Generating a Linear Approximation to the Abatement Function

A trading program for pollution involves the existence of a tradable commodity that is able to measure the emissions or the discharges (Stephenson, Norris and Shabman 1998). In the context of water quality trading, it has been argued that the characteristics of nonpoint source represent barriers to the quantification of the emissions (Malik et al. 1994). Therefore the development of a tradable commodity by estimating a system of points that captures the abatement actions’

efficiency in reducing ambient pollution offers a possible solution to this problem. In the context of watershed pollution, different abatement actions have different impacts on edge-of-field abated emissions, and identical reductions in the edge-of-field emissions might have different impact on the ambient pollution level.

A well designed system of points needs to account for all these characteristics. In this context, Kling (2011) proposed a point based trading system where agricultural producers would be required to implement abatement actions that accrue enough points per acre to meet a

predetermined standard. The point values assigned to each abatement practice approximate: (a) how effective an abatement practice is in reducing the edge-of-field emissions and (b) the impact of the edge-of-field reduced emissions on the ambient water quality. Since the abatement function ( is approximated as a linear combination of the abatement actions impact measured at edge-of-field level and delivery coefficients, and the field level reduced emissions depend on the abatement action, without any loss, the abatement function can be written as a function of the vector of abatement actions :

Next, assuming that there are nonlinearities at the field level, the edge-of-field reductions are approximated as ≅ ∑ , where measure the impact of abatement action given field . The impact of field ′ edge-of field reductions on ambient water quality is

≅ ∑ ∑ , where , referred hereon as “point

coefficient”, gives the number of points assigned to the abatement action given field i. Since the point values are defined in terms of abatement, they can be interpreted as the marginal contribution to the total abatement of a particular field given that the abatement action is taken. Finally, the linear approximation of the abatement function can be re-written as:

≅ ∑ ∑ ∑ ∗ (15)

where is a column vector of point values or coefficients to be estimated, and is a row vector of abatement actions.

The above linear approximation of the abatement function is made around the baseline emissions. Alternatively, the linear approximation can be made around the optimal solution (i.e. the optimal vector of abatement actions that achieves the desired abatement goal). In this case, the point coefficients can be interpreted as abatement impact relative to the optimal solution.

3.3.1. An empirical approach for estimating the vector of point coefficients a

To estimate the point coefficients for each abatement action and each field, I employ a multistep procedure using the special features of a watershed-based hydrological model, SWAT. In SWAT, a watershed is delineated into subbasins and further on into smaller fields units called hydrological response units (HRU). As a result, a watershed can contain thousands of fields. My method to estimate the point coefficients is to generate sets of random allocations of

unique watershed configuration. The impacts on the ambient level of water quality, in terms of mean annual abatement loadings of nitrogen or phosphorus, are obtained by running the SWAT model for each configuration.15 _{The water quality outcomes measured in abatement levels}

( ) are then combined with the vectors of abatement actions’ assignments ( ) to estimate the vector of point coefficients, , by combining the results of a series ordinary least square

estimations min ′ .16

Often cases the number of fields (HRU) in a watershed is large, it is challenging to generate a sufficient number of watershed configurations to estimate NxJ point coefficients.17

My approach to estimating point values takes advantage of the outputs generated by SWAT to break the above estimation into several steps. (a) estimate the point values at the subbasin level using the ambient levels measure at the watershed exit, (b) estimate point values at the field level using the field provided outputs, and (b) combine the results to obtain field specific point coefficients for each abatement action. Combining the two sets of results allows me to estimate the field specific point coefficients for each abatement action but also to estimate the delivery coefficients.

SWAT computes the ambient emissions at each subbasin outlet as a function of the component fields’ emissions. Next, the ambient emissions from each subbasin are routed into a nonlinear and nonseparable way to determine the ambient water quality at the main outlet. Hence, SWAT provides emissions outputs at the field, subbasin, and main watershed level.

15 _{The abatement levels are obtained by subtracting the impacts on the ambient levels from the}

baseline emissions.

16 _{is a 1 column vector, is a matrix is a column vector}

17 _{In this case should be greater than , where is the number of fields and is the number}

Three different set of point coefficients are estimated using a multistep procedure described in Appendix A. The sets of point coefficients differ in the degree of approximation. More specifically, the first set of point coefficients are field specific (i.e., for each field, I

estimate a Jx1 point coefficients), the second set of point coefficients are subbasin specific (i.e. a given abatement actions has the same number of points for any field in a subbasin, and finally, the last set of point coefficients is watershed specific.

The obtained point values implicitly contain information on the trading ratios across different locations within the watershed as well as the trading ratios between different abatement actions, hence any trading based on the point coefficients will be made on a one-to-one base.

Once the environmental agency determines the point values that are credited to a particular abatement action in a specific field, he is able to compute the total point values associated with any water quality target. While the command and control policy is not affected by the total number of points, in the case of a performance standard and of a tradable credit program, the total point value chosen by the regulator will directly affect the total abatement level achieved at the watershed level.

For the performance standard policy, the regulator needs to choose the appropriate farm- level point requirements. Under the trading approach, credits or the point coefficients generated by abatement actions are tradable, on a one-to-one basis, across the watershed. As a result, a farmer solves:

, , , . . ∑ (16) where is a binary variable that takes a value of 1 if the abatement action j is chosen, the abatement actions, is the number of points to be traded, the point values assigned to

abatement action j given field i, and is the field level constraint assigned by the regulator to field i. The point price is determined in a points market equilibrium by ∑ 0

This trading approach can be conceptually viewed as a combination of an emissions permit and ambient permit system (Rabotyagov et al. 2012). Under an emissions permit system rights are defined in term of what firms emit. Under an ambient permit system, right are defined in terms of pollution contribution to a receptor (Montgomery 1972; Baumol and Oates 1988). In this case point credits are specified at farm (field) level allowing the trade to occur on a one-to- one basis. Next, a point value approximates the impact of an abatement action on the total level of abated pollution measured at a single pollution receptor (watershed outlet). Trading ratios that account both for location and the abatement actions tradeoffs are embedded into the point

coefficients.

The point-credit approximation procedure can also be adapted (a) for a single pollutant market; (b) for multiple pollutant markets where a separate system of points is estimated for each pollutant, and (c) to extend the single pollutant market by including the participation in a carbon market.