ono (2003), we assume that all the tax incomes are spent on environment. For
simplicity, we assume that all the revenue of the government is spent on protecting environment.
The environmental sector’s budget is
( )( ) ( ) ( ) ( ) ( ).tYtNtwtKtr eeek =++ δ (12)
According to Zhang (2011), the environmental sector employs labor and uses capital in such
a way that the purification rate achieves its maximum under the given budget constraint. The
sector’s optimal problem is given by
( )tQeMax s.t.: ( )( ) ( ) ( ) ( ) ( ).tYtNtwtKtr eeek =++ δ
The optimal solution is
( )( ) ( ) ( ) ( ) ( ) ( ),, tYtNtwtYtKtr eeeek βαδ ==+ (13)
9
where
.,
ee
e
ee
e
βα
β
β
βα
α
α
+
≡
+
≡
The time preference and the propensity to hold wealth
Following Zhang (2012), we introduce preference change through making the propensity to own
wealth and propensity to consume endogenous variables. Zhang’s approach is influenced by the
traditional approach to preference change in economic theory. To illustrate the approach in this
study, we consider a traditional modeling framework by Chang et al. (2011) in which the
representative household maximizes the following discounted lifetime utility with perfect
foresight
( ) ( ) ,,
0∫
∞ − tdemcu tρ
in which u is the utility function, c is consumption, and m is holdings of real money
balances. The time preference ( )tρ is endogenously determined (see also, Uzawa, 1968;
Epstein, 1987; Obstfeld, 1990; and Shi and Epstein, 1993). The variable changes as follows
( ) ( )( ) ,
0∫ ∆=
t
sdsutρ
where 0>∆ is an instantaneous subjective discount rate at time ,s which satisfies ,0' >∆
,0" >∆ and .0' >∆−∆ u We have
( ) ( )( ).tut ∆=ρ&
There are many other studies with endogenous time preference (for instance, Dornbusch and
Frenkel, 1973; Persson and Svensson, 1985; Blanchard and Fischer, 1989; Orphanides and
Solow, 1990; Das, 2003; Hirose and Ikeda, 2008). Although this study does not follow the
Ramsey approach in modeling behavior of household, we will adapt the ideas about time
preference within the Ramsey framework. The time preference in the traditional approach is
related to real wealth or/and current consumption. This study treats the propensity to save as a
function of the wage rate and wealth. Following Zhang (2012), the dynamics of the propensity to
save is
( ) ( ) ( ),0 tktwt kw λλλλ ++= (14)
where ,0>λ ,wλ and kλ are parameters. When ,0== kw λλ ( )t0λ is constant. If we follow
Uzawa’s idea, then it is reasonable to assume 0>wλ and .0=kλ If we follow the assumption
that the rate of time preference is positively related to wealth, for instance, accepted by Smithin
(2004) and Kam and Mohsin (2006), then 0=wλ and .0>kλ
10
The habit formation and the propensity to consume
The modeling of the propensity to save is influenced by the literature of time preference change.
