## Abstract

This paper develops a weighted continuous state space approach to convergence analysis and makes a first empirical attempt at examining the heterogeneity and convergence of China’s marginal abatement cost (MAC) of CO_{2} using a dataset of 286 cities during the years 2002–2013. The results show clear patterns of convergence in China’s MAC but substantial heterogeneity remains in the long run. Both joint and conditional emission-weighted distributions suggest less heterogeneity in abatement cost but greater potential for low-cost abatement opportunities than it appears in the commonly-presented unweighted distributions. The findings provide strong scientific support for China to take more proactive and aggressive measures to mitigate CO_{2} emissions.

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## Notes

Early nonparametric models use radial measures, which reflect the necessary proportional improvements of relevant factors (inputs/outputs) for the evaluated decision management unit (DMU) to reach the frontier. Yet radial measures tend to overestimate energy efficiency, because it ignores the non-radial input/output slacks. To overcome this issue, the non-radial SBM was proposed to treat non-proportional improvements (Tone 2001).

The approach inevitably underestimates total energy consumption due to notably, the omission of natural gas. The underestimation of energy consumption is likely to generate an upwards bias in the point estimates MAC across the sample. However, as natural gas only accounts for a very small proportion of China’s total energy consumption, we consider the bias to be insignificant. In addition, the bias is to shift the entire MAC distribution upwards while the impact on the dispersion and dynamic convergence or divergence characteristics of MAC distribution (i.e. the focus of the paper) is likely to be even less significant.

For simplicity, we are assuming a homogeneous curvature of the MAC curve of PAA cities; however, this may not be true. For a discussion of the implications of heterogeneous curvatures in MAC curves, please refer to Hailu and Ma (2017).

The convergence/divergence analysis using the state space approach is a descriptive rather than an inferential model such that it is unable to discern causal effects. Discussion of potential factors that influence heterogeneity and convergence is most likely speculative. Identifying causal effects will have to involve inferential statistics which is beyond the scope of this paper.

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## Acknowledgements

The research received financial support under a number of funding schemes including the ECR Fellowship Support Award at the University of Western Australia, China National Social Science Foundation (No. 15ZDA054), Humanities and Social Sciences of Ministry of Education Planning Fund (No. 16YJA790050) and the National Natural Science Foundation (No. 71333007).

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## Appendix: Estimating of the MAC of Carbon Dioxide

### Appendix: Estimating of the MAC of Carbon Dioxide

Assuming that there are *I* cities and each city uses *N* inputs \( \varvec{x} = \left( {x_{1} , \ldots ,x_{N} } \right) \in R^{N} \) to produce one good output *y* and one undesirable output *b*, we can then define a production possibility set (PPS) as follows:

where \( \varvec{X} \), \( \varvec{Y} \) and \( \varvec{B} \) are *N* × *I*, 1 × *I* and 1 × *I* matrices of input and output data for all *I* cities. \( \varvec{\lambda}\in R^{I} \) is a non-negative intensity vector, indicating that the above definition corresponds to the constant returns to scale (CRS) technology.

Following Cooper et al. (2007), we can further define the non-radial SBM model:

where \( s_{i,n}^{x} \) and \( s_{i}^{b} \) denote the slack variables of input *n* and the undesirable output, which represent excesses in input *n* and the undesirable output. *s*
^{y}_{
i}
is a slack variable of the good output which corresponds to shortage in the good output. \( \varvec{X}_{\varvec{n}} \) is the *n*th row of \( \varvec{X} \). The objective value of Eq. (A1) satisfies \( 0 < \rho_{i}^{ *} < 1 \). A DMU (*x*_{i}, *y*_{i}, *b*_{i}) is efficient if and only if \( \rho_{i}^{ *} = 1 \) ,indicating that all slack variables are zero. Since Eq. (A1) is a nonlinear programming model, the dual linear program of Eq. (A1) can be specified as follows:

The dual variables \( \varvec{p}^{\varvec{x}} \in R^{N} , \)*p*^{y} ∊ *R*^{M}, and *p*^{y} ∊ *R*^{K} are the relative shadow prices of input, desirable output and undesirable output, respectively. Assuming that the absolute shadow price of desirable output (gross city product, GCP) equals to its market price (i.e. one), the absolute shadow price of the undesirable output (CO_{2} emissions) can be estimated through:

Therefore, the shadow price of undesirable output can be interpreted as the marginal rate of transformation between undesirable output and desirable output. It shows the tradeoff between desirable and undesirable output. In this paper, we include three inputs (labor, capital, and energy consumption), one desirable output (GCP) and one undesirable output (CO_{2} emissions).

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Wu, J., Ma, C. The Convergence of China’s Marginal Abatement Cost of CO_{2}: An Emission-Weighted Continuous State Space Approach.
*Environ Resource Econ* **72**, 1099–1119 (2019). https://doi.org/10.1007/s10640-018-0240-1

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DOI: https://doi.org/10.1007/s10640-018-0240-1

### Keywords

- Marginal abatement cost
- Carbon emissions
- Convergence
- State space approach
- Continuous dynamic distribution approach
- China