The Convergence of China’s Marginal Abatement Cost of CO₂: An Emission-Weighted Continuous State Space Approach

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 CO₂ 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 CO₂ emissions.

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:

$$ P = \left\{ {(\varvec{x},y,b)|x \ge \varvec{X\lambda },y \le \varvec{Y\lambda },b \ge \varvec{B\lambda },\quad \lambda \ge 0} \right\} $$

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:

$$ \rho_{i}^{*} = \hbox{min} \frac{{1 - \frac{1}{N}\mathop \sum \nolimits_{n = 1}^{N} \frac{{s_{i,n}^{x} }}{{x_{i,n} }}}}{{1 + \frac{1}{M + K}\left( {\frac{{s_{i}^{y} }}{{y_{i} }} + \frac{{s_{i}^{b} }}{{b_{i} }}} \right)}} $$

(A1)

$$ \begin{array}{*{20}l} {{\text{s}}.{\text{t}}.\;x_{i,n} = \varvec{X}_{n}\varvec{\lambda}+ s_{i,n}^{x} ,\quad \forall n;\quad y_{i} = \varvec{Y\lambda } - s_{i}^{y} ;} \hfill \\ {b_{i} = \varvec{Y\lambda } + s_{i}^{b} ;s_{i,n}^{x} \ge 0,s_{i}^{y} \ge 0,s_{i}^{b} \ge 0,\varvec{\lambda}\ge 0,\quad {\text{i}} = 1, \ldots {\text{I}}} \hfill \\ \end{array} $$

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 nth 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:

$$ {\text{Max}}\;p^{y} y_{i} - p^{x} x_{i} - p^{b} b_{i} $$

(A2)

$$ \begin{array}{*{20}l} {{\text{s}}.{\text{t}}.\;p^{y} \varvec{Y} - \mathop \sum \limits_{n = 1}^{N} \varvec{p}^{\varvec{x}} \varvec{X}_{\varvec{n}} - p^{b} \varvec{B} \le 0} \hfill \\ {\varvec{p}^{\varvec{x}} > \frac{1}{m}\left( {\frac{1}{{\varvec{x}_{\varvec{i}} }}} \right)} \hfill \\ {p^{y} \ge \frac{{1 + p^{y} y_{i} - \varvec{p}^{\varvec{x}} \varvec{x}_{\varvec{i}} - p^{b} b_{i} }}{M + K}\left( {\frac{1}{{y_{i} }}} \right)} \hfill \\ {p^{b} \ge \frac{{1 + p^{y} y_{i} - \varvec{p}^{\varvec{x}} \varvec{x}_{\varvec{i}} - p^{b} b_{i} }}{M + K}\left( {\frac{1}{{b_{i} }}} \right)} \hfill \\ \end{array} $$

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 (CO₂ emissions) can be estimated through:

$$ SP^{b} = SP^{y} \times \frac{{p^{b} }}{{p^{y} }} $$

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 (CO₂ emissions).