In order to model how the propensity to consume, we will adapt the basic ideas in the habit
formation approach to our framework. To illustrate the ideas in the traditional approach, we
introduce the following habit formation (e.g., Alvarez-Cuadrado et al., 2004; and Gómez,
2008)
( ) ( ) ( ) ( ) ,10,0,10 ≤≤>= −
∞−
−
∫ φρρ
φφ sdsCsCet
t
tsh
h
where ( )tC is the consumer’s consumption and ( )tC is the economy-wide average
consumption. A larger value for 0h implies that the household puts lower weights to more
distant values of the levels of consumption. Taking the derivatives the equation with respect to
time yields
( ) ( ) ( ) ( )[ ].10 tsCsCt hhh& −= −φφ
If ,0=φ the habit formation corresponds to the model with external habits. If ,1=φ the
habit formation corresponds to the model with internal habits. If ,10 << φ habits arise from
both the consumer’s and average past consumption. There are other models with habit formation
(Deaton and Muellbauer, 1980; Carroll, 2000; Fuhrer, 2000; Kozicki and Tinsley, 2002;
Amano and Laubach, 2004; Carroll et al., 1997; Corrado and Holly, 2011). Following Zhang
(2012), this study also applies the concept of habit stock to analyze how the past consumption
affects the current preference. The habit formation is specified as
( ) ( ) ( )[ ].0 ttct hhh& −= (15)
Equation (15) corresponds to the model with internal habits. If the current consumption is
higher than the level of the habit stock, then the level of habit stock will rise, and vice versa. The
propensity to consume is relate to the habit stock as follows
( ) ( ) ( ),0 ttwt hw hξξξξ ++= (16)
where ,0>ξ wξ and 0≥hξ are parameters. If 0=wξ and ,0=hξ ( )t0ξ is constant. The term
( )tywξ implies that the propensity to consume is affected by the wage rate. If ,0)(<>wξ then a
rise in the wage rate enhances (lowers) ( ).0 tξ It is reasonable to assume .0≥wξ The term
( )th hξ shows that if ( )th rises, the propensity to consume will rise, and vice versa.
We have thus built the dynamic model. We now examine dynamics of the model.
3. The Motion of the Economic System
The appendix confirms that the motion of the economic system is given by three autonomous
differential equations with ( ) ( )ttz h, and ( ).tE as the variables, where ( )tz is a new variable
defined by
11
( ) ( )( ) .ktr
tw
tz
δ+
≡
The following lemma shows that once we solve the time-invariant system, we know the
values of all the other variables in the economy at any point of time.
Lemma 1
The motion of the three variables, ( ) ( )ttz h, and ( ),tE is obtained by solving the following
three autonomous differential equations with
( ) ( ) ( ) ( )( ),,, tEttztz z h& Λ=
( ) ( ) ( ) ( )( ),,, tEttzt c hh& Λ=
( ) ( ) ( ) ( )( ),,, tEttztE E h& Λ= (17)
in which ,zΛ ,cΛ and EΛ are functions of ( ) ( )ttz h, and ( )tE defined in the appendix. All the
other variables are solved as functions of ( ) ( )ttz h, and ( )tE as follows: ( )tr and ( )tw by (A2)
→ ( )t0ξ by (A16) → ( )t0λ by (A15) → ( )tλ and ( )tξ by (5) → ( )tKi and ( )tKe by (A7) →
( ) ( ) ( )tKtKtK ei += → ( ) ( ) NtKtk /= → ( )tNi and ( )tN x by (A1) → ( )tFi by (1) → ( )tQe
by (10) → ( )tŷ by (3) → ( )tc and ( )ts by (10).
From the procedure in Lemma 1 we can get the value of any variable at any point of time as
functions of ( ) ( )ttz h, and ( ).tE The three dimensional autonomous differential equations are
nonlinear. It is almost impossible to get analytical solution of the time-invariant system.
Nevertheless, we can use a common computer to follow the motion of the three-dimensional
time-invariant system. To simulate the model, we choose the following parameter values
.03.0,05.0,1.0,1.0,05.0,05.0
,05.0,05.0,40.0,10.0,2.0,1.0,02.0,01.0
,6.0,05.0,05.0,4.0,4.0,3.0,5.0,1,5
0
0
======
=======−=
=−=−=======
kcfwk
cihwkw
eieeisi bbAAN
δθθθττ
ττξξξλλ
λβαα
h (18)
In the remainder of this study, the depreciation rate is fixed as .03.0=kδ The population
is chosen 10 and the total available time is unity. In our neoclassical model the population
size has no impact on the per-capita variables, even though it affects the aggregate variable
levels. The chosen values of the available time and the population will not affect our main
conclusions. The total productivity and the output elasticity of capital of the capital goods
sector are respectively ,1.1 ,35.0 and the total productivity and the output elasticity of capital
of the capital goods sector are respectively 9.0 and .30.0 It should be noted that both in
theoretical simulations and empirical studies the output elasticity of capital in the Cobb-Douglas
production is often valued approximately equal to 3.0 and the value of the total productivity is
chosen to be close to unity (e.g., Miles and Scott, 2005; Abel, Bernanke, Croushore, 2007).
