ISSN 1479-3889 (print), 1479-3897 (online) International Journal of Nonlinear Science Vol. 1 (2006) No. 3, pp. 149-154

Renewable Resources, Technological Progress and Economic Growth Honglin Yang , Lixin Tian 1 , Zhanwen Ding Institute of System Engineering, Jiangsu University, Zhenjiang, Jiangsu, 212013, China (Received 28 February 2006, accepted 7 March 2006 )

Abstract: We introduce a renewable nature resource to an economic growth model with creative destruction. Then, we study the optimum growth path of economy. The result shows that positive growth and negative growth are both possible along the long-run balanced growth paths. Moreover, when the optimum growth rate is positive, we analyze how the optimum growth rate is affected by the relevant variables of the model. Keywords: renewable resources; technological progress; economic growth

1

Introduction

As we known, the production in any economy may not be independent of natural resources (renewable or unrenewable). However, most economic growth theories regard economic growth as the function of capital, technology, employment, bank rate and rule. All resources could be mutually substituted or substituted by other production elements [1]. In other words, resource is one of the affecting factors, but not a determinative factor. However, resources play an important role in our economy, and there would be no growth without the resource input. Then the question is that how the natural resources affect the modern economic growth. Numerous literatures have studied this problem. The results show that economy would keep positive growth rate even the stock of the resource is finite[2-5]. But technological progress is necessary. Concerning the description of technological progress, there are mainly two modes. One is called “level innovation”, reflected in the enlargement of categories; the other is called “vertical innovation”, reflected in the improvement of quality and productivity. Andre Grimaud and Luc Rogue [6] studied the relationship among technology advancement (“vertical innovation” mode), unrenewable resource and economic growth. Some scholars also investigated the relationship between energy resource and the economic growth of China. Sun and Tian [7-10] set up a new system named the energy resource system which is a third order autonomous system, and studied the dynamics behavior, Hopf bifurcation, chaotic attractor, chaotic synchronization and chaotic control of the energy resource system. Hopf bifurcation, chaotic attractor, chaotic synchronization and chaotic control are hot topics in nonlinear science(see [11-12]). [13] focused on constructing the production function based on energy and capital. Researches imply that high uncertainty of substitution elasticity for capital and energy is found. Such uncertainties indicate that large capital should be input to improve energy efficiency and explore new resources in the future economic development of China. [14] with the capital, energy and labor as input, a China’s trans-log function with time trend is built. Moreover, the elasticity of productivity, substitution elasticity and the bias of technological progress of different inputs are studied. It also pointed out the main character and transformation trend of economic energy system of China. As for the present development of resource and economy in China, the restriction of resources was not ultimately lifted [15]. Therefore, It has become a popular issue in economics to study how to realize the 1

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sustainable use of resources and development of economy. [16] introduced the restraint of energy utilization equation into the economic growth models, which discussed how to realize sustainable growth of economy and sustainable development of society under the condition of sustainable utilization of energy. In this paper, we focus on the relationship among technological progress with the mode of ”vertical innovation”, renewable resources and economic growth. Furthermore, we discuss the steady-state economic growth and how the growth rate is affected by the the parameters of the model.

2

The model

Suppose there are three sectors in the economy: final production sector, intermediate production sector and R&D sector. At each date t, the final output is produced by a competitive sector according to Yt = At xαt Rt1−α

(1)

