Long-term economic growth under environmental pressure: An optimal path

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Long-term economic growth under environmental pressure: An optimal path Feng Dai a,∗ , Pengpeng Li b , Ling Liang a a b

Zhengzhou Information Engineering University, China Henan Agriculture University, China

a r t i c l e

i n f o

Article history: Received 9 July 2014 Accepted 27 March 2015 Available online xxx JEL classification: C53 E17 O47

a b s t r a c t This paper presents a model, based on the advance-retreat course (ARC) model (Dai, Liang, & Wu, 2013; Dai, Liu, & Liang, 2013), of long-term economic growth under environmental pressure. The model is used to explain economic convergence and divergence; construct an optimal long-term growth model for basic, emerging and real total output; derive an optimal growth accounting equation; indicate the optimal paths of long-term growth and economic structure change; analyze empirically the growth for U.S. and China. Among the findings are that emerging industries contribute significantly to real output in the long term; that economic diversification can increase real output and promote long-term growth. The paper suggests policy orientations that are needed to avoid economic collapse. © 2015 The Board of Trustees of the University of Illinois. Published by Elsevier B.V. All rights reserved.

Keywords: Long-term economic growth Environmental pressure Optimal path Convergence Divergence

1. Introduction The economic growth process embodies the economic state and influences the quality of human life. Economists hope to understand and grasp the process and character of economic development through their research on long-term economic growth and then to control any economic fluctuations and ensure long-term economic growth. There has been much outstanding research in economic growth, such as the real business cycle theory (Kydland & Prescott, 1982; Plosser, 1989), the new economics of growth (Lucas, 1988; Romer, 1986, 1990), the R&D-Based Theory in Economic Growth (Jones, 1995, 1998; Solow, 1956, 1957), long-term convergence and longterm regional economic growth (Barro and Sala-i-Martin, 1992, 1995). In addition, Phelps (1966a, 1966b, 1972) proposes the

∗ Corresponding author at: Department of Management Science, Zhengzhou Information Engineering University, Building 75-1-701, # 5, Jian-Xue Street, Wen-Hua Road, Zhengzhou, Henan 450002, China. Tel.: +86 0371 81630975; fax: +86 0371 81630975. E-mail addresses: [email protected] (F. Dai), [email protected] (P. Li), liang [email protected] (L. Liang).

“Golden Rules of Economic Growth”, analyze the relationship among money, public expenditure and labor supply and explains the problems of economic stagflation and the growing unemployment. Pasinetti (1983) presents an original theoretical treatment of the problems of maintaining full employment in a multisector economic system with a growing population and different rates of technical progress in different sectors. Mowery and Rosenberg (1991) demonstrate the importance of a historical perspective in understanding the role of technological innovation in the economy. De La Croix and Michel (2002) provide an in-depth treatment of the overlapping generations model in economics, and they incorporate production into the model. De La Grandville (2009) provides a fascinating introduction to the theory of economic growth and shows how many of the results from this field are of paramount importance for society. In recent years, economists have become more concerned with the influence of economic environmental factors, such as economic policy, natural resources and labor, on long-term growth and convergence. Maarten (2007) notes that the convergence rate is affected when the intraregional aspects of agglomeration are taken into account. Maasoumi and Wang (2008) investigate economic convergence in China and find that the policies, reforms and other differences in the region will influence the convergence.

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Dimitrios (2008) discusses how the interactions between capital accumulation, endogenous longevity and environmental quality determine both the long-run growth rate of the economy and the pattern of convergence toward the balanced growth path. Myers (2009) believes that the labor market is interacting with changes in output and production. Fung (2009) tests for convergence in financial development and economic growth; the results show strong evidence for conditional convergence and show that some low-income countries with a relatively under-developed financial sector are more likely to be trapped in poverty. This finding explains the observed “great divergence” between the poor and the rich countries. The above studies show that environmental factors, including economic structure, policy, natural resources, and capital and labor, can foster both economic convergence and divergence. Indeed, pressure (or resistance) from environmental factors – pollution caused by industrial production, competitive pressures arising from merchandize trade, resource depletion associated with energy consumption, capital and financial risk, etc. – can change the economic growth process. Thus, environmental pressure must be considered when analyzing long-term economic growth. In another hand, the study of Hill (1997) shows that the concerns about environmental issues and pressure on manufacturing firms to decrease their environmental impact have both intensified since the early 1980s. Lorek (2001) thinks that growing resource consumption goes together with growing environmental pressures and vice versa, although not necessarily proportionally. Azad and Ancev (2010) point out that growing public concern about the health of rivers and wetlands, and the ecosystems they support, puts pressure on large water users-such as the irrigation industryto find ways to use less water. Erdem (2012) indicates that rapid growth in energy consumption influences on the one hand energy prices and endangers energy supply security; on the other hand, it distresses ecological balances. This means that the challenge for long-growth is increasing, and can also be taken as the increase of environmental pressure. In fact, environmental pressure arises mainly from physical factors, such as environmental disruption and lack of natural resources. However, social factors, such as cultural background, political instability, damage due to war, economic systems, laws and regulations, market structure, financial events, and financial order, also have important effects on economic growth that cannot be ignored. Although some social factors can sometimes play a positive role in promoting growth, over time they may become not apply to economy and create environmental pressures that impede social progress and economic growth and cause reduced output. Social and economic change can not only improve the efficient use of natural resources and the natural environment but also perfect or improve social factors, and the more that social factors and resources accumulate, the more complex they become. Therefore, a consideration of the economic effects of environmental pressure should include both natural and social environmental pressure. The existing literature has focused on the former. Concerning environmental pressure and its impact on the economy and combining both types of environmental pressure in a model of economic growth, this paper discusses long-term economic growth under environmental pressure; presents an optimal growth model and uses it to illustrate optimal economic structure and show the optimal path of long-term growth; analyzes convergence and divergence in long-term growth; and studies empirically the U.S. and China’s economic growth. Most importantly, the paper shows that emerging industries contribute significantly to real output, that economic diversification leads to increased real output and long-term growth, and that divergence ultimately arises in response to environmental pressure.