Although the chosen values of the preference parameters are not empirically based, we choose
the coefficients associated with the wage and wealth very small so that we may effectively
analyze the effects of changes in these coefficients on the economic structure. We now specify
the initial conditions to see how the variables change over time. To follow the motion of the
system, we choose the following initial conditions:
12
( ) ( ) ( ) .3.10,120,5.60 === hEz
Figure 1 plots the simulation result. We first observe that the habit stock of leisure time
falls while the habit stock of consumer goods rise till they respectively approach the leisure
time and the consumption level of consumer goods. This happens as the two stocks are
initially different from their corresponding variables. The work time, total labor supply and
labor inputs of the two sectors are increased over time. The total capital and capital input of
the consumer goods sector fall, while the capital input of the capital goods sector rises. The
price and wage rate fall slightly, while the rate of interest rises. The propensity to consume
rises, while the propensities to use leisure time and to save are affected only slightly. The
GDP and the output level of the consumer goods sector fall while that of the capital goods
sector rises. It should be noted that it takes much less time for the leisure time to converge to
its habit stock level than the consumption level of consumer goods to its habit stock.
Figure 1 shows that the variables tend to move towards stationary states. This implies the
existence of an equilibrium point. Our simulation identifies the equilibrium values of these
variables as follows
.09.1,72.0,27.0
,65.0,25.0,83.0,11.0,05.3,69.11
,50.0,50.4,83.0,53.0,31.5,96.10,75.14
00
====
======
=======
c
ei
eieei
c
wrKK
NNYQFEK
hλξ
λξ
It is straightforward to get the following three eigenvalues
.05.0,03.0137.0 −±− i
As the three eigenvalues have real negative parts, the equilibrium point is locally stable.
Hence, the system always approaches its equilibrium if it is not far from the equilibrium point.
This is important as it guarantees the validity of comparative dynamic analysis for transitional
paths.
4. Comparative Dynamic Analysis
From the analysis in the previous section we know that that the economic system has a unique
locally stable equilibrium. This guarantees that we can make comparative dynamic analysis. This
section conducts comparative dynamic analysis with regard to some parameters. It should be
remarked that because the system contains many variables which nonlinearly interact with each
other in a very complicated manner over time, it is not easy to accurately interpret how all these
variables interact over time.
13
Figure 1 – The Motion of the Economic System
Lower weights being put to more distant values of the levels of consumption
We first study the case where the household puts lower weights to more distant values of the
levels of consumption in the following way: .3.01.0:0 ⇒h The rise in the parameter also
means that the habit stock and the current level of consumption mutually converge faster. Figure
2 diagrams the simulation results. In this study we use the variable ( )tx∆ to stand for the change
rate of the variable, ( ),tx in percentage due to changes in some parameter value. Indeed, the
disturbance in the speed of adjustment will not affect the equilibrium of the dynamic system.
Nevertheless, the transitional paths towards the equilibrium points of the variables are strongly
perturbed. As the household puts lower weights to more distant values of the levels of
consumption, initially the transitional path of the habit stock is deviated from the original path.
As the speed is sped up, the path of the stock habit becomes lower than its original path. As the
habit stock becomes lower, the consumption level also falls initially in association with falling in
the propensity to consume. The disturbance causes the propensity to consume to fall and the
propensity to save to rise. As the relative propensity to save is λ increased, the national wealth is
augmented. The disturbance in the national wealth enables the two sectors to employ more
capital. The labor distribution path is also shifted. The labor force is shifted initially from the
industrial sector to the environmental sector, but subsequently the direction is opposite before the
labor distribution comes to its original equilibrium point. The wage rate is enhanced in
association with falling in the rate of interest. The level of pollution falls initially, but rise
subsequently. The output levels of the two sectors and the total tax income are enhanced before
they come back to their original equilibrium levels.