where Yt is the output at time t, which is used for consumption.At is the level of technology at time t,andxt is the amount of intermediate product. To make it simple, we suppose there is only one intermediate production. Rt is the amount of resource plunged into production by the final sector. α, 1 − α express the output elasticity of intermediate production and the resource, 0 < α < 1. Assuming that the labor (L) supply is fixed, we standardize the total flow of labor to one (L = 1). The labor has two competing uses. First, it can produce intermediate production. Second, it can be used for research. Suppose the labor used for research is nt at time t, and the labor used for produce intermediate production is xt (For convenience, if one unit of labor is plunged into intermediate production sector, then one unit of intermediate production is produced. That is to say, the producing technology of intermediate production is one-for-one, xt is also the amount of intermediate production). Then we have, at each time t: xt + nt = 1. Suppose that one unit of labor is used for research, innovations arrive randomly with the Poisson distribution [17] with parameterλ, λ > 0. Each technology innovation would replace the former one, i.e. Aτ = γAτ −1 , γ > 1for all τ . At the period of [t, t + ∆t], the probability of technological innovation happened is λnt ∆t, while the probability of technological innovation unhappened is 1 − λnt ∆t, then the expectation of A at t + ∆t is E(At+∆t ) = λnt ∆tγAt + (1 − λnt ∆t)At = At + λ(γ − 1)nt At ∆t We then set ∆t → 0,

A˙ t = (γ − 1)λnt At

(2)

The technological innovation we are discussing here is ”vertical innovation”, which is embodied by intermediate production. We may assume that different quality levels of intermediate production will be completely replaced. Therefore, if a higher quality is found, the lower level will be expelled, which is known as the process of ”creative destruction” in economics. Furthermore, we may assume before the substitution of the production by higher quality level, the product was monopolized by the inventor and provided to the final production sector. Suppose the stock of resource is St at time t, the renewable rate of resource is σ, then the dynamic equation of the stock of resource at t is S˙ t = σSt − Rt (3) Because the output of the economic system is used for consumption, the consumption at t is Ct , then C 1−θ −1

Ct = Yt . Note that the utility function of consumer is U (C) = t1−θ , where θ > 0, expressed the elasticity of marginal utility and 1θ is the intertemporal elasticity of substitution. The utility discount rate at different time is constant ρ > 0. Then under the infinite field, the problem of the social planner is to choose nt and Rt , in order to maximize the utility Z ∞ 1 M ax (At (1 − nt )α Rt1−α )1−θ e−ρt dt 1 − θ 0 IJNS email for contribution: [email protected]

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151

s.t. A˙ t = (γ − 1)λnt At S˙ t = σSt − Rt

3 Analysis of steady-state optimal growth Based on the growth theory in modern economics, we may find that the economic growth in most countries has the character of steady-state in long period of time, i.e. the growth rate of all per variable is a constant. Therefore, we focus on discussing the case of steady-state growth in long period of time. In the above optimization problem, the control variable is nt and Rt , and the state variable is At and St . We construct the Hamilton function H=

1 (1−α)(1−θ) [A1−θ (1 − nt )α(1−θ) Rt − 1] + µ1 λ(γ − 1)nt At + µ2 (σSt − Rt ) t 1−θ

where µ1 , µ2 are the shadow price of technological progress and resource. The first order conditions are ∂H (1−α)(1−θ) = αA1−θ (1 − nt )α(1−θ)−1 Rt + µ1 λ(γ − 1)At = 0 t ∂nt

(4)

∂H (1−α)(1−θ)−1 = (1 − α)A1−θ (1 − nt )α(1−θ) Rt − µ2 = 0 t ∂Rt

(5)

µ˙ 1 = ρµ1 −

∂H α(1−θ) (1−α)(1−θ) = ρµ1 − A−θ Rt − µ1 λ(γ − 1)nt t (1 − nt ) ∂At

(6)

∂H = ρµ2 − σµ2 ∂St

(7)

µ˙ 2 = ρµ2 − Concluded from (4)∼(7), we may have

(1−α)(1−θ)

α(1−θ)−1 R αA−θ t (1 − nt ) t µ1 = λ(γ − 1)

(8) (1−α)(1−θ)−1

µ2 = (1 − α)A1−θ (1 − nt )α(1−θ) Rt t gµ 1 = ρ −

λ(γ − 1) λ(γ − 1)(1 − α) + nt α α gµ2 = ρ − σ

(9) (10) (11)

Since gA = λ(γ − 1)nt ,xt = 1 − nt , from (8) and (9), we have gµ1 = −θgA + (1 − α)(1 − θ)gR = −θλ(γ − 1)nt + (1 − α)(1 − θ)gR

