Estimation of Wastewater Treatment Objectives through Maximum Entropy

JOURNAL OF ENVIRONMENTAL ECONOMICS AND MANAGEMENT ARTICLE NO.

32, 293]308 Ž1997.

EE970969

Estimation of Wastewater Treatment Objectives through Maximum Entropy Linda FernandezU Department of Agricultural and Resource Economics, 207 Giannini Hall, Uni¨ ersity of California at Berkeley, Berkeley, California 94720-3310 Received October 17, 1995; revised May 1, 1996 The paper examines how a public wastewater treatment plant balances objectives of cost minimization and pollution prevention. The parameters of the objective function and state equation in an optimal control model are estimated using maximum entropy and time series observations of water quality and expenditures for wastewater treatment. The estimation method does not require restrictions needed by other techniques used to estimate nonlinear, ill-posed problems. The parameter estimates indicate that the treatment plant emphasizes water quality enhancement over cost minimization. Results from the sensitivity analysis show that the plant favors conventional treatment over pollution prevention. Q 1997 Academic Press

I. INTRODUCTION In urban areas wastewater treatment plants treat fluid waste from domestic, commercial, and industrial sectors. Most municipal treatment plants are publicly owned and operated. Publicly owned treatment works ŽPOTWs. operate under environmental quality laws. These regulations in the United States are part of the federal Clean Water Act for water quality standards in waterways receiving the treated waste. The waterways support numerous beneficial uses including water supply, recreation, commercial fishing, and marine habitat. Due to differing financial