The environmental tax rate on consumption being enhanced
We now enhance the environmental tax rate on consumption as follows: .07.005.0: ⇒cτ The
impacts are plotted in Figure 4. As the tax rate on consumption is increased, the consumption
25 50 75 100
12
13
14
15
25 50 75 100
11
11.3
11.6
11.9
0 25 50 75 100
0.82
0.83
0.84
0 25 50 75 100
3
4
5
0 25 50 75 100
0.72
0.78
0.82
0.86
0 25 50 75 100
0.5
0.55
0.6
0.65
25 50 75 100
1.1
1.2
1.3
25 50 75 100
0.255
0.26
0.265
0.27
25 50 75 100
0.1
0.104
0.108
h
K
0ξ
λ
r
t t
c
t
t
t t
t
t t
E
iN
eK
iF
eY
w
eQ
eN
0λ
ξ
14
Figure 2 – The Household Puts Lower Weights on More Distant Values of Consumption
level and the habit stock of consumption are lessened. The lowered habit stock diminishes the
propensity to consumption, which implies augmenting in the propensity to save. The national
wealth is increased as the propensity to save is increased. More capital and labor resources are
located to the environmental sector. The environment is improved. Both the rate of interest
and wage rate are increased. It should be noted that in the Solow-type neoclassical growth
theory without endogenous environment, the rate of interest and wage rate are changed in the
opposite directions. In our model the two variables are changed in the same direction because
the environmental change affects the productivity. In our simulation case the improved
environment augments the productivity of the industrial sector. This leads to the same change
direction in the wage rate and the rate of interest. The net consequence of the rising national
wealth and capital input of the environmental sector leads to lowering in the capital input of
the industrial sector. As the capital and labor resources located to the industrial sector are
lowered, the output level of the industrial sector is reduced.
Wealth more strongly affecting the propensity to save
How the propensity to save may influence economic growth and development is a main question
in economics. It is well known that Adam Smith and Keynes have the opposite opinions about
the effects of a change in the saving propensity. Adam Smith holds that a rise in the propensity to
save will encourage long-run economic growth as to save more means more capital in the
economy, while Keynes argues that to save less means to create more job opportunities and
economic growth will be encouraged. In modern economics there is no convergence in empirical
studies about the impact of the propensity to save. Moreover, there are only few formal models
of economic growth with endogenous preference for saving. Our model explicitly introduces
endogenous propensity to save.
30 60 90
0
0.15
0.3
0.45
30 60 90�0.04
0.02
0.08
0.14
0 30 60 90
0
0.2
0.4
0 30 60 90
0
0.4
0.8
1.2
30 60 90
�0.15
�0.1
�0.05
0
0 30 60 90
0
0.1
0.2
0.3
30 60 90
�1.2
�0.8
�0.4
0
30 60 90
�6
�4
�2
0
30 60 90
�1.2
�0.8
�0.4
0
t
K∆
h∆
0ξ∆
eY∆
c∆
eK∆
iK∆
t t
t t
t
t
t t
0λ∆
iN∆
E∆
r∆
w∆ i
F∆
eQ∆
eN∆
λ∆
ξ∆
15
Figure 3 – A Rise in the Tax Rate on Consumption
We now examine how the economic system responds to the following exogenous
disturbances: .03.002.0: ⇒kλ The change augments the propensity to save. The relative
propensity is increased, while the relative propensity to consume is lessened. The change in
the preference results in the increase of national wealth. The two sectors’ capital inputs are
thus increased. More capital supply leads to lower rate of interest and rising wage rate. Labor
force is shifted from the environmental sector to the industrial sector. The net result of
lessened labor input and augmented capital input of the environmental sector is the rising
output level of the sector. As the increased production and consumption pollute the
environment more severely than before, the more efforts in cleaning environment do not
improve the environment.
