## Abstract

China is facing the challenge of climate change and environmental protection in line with the promotion of sustainable development goals. Climate, environmental, and economic policy can each individually impact the effectiveness of other policies in the process of implementation. Understanding synergies among policy measures and identifying the co-benefits of environmental sustainable development generated by them is the focus of the paper. Because of existing restrictions in current research methods, the research community lacks the capacity to detect the interrelation among environmental, economic, and social development. Co-benefit research has gradually become an important area on sustainable development. Based on endogenous economic growth theory, we build a modified endogenous growth model to determine the optimal rates of economic growth, environmental pollution, energy input, and other factors in the context of co-benefits. Further empirical analysis is given which is missing in previous studies in this field. The empirical analysis reveals that China’s actual GDP, consumption, energy input, and physical capital growth rates are much higher than the optimal growth rate. However, the human capital and environmental pollution control growth rates are slower than the optimal values. Although China has a rapid growth in economy, the optimal level of comprehensive co-benefits has not been reached yet.

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## Acknowledgements

We acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 71774033) and Fudan Tyndall Centre (No. JIH6286001).

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## Appendix: Derivation details of the model

### Appendix: Derivation details of the model

### Appendix 1: Optimal growth rates

The objective function of the dynamic model can be established as follows

The current-value Hamiltonian is

where *C*, *R*, *μ*, and *V* are control variables, and *K*, *H*, *X*, and *S* are state variables.

We differentiate \(\tilde{H}\) partially with respect to the four control variables *C, R, μ,* and *V*, and set the results equal to zero:

The Euler equations of the four state variables are

Let g be the growth rate of each variable, that is,

From (40),

Assume that the technology level of environmental protection remains the same, that is, \(Q^{{\prime }} \left( V \right)\) is a constant. From (43),

Take derivative of (50) with respect to *t* on both sides, we have

From (44),

At steady state, the growth rates of all the variables are constant. So \(\dot{\lambda }_{k}\)/\(\lambda_{K}\) is a constant. \(\delta\) and \(\rho\) are constants, so \(\left( {1 + \frac{{\lambda_{X} }}{{\lambda_{K} }}\gamma } \right)\alpha \frac{Y}{K}\) is a constant. Take derivative of \(\left( {1 + \frac{{\lambda_{X} }}{{\lambda_{K} }}\gamma } \right)\alpha \frac{Y}{K}\) with respect to *t*. We have

From (41),

\(\eta \mu\)/\(\beta\) is constant, so the left-hand side of (54) is constant. Take derivative of (54) with respect to *t*.

From (42),

1/(\(1 - \alpha - \beta\)) is constant, so the left-hand side of (56) is constant. Take derivative of (56) with respect to *t*.

From (46),

\(\dot{\lambda }_{X}\)/\(\lambda_{X}\), \(\varepsilon\), and \(\rho\) are constants, so \(X^{\varphi }\)/\(\lambda_{X}\) is constant. Take derivative of (58) with respect to *t*.

From (47),

Take derivative of (60) with respect to *t*.

Divide Eq. (5) by *S* on both sides, we have

\(g_{S}\) is a constant, so take derivative of (62) with respect to *t*.

At steady state, from the relation of output, consumption, and physical capital, we have

Combine (49) to (57), (64), and (65), we have

Finally, from (59),

From the above, the optimal growth rate functions of physical capital, nonrenewable resources, environmental pollution, consumption, and human capital can be expressed as Eqs. (66)–(69).

### Appendix 2: The modified Uzawa–Lucas two-sector model

Suppose the consumption utility function is

Then, the traditional Uzawa–Lucas two-sector model is

where *Y* is the production, *C* is the consumption, *K* is the physical capital stock, *H* is the human capital stock, *μ* is the proportion of human capital investing into the physical production department, and *α* is the output elasticity of the physical capital. The physical capital and human capital discount \(\delta\) are assumed at the same rate. Then, since our production function involves three variables, the modified Uzawa–Lucas two-sector model is introduced

From the amended model, *η* can be estimated. According to Barro and Canestrelli (2005), the Hamiltonian function of this amended model is

The first-order conditions of the controlled variables *C* and \(\mu\) are

By using the equation \(\dot{\lambda }_{k} = - \partial J/\partial K\), the following equation is obtained

By using the equation \(\dot{\lambda }_{H} = - \partial J/\partial H\), the following equation is obtained

Differentiating Eq. (76) with respect to time and combining (79), the growth rate of consumption is given by

Then, subtracting Eq. (80) by \(\gamma_{k} = \frac{{\dot{K}}}{K} = AK^{\alpha - 1} \left( {\mu H} \right)^{\beta } R^{1 - \alpha - \beta } - \frac{C}{K} - \delta\), we have

By differentiating (76) with respect to time and combining the utility function (70) and (79), the growth rate of \(\mu\) is given by

Since Eqs. (80)–(82) are time independent, their first-order derivatives with respect to time are equal to zero. Then we have the steady-state solution. According to the conclusions from Uzawa–Lucas model, the steady-state rate of human capital return, \(r^{*}\), is equal to the net marginal production of *K* in the product sector and the net marginal production of *H* in the human capital sector. Moreover, \(r^{*}\) is equal to the difference of the technological parameter *η* and the human capital depreciation rate, *δx*

Equation (83) measures the technological parameter for the education sector *η*.

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Gao, S., Jiang, P. Detecting and understanding co-benefits generated in tackling climate change and environmental degradation in China.
*Environ Dev Sustain* **22, **4589–4618 (2020). https://doi.org/10.1007/s10668-019-00399-0

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### Keywords

- Co-benefits
- Climate change
- Environmental protection
- Endogenous economic growth
- China