2. Foundation 2.1. Categorized production function Industries can generally be divided into two categories: traditional and emerging. Traditional industries are those that mostly involve labor and basic manufacturing, while emerging industries are those that mostly involve new science and technology. Traditional industries require large quantities of labor and equipment – resources that constitute the foundation of traditional industry. In a traditional industry, capital often takes a material form (e.g., equipment or buildings), while labor involves the efforts of workers with standardized skills. Technological progress is measured by the technologies embodied in capital equipment, final goods and services. Traditional industries usually employ advanced processing techniques and complete equipment systems and enjoy stable product markets. Traditional industries often require a higher cost of capital and better technology. In addition, technology levels in traditional industries tend to remain stable for long periods of time. In contrast, powerful technology is fundamental to emerging industries. In an emerging industry, capital may take a material or immaterial form; it may include equipment, patents, software, intangible assets and workers with standardized professional skills. Technology develops rapidly in emerging industries; thus, overall technological levels tend to evolve quickly. For the sake of convenience, capital inputs will be expressed as the value of capital required in production, labor inputs as the number of workers required in production and technology inputs as the cost of research and development. Thus, the production function (Barro & Sala-i-Martin, 1995; Solow, 1956, 1957) for an economy can be expressed as Y = A·F(K,L), where Y is GDP, K is capital, L is labor, and A is multifactor productivity. For given quantities of capital and labor, improvements in technology will yield increased output. Thus, economies with more advanced technology exhibit greater productive efficiency. Because capital, labor and technology change over time (K = K(t), L = L(t), A = A(t)), the technology level A(t), assuming  differentiability, can also be expressed as dA(t)/h(t); thus, A(t) = h(t)dt = H(t) + a, where a is a constant. Therefore, output can be expressed as: Y = Y1 + Y2

(1)

where Y1 = a·F[K(t), L(t)] and Y2 = H(t)·F[K(t), L(t)]. In Model (1), the technology level associated with output Y1 = a·F[K(t), L(t)], a, is a constant, signifying a stable technology level, which is a characteristic of traditional industries. Therefore, Y1 , the output of traditional industries, is referred to as basic output. The technology level associated with output Y2 = H(t)·F[K(t), L(t)], H(t), is a function of time, signifying that the level of technology is variable, which is a feature of emerging industries. Therefore, Y2 , the output of emerging industries, is referred to as emerging output. We refer to model (1) as the categorized production function (CPF) for traditional and emerging industries, where A(t) = H(t) + a is categorized total factor productivity. Model (1) indicates that traditional industries have two inputs, capital and labor, whereas emerging industries have three inputs, capital, labor and technology. Model (1) can be concisely expressed as: Y =+ where  = a·F[K(t), L(t)] is the production function for traditional industries,  = ·q(t) is the production function for emerging industries, and q(t)=H(t)/a is the ratio of the technology level of emerging industries to that of traditional industries, indicating the degree of innovation, or innovation efficiency, of the former. Innovation efficiency is a dimensionless quantity that expresses the advantage in productive efficiency of emerging over traditional industries. Here,

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innovation encompasses all of the benefits of technological and scientific progress. In general, traditional and emerging industries have different capital and labor requirements. Model (1), however, shows that the input factors of both traditional and emerging industries stem from the economy’s overall quantities of capital and labor. This finding can be explained as follows. Each unit of capital can be divided into two parts: one part used in traditional industries and the other used in emerging industries. Similarly, each unit of labor can be divided into two skill types: one applicable to traditional industries and the other applicable to emerging industries. Thus, capital and labor can flow between traditional and emerging industries. When productive efficiency increases in emerging industries, capital and labor will flow toward those industries. Model (1) illustrates that, for the economy as a whole, part of total output is produced by traditional industries, and the remainder is produced by emerging industries. If emerging industries are low in innovation efficiency, economic output mainly comes from traditional industries. If emerging industries are relatively high in innovation efficiency, economic output mainly comes from emerging industries. During periods of the latter type, the economy will likely be highly developed, because growth is largely driven by innovation. If more high and new technologies are introduced to a traditional industry, and the output growth is mainly promoted by the innovations, then the industry is transformed into an emerging industry. 2.2. An economic growth model with environmental pressure Economic growth requires productive inputs and consumes a variety of economic resources. However, it is important to note that economic growth can be hindered by various factors, including resource scarcity, market competition, investment risk, financial risk, environmental crises, social unrest, natural disasters, disease and war, all of which generate environmental pressure (or resistance) to economic growth. In general, the environmental pressure is defined as the integrated (natural and social) force hampering the socio-economic progress. It is also worth noting that, in addition to promoting economic growth, innovation may itself generate environmental pressure; it may increase resource consumption, environmental pollution or investment risk, which increase consumption of real output and thus raise innovation costs. The consumption of social and economic resources due to environmental pressure generated by economic growth is referred to as exogenous or environmental cost. We denote the environmental cost of basic output as  and the environmentalcost of emerging output as . Environmental costs arise from environmental pressure related to factor inputs. When factor inputs change, environmental costs also change. According to Reed (2001) and Schoenberg, Peng, and Woods (2003), the ratio of the change in environmental costs to the change in factor inputs is a power function of current inputs; that is, d d = vϕ and = w ϕ d d where ϕ > 0 indicates the existence of environmental pressure, and v and w are the environmental cost coefficients for basic and emerging output, respectively, expressed simply as basic and emerging cost coefficients. Hence, the basic environmental cost and emerging environmental cost are, respectively: =

v 

 and  =

w   

where  = ϕ + 1 is called the environmental pressure index. Without loss of generality, let the constants of integration equal zero.

