Growth , Technology , and Environmental Change-- Nonlinearity and Non-constant Returns

This paper proposes a growth model with endogenous technology and environmental change. The economy consists of two sectors, production and environmental. The production sector produces goods with knowledge, labor, and capital as inputs under perfect competitive conditions. Knowledge accumulates through learning by doing. The environment is affected by production, consumption, the environmental sector’s production efficiency, and the nature’s purification. The simple model shows that it is difficult to explicitly judge the impact of factors such as environmental policy, knowledge accumulation efficiency and preference change on the environment.


INTRODUCTION
Whether or not continued global economic growth is comparable with global environmental changes has become one of the major challenges in the future.It may be argued that in order to examine environmental issues, it is necessary to construct the models with genuine dynamic interactions between distribution of production factors, production, consumption, technology, and environment.The purpose of this study is to construct a macroeconomic model to analyze interdependence among economic development, environmental changes, and knowledge accumulation.
Both production and consumption may pollute the environment.Growth often implies worsened environ- mental conditions.But growth also implies a higher material standard of living, which will, through the demand for a better environment induces changes in the structure of the economy to improve the environment.As society accumulates more capital and makes progresses in technology, more resources may be used to protect, if not improve, the environment.It is well observed that a country in the beginning of its economic development will be experiencing a worsening of the environment, while a country in which growth has taken place over a longer period of time will be adjusting its patterns of growth in such a way that the environment in fact improves.There are dynamic tradeoffs among economic growth, consumption, pollution, and human efforts of protecting environment.Tradeoffs between consumption and pollution have been extensively analyzed since the seminal papers of Plouder (1972) and Forster (1973).There is a large amount of literature available on issues of interdependence between economic growth and environment (Smith, 1972; Fisher and Peterson, 1976;  Maget, 1978; Kanemoto, 1980; Tietenberg, 1988;   Krutilla, 1991; Falk and Mendelsohn, 1993; Barrett,  1991).It has become clear that it is not easy to analytically examine the economic growth with endogen- ous pollutant accumulation and environment policy.On the basis of these efforts, the purpose of this study is to show how the environment may interact with techno- logical change and economic growth within a perfect competitive economy under the government's interven- tion in environment protection.
Economic growth and improved living standards are dependent on people's ability to create and Utilize knowledge.On the other hand, knowledge accumulation is sustainable only with some economic bases.There is interdependence between knowledge creation and utili- zation and economic growth.The idea of endogenous knowledge growth is not new and it has been incorporated into economic analysis for a long time.It may be said that the modeling of interaction between economic growth and knowledge accumulation was initiated with Arrow's paper on learning by doing (Arrow, 1962) and Uzawa's paper on education and growth (Uzawa, 1965).There have been an increasing number of publications on relations between knowledge accumulation and economic develop- (e.g.Sato, 1996; Romer, 1986; Lucas, 1988; Matsuyama,   1999; Zhang, 1999, 2000).These approaches have provided insights into the complexity of modern economic development.But only a few models explicitly take account of environmental issues in growth models with endogenous capital and knowledge.
This study proposes a dynamic model to examine the issue of interdependence among economic growth, technological change, pollution, and government environ- mental policy.The government's environmental policy is to maximize the consumer's utility by allocating labor and capital resources for environment protection.Capital accumulation is endogenously determined.Pollutant accumulation speed is dependent on the production level, the level of consumption, natural purification power, and human efforts of purifying environment.The remainder of this paper is organized as follows.The second section defines the growth model with endogenous technological change and capital and pollutant accumu- lation.The third section analyzes the properties of the dynamic system.The fourth, fifth, and sixth sections, respectively, examine the effects of changes in the environmental policy, knowledge accumulation efficiency, and preference structure on the economic structure and environment.The seventh section concludes the study.