The environmental sector improving its productivity
We now deal with the impact of the following productivity enhancement in the environmental
sector: .6.05.0: ⇒eA An immediate result of the productivity improvement is augments
of the environmental sector’s output level. The environment is provided as the economy is
more effectively in cleaning the environment. The labor distribution is slightly changed. The
improved environment enhances the productivities of the industrial sector. Although the
relative propensity to save is lowered, the national wealth is augmented. The two sectors employ
more capital inputs irrespective of rising in the cost of capital. The wage rate, habit stock of
consumption and the consumption level are raised.
30 60 90
0
4
8
12
30 60 90
�9
�6
�3
0
30 60 90
0
0.16
0.32
0.48
30 60 90
�1.5
3
7.5
12
30 60 90
0
4
8
12
0 30 60 90
0.04
0.07
0.1
0.13
30 60 90
�1.8
�1.2
�0.6
0
30 60 90
�1.2
�0.8
�0.4
0
0 30 60 90
0
0.3
0.6
0.9
t
K∆
h∆
0ξ∆
eY∆
c∆
eK∆
iK∆
t
t
t t
t
t
t
t
0λ∆
iN∆
E∆
r∆
w∆
iF∆
eQ∆
eN∆
λ∆
ξ∆
16
Figure 4 – The Propensity to Save Being More Strongly Affected by Wealth
Figure 5. The Environmental Sector Improving Its Productivity
Raising the tax rate on the industrial sector
Let us now change the tax rate on the output level of the industrial sector as follows:
.07.005.0: ⇒iτ The raised tax rate lowers the output level of the industrial sector. As the
tax revenue is increased, the environmental sector has more resources to employ more capital
30 60 90�0.9
0.2
1.3
2.4
30 60 90
�0.2
0.2
0.6
1
0 30 60 90
0
1
2
30 60 90
0
2
4
6
30 60 90
�0.9
�0.6
�0.3
0
30 60 90
3
4
5
30 60 90
�3
�2
�1
0
30 60 90
�3
�2
�1
0
1
30 60 90
�5
�3
�1
0
0 30 60 90
0
6.5
13
19.5
30 60 90
�15
�10
�5
0
0 30 60 90
0
0.45
0.9
1.35
0 30 60 90
0
0.4
0.8
1.2
30 60 90
�0.025
�0.015
�0.005
0.005
30 60 90�0.04
0
0.04
0.08
0 30 60 90
0
0.08
0.16
0.24
0 30 60 90
0
0.45
0.9
1.35
0 30 60 90
0
0.065
0.13
0.195
t
K∆
h∆
0ξ∆
eY∆
c∆
eK∆
iK∆
t t
t
t
t
t
t
t
0λ∆ iN∆
E∆
r∆
w∆
iF∆
∆
eN∆ λ∆
ξ∆
t
K∆
h∆
0ξ∆
eY∆
c∆
eK∆
iK∆
t
t
t
t
t
t
t t
0λ∆
iN∆
E∆
r∆
w∆
iF∆
∆
eN∆ λ∆
ξ∆
17
and labor inputs. The resources are shifted from the industrial sector to the environmental
sector. The environment is improved. The industrial sector’s productivity is enhanced. As the
productivity is enhanced only slightly in initial stage, the wage rate is reduced. As the
productivity is further increased, the wage rate is increased. The reduced national wealth is
associated with rising cost of capital. The relative propensity to save rises initially, but
subsequently falls. The consumption level and habit stock of goods are increased.
