Multiple Interactive Pollutants in Water Quality Trading

Abstract

Efficient environmental management calls for the consideration of multiple pollutants, for which two main types of transferable discharge permit (TDP) program have been described: separate permits that manage each pollutant individually in separate markets, with each permit based on the quantity of the pollutant or its environmental effects, and weighted-sum permits that aggregate several pollutants as a single commodity to be traded in a single market. In this paper, we perform a mathematical analysis of TDP programs for multiple pollutants that jointly affect the environment (i.e., interactive pollutants) and demonstrate the practicality of this approach for cost-efficient maintenance of river water quality. For interactive pollutants, the relative weighting factors are functions of the water quality impacts, marginal damage function, and marginal treatment costs at optimality. We derive the optimal set of weighting factors required by this approach for important scenarios for multiple interactive pollutants and propose using an analytical elasticity of substitution function to estimate damage functions for these scenarios. We evaluate the applicability of this approach using a hypothetical example that considers two interactive pollutants. We compare the weighted-sum permit approach for interactive pollutants with individual permit systems and TDP programs for multiple additive pollutants. We conclude by discussing practical considerations and implementation issues that result from the application of weighted-sum permit programs.

Appendices

Appendix 1 Kuhn-Tucker Conditions for the General Least-Cost Model

By introducing the multipliers λ _ir and ϕ _r for the constraints in Eqs. 2 and 3, respectively, we can create the following Lagrangian function:

$$ L=\sum\limits_{j=1}^m {C_j (e_{1j},\ldots,e_{nj})}+\sum\limits_{i=1}^{n-1} {\sum\limits_{r=1}^k {\lambda _{ir} \left(\sum\limits_{j=1}^m {a_{ij}^r e_{ij} } -\overline {S_{ir} } \right)+\sum\limits_{r=1}^k {\lambda _{nr} (\sum\limits_{j=1}^m {\sum\limits_{i=1}^n {b_{ij}^r e_{ij} } } -} } } \overline {S_{nr}})+\\ \sum\limits_{r=1}^k {\phi _r \left[ {f_r (\sum\limits_{j=1}^m {a_{1j}^r e_{1j},..,\sum\limits_{j=1}^m {a_{n-1j}^r e_{n-1j}, \sum\limits_{j=1}^m {\sum\limits_{i=1}^n {b_{ij}^r e_{ij} } } )-\overline {D_r } } } } \right]} $$

(A1-1)

Note that Eq. 4 in the text is an appropriate nonnegativity condition that we adopted to ensure that the emissions cannot be less than zero. Therefore, accounting for nonnegativity conditions in Eq. A1-1 is not necessary because under Kuhn-Tucker optimality conditions, the Lagrange multiplier cannot be negative. If the Lagrange multiplier is nonnegative, then we know that the associated constraint is binding under these conditions (see, e.g., Chiang 1984).

If f _r is a differentiable and concave function, the following Kuhn-Tucker conditions are necessary and sufficient for an optimal solution:

$$ \frac{\partial L}{\partial e_{ij} }=\frac{\partial C_j (e_{1j}, \ldots,e_{nj} )}{\partial e_{ij} }+\sum\limits_{r=1}^k {\lambda _{ir} a_{ij}^r } +\sum\limits_{r=1}^k {\lambda _{nr} b_{ij}^r } +\sum\limits_{r=1}^k {\phi _r \left[ {\frac{\partial f_r }{\partial S_{ir} }a_{ij}^r +\frac{\partial f_r }{\partial S_{nr} }b_{ij}^r } \right]} \ge 0\,\&\,e_{ij} \ge 0 \quad i=\left\{ {1,\ldots,n-1} \right\},\,j=\left\{ {1,\ldots,m} \right\} $$

(A1-2)

Then

$$ \sum\limits_{i=1}^{n-1} {\sum\limits_{j=1}^m {e_{ij}}} \left(\frac{\partial C_j (e_{1j}, \ldots,e_{nj} )}{\partial e_{ij} }+\sum\limits_{r=1}^k {\lambda _{ir} a_{ij}^r } +\sum\limits_{r=1}^k {\lambda _{nr} b_{ij}^r } +\sum\limits_{r=1}^k {\phi _r \left[ {\frac{\partial f_r }{\partial S_{ir} }a_{ij}^r +\frac{\partial f_r }{\partial S_{nr} }b_{ij}^r } \right]}\right)=0 $$