3

Following Sanchez, Gonzalez-Estevez, Lopez-Ruiz, and Cosenza (2007), we assume that the authority can adjust the environmental pressure index through economic policy. Specifically, it can reduce the environmental pressure index through free and open (free-market and non-protectionist) policy and increase the environmental pressure index through a closed and protective policy, but it cannot eliminate environmental pressure altogether, that is,  > 1. The formulas for environmental costs indicate that an increase in productive inputs will be accompanied by an increase in environmental pressure or environmental cost. Generally, the environment for sustained growth is likely to improve gradually. However, in achieving long-term growth and development, an economy faces a growing number of challenges of increasing complexity, challenges (which arise from such issues as institutional policy, financial risk, environmental crisis, social unrest, natural disaster, disease and war) that will be increasingly difficult to address under current approaches. Indeed, traditional and emerging industries may face different environmental pressures, and governments may accordingly implement different policies to address development and growth in the case of each industry type. For convenience, however, we assume equality between the environmental pressure indices for traditional and emerging industries. Given that economic growth cannot be neatly separated from environmental pressure and resulting environmental costs, real output based on Model (1) can be expressed as: Y =  +  − ( + ) =  · G

(2)

1 + q − (1/)(v + wq )−1

where G = is total factor productivity with environmental pressure. Model (2), which can be referred to as normal growth model with environmental pressure, is a normal advance-retreat course (ARC) model (Dai, Liang, et al., 2013; Dai, Liu, et al., 2013). According to the Cobb–Douglas production function, output can be expressed as  = a·F[K(t), L(t)] = 0 K˛ Lˇ in Model (2), where K and L represent capital and labor, respectively, 0 = a·c0 , c0 are initial values, and ˛, ˇ > 0. Emerging industry output can then be expressed as  =  0 K˛ Lˇ q. Model (2) indicates that during normal economic growth, environmental pressure increases as output increases, which is a process that continues until the economy goes into recession. Only successful policies and institutional reform can generate a new economic environment, relieve the existing environmental pressure and initiate a new cycle of economic growth. As the economy continues to grow, however, additional environmental pressure accrues. Hence, the ARC Model (2) reflects the cyclical features of economic growth. 2.3. Types of economic growth In Model (2), according to the environmental pressure index  < 1,  = 1 and  > 1, we can divide economic growth into three basic types. - Infinite-growth type. When  < 1, economic growth is infinite. At this point, with the increase of the basic output  and innovation q, the real output will increase and will tend toward infinity. In reality, the economy cannot grow indefinitely. Therefore, infinite economic growth is unrealistic. - Critical-growth type. When  = 1, economic growth is critical. If max{v, w} < 1, with the increase of basic output and innovation, the real output will increase and tend to be infinite. However, the critical-growth economy is just an ideal mode and is not always correct. - Consumption-growth type. When  > 1, economic growth is consumptive. With the increase of basic output and innovation, the environmental cost and environmental pressure will gradually increase. When the environmental cost and pressure increase to

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a certain degree, an economic recession occurs. Therefore, consumptive economic growth is the most common form and is the type of economic growth that is confronted by governments and enterprises. In the following section, we focus on the consumptive economic growth type, which is  > 1. 3. Methods In the following discussion, we assume  = 0 K˛ Lˇ and ˛ + ˇ = 1 in Model (2) based on Cobb–Douglas production function. 3.1. Optimal output growth In Model (2), if innovation remains unchanged, let the derivative dY/d = 1 + q − (v + wq )−1 = 0. Then, we obtain optimal basic output as follows:

 + =

1+q

1/(−1) and  + = + · q

v + wq

(3)

Because d2 Y/d2 = −( − 1)(v + wq )−2 < 0, basic output in Eq. (3) implies that real output in Model (2) is a maximum. + is optimal basic output; correspondingly,  + is optimal emerging output. In Eq. (3), if we let + = 0 (K+ )˛ (L+ )ˇ (where K+ is the optimal quantity of capital and L+ is the optimal quantity of labor), then ˙ + /+ = ˛(K˙ + /K + ) + ˇ(L˙ + /L+ ) = (q{v − wq−1 [ + ˙ We can then compute ( − 1)q]}/[( − 1)(1 + q)(v + wq )])(q/q). the optimal employment growth rate for given growth rates of capital and innovation, and vice versa. Substituting Eq. (3) into Model (2), we obtain the optimal output growth model under environmental pressure as follows: Y + = + G+ =



1−

  (1 + q) 1/(−1) 1 

(4)

where + = 0 (K+ )˛ (L+ )ˇ , G+ = (1 − (1/))(1 + q) is optimal total factor productivity with environmental pressure. Using Eq. (3) and Model (4), the optimal growth accounting equation under environmental pressure is:



q −1



v − wq−1 (1 + q)(v + wq )



q˙ q

(5)

