An Optimal Emission Mechanism of Sustainability of China: How to Achieve a Win-Win Solution Between Economy and Environment?

2.1 Theoretical Review

Suppose that there is a manufacturing system (e.g., an industrial branch, a factory, a plant, etc.) produces useful goods. It has a negative environment impact. The fixed assets or equipment in it will be more effective due to materialized technological progress. And at the same time, negative effects will reduce. The following modification of the one-sector integral model with controllable equipment renovation can describe the manufacturing system above^[15]:

with the following additional balance for environment contamination:

In the equations above, Q(t) is the total output at time t, P (t) is the total required labor at time t, R(t) is the amount of pollutant emission per time unit (the level of environment contamination). β(λ(τ),τ,t) is the specific productivity, r(λ(τ),τ,t) is the amount of waste produced by one EU (equipment unit) per time unit, λ(t) represents the EU, m(t) is the quantity of new EU entered into the system. a(t) is the time limit of EUs use.

1) Optimization with penalty for pollutant emissions (factory)

Assume that the penalty r(·) can be imposed only for exceeding some admissible level R_max for pollutant emissions and the penalty is proportional to the R_max exceeding. Then the objective functional is of the form:

where

In (4), ρ(t) is a discounting multiplier, ρ′ < 0, 0 < m(t) < M (t), 0 < λ(t) < L(t), a′ (t) ≥a₀, a(t) < t.

So solving problem of maximizing the efficiency of factories changed into solve the optimization problem of (4).

2) Technology Optimization in economic-ecological system (government).

Intensification of the economic-ecological system (EES) resources and conservation and enhancement of the environment are the two inherent and contradictory criteria in EES control problems. As a rule, these criteria have different priorities at different management levels. Therefore, it is inevitable to analyze a multi-level hierarchical control system in solving EES control problems. The simplest form is a double control structure described below.

First, consider an optimal control problem of the EES (“Central” - “Factory”). “Central” is on the upper hierarchical level and controls the “Factory” via penalty rules. Factories need to pay a fine for exceeding a prescribed environment contamination level. Function which represents “Central”’s goal is as follows:

This time, form the optimal control strategy:

(i) “Central” optimal strategy

An optimal penalty r^∗ is less than r_max on the intervals when R*(t) > R_max(t). And r^∗ is determined optimally by taking the “Factory” expenditures into account.

(ii) “Factory” optimal strategy

An accelerated technological renovation with the environmental penalty r^∗ on the time intervals when the “Factory” pollutant emission R*(t) exceeds R_max(t).

The switch to an optimal “Factory” renovation rate determined by technological change only when the R*(t) does not exceed the admissible level.

2.2 Model Settings

It is difficult to obtain a continuous record for industry in practice. So we need to discrete the integral above, to make this model feasible. We also redefine the variables in (4), (6).

In (4), we use the technical input to measure the technological progress of industry. Depreciation rate is not to be considered. Industrial GDP (denoted by G) is chosen to replace Q(t).

Our object is the impact of the technical input in environment, so here we choose R&D funding instead of the cost λ (t) m (t), denoted by M (t). In summary, the discretized equation (4) is

In the same way, the discretized equation (6) is

The utility function determines the results of the “Central” objective function, which is pivotal. It needs to meet the requirements that “Central” objective function requires:

The utility function in Environmental Pollution Abatement and Economic Growth: Model and Evidence from China (Huang J N, Chen S H) can be used:

In this paper, x = G(t) −M (t), P = R(t). So when n = 1:

The tests whether the utility function satisfies the requirements of the objective function of “central” are as follows:

It shows that the marginal utility of (G −m) is always positive.

It shows that marginal utility is diminishing. Then take −σ(G −m)^−σ−1′s partial respect to σ:

When σ<1ln(G−m),∂∂σ(∂2I0∂(G−m)2)<0.. It means ∂2I0∂(G−m)2 decreases with σ. The smaller σ is, the faster the marginal utility decreases.

When σ>1ln(G−m),∂∂σ(∂2I0∂(G−m)2)>0.. It means ∂2I0∂(G−m)2 increases with σ. The bigger σ is, the faster the marginal utility decreases.

It shows that marginal utility of environment pollution is negative. Take ∂2I0∂R2’s partial respect to θ:

∂2I0∂R2 decreases with θ, that is the bigger θ is, the curve of ∂2I0∂R2= −ϕθ⁢Rθ−1 flatter, and the faster the marginal utility of R decreases. Obviously, “Central” hopes that the marginal disutility of R diminishes as faster as possible. Equation (9) meets the requirements, can be used as a utility function of the model. Thus the “central” objective function is

G(t)−m(t) represents the profit without punishment. It has a positive utility to the country. R represents the pollution. It has a negative utility to the country. So the emission R(t) is chosen to measure pollution. What is most important is ϕ which represents a penalty threshold value. It shows whether government pays more attention to the economy or the environment. The smaller ϕ is, the government pays more attention to the economy than the environment.