(A1-3)

Here we have made some minor simplifications. Given Kuhn-Tucker conditions, the product of the multiplier and the constraint should equal 0 for all i and r, not just for the sum over all i and r. We have condensed the required equations to reduce the length of the paper, but this does not change our results. Based on this explanation, Eqs. A1-3, A1-5, A1-7, A1-9, and A1-11 are derived from the desired Lagrangian function (A1-1).

$$ \frac{\partial L}{\partial e_{nj} }=\frac{\partial C_j (e_{1j},\ldots,e_{nj} )}{\partial e_{nj} }+\sum\limits_{r=1}^k {\lambda _{nr} b_{nj}^r } +\sum\limits_{r=1}^k {\phi _r \frac{\partial f_r }{\partial S_{nr} }b_{nj}^r } \ge 0\,\&\, e_{nj} \ge 0: \quad j=\left\{ {1,\ldots,m} \right\} $$

(A1-4)

Then

$$ \sum\limits_{j=1}^m {e_{nj}} \left(\frac{\partial C_j (e_{1j},\ldots,e_{nj} )}{\partial e_{nj} }+\sum\limits_{r=1}^k {\lambda _{nr} b_{nj}^r } +\sum\limits_{r=1}^k {\phi _r \frac{\partial f_r }{\partial S_{nr} }b_{nj}^r }\right)=0 $$

(A1-5)

$$ \frac{\partial L}{\partial \lambda _{ir} }=-\left(\sum\limits_{j=1}^m {a_{ij}^r e_{ij} } -\overline {S_{ir} } \right)\ge 0\,\&\,\lambda _{ir} \ge 0 \quad i=\left\{ {1,\ldots,n-1} \right\}, r=\left\{ {1,\ldots,k} \right\} $$

(A1-6)

$$ \sum\limits_{i=1}^{n-1} {\sum\limits_{r=1}^k {\lambda _{ir} \left(\overline {S_{ir} } -\sum\limits_{j=1}^m {a_{ij}^r e_{ij} }\right)} } =0 $$

(A1-7)

$$ \frac{\partial L}{\partial \lambda _{nr} }=-\left(\sum\limits_{i=1}^n {\sum\limits_{j=1}^m {b_{ij}^r e_{ij} } -\overline {S_{nr}}}\right)L\ge 0\,\&\,\lambda _{nr} \ge 0 \quad r=\left\{ {1,\ldots,k} \right\} $$

(A1-8)

$$ \sum\limits_{r=1}^k {\lambda _{nr} \left(\overline {S_{nr} } -\sum\limits_{i=1}^n {\sum\limits_{j=1}^m {b_{ij}^r e_{ij}}}\right)}=0 $$

(A1-9)

$$\frac{\partial L}{\partial \phi _r }=-\left(f_r \left(\sum\limits_{j=1}^ma_{1j}^r e_{1j},..,\sum\limits_{j=1}^m a_{n-1j}^r e_{n-1j},\sum\limits_{j=1}^m \sum\limits_{i=1}^n {b_{ij}^re_{ij}}\right)-\overline {D_r } \right)\ge 0\,\&\,\phi _r \ge 0\quad r=\left\{ {1,\ldots,k} \right\}$$

(A1-10)

$$\sum\limits_{r=1}^k {\phi _r \left(\overline {D_r } -f_r\left(\sum\limits_{j=1}^m a_{1j}^r e_{1j} ,..,\sum\limits_{j=1}^m{a_{n-1j}^r e_{n-1j} ,\sum\limits_{j=1}^m{\sum\limits_{i=1}^n{b_{ij}^r e_{ij}}}}\right)\right)}=0$$

(A1-11)

Appendix 2 Kuhn-Tucker Conditions for General Weighted-Sum Permit Markets

Lence (1991) showed that for multiple pollutants, the solution for the model based on Eqs. 1, 7, and 4 is the same as the solution for an individual polluter seeking to minimize the following model:

$$ \hbox{Minimize}: C_j (e_{1j} ,\ldots,e_{nj} )+p(l_j -l_j^0 ) $$

(A2-1)

$$ \hbox{Subject to}: l_j -\sum\limits_{i=1}^n {w_{ij} e_{ij} } \ge 0 $$

(A2-2)

$$ e_{ij} \ge 0 \quad i=\left\{ {1,\ldots,n} \right\} $$

(A2-3)