Eq. (5) indicates that optimal output is increasing when cur1/(−1) and decreasing when current rent innovation q > q+ = (v/w) 1/(−1) 1/(−1) innovation q > q+ = (v/w) . Therefore, q+ = (v/w) may be regarded as the innovation-balance point for optimal growth. We observe, given the innovation-balance point, that an increase in the basic cost coefficient, or a decrease in the emerging cost coefficient, will enlarge the innovation growth space. In addition, the innovation growth space can also be enlarged through a free and open policy, because the environmental pressure index will thereby be reduced. 3.2. Steady-state growth In Model (4), define y ≡ Y+ /L+ as real output per capita, and k ≡ K+ /L+ as capital per capita. Model (4) can then be written as: y = 0 k˛ G+ .

(6)

And the growth accounting equation is: y˙ G˙ + k˙ =˛ + + y G k

 y˙ ∗ y

 ∗

=

(7)

k˙ k

1 = 1−˛



G˙ + G+

∗

(8)

From Eq. (8), output per capita and capital per capita will grow ∗ ∗ at the rate of (1/(1 − ˛))(G˙ + /G+ ) , and changes in (G˙ + /G+ ) will ∗ + + ˙ influence the steady-state growth process. If (G /G ) = g(g > 0) ∗ ∗ ˙ ˙ is a constant, Eq. (8) can be written as (y/y) = (k/k) = g/(1 − ˛). We therefore have the following steady-state growth models: y∗ = y0∗ e(g/(1−˛))t

(9)

k∗ = k0∗ e(g/(1−˛))t

(10)

where y* and k* are steady-state output per capita and steady-state capital per capita, respectively. The phenomena of current y approaching y* and current k approaching k*, are referred to as convergence; the process of the convergence is called transition; and the current y moving away from y*, and the current k moving away from k*, is called divergence. Eqs. (9) and (10) indicate the steady-state growth goals of real output per capita and of capital per capita, respectively. To meet the second-order condition for profit-maximization (Mai & Shieh, 1984; Lin, 2004), ˛ + ˇ < 1, instead of ˛ + ˇ = 1, is usually assumed in Model (2) or (4). The relevant discussion is presented in Appendix A. 3.3. Long-term growth and diversity in economy In Model (4), as innovation increases, the limiting value of optimal output under environmental pressure is: lim Y + =

v + wq

Y˙ + K˙ + L˙ + G˙ + =˛ + +ˇ + + + = Y+ K L G

According to the literature (Barro & Sala-i-Martin, 1995), if y/k does not change, economic growth is in steady-state. We denote ∗ ∗ ˙ ˙ this (y/k)*. Because d(y/k)∗ /dt = 0, we obtain (y/y) = (k/k) , and Eq. (7) can be expressed as:

q→∞



1−

1 

  1 1/(−1) w

(11)

Eq. (11) implies that the target of long-term growth is raised if the environmental cost coefficient of emerging output, that is the environmental cost of emerging output, is reduced. From Eqs. (5) and (11), we observe that, given suitable economic environment, long-term optimal output growth can be classified into three stages. In the first stage, the economy is in a state of healthy expansion: cur1/(−1) and optimal output grows. rent innovation is q < q+ = (v/w) In the second stage, the economy is in a normal recession: current 1/(−1) and optimal output falls. In the innovation is q > q+ = (v/w) third stage, the economy is in a state of balanced growth, reaching a stable limiting value when innovation efficiency is sufficiently large. The path, including the three stages, is an optimal growth path. ˙ is finite, then If the innovation growth rate, s ≡ q/q, lim (Y˙ + /Y ) = lim [(q/( − 1))((v − wq−1 )/(1 + q) · (v + wq ))] q→+∞

q→+∞

s = 0 from Eq. (5). Thus, we see that along an optimal path, real output eventually reaches a stable state. We now consider whether economic diversification contributes to long-term growth. In general, economic diversification refers to industrial segmentation caused by the segmentation of products and markets. From Appendix B, we know that the more subdivided industry is, the higher are the real output. In fact, if industries can share all of the benefits of technological progress and innovation growth, then industrial subdivision can disperse environmental pressure and market risk. Individual industries will be more specialized and better able to apply specialized technologies to their production processes and products, and the economy is diversified. Consequently, these industries will face less market risk and lower environmental pressure. Industries can thus develop more

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specialized markets, incur smaller environmental costs, and realize higher returns and higher real output growth. Here, we do not take the cost of subdivision into account. Nevertheless, the contributions of industrial segmentation to output growth should not be ignored. Appendix B also shows that, if the environmental pressure index does not change, industrial subdivision will increase the limiting value of optimal output and promote long-term economic growth. 3.4. Convergence and divergence In Model (2), define y¯ ≡ Y/L as real output per capita and k¯ ≡ K/L as capital per capita. Model (2) can then be written: y¯ = 0 k¯ ˛ G

(12)

For steady-state y* and k*, y¯ approaching y* or k¯ approaching k* is convergence, and y¯ moving away from y* or k¯ moving away from k* is divergence. Unlike most current economic growth models, Model (12) incorporates not only labor, capital and innovation, but also environmental pressure. Because economic policy and structure reforms can affect environmental pressure, Model (12) has implications for economic policy. Importantly, Model (12) indicates that economic growth has environmental costs. Increases in basic and emerging output cause real output to grow, which in turn causes environmental costs of basic and emerging output to gradually rise. Comparing Models (9) and (12), we see that when environmental costs are low, real per capita output stays close to steady-state per capita output while basic and emerging output both rise, and convergence occurs. However, when environmental costs are high and real per capita output starts to deviate from steady-state per capita output, divergence ensues. Thus, environmental pressure can cause divergence. 4. Results Eq. (3) implies that technology and innovation permeate all aspects of traditional industries; basic output shrinks as innovation increases, and real output growth comes to depend mainly on emerging output growth. We therefore obtain Conclusion 1.

