Emission trading for air pollution hot spots: getting the permit market right

Abstract

Conventional emission permit markets are inefficient for non-uniformly mixed pollutants that create geographic ‘hot spots’ of different ambient emission concentrations and environmental damage. Economically efficient ambient concentration contribution markets involve difficult interactions among multiple markets that makes them practically infeasible. Extending economic theory by Muller and Mendelsohn (Am Econ Rev 99(5):1714–1739, 2009. doi:10.1257/aer.99.5.1714) and others about ‘getting the prices right’ through bilateral trading ratios, this paper introduces theoretical simplifications and a novel type of single permit market with a hybrid price-quantity instrument that addresses the dual heterogeneity of firm-specific abatement costs and regional variation in damage. This paper shows how to ‘get the market right’ robustly through simplicity, liquidity, and gradualism. Analytic solutions and simulation results demonstrate the feasibility of the novel market concept. Also discussed is the potential applicability of the market design to interstate trading in the United States in the wake of the recently implemented Cross-State Air Pollution Rule.

Appendices

Appendix 1: Quasi-linear optimization

This paper makes use of a novel approach to solve the problem of optimizing a non-linear policy objective function. There is a class of policy problems that are characterized by (a) a policy objective function that is non-linear (e.g., a dose–response function); (b) a macro-economic policy instrument (e.g., a country-wide emission cap) and a set of corresponding micro-economic policy instruments (e.g., firm-level price factors); (c) additive separability of the micro-economic instruments; (d) monotonicity of the effect of the macro-economic variable with respect to the objective function; and (e) the ability to use the macro-economic policy instrument to drive a gradual adjustment process over time. The “quasi-linear optimization” approach introduced in this paper replaces a complex non-linear optimization problem (which yields an optimal solution in a single step) with an equivalent quasi-linear optimization problem that is much easier to solve, but only through gradual convergence to the optimum over time. Specifically, consider the problem of minimizing the single-valued quadratic objective function

$$\begin{aligned} F({{\mathbf {x}}},z)= & {} \left[ \sum _i f_i(x_i,z)\right] ^2+\sum _i g_i(x_i,z)\nonumber \\\equiv & {} f({{\mathbf {x}}},z)^2-g({{\mathbf {x}}},z)\nonumber \end{aligned}$$

where ${{\mathbf {x}}}=(x_1,\ldots ,x_n)$ is a vector of n micro-economic policy instruments that satisfy $\partial ^2 F/\partial x_i\partial x_j=0$, z is a macro-economic policy instrument that satisfies $\partial F/\partial z<0$ for any $z>z^*$, and $\alpha$ is a scaling parameter. Many non-linear policy problems can be approximated reasonably well as quadratic functions, at least near the optimal solution. Solving the system of first-order conditions

$$ \forall i:f({{\mathbf {x}}},z)\frac{\partial f_i}{\partial x_i}+\frac{\partial g_i}{\partial x_i}=0 $$

and

$$ f({{\mathbf {x}}},z)\sum _i\frac{\partial f_i}{\partial z}+\sum _i\frac{\partial g_i}{\partial z}=0\quad $$

yields the optimal policy $({{\mathbf {x}}}^*,z^*)$. However, this system of $n+1$ equations in $n+1$ unknowns can be very difficult to solve. Even though the partial derivatives may only depend on $x_i$ separately, the appearance of $f({{\mathbf {x}}},z)$ interacts the $x_i$ in each equation with all other $x_{j\ne i}$.

Quasi-linear optimization involves minimizing the objective function

$$ \tilde{F}\equiv f({{\mathbf {x}}}_{t-1},z_{t-1}) f({{\mathbf {x}}}_t,z_t) +g({{\mathbf {x}}}_t,z_t) $$

gradually over time t. Objective function $\tilde{F}$ approximates F when $|z_t-z_{t-1}|$ is small. In each period t, the n first-order conditions $\partial f_i/\partial x_{t,i}=0$ deliver a set of micro-economic policies ${{\mathbf {x}}}^*_t$ that are optimal in that time period conditional on the choice of macro-economic policy $z_t$ that is governed exogenously. Because of the independence of the micro-economic policy instruments, these n first-order equations can be solved individually for the optimal policy $x^*_{i,t}$ at time t. Note that $f({{\mathbf {x}}}_{t-1},z_{t-1})$ is simply a scalar evaluated in the previous time period. The policy maker gradually reduces $z_t$ over time, starting from the initial policy-less state $z_0$. Decreasing $z_t$ in small steps, the optimization procedure stops when decreasing $z_t$ yields no further reduction in $\tilde{F}({{\mathbf {x}}}^*_t,z_t)$, i.e., when $\partial \tilde{F}({{\mathbf {x}}}^*,z)/\partial z=0$.

Appendix 2: Pollution-plant centrality

The regulator’s main function in overseeing the permit market is to announce new $\phi _{i,t}=Q_{i,t}/\bar{Q}_{i,t}$ at the beginning of each trading period t, based on measurements from the previous period. As ambient concentrations $A_j$ depend on contributions from all plants, the hazard summation factor $Q_{i,t}$ can be written as

$$\begin{aligned} Q_{i,t}= & {} \omega \sum _j P_j A_{j,t-1} d_{ij}\nonumber \\= & {} \omega ^2 \sum _j P_j d_{ij} \sum _k d_{kj} E_{k,t-1}\nonumber \end{aligned}$$

Emissions from other plants affect plant i through the network of dispersion coefficients $d_{ij}$. Firms are interconnected participants in the permit market where actions by one firm such as increasing or decreasing emissions may influence neighbouring plants by raising or lowering their particular hazard factor $\phi _i$.

It is possible to capture the degree of interconnectedness through empirical measures of centrality. Such measures are widely used in the theory of social and economic networks (Jackson 2008), and they are also used in applications such as Google’s web page ranking. In matrix notation, ${{\mathbf {Q}}}=\omega ^2 {{\mathbf {D}}} {{\mathbf {P}}} {{{\mathbf {D}}}}^{{{\mathsf {T}}}} {{\mathbf {E}}}$, where ${{\mathbf {Q}}}$ and ${{\mathbf {E}}}$ are $n\times 1$ vectors of each plant’s (non-normalized) hazard factor and emission level, ${{\mathbf {D}}}$ is an $n\times m$ matrix of dispersion coefficients, ${{\mathbf {P}}}$ is an $m\times m$ diagonal matrix with region factors $P_j$, and the ${}^{{{\mathsf {T}}}}$ superscript indicates the matrix transpose. ${{\mathbf {S}}}\equiv {{\mathbf {D}}} {{\mathbf {P}}} {{{\mathbf {D}}}}^{{{\mathsf {T}}}}$ is a symmetric $n \times n$ matrix that captures the inter-firm influences weighted by ${{\mathbf {P}}}$. Matrix ${{\mathbf {S}}}$ can be normalized through the transformation ${{\mathbf {U}}}\equiv {{\mathbf {R}}}{{\mathbf {S}}}{{\mathbf {R}}}-{{\mathbf {I}}}$ where ${{\mathbf {R}}}=[{{\rm diag}}({{\mathbf {S}}})]^{-1/2}$ and ${{\mathbf {I}}}$ is the identity matrix; this transformation is the same procedure that is used to compute a Pearson correlation matrix from a variance-covariance matrix. Thus, all elements in matrix ${{\mathbf {U}}}$ identify the relative impact of one plant on another, scaled to the range [0, 1[. Subtracting ${{\mathbf {I}}}$ ensures that the diagonal elements in ${{\mathbf {U}}}$ are all zero, as is required for adjacency matrices that are used to analyze networks.

The influence of an individual plant in the emission market network can be captured by eigenvector centrality, originally proposed by Bonacich (1972). The largest eigenvalue $\bar{\lambda }$ that solves the eigenvector equation ${{\mathbf {U}}}{{\mathbf {v}}}=\lambda {{\mathbf {v}}}$ is associated with the principal eigenvector $\bar{{{\mathbf {v}}}}$, which can be obtained numerically through power iteration even for very large matrices. The centrality measure can be suitably disaggregated into plants with zero or postive centrality. For the $n^{+}\le n$ firms with positive centrality, rescaling makes the measures easier to interpret. Let $\chi _i\equiv (\bar{v}_i/\sum _k \bar{v}_k)/(1/n^{+})$ denote plant centrality with an average of 1. Values larger than 1 indicate above-average centrality, and values below 1 indicae below-average centrality.

The plant-specific hazard factors $\phi _i$ will be strongly rank-correlated with the centrality measures $\chi _i$. The firm centrality measure is useful in the context of spatially interacting plants because it readily identifies which plants affect their neighbours more, and which less. For example, a centrality measure of $\chi _i=1.5$ suggests that plant i has 50 % stronger influence on neighbouring plants than the average plant. By comparison, a plant with $\chi _i=0.5$ is much less spatially connected, influencing other firms 50 % less than average. Plants that are far from any other plants have a $\chi _i$ equal or close to zero.