THE MODEL
We consider an economic system, which consists of two sectors, production and environmental.The production sector is similar to the standard one-sector neoclassical growth model (e.g.Zhang, 1999).Only one commodity is produced in the system.The commodity is assumed to be composed of homogeneous quality, and to be produced by employing three factors of production, namely knowledge, labor, and capital.At this initial stage, we neglect dynamics of population, assuming that the population is constant.The population is employed by the two sectors.The labor distribution is determined by the market mechanism.The environmental sector employs labor and capital to purify environment.The government finances the environment sector through taxing the production sector.
It is assumed that the labor and capital markets are perfectly competitive and the labor and capital are always fully employed.We introduce: N--the fixed labor force; K(t) and F(t)--the total capital and the output at time t; Ni(t) and Ki(t)--the labor force and capital stocks employed by the production sector; Ne(t) and Ke(t)--the labor force and capital stocks employed by the environmental sector; C(t)--the consumption level of goods; E(t)--the level of pollutant stocks; r(t) and w(t)--the rate of interest and the wage rate, respectively; and z--the fixed tax rate, 0 < -< 1.
There are three factor inputs, knowledge, capital and labor, in economic production.We assume that the environmental quality may affect productivity of production units such as hotels, restaurants, and hospitals and deteriorate machines.We specify production function as follows F(t) ZmK.N exp(-hpE), a +/3 1, a,/3 > O, m, hp >-0   (2.1) where Z(t) is the level of knowledge at time and rn is the knowledge utilization efficiency parameter of the production sector.We introduce knowledge stock Z(t) of the system.In this study, the concept of knowledge refers to disembodied knowledge.Knowledge means ideas and theories, which exist, for instance in books and journals.They are free for anyone to utilize.Knowledge has the characteristics of public good in the sense that utilization of knowledge by any economic sector will not affect that by any other sectors.New theories in mathematics, theoretical physics, economics, philosophy and the like are accessible to the public, almost as soon as they are discovered.Knowledge is not a direct input to production, but may affect human capital (which is an input to production).We assume that knowledge may indirectly affect economic production in the way that human capital accumulation is affected by knowledge and human capital is a direct input to production.The term, exp(-hpE), in F(t)   means that productivity is negatively related to the pollution level.In this study, we neglect possible impact of the environment on productivity, i.e. hp 0.
It can be seen that this omission will not significantly affect our analytical results.
We select the commodity to serve as numeraire.The marginal conditions are given by The income Y from the interest and wage payments at time is given by Y rK + wN. (2.3) We now describe the dynamics of the stock E(t) of pollutants.We assume that pollutants are created through two sources, production and consumption.Pollutants may be reduced by two ways.The nature may treat certain pollutants in a similar way to that of waste treatment plants.Some of the pollutants may naturally disappear without any human efforts.Pollutants may be treated by using capital and labor.We specify the dynamics of the stock of pollutants as follows dE dt qfF -+-qcC Qe qoE (2.4) in which qf, qc, and qo are positive parameters and Qe(t) f(E)ZnKN (2.5) where u and v are positive parameters, n is the knowledge utilization efficiency parameter of the environmental sector, and fiE) (-->0) is a function of E. The term qfF means that pollutants that are produced during production processes are linearly positively proportional to the output level (Gruver, 1976; Fisher and Peterson, 1976; Stephens,  1976).The term qcC means that in consuming one unit of the good, the quantity qC is left as waste.The parameter q depends on the technology and environmental sense of consumers.The parameter qo is called the rate of natural purification.The term qoE measures the rate that the nature purifies environment.The term ZnKN in ae means that the purification rate of environment is positively related to knowledge utilization efficiency, capital and labor inputs (M/iler, 1974).The function fiE) in Qe(t)--f(E)ZnKN implies that the purification efficiency is dependent on the scale of pollutants at time t.It is not easy to generally specify how the purification efficiency is related to the scale of pollutants.For simplicity, we specify f as follows f(E)= qeEv where qe > 0 and v > 0 are parameters.The function has the following properties f(0) 0, !imf(E) oo df d2f in which , , and A, respectively, are the propensities to enjoy environment, to consume goods, and to save.
Consumers get income Y from the interest and the wage payments.They can also sell their properties, which are equal to K, to purchase consumption goods and make investment.The total available budget for savings and consumption is thus equal to Y*=Y+K.
We assume that the consumers pay the depreciation of capital goods, which they own.The total amount is equal to 6kK where 6k is the depreciation rate of physical capital.
At each point of time, the consumers would distribute among savings (S), consumption of goods (C), and payment for depreciation (K) where 6k is the fixed depreciation rate of capital.The budget constraint is thus given by C + rkK + S Y* Y + K. (2.8) The households determine C and S with the level of E* as given.Maximizing U subject to Eq. (2.8) yields C pY + (1 6)pK, S ApY + (1 )ApK (2.9) where p 1/(: + A).
It is assumed that the savings is equal to investment.The change in the households' wealth is equal to the net savings minus the wealth sold at time t, i.e.
Obviously, when E is very large, the specified functional form is problematic.At this initial stage of investigation, we accept the above-specified form.In order to describe the behavior of households, we define a variable E* E0 E (2.6) where E0 is called the threshold of pollution level.For instance, consumption of nuclear-generated electricity brings about the creation of radionuclides that cause death or severe mutation, when threshold concen- trations are exceeded.Electricity production using coal creates atmospheric CO2 concentrations which, at sufficiently high levels, may cause dramatic changes.We assume that the critical level is known.This assumption may be relaxed (Cropper, 1976; Smith,  1972; Clarke and Reed, 1994).
We assume that the disutility that the society experiences from pollution is a continuous function of the environmental pollution stock.It is assumed that the utility level U(t) that a typical household obtains is dependent on the consumption level C(t) of commo- dity, the environmental condition E*(t) and the net savings S(t).The utility function is specified as follows dK --S-K.dt Substituting S in Eq. (2.9) into the above equation yields dK XpY-( + 6A)pK.
(2.10) dt We now determine how the government determines the number of labor force and the level of capital employed for purifying pollution.The government budget is given by rKe + wNe 'F.(2.11)We assume that the government will employ the labor force and capital stocks for purifying the environment in such a way that the purification rate achieves its maximum under the given budget constraint.The government's optimal problem is given by Max ae f(E)ZnKN s.t." rKe -1-wNe 'F.