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Conclusion 2 indicates an optimal economic growth path, that is a cycle of optimal economic growth under environmental pressure. Then, based on Appendix B, we have Conclusion 3 below. 4.3. Conclusion 3 If the environmental pressure index and environmental cost coefficients are unchanged, all the subdivided industries will share the benefits of technological progress and innovation growth. Thus, economic diversification will increase real output and promote long-term growth. Conclusion 3 indicates the path of economic structural reform that is needed to promote long-term growth. In Model (12), when economic growth is in its early or middle stage, environmental costs are small and convergence occurs; when economic growth is in its final stage, environmental costs are large and divergence results. We thus obtain Conclusion 4. 4.4. Conclusion 4 High environmental pressure causes divergence. The economy is thus pushed into recession. Conclusion 4 indicates that economic recession can result from high environmental costs. According to Models (2) and (12), in the final stage of normal growth, high environmental pressure makes it increasingly difficult for the economic system and economic policy to stimulate capital investment, technological progress and innovation growth. Divergence develops and economic output starts declining rapidly. If basic output follows Eq. (3) and real output follows Model (4), then real output will tend to approach a stable value instead of declining quickly. Therefore, from Conclusions 2, 3 and 4, Conclusion 5 follows. 4.5. Conclusion 5 When real output follows a normal growth path and no reform measures are implemented, the economic recession may be collapsed in case it occurs. If basic output and real output are optimal, then economic collapse can be avoided. Conclusion 5 indicates there are two different types of recessions and presents a way to avoid economic collapse.

4.1. Conclusion 1 Given economic process, with technological progress and innovation growth, traditional industries are either transformed into emerging industries through the introduction of innovations or shrink and eventually disappear. Emerging industries then contribute increasingly significantly to real output. Conclusion 1 indicates a structural change in the process of real output growth, and that innovation goods will become the dominants in future economic growth, and the investments on the specific targeting innovation goods will become the dominant ways to invest. As we know from Model (4) and Eqs. (3), (5) and (11), there are three stages of optimal economic growth under environmental pressure: (i) optimal output growth, (ii) normal (optimal output) recession, and (iii) the approach of optimal output growth to its stable limiting value. We therefore arrive at Conclusion 2, as follows. 4.2. Conclusion 2 If the environmental pressure index does not change, and basic output and emerging output are optimal, then optimal output will first grow, then decline and finally attain a stable limiting value with continuous innovation growth.

5. Discussion 5.1. Economic growth Solow (1957) introduced technical progress into production function, thereby establishing the famous Solow growth model, which converges technical progress into the production function as a means of analyzing long-term economic growth characteristics. Romer (1986, 1990) studies a model of long-run growth in which knowledge is assumed to be an input in production that has increasing marginal productivity, and a model in which growth is driven by technological change that arises from intentional investment decisions made by profit maximizing agents. Model (2) is simplified to the Solow growth model and the AK model (Jones, 1998; Romer, 1986) if v = w = 0, and to the Cobb–Douglas production function if v = w = 0 and  0 = 0. On the other hand, if  = 1 and v= / 0 and w = / 0, the environmental costs are linear, and thus y = (1 − v) + (1 − w), which can be discussed according to works on the Solow model. King and Rebelo (1990) show that national taxation can substantially affect long run growth rates. Alesina and Rodrik (1994) study the relationship between politics and economic growth in a simple model of endogenous growth with distributive conflict among

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agents endowed with varying capital/labor shares. Model (2) indicates the quantitative relationship between macroeconomic policy and output growth. Rebelo (1991) describes a class of models in which the heterogeneity in growth experiences can be the result of cross-country differences in government policy. By the EPI, Model (2) can describe the heterogeneity and differences in government policy. Peretto (1998) discusses a model where increasing returns generate long-run growth but where the scale effect is absent, and predicts that steady-state productivity growth does not depend on population size because an increase in population size leads to entry. Model (2) highlights the impacts of capital, labor, innovation and environmental pressure on economic growth. Dinopoulos and Unel (2011) develop a fully endogenous, variety-expansion growth model with firm-specific quality heterogeneity, and this is in accord with the diversification in Conclusion 3. Garces and Daim (2012) conclude that technological innovation positively affects the U.S. economy in the long run. Eq. (4) indicates the optimal effect of technological innovation on real output. Desmet and Parente (2012) puts forth a theory of the industrial revolution whereby an economy transitions from Malthusian stagnation to modern economic growth as firms implement cost-reducing production technologies. We can show the process of long-term growth and structure change by Eqs. (4) and (11), and Models (2), (6) and (12). The current literatures cited above address the effects of technological change, political conditions, population size and other factors on growth, but we need to consider the impact of environmental pressure. Because economic pressure arising from environmental factors significantly affects output and growth, this pressure must not be ignored; rather, it is an important parameter that must be introduced into growth models and correlation analysis. The ARC Model (2) represents a normal growth model, but with environmental pressure incorporated into it as a key feature. 5.2. The optimal growth Here, we discuss the optimal growth paths of basic, emerging and real output, as implied by Eqs. (3) and (4) and depicted in Fig. 1. Fig. 1 shows the evolutionary process of an optimal growth cycle. In the early stage of the cycle, (0, q1 ), optimal basic output is high and grows at a fast rate. Thus, basic output makes a major contribution to real output at this stage. In the middle