When the objective utility function of “Central” is positive, the entire industrial production brings a positive utility to the country. When it is negative, the utility is negative. More efforts should be made on controlling pollution and improving industrial technology such as purchasing pollution controls and increasing technical input in environment.

Thus, the model has been established. It is a model about the industrial GDP, technical input in industry and pollution penalty function. The purpose of it is: When given an expected industrial GDP, finding the proper technical input, obtaining industrial pollutant emissions standards and the penalty on excessive pollution factories with the aim of achieving “Central”’s maximum utility and “Factory”’s maximum benefits.

3.1 Analyze of Technical Input in Environment, Industrial GDP and Environment Investment

We got the industrial GDP data, R&D funding data with the year 2000 as the base period. R&D funding, industrial GDP, environmental protection investment and investment in industrial pollution is increasing. R = 0.97 shows a high correlation between R&D funding and industrial GDP, and the correlation is significant. With the increase of R&D funding, the scientific and technological level of industrial will be greatly improved, and industrial GDP will increase naturally. When the R&D funding amount exceeds a certain value, its impact on the industry’s GDP will gradually diminish. At this point, factories will not spend more capital to improve technique, because it does not bring a corresponding income. We obtained: To some extent, industrial technological innovation can promote the development of industrial.

3.2 Analyze of the Growth Rate of R&D Funding, Industrial GDP and Environment Investment

In order to study if there is a certain relationship between the growth rate of R&D funding and industrial GDP growth rate, R&D funding growth rate, industrial GDP growth rate, environmental protection investment growth rate, ratio of R&D funding to industrial GDP and other industry data are given.

Clearly, the growth rates maintain at a high level, especially R&D funding. The increasing ratio of R&D funding to industrial GDP shows that China is in a transition from labor-intensive to technology-intensive. According to the data, there is no correlation among industrial GDP growth rate, R&D funding growth rate and environmental protection investment growth rate. In other words, the determination of industrial GDP growth rate does not affect the determination of R&D funding growth rate. Environmental investment and R&D investment is irrelevant. It is worth mentioned that a sudden increase in investment in environmental protection to 82.64928 in 1999, the growth rate of is 584.33%, which is inseparable from the flood in 1998. This shows that China’s current environmental monitoring still lags. Investment in technological innovation is a very important in the transition period. But not the more investment the better, technology has its specific development process.

3.3 Results and Analyze

Through the analysis of the growth rate of science and technology investment, industrial GDP and environmental investment, government got the way that how to set the growth of technical input in industry, emission standards and penalty when given an expected GDP growth rate. This set of values makes “factory” and “central” optimal at the same time. The data is as shown in Tables 3 and 4.

From Table 3, when ϕ = 0.1, 1 unit of positive utility (generated by economic profit) corresponds to 10 units of disutility (generated by environmental pollution). This time, government has a serious balance bias in economic development. So under various GDP growth rate conditions, there is not so much difference among the optimal technical input growth, the penalties and the emissions standard. That is, it is invalid to increase investment in technological innovation and the penalties. When ϕ = 0.5, 1 unit of positive utility corresponds to 2 units of disutility. This time, government has a slight balance bias in economic development. So under various GDP growth rate conditions, increasing technical input and the penalties is useful, but the total amount is not significant. When ϕ = 1, 1 unit of positive utility corresponds to 1 unit of disutility. This time, government has a balance in economic and environment. When the GDP growth rate at the range of 3%∼19%, gaining technical input and increasing penalty is has slight benefit for economy development and emissions decrease. When the expected GDP growth rate is 21.99%, the emissions increase. It takes some time to get a higher technology level would be a proper explanation. When ϕ = 1.25 or 1.75, government has a balance bias in environment. The penalty for unit emission and the change of technical input changes greater, and the penalty for unit emission play a larger role. There is better control of emission by reducing technical input to some degree.

Tables 3 and 4 are the results obtained by Matlab. Optimized solutions of the two objective functions under different constraints are show in these two tables. The two tables are essentially identical. ϕ represents a penalty tendency, which measures government’s emphasis on the environment in some sense. What can be seen in Table 4 is that different values of ϕ correspond to different combinations of penalties and the growth rate of technical input, thought in the condition of approximate GDP. The emission combinations (including wastewater, waste gas, and Solid Waste gas) corresponded by different technical input growth are different significantly. A scientific setting of penalties is valid to reach equilibrium between environmental protection and economic growth. The optimization results in Table 3 and Table 4 are represented visually in the following three-dimensional figures:

From the figures above, the technical input growth rate is one of the factors that reduce emissions punishment, and the impact of the technology investment growth rate is more significant when industrial GDP growth rate is in a lower degree (Figure 1(a)). The solid waste emissions punishment is proportional to the growth rate of technical input inversely, and proportional to industrial GDP growth rate directly (Figure 1(c)). Decreasing technical input can control the wastewater emissions and waste gas emissions effectively at the same industrial GDP growth rate (Figure 1(d) and Figure 1(e)). Solid waste emissions reduce with the improvement of industrial GDP growth rate, but the impact of technical input growth rate is weak.

Figure 1

The relationships among growth of industrial GDP, technical input growth rate and emissions