If the total permit is chosen so that

$$ \sum\limits_{j=1}^m {l_j^0 } =L_{ws}^0 $$

(A2-4)

market equilibrium exists in the permit system if there are nonnegative prices for all pollutants such as p ^* _i , and then the equilibrium levels of e ^* _ij and l ^* _ij solve the polluter’s minimization problem, and the market-clearing conditions are as follows:

$$ \sum\limits_{j=1}^m {(l_{ij}^\ast } -l_{ij}^0 )\le 0 \Rightarrow p_i^\ast \sum\limits_{j=1}^m {(l_{ij}^\ast } -l_{ij}^0 )=0\hbox{ for}\;\forall _i $$

(A2-5)

The optimal discharge matrix, E ^ws _i , defines a permit-constraint joint-cost minimum for shadow price λ^ws≥0 such that

$$L=\sum\limits_{j=1}^n {C_j (e_{1j} ,\ldots,e_{nj})} +\lambda^{ws}\left(\sum\limits_{j=1}^m \sum\limits_{i=1}^n {w_{ij} e_{ij}}-L_{ws}^0\right)$$

(A2-6)

$$ \frac{\partial L}{\partial e_{ij} }=\frac{\partial C_j (e_{1j}^{ws} ,\ldots,e_{nj}^{ws} )}{\partial e_{ij} }+\lambda ^{ws}w_{ij} \ge 0; \quad \sum\limits_{i=1}^n {\sum\limits_{j=1}^m {e_{ij}^{ws} \left[ {\frac{\partial C_j (e_{1j}^{ws} ,\ldots,e_{nj}^{ws} )}{\partial e_{ij} }+\lambda ^{ws}w_{ij} } \right]}}=0 $$

(A2-7)

$$ \frac{\partial L}{\partial \lambda ^{ws}}=L_{ws}^0-\sum\limits_{j=1}^m {\sum\limits_{i=1}^n {w_{ij} e_{ij}^{ws}{}_{ij}}}0; \quad \lambda ^{ws}\left[ {L_{ws}^0 -\sum\limits_{j=1}^m {\sum\limits_{i=1}^n {w_{ij} e_{ij}^{ws} } } } \right]\ge 0 $$

(A2-8)

Appendix 3 Kuhn-Tucker Conditions for the Least-Cost Model in Scenario 1: Multiple Interactive Pollutants with Only an Interaction Function

By using the same Kuhn-Tucker conditions as in Appendix 2 and defining an efficient emission matrix for multiple interactive pollutants (denoted “imp”), E ^imp _i (E ^imp _i = (e ^imp _i1 ,...,e ^imp _ij ,...,e ^imp _im )), and the relevant Lagrangian multiplier, ϕ ^imp _r , we obtain the following relationships:

$$L=\sum\limits_{j=1}^J {C_j (e_{1j}^{imp} ,\ldots,e_{nj}^{imp})}+\sum\limits_{r=1}^k \phi _r^{imp} \left[{f_r \left(\sum\limits_{j=1}^ma_{1j}^r e_{1j}^{imp} ,..,\sum\limits_{j=1}^m a_{nj}^re_{nj}^{imp}\right)-\overline D_r} \right]$$

(A3-1)

$$ \frac{\partial L}{\partial e_{ij} }=\frac{\partial C_j (e_{1j}^{imp} ,\ldots,e_{nj}^{imp} )}{\partial e_{ij} }+\sum\limits_{r=1}^k {\phi _r^{imp} \frac{\partial f_r }{\partial S_{ir} }a_{ij}^r } \ge 0\;\&\;e_{ij}^{imp} \ge 0 \quad i=\left\{ {1,\ldots,n} \right\}, j=\left\{ {1,\ldots,m} \right\} $$

(A3-2)