Fig. 1. The optimal growth paths of basic, emerging and real output. Note: let ˙ = 0.06, q = q0 est . Then, according to Eq. (3)  = 1.35, v = 0.1, w = 0.05, q0 = 0.1, s = q/q and Model (4), figure shows the optimal growth paths of basic, emerging and real output, and the change in their structure.

stage, (q1 , q2 ), emerging output surpasses basic output and grows rapidly as basic output declines. During this stage, emerging output replaces basic output as the main driver of real output growth. In the final stage, (q2 , −), emerging output may continue to grow somewhat as real output declines, but eventually emerging output declines. Ultimately, emerging and real output approach their limiting values. Because technological progress and innovation growth, which underly emerging output growth, continuously evolve in response to market conditions, technological progress and innovation growth can support stable emerging output, and ultimately stable real output. In addition, Fig. 1 shows the role of technological progress and innovation growth in long-term growth. It is worth noting that the environmental pressure index in Fig. 1 is a constant. As real output approaches its limiting value, if authorities could implement economic and structural reforms that effectively decrease the environmental pressure index – reducing the environmental cost coefficient for emerging industry and creating a freer economic environment – then the economy may embark on a new optimal growth cycle. 5.3. Convergence and divergence The empirical research shows that for similar economies, convergence is ubiquitous (Barro & Sala-i-Martin, 1992). How do we explain the findings of Fung (2009)? The world economy and the economic environment are constantly changing. Thus, we must ask whether convergence is sustainable over time, what prospects exist for convergence under environmental pressure, and whether divergence is likely to come into play. If convergence or divergence is related to environmental pressure, what is the relationship among them? These questions are discussed below. To explain why the transition and convergence of real output per capita vary depending on the degree of environmental pressure, environmental pressure indices are separately given as  1 = 1,  2 = 1.60 and  3 = 1.7 in Model (12). Fig. 2 shows the processes of transition, convergence and divergence of real output per capita.

Fig. 2. The convergence and divergence of real output per capita under environmental pressure. Note: without a loss of generality, let ˛ = 1/2 and L = 1 (labor) in Model (12). In this figure, y* is steady-state output per capita. When  1 = 1, the corresponding real output per capita, y1 , will gradually converge to steady-state y*. When  2 = 1.6, y2 maintains convergence for an extended time, but ultimately slowly diverges. When  3 = 1.7, after a short convergence, y3 diverges from y*, and the divergence gradually speeds up.

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Fig. 3. Real GDP data and relevant curves. Note: all of Real GDP and relevant curves are plotted on a logarithmic scale. (a) GDP is in billions of U.S. dollars. Y* = 6.3 + 0.048868t is the logarithm of the steady-state output, where 4.8868% is the average of the U.S. Federal Funds Rate (1941–2003). The initial data source is the 2004 Report Spreadsheet Tables (http://www.gpoaccess.gov/usbudget, B-73. Bond yields and interest rates, 1929–2003). (b) GDP is in hundred millions of RMB. Y* = 8.9 + 0.125t is the steady-state output, and estimated on China’s GDP data (1996 and 2010, the peak data).

As observed in Fig. 2, when environmental pressure index  1 = 1, real output per capita converges gradually to its steady-state value after a transition. This result is consistent with general convergence (Barro & Sala-i-Martin, 1995). In addition, as shown in Fig. 2, if  2 = 1.60, then divergence gradually sets in after real output per capita maintains convergence for an extended period. If  3 = 1.7, after a brief convergence, real output per capita diverges from its steady-state; and then divergence

innovation growth rate s = 0.097.3 Thus, Model (2) is expressed as Y (t) = 0 e0.11t +  0 e0.207t − (0 e0.11t + 0 e0.207t ), which can be estimated by the regression method. When the error is small and the coefficient of determination is large, the fitting result based on min >1

0 = 35.7489657770667578; 0 = 3.3063351739343; gradually speeds up. Therefore, the greater the environmental pressure, the more early the real output per capita diverges from its steady-state value and the more pronounced the divergence. Notably, the initial transition paths are all similar, whether environmental pressure is large or small. The reason for the similarity is that in the initial stage, environmental pressure does not yet play a significant role. These results cannot be replicated using current growth models.

0 = 0.00012404265482953;

6.1. Using Model (2) to fit the U.S. GDP The real GDP data used in this study were obtained from the United States White House website1 and recorded as D(t), t = 1940, . . ., 2012. Model (2) is used to fit the U.S. GDP data below employing the fitting function Fit[Y(t), D(t), t] in the MAPLE software system. In Model (2), if basic output is expressed as  = 0 K˛ Lˇ = 0 et , where  is the basic output growth rate. We let q = q0 est , the basic output growth rate  = 0.1102 and that the economic

1

http://www.whitehouse.gov. Direct capital average growth rate (1940–2010) is 6.7%, data source: U.S. White House Web, http://www.whitehouse.gov. Average growth rate of employment labor 2

2

[Yˆ (i) − D(i)] is:

i=1940

0 = 0.30889078002588355;

q0 = 0.002780214901;

 = 1.412.