Figure 6 – Raising the Tax Rate on the Industrial Sector
5. Concluding Remarks
The paper constructed an economic growth model with environmental change and preference
formation. The paper is focused on dynamic interactions among capital accumulation,
environmental change, habit formation, preference change, and division of labor in perfectly
competitive markets with environmental taxes on production, wealth income, wage income and
consumption. The model integrated the dynamic economic mechanisms in the neoclassical
growth theory, the environmental dynamics in traditional models of environmental economics,
and the literature of economic growth with habit formation and within a comprehensive
framework. We could have synthesized the different ideas in a few main streams of economic
theory as we applied an alternative approach to household behavior initially proposed by Zhang
(1993). We showed that the motion of the economic system is given by three nonlinear
autonomous differential equations. We simulated the time-invariant system.
The simulation demonstrates some dynamic interactions among the economic variables
which can be predicted neither by the neoclassical growth theory nor by the traditional
economic models of environmental change. For instance, we examined the effects that the
household puts lower weights to more distant values of the levels of consumption. If the past
consumption has weaker impact on the current consumption, although the long-term equilibrium
of the dynamic system will not be affected, the transitional paths are shifted as follows: initially
the transitional path of the stock habit becomes lower than its original path; the consumption
30 60 90
0
5
10
30 60 90
�9
�6
�3
0
30 60 90�0.1
0
0.1
0.2
0.3
30 60 90
0
3
6
9
30 60 90
0
3
6
9
30 60 90
0
0.05
0.1
0.15
30 60 90�0.1
�0.02
0.06
0.14
0 30 60 90
0
0.4
0.8
1.2
0 30 60 90
0
0.5
1
1.5
t
K∆
h∆
0ξ∆
eY∆
c∆
eK∆
iK∆
t
t
t t
t
t
t t
0λ∆
iN∆
E∆
r∆
w∆
iF∆
∆
eN∆
λ∆
ξ∆
18
level falls initially in association with falling in the propensity to consume; the exogenous
disturbance causes the propensity to consume to fall and the propensity to save to rise; the
national wealth and capital inputs of the two sectors are augmented; the labor force is shifted
initially from the industrial sector to the environmental sector, but subsequently the direction is
opposite before the labor distribution comes to its original equilibrium point; the wage rate is
enhanced in association with falling in the rate of interest; the level of pollution falls initially, but
rise subsequently; the output levels of the two sectors and the total tax income are enhanced
before they come back to their original paths.
As the model is based on the basic ideas in some economic theories, it is straightforward to
extend the model in some directions. For instance, we may introduce leisure time as an
endogenous variable. Munro (2009: 3) observes: “In the unitary model, the household acts as
if it is a single individual maximizing a single utility function in the face of one budget
constraint. It is a simplifying modeling assumption that is widely used in most branches of
economics, but it is wrong. The fact that the unitary model is inaccurate is well-known and
has been known for many years now.” It is necessary to model family structure and economic
structure for understanding relations among growth, environmental change and preference
change (see, for instance, Dinda, 2004; Hamilton and Zilberman, 2006).
Appendix: Identifying the Three Autonomous Differential Equations
We now find the three autonomous differential equations and confirm the procedure in the
lemma. First, from (2) and (15), we solve
,
e
ee
i
ii
N
K
N
K
z
ββ
=≡ (A1)
where ,/ jjj αββ ≡ ., eij = Substituting (1) into (2) yields
( ) ( ) ,,,,
i
i
i
i
i
iii
k
iiii
zA
Ezw
z
A
Ezr α
α
β
β
β
β
δ
βα Γ
=−
Γ
= (A2)
where we use (A1). From (8) and (A1), we solve
.NzKK eeii =+ ββ (A3)
Insert (5) in (7)
( ) .ˆ ik FKKyN =+−+ δλξ (A4)
Put (3) in (A4)
( )( )[ ] ( ) ,1 iwk FwNKr =++−++ τλξδτλξ (A5)
where kk ττ −≡ 1 and .1 ww ττ −≡ Replacing iF in (A5) with ( ) iiiki KrF ταδ /+= from (2),
we acquire
19
( )( )[ ] ( ) ( ) .1
ii
i
kwk
K
rwNKr
τα
δτλξδτλξ +=++−++ (A6)
From ,KKK ei =+ (A6) and (A3), we solve
,,, ei
e
ii
ei KKK
KNz
KNK +=
−
==
β
β
φ (A7)
where
( ) ( )
( )
( ) ( ) ( ) ( ) ( )
( ) ( )[ ] ./,~
,11,,1,
,~,,,
ieiik
keikwek
rEz
rrEzwzrEz
z
Ez
βδβταδδφ
τββτφτβτφ
φφλξ
δφλξ
λξφ
ξ
ξξ
ξξ
ξ
−++≡
+−+≡++≡
++
−+
≡
From the definition of ξ and ,λ we have
,
~
00
00
λξ
λξτ
λξ
+
+
=+ (A8)
where ( ).1/1~ cττ +≡ Insert (A8) in the definition of φ
( ) ( ) ( )
( ) ( )
.~~
~
,,,
0000
0000
00 λξφφλξτ
λξδφλξτ
λξφ
ξξ
ξ
+++
+−+
=
z
Ez (A9)
From (14), we acquire
.0 KN
wN
k
w
k
+
+
=
λ
λλ
λ
λ
(A10)
Put (A7) in (A8)
,
~
00 φβφλ += (A11)
where
( ) ( ) .~,,0
e
kie
e
k
w
z
wEz
β
λββ
β
β
λ
λλφ
−
≡++= (A12)
Put (A9) in (A11)
20
( ) ( )
( ) ( )
,~~
~~~
0000
0000
00 λξφφλξτ
λξδβφβλξτ
φλ
ξξ
ξ
+++
+−+
+=
z
(A13)
We rewrite (A13) as follows
,0201
2
0 =−+ ωλωλ (A14)
where
( )
( ) .~
~~~~~
,,
,~
~~~~~
,,
000000
02
0000
01
ξξ
ξξξ
ξξ
ξξξξξ
φφ
ξφβτξδβξφφφφξτ
ξω
φφ
φβδβφφφφξφφξτ
ξω
+
+−+
≡
+
−+−−+
≡
z
Ez
z
Ez
We solve (A14) with 0λ as the variable
( ) .
2
4
,, 2
2
11
00
ωωω
ξλ
+±−
=Ez (A15)
We have two solutions from the above equation. In our simulation case the solution
( )
2
4
,, 2
2
11
000
ωωω
χξλ
++−
=z
is meaningful. From (16), we have
( ) .,,0 hh hw wEz ξξξξ ++= (A16)
We solve all the variables as functions of ,z ,E and h as follows: r and w by (A2) → 0ξ by
(A16) → 0λ by (A15) → λ and ξ by (5) → iK and eK by (A7) → ei KKK += →
NKk /= → iN and xN by (A1) → iF by (1) → eQ by (10) → ŷ by (3) → c and s by
(10). Here, we express the function for wealth obtained by this procedure as ( ).,, hEzk Φ=
From (11), (3), (15) and (17) and the procedure to determine the variables as functions of ,z
,E and ,h we have the following three differential equations
( ) ,ˆ,, kyEzk −≡Λ= λh& (A17)
( ) ,,, 0 EQCFEzE ecifE θθθ −−+≡Λ= h&
( ) ( ) ( )[ ].,, 0 ttcEzc hhhh& −≡Λ= (A18)
We do not provide the expressions because they are tedious. Taking derivatives of
( )h,, Ezk Φ= with respect to time yields
21
,
h
&&
∂
Φ∂
Λ+
∂
Φ∂
Λ+
∂
Φ∂
= cE
E
z
z
k (A19)
where we also apply (A18). Injecting (A17) in (A19) yields
( ) .,,
1−
∂
Φ∂
∂
Φ∂
Λ−
∂
Φ∂
Λ−Λ≡Λ=
zE
Ezz
c
cEz
h
h& (A20)
We thus proved Lemma 1.
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