Then

$$\sum\limits_{i=1}^n \sum\limits_{j=1}^m e_{ij}^{imp}\left(\frac{\partial C_j (e_{1j}^{imp} ,\ldots,e_{nj}^{imp})}{\partial e_{ij} }+\sum\limits_{r=1}^k \phi _r^{imp}\frac{\partial f_r}{\partial S_{ir} }a_{ij}^r\right)=0$$

(A3-3)

We can also rewrite the above relationship as follows:

$$\sum\limits_{j=1}^m>e_{ij}^{imp}\left(\frac{\partial C_j (e_{1j}^{imp}>,\ldots,e_{nj}^{imp})}{\partial e_{ij}}+\sum\limits_{r=1}^k {\phi >_r^{imp} \frac{\partial f_r}{\partial >S_{ir}}a_{ij}^r}\right)=0\,\hbox{ for }\,i=\left\{{1,\ldots,n} >\right\}$$

(A3-4)

$$\frac{\partial L}{\partial \phi >_r}=-(f_r \left(\sum\limits_{j=1}^ma_{1j}^r >e_{1j}^{imp},..,\sum\limits_{j=1}^m a_{nj}^r e_{nj}^{imp} )-\overline>D_r\right)\ge 0\;\&\;\phi _r \ge 0 \quad r=\left\{ {1,\ldots,k}>\right\}$$

(A3-5)

$$\sum\limits_{r=1}^k {\phi _r^{imp} \left(\overline D _r -f_r\left(\sum\limits_{j=1}^m a_{1j}^r e_{1j}^{imp},..,\sum\limits_{j=1}^ma_{nj}^r e_{nj}^{imp}\right)\right)}=0$$

(A3-6)

Appendix 4 Kuhn-Tucker Conditions for the Least-Cost Model in Scenario 2: Multiple Interactive Pollutants Model in Scenario 1 with Individual Pollutant Constraints

By using the same Kuhn-Tucker conditions as in Appendix 3 and defining an efficient emission matrix for multiple interactive pollutants (denoted “ imp”), E ^imp _i (E ^imp _i = (e ^imp _i1 ,...,e ^imp _ij ,...,e ^imp _im )), and the relevant Lagrangian multiplier, ϕ ^imp _r , as well as a multiplier for individual constraints for the form in Eq. 2, we obtain the following relationships for scenario 2:

$$L=\sum\limits_{j=1}^J {C_j (e_{1j}^{imp} ,\ldots,e_{nj}^{imp})}+\sum\limits_{i=1}^n {\sum\limits_{r=1}^k {\lambda _{ir}^{imp}\left(\sum\limits_{j=1}^m {a_{ij}^r e_{ij}^{imp}}-\overline S_{ir}\right)+}} \sum\limits_{r=1}^k \phi _r^{imp} \left[f_r\left(\sum\limits_{j=1}^m a_{1j}^r e_{1j}^{imp},..,\sum\limits_{j=1}^ma_{nj}^r e_{nj}^{imp}\right)-\overline D _r\right]$$

(A4-1)

$$ \frac{\partial L}{\partial e_{ij} }=\frac{\partial C_j (e_{1j}^{imp} ,\ldots,e_{nj}^{imp} )}{\partial e_{ij} }+\sum\limits_{r=1}^k {\lambda _{ir}^{imp} a_{ij}^r } +\sum\limits_{r=1}^k {\phi _r^{imp} \frac{\partial f_r }{\partial S_{ir} }a_{ij}^r } \ge 0\;\&\;e_{ij}^{imp} \ge 0 \quad i=\left\{ {1,\ldots,n} \right\}, j=\left\{ {1,\ldots,m} \right\} $$

(A4-2)

Then

$$\sum\limits_{i=1}^n \sum\limits_{j=1}^me_{ij}^{imp}\left(\frac{\partial C_j (e_{1j}^{imp},\ldots,e_{nj}^{imp})}{\partial e_{ij} }\,+\,\sum\limits_{r=1}^k{\lambda_{ir}^{imp} a_{ij}^r}\,+\,\sum\limits_{r=1}^k \phi _r^{imp}\frac{\partial f_r}{\partial S_{ir}}a_{ij}^r\right)=0$$

(A4-3)