(12)

gµ2 = (1 − θ)gA + [(1 − α)(1 − θ) − 1]gR = (1 − θ)λ(γ − 1)nt − (θ + α − αθ)gR

(13)

Followed by (10)∼(13) ρ−

λ(γ − 1) λ(γ − 1)(1 − α + θα) + nt = (1 − α)(1 − θ)gR α α ρ − σ = (1 − θ)λ(γ − 1)nt − (θ + α − αθ)gR

Now we solve the equation group combined by (14) and (15), then µ ¶ ρ − σ(1 − θ)(1 − α) α 1− +1−α nt = θ λ(γ − 1) 1 gR = [λ(γ − 1)(1 − θ) − ρ + σ(1 + θα − α)] θ IJNS homepage:http://www.nonlinearscience.org.uk/

(14) (15)

(16) (17)

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By (16), (17) and gY = gA + (1 − α)gR , gA = λ(γ − 1)nt we may obtain 1 gA = λ(γ − 1)nt = [ασ(1 − θ)(1 − α) − αρ + λ(γ − 1)(α + θ − θα)] θ

(18)

1 gC = gY = gA + (1 − α)gR = [σ(1 − α) + λ(γ − 1) − ρ] θ

(19)

From (3), gS = σ −

Rt St ,

since gS is constant in steady-state growth, then

Rt St

is constant. Thereby,

1 gS = gR = [λ(γ − 1)(1 − θ) − ρ + σ(1 + θα − α)] θ

(20)

According to the first transversality condition lim µ1 At e−ρt = 0, implies t→∞

ρ−

λ(γ − 1) λ(γ − 1)(1 − α) + nt + λ(γ − 1)nt − ρ = 0 α α

and nt < 1, we obtain the first transversality condition, θ >1−

ρ λ(γ − 1) + σ(1 − α)

(21)

Followed by the second transversality lim µ2 St e−ρt = 0, then t→∞

1 ρ − σ + [λ(γ − 1)(1 − θ) − ρ + σ(1 + θα − α)] − ρ = 0 θ implies gR − σ < 0, consequently the first transversality condition (21) still holds. In order to keep nt > 0, further require µ ¶ α ρ−σ θ> (22) 1 − α λ(γ − 1) − ασ In this way, we obtain the steady-state value of nt and the steady-state growth rate of every variable. Next, we discuss more about the economic steady-state growth ³ ´ rate. ρ−σ ρ α If ρ < λ(γ − 1) + σ(1 − α), then 1−α λ(γ−1)−ασ < 0 < 1 − λ(γ−1)+σ(1−α) , if and only if

ρ , we have 0 < nt < 1, which means the path for optimal steady-state growth exists. θ > 1 − λ(γ−1)+σ(1−α) Meanwhile, gY > 0, i.e. the optimal growth rate is positive along the steady-state ³ optimal ´growth path. ρ ρ−σ α , if and only if If ρ > λ(γ − 1) + σ(1 − α), then 1 − λ(γ−1)+σ(1−α) < 0 < 1−α λ(γ−1)−ασ ³ ´ ρ−σ α θ > 1−α λ(γ−1)−ασ , we have 0 < nt < 1, which means the path for optimal steady-state growth exists. Meanwhile, gY < 0, i.e. the optimal growth rate is negative along the steady-state optimal growth path.