The real GDP data, Model (2)-based U.S. GDP fitting curves, the steady-state output and optimal real output are depicted in Fig. 3a. Yˆ (t) is the Model (2)-based U.S. GDP fitting curve, the fitting errors are:

2012

ε1 =

1 ¯1 D

t=1940

2

[D(t)−Yˆ (t)]

= 0.052168;

2012−1939

with coefficient of

determinationR12

 = 0.977227. 2012

6. Empirical Research

2012

ε2 =

1 ¯2 D

t=1965

2

[D(t)−Yˆ (t)]

2012−1964

= 0.0065326;

with coefficient of

2012

¯1 =( D(t))/(2012 − determination R22 = 0.993832.where D t=1940 2012 ¯ 1939), D2 = ( t=1965 D(t))/(2012 − 1964). In Fig. 3, the results for real GDP show that Model (2) is a much better fit after 1964 than before, this means that the environmental pressure on U.S. economy increases. The figure also shows that around 2010, the U.S. economy peaked and started showing signs of decline and that a tendency to decline continued after 2012.

force (1940–2010) is 4.3%, data source: U.S. Bureau of Labor Statistics Web, http:// data.bls.gov/pdq/SurveyOutputServlet. 3 The growth rate of utility patent applications in U.S (1969–2010), data source: U.S. Patent and Trademark Office Web, http://www.uspto.gov/web/offices.

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6.2. Using Model (2) to fit the China’s GDP Similar to the empirical for U.S. GDP, empirical research based on China’s GDP4 is presented in the section. Where, the basic output growth rate  = 0.215 and that the economic innovation growth rate s = 0.23.6 Thus, Model (2) is expressed as Y (t) = 0 e0.21t +  0 e0.44t − (0 e0.21t + 0 e0.44t ). When the error is small and the coefficient of determination is large, the fitting result is: 0 = 1415.64433421687431;  0 = 0.594398368941229105; q0 = 0.0003343984151; 0 = 104.50662279931; 0 = 0.00031813997053428;  = 1.413.7 The real GDP data, Model (2)-based China’s GDP fitting curves, the steady-state output and optimal real output are depicted in Fig. 3b. The fitting error is:

1 ε= D

2013 t=1978

2

[D(t)−Yˆ (t)]

2013−1977

= 0.07113921495;

with coefficient of determinationR2 = 0.9831491908; ¯ = where D





2013 D(t) t=1978

(12)

/(2013 − 1978), Yˆ (t) is the Model (2)-

based China’s GDP fitting function. Fig. 3b shows that China’s GDP is peaked at T = 2012.86, and started showing signs at and after 2013. In fact, China’s GDP growth is slowing in 2013. 6.3. Analysis Fig. 3 shows that both American and Chinese economic growths are now under increasing environmental pressure. In detail, Fig. 3a shows that the American economy has grown throughout this period along a steady-state growth path. After a long-term convergence of approximately 70 years, divergence began in 2008. This divergence suggests that the American economy is facing a momentous decision. In contrast to real U.S. GDP and normal growth based on Model (2), optimal output is now higher, which implies that policymakers should seek to achieve growth along the optimal path. If no significant changes in the economy occur, the future recovery of the American economy may be weak. According to the results in this paper, if the goal is to promote longterm growth, the U.S. government should make full use of the large technological reserves it has accumulated. Accordingly, the government should implement policies that promote construction of an economic system that is based on innovation goods, development of emerging industries, changes in the economic structure and system, and the start of a new economic growth cycle. Fig. 3b shows that the Chinese economy has grown throughout this period along a steady-state growth path. After a long-term convergence of approximately 34 years, divergence began in 2012. This divergence suggests that the Chinese government is faced with a choice of economic change. According to Conclusions 1, 4 and 5, both U.S. and China require a major change in order to maintain a sustained growth. The ways of causing the change include below:

- Policy change. In order to reduce the environmental pressure and cost in economic growth, changes need to take place in economic policy. This helps to improve the production and product structure, and form the new mode for economic growth. - Technological innovation. Technical innovations may revolutionize the mode of production, prompted a new long-term economic growth. - Structural change. The effective segmentation based on existing industries may invent or stimulate the new market demands and promote economic growth. Literature (Dai, Liu, et al., 2013) gives the relevant analysis on industry segmentation. - Comprehensive reform. The reforms of economic policy, as well as technology and market structure, may cause a brand-new mode of production, and prompte a new round of sustainable economic growth. Fig. 3a and b indicate that both U.S. and China’s current outputs are much lower than their optimal outputs; this means there are huge rooms for U.S. and China’s economic growth. The key is how to achieve the potential outputs by using the changes. 7. Conclusions Using the Solow growth model and normal advance-retreat course (ARC) model, this paper considers long-term economic growth under environmental pressure and achieves the following: (i) Presents the normal growth model under environmental pressure. Economic growth must utilize economic and natural resources to confront environmental pressure. This paper discusses long-term economic growth and presents the normal economic growth model under environmental pressure. Based on differing degrees of environmental pressure, economic growth is divided into three main types: infinite-growth, critical-growth and consumption-growth. Critical-growth is a special case of economic growth, while consumption-growth is the most common case. (ii) Builds an optimal output growth model under environmental pressure. Using normal model of economic growth under environmental pressure, the paper formulates optimal basic, emerging and real output growth models. The optimal models indicate that to maximize real economic output, basic output and innovation should be properly proportioned. (iii) Proposes an optimal growth accounting equation. Based on the optimal output growth models, this paper proposes an optimal growth accounting equation under environmental pressure. The equation indicates that the optimal path of long-term economic growth includes three stages: healthy growth, normal recession, and output approaching a stable limiting value. (iv) Completes the empirical researches. By using the real GDP data of U.S. and China, this paper completes the empirical researches for economic growth. The results show that both U.S. and China require the major changes in order to maintain a sustained growth. Based on the above analysis, this paper reaches the following conclusions:

4 The real GDP data were obtained from the Web of National bureau of statistics of China (http://data.stats.gov.cn/workspace/index?m=hgnd), and recorded as D(t), t = 1978, . . ., 2013. 5 Total social fixed asset investment growth rate (1980–2012) is 21.34599729%, data source: National bureau of statistics of China. 6 Domestic patent grant growth rate (1995–2012) is 22.76318205%, data source: National bureau of statistics of China. 7 The environmental pressure index of China is  = 1.413, this mean that China’s economic growth is under larger environmental pressure than U.S.

- Change in the structure of real output. In long-term economic growth, with technological progress and innovation growth, traditional industries shrink and eventually disappear, while emerging industries expand and come to account for an increasing share of real output. - Paths for reforming economic structure. If the environmental pressure index and the environmental cost coefficients are

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unchanged, and all the subdivided industries share the benefits of technological progress and innovation growth, then economic diversification will increase real output and promote long-term economic growth. - Future investment way. Innovation goods will become the dominants in future economic growth, and the investments on the specific targeting innovation goods will become the dominant ways to invest. We can raise the long-term target of economic growth by reducing the environmental costs of emerging outputs. - Economic recession is caused by high environmental costs. High environmental costs caused by high environmental pressure lead to divergence, the economy is then pushed into recession. - Economic collapse can be avoided. If basic and real output growths are optimal, then economic collapse is avoidable.

9

wq )−1 n , and real output for the economy as a whole (real output summed over all subdivided industries) is:



Y¯ n = n · Yn =  1 + q −

1 n−1

1 (v + wq )−1 

(B.2)

Comparing Models (B.1) and (B.2), and noting that n > m, we obtain Inequality (B.3) as follows:



Y¯ n − Y¯ m = 1 −

 m −1  n

1 m−1

(v + wq ) > 0

(B.3)

Inequality (B.3) implies that industrial subdivision can increase real output for the economy as a whole. If we set the derivative dY¯ /dn = 1 + q − (1/n−1 )(v +  wq )−1 = 0 in Model (2), then optimal basic output is 1/(−1)

Another conclusion derived from this paper is that differences in environmental pressure cause large gaps in wealth and real output between rich and poor countries (as well as between the rich and poor within countries) and that these gaps increase over time.

+ = n((1 + q)/(v + w · q )) . Substituting this expression into Model (B.2), we obtain optimal output for the economy as a whole after subdivision as follows:



Y¯ n+ = n 1 −

Appendix A. If ˛ + ˇ < 1 in Eq. (4), we have ˛ + ˇ = 1 − ı (ı > 0). Then  = 0 k˛ L1−ı . Model (4) can then be expressed as y = 0 k˛ L−ı G+ . The growth accounting Eq. (7) can be expressed as: y˙ G˙ + k˙ L˙ =˛ + + −ı y G L k

(A.1)

If y/k does not change (i.e., (y/k)*), from d(y/k)*/dt = 0, we obtain (y/y)* = (k/k)*, and Eq. (A.1) can be expressed as follows:

 y˙ ∗ y

 ∗

=

k˙ k

=

1 1−˛



L˙ G˙ + −ı G+ L

∗

(A.2)

From Eq. (A.2), output per capita and capital per capita will ∗ ˙ grow at the rate of (1/(1 − ˛))((G˙ + /G+ ) − ı(L/L)) , and changes in ∗ + + ˙ ˙ ((G /G ) − ı(L/L)) will influence the steady-state growth process. ˙ According to Model (4), when G˙ + /G+ = ı(L/L) + g (where g > 0 is a constant), we have G˙ + L˙ −ı =g G+ L

(A.3)

We also obtain steady-state growth Models (9) and (10) based on Eq. (A.3). Appendix B. If real output in an economy follows Model (2), the environmental pressure index and the innovation and environmental cost coefficients are unchanged, we assume m industries, where the value of basic output in each industry is identical, but each industry produces a unique product. We also assume that each industry shares the benefits of technological progress and innovation growth. For each industry, basic output is m ≡ (1/m) and emerging output is  m ≡ m · q = (1/m). Therefore, according to Model (2), real output for each industry is Ym = m · Gm , Gm = (1 + q) − (1/)(v + wq )−1 m , and real output for the economy as a whole (real output summed over all industries) is



Y¯ m = m · Ym =  1 + q −

1 m−1

1 (v + wq )−1 

(B.1)

We now subdivide the economy into n (n > m) industries, where the value of basic output is identical for all industries, but each industry produces a unique product. For each subdivided industry basic output is n ≡ (1/n) and emerging output is  n ≡ n · q = (1/n). Therefore, according to Model (2), real output for each subdivided industry is Yn = n ·Gn , Gn = (1 + q) − (1/)(v +

1 

  (1 + q) 1/(−1)

(B.4)

v + wq

1/(−1) From Model (B.4), we have lim Y¯ + = n(1/w) , which q→∞

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