C+S-K+BK=F.
We assume that the labor and capital are fully employed Ki + Ke K, Ni + Ne N. (2.14) There are different ways of creating new knowledge.In the economic literature, processes of knowledge creation through learning by doing and pure and applied research are well modeled.In this study, for simplicity, we assume that knowledge accumulation is through learning by doing.We may introduce research and development activities in the way as in Zhang (1999).We propose the following possible dynamics of knowledge dZ 7.iF dt Z 6zZ (2.15) in which 7.i, e, and 6z are parameters.We require 7.i, and 6z to be non-negative.We interpret 7.iF/Z as the contribution to knowledge accumulation through the production sector's learning by doing.In order to explain Eq. ( 2.15), we consider a case in which knowledge is a function of the total production output during a certain historical period Z(t) al F( O)dO +a3 in which a l, a2 and a3 are positive parameters.The above mentioned equation implies that the knowledge accumulation through learning by doing exhibits decreasing (increasing) returns to scale in the case of a2 < (>)1.We interpret al and a3 as the measurements of the efficiency of learning by doing by the production sector.Taking the derivatives of the equation yields dZ 7.iF dt Z in which 7.i ala2 and e 1 a2.Adding the depreciation part to the above equation yields Eq. (2.15).
We have thus defined the model.The model has 14 endogenous variables, Z, Ki, Kn, Ni, Nn, E*, K, C, S, E, r, w, Y, F, and U.It is easy to check that the system has the same number of independent equations.We now examine the behavior of the system.

PROPERTIES OF THE DYNAMIC SYSTEM
First, we show that the dynamics can be represented by a three-dimensional differential equations system.Then, we provide conditions for existence of equilibria and for stability.

Ni
(3.6)By Y F, Eq. (3.6) and Ni + Nn N, we solve Ni tiN, Ne teN Then, for a given tax rate, the labor distribution of the two sectors are linearly and positively in proportion to the total population at any point of time.
Summarizing the discussion, we get the following lemma.
LEMMA 3.1 For any given positive levels of Z(t), K(t), and E(t) at any given point of time, all the variables in the system can be expressed as functions of Z(t), K(t), and and (2.15), we represent the dynamics of the economic system in terms of the following three differential equations: It is direct to check that the dynamic system has a unique equilibrium, given by where A --= p/3( + ASk)+ 6zX.
If x < 0, then Re{ 4'j < 0, j-1,2.In this case, the unique equilibrium is stable.If x > 0, Re 1 < 0. In this case, the unique equilibrium is unstable.PROPOSITION 3.1 In the case of x < (>)0, the dynamic system has a unique stable (unstable) equilibrium.
The stability of the system is determined by the parameter x =-rn/-s-1.As m is the production sector's knowledge utilization efficiency parameter and s is the return to scale effects of knowledge in knowledge accumulation, we may interpret x as the measurement of return to scale effects of knowledge in the whole system.We may thus make the following interpretation of the parameter x.We say that the knowledge utilization and creation of the production sector exhibits increasing (decreasing) return to scale effects in the dynamic system when x > (<)0.The above proposition simply says that if the knowledge utilization and creation of the production sector exhibits increasing (decreasing) return to scale effects, then the dynamic system is unstable (stable).This conclusion is intuitively acceptable.
In the remainder of this study, we examine the impact of changes in some parameters on the long-run equilibrium.
THE TAX POLICY AND THE LONG-RUN EQUILIBRIUM AiZmK a t-(1 6k)qcpK AeZnEVK u + qoE (3.9)This section examines the impact of changes in the tax rate, z, on the system.Taking derivatives of Eq. (3.9) with respect toyields in which x=- ----s-1.m We require x # 0. We thus have a unique equilibrium.By the first equation, we explicitly solve Z.So the second one gives the value of K.In the last equation in Eq. (3.9), the right-hand side is constant (because we have solved K and Z).It is direct to check that the last equation (with E as a single variable) has a unique solution for 0 < E < +oo.
The three eigenvalues, 4'2, J 1,2, 3, are given by /_ 2 It is important to examine how a shift in the preference structure may affect the pollution issue.We now examine how changes in the propensity A to hold wealth affect the system.It should be remarked that an increase in the propensity to save implies a decrease in the propensity : to consume goods.Taking derivatives of Eq. (3.9) with respect to A yields An increase in the propensity to save increases (reduces) the level of knowledge in the case of x < (> )0.An increase in A increases (reduces) the level of capital stocks in the case of am/x[3 < (>)1.It is not easy to explicitly judge the sign of dE/dA.

CONCLUDING REMARKS
This study proposed a dynamic model to examine the issues related to interdependence between economic growth, technological change, pollution and government environmental policy under perfectly competitive mar- kets.In our model, knowledge accumulation is through learning by doing.Pollutant accumulation is dependent on the production, consumption, natural purification power, and human efforts to purify the environment.We showed that the dynamic system has a unique equilibrium.The unique equilibrium is either stable or unstable, depending on whether the system exhibits decreasing or increasing returns to scale.We also examined the effects of changes in some parameters on the long-run economic structure.
We may extend the model in different ways.For instance, we may extend the one-sector model to a model with multiple sectors.We may assume that the tax rate is an endogenous variable by specifying -as a function of F and E at any point of time as -= H(F,E,t) where measures the impact of the households' preferences on the government's tax policy and other factors.It may be reasonable to require that for a given level of F, an increase in E tends to increase the tax rate, i.e.He --> 0.
But whether an increase in the output will increase the tax rate is difficult to predict.That is, HF may be either positive or negative, depending on the social and environment consciousness of the society under conside- ration.We may also assume that the government's budget for environment protection is dependent on consumption level.In this case, the household budget constraint and the government budget constraint are changed.