4

Character of steady-state

In this section, we focus on the condition for keeping positive growth rate in long period of economic steady-state growth and influence given by every parameter. As noted above, growth is positive if and only if ρ < λ(γ − 1) + σ(1 − α). Under this case, the effects of each parameter on nt and the steady-state growth rate are stated in Table 1. α[ρ−σ(1−θ)(1−α)] ∂nt t Since ∂n , ∂γ = α[ρ−σ(1−θ)(1−α)] , their symbol is determined by ρ and σ(1 − ∂λ = θλ2 (γ−1) θλ(γ−1)2 ∂nt t θ)(1 − α). Especially, if σ = 0, i.e. the resource is unrenewable, then ∂n ∂λ > 0, ∂γ > 0. Since the increase of λ or γ, which means the improvement of the efficiency of R&D sector. Consequently, it will attract more labor to R&D sector. In this sense, due to the influence of the renewable rate of the resource, it becomes ∂nt t uncertain. But if θ > 1, we have ∂n ∂λ > 0, ∂γ > 0. That is to say, nt is influenced not only by λ or γ, but also by the renewable rate of resource and the elasticity of marginal utility. Note that the λ(γ − 1) in the model expressed the efficiency of R&D sector, thus the influence of λ and γ to economy are absolutely similar. So we take γ for example. An increase in γ, which means the

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∂nt ∂x

x=λ Symbol uncertainty

Table 1: Properties of the optimal path x=γ x=ρ x=θ Symbol uncertainty 0

>0

0 > 0, if θ < 1 < 0, if θ > 1

>0 > 0, if θ < 1 < 0, if θ > 1

0, if θ < 0, if θ >0 >0

153

1 1

improvement of efficiency of R&D sector, would improve the technology progress rate gA . Moreover, the economic growth rate gY would be improved. However, the influence from γ to increase rate of resource gR is uncertain. Actually, it is relevant to marginal elasticityθ. If θ > 1, the increase of γ will cause the decrease of gR . That’s because θ > 1 means that consumers deriver relatively more utility from a uniform path of consumption. Thus, the consumer will choose a lower growth of consumption, and since R gY = gA + (1 − α)gR , the decrease of gR will delay the increase of gY . Accordingly, ∂g ∂γ < 0. An increase in discount rate ρ means that households obtain more utility from current consumption relative to future consumption. Then investment in R&D, which implies a sacrifice in current consumption for the sake of future consumption, will not attract them. As a result, nt must decrease. Furthermore, gA will decrease. Moreover, the higher discount rate means the consumer prefer present consumption, so it means a lower growth of consumption, and thus a lower output growth. Therefore, in order to keep more consumption, the producer should have more output. Hence, the producer will extract more resources, resulting in the decrease of gS and gR . An increase in the elasticity of marginal utility θ means households will derive more utility from a uniform consumption path. Accordingly, the consumer will refuse to deviate from this consumption mode, and will not invest in R&D sector (investing would imply a higher consumption future). As a result, nt decreases. Furthermore, gA will decrease. Meanwhile, in order to achieve a flatter consumption path, the consumer would choose a lower growth of consumption and a lower output growth, and therefore a lower growth rate of the resource extraction. An increase in the resource renewable rate σ will lead to the increase of gS and gR . At the same time, the output growth rate gY would increase. The influence of nt and gA from σ is relevant to the marginal elasticity θ. If θ > 1, the increase of σ will lead to the decrease of nt and gA .

5 Conclusion In this paper, we introduce the renewable resource to the economic growth model with creative destruction, and focus on the optimal steady-state economic growth path. Results indicate that positive growth and negative growth are both possible along the long-run balanced growth paths, and we gave the conditions under which growth is positive along this path. When the economy has steady-state positive growth rate, the efficiency of R&D sectorλ,γ and the renewable rate σ have positive effects on economic growth, while the discount rate ρ and marginal elasticity θ have negative effects on the economic growth. According to the analysis in this paper, the government may indirectly take corresponding policy to adjust the parameter in the model to maintain the economic growth at a high speed. Through the relevant financial policy, the utility discount rate could be tuned up by economic lever. The actual renewable rate could be increased by improving the recycling and reusing rate. Finally, the technology parameter could be aggrandized by the devotion of R&D sector and exploitation of human capital.

Acknowledgements Researches were supported by the National Nature Science Foundation of China ( No. 90210004 ) and National Social Science Foundation of China ( No. 02BJY051 ) and Social Science Foundation of Jiangsu Province of China ( No. 05SJD790061 ). IJNS homepage:http://www.nonlinearscience.org.uk/

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