We can also rewrite the above relationships in condensed form as follows:

$$\sum\limits_{j=1}^m>e_{ij}^{imp} \left(\frac{\partial C_j(e_{1j}^{imp}>,\ldots,e_{nj}^{imp})}{\partial e_{ij}}+\sum\limits_{r=1}^k{\lambda_{ir}>^{imp} a_{ij}^r}+\sum\limits_{r=1}^k \phi_r^{imp} \frac{\partial > f_r}{\partial S_{ir}}a_{ij}^r \right)=0\hbox{ for }i=\left\{{1,\ldots,n}>\right\}$$

(A4-4)

$$\frac{\partial L}{\partial \phi _r }=-\left(f_r\left(\sum\limits_{j=1}^m a_{1j}^r e_{1j}^{imp},..,\sum\limits_{j=1}^m a_{nj}^re_{nj}^{imp}\right)-L_{dr}^0\right)\ge 0\;\&\;\phi _r^{imp} \ge0 \quad r=\left\{{1,\ldots,k} \right\}$$

(A4-5)

$$ \sum\limits_{r=1}^k {\phi _r^{imp}\left(\overline D _r -f_r \left(\sum\limits_{j=1}^m {a_{1j}^re_{1j}^{imp}} ,..,\sum\limits_{j=1}^m {a_{nj}^r e_{nj}^{imp}}\right)\right)=0} $$

(A4-6)

$$\frac{\partial L}{\partial \lambda _{ir} }=-\left(\sum\limits_{j=1}^m{a_{ij}^r e_{ij}^{imp} } -\overline S _{ir}\right)\ge 0\;\&\;\lambda_{ir}^{imp} \ge 0 \quad i=\left\{ {1,\ldots,n} \right\}, r=\left\{{1,\ldots,k} \right\}$$

(A4-7)

$$\sum\limits_{i=1}^n {\sum\limits_{r=1}^k {\lambda _{ir}^{imp}\left(\overline S _{ir} -\sum\limits_{j=1}^m {a_{ij}^r e_{ij}^{imp}}\right)}} =0$$

(A4-8)

Appendix 5 Notation

i = {1,..., n}	Number of different pollutants
j = {1,..., m}	Number of polluters
r = {1,..., k}	Number of checkpoints in the body of water
q = {1,...,p}	Number of products manufactured by the polluter
a r ij	Water quality impact factor indicating the contribution of 1 unit of pollutant i emitted by polluter j to the concentration of pollutant i at checkpoint r (1/l)
b r ij	Water quality impact factor indicating the contribution of 1 unit of pollutant i emitted by polluter j to the concentration of a water quality indicator that is indirectly affected by pollutant i (e.g., the reduction in dissolved oxygen content) at checkpoint r (1/l)
S ir	Concentration of pollutant i at checkpoint r, mg/L
e ij	Emission of pollutant i by polluter j, mg (pollution load)
$\overline S _{ir}$	Target level for the concentration of pollutant i at checkpoint r, mg/L
f r	Damage function at checkpoint r
$\overline D _r$	Target damage level at checkpoint r
C j	Total pollution-control cost function for polluter j
es r	Elasticity of substitution at checkpoint r
ρ r	A parameter related to the elasticity of substitution
N	Nitrogen pollutant
P	Phosphorus pollutant
Al r	Algal level, in terms of either mass of algae or cl a
B r	A parameter of the damage function at checkpoint r
μ r	A parameter of the damage function at checkpoint r
$\overline {Al} _r$	Target algal level (production function) at checkpoint r
L	Lagrangian function
ϕ r , γ, and λ ir	Lagrangian multipliers for different constraints
l ij	Pollution permit issued by the regulator for pollutant i of polluter j
l 0 ij	Initial pollution permit issued by the regulator for pollutant i of polluter j
L 0 i	Total initial pollution permits for pollutant i
L 0 ws	The number of initial weighted-sum permits issued by the regulator
w ij	Weighting factor for pollutant i emitted by polluter j
**	Indicates the optimal values for multiple polluters
*	Indicates the optimal values for a single polluter
e ws ij	Optimal emission of pollutant i emitted by polluter j in the weighting system
G j	Production cost function for the jth multiproduct polluter
e imp ij	Optimal emission of multiple interactive pollutant i emitted by polluter j in scenarios 1 and 2
p i	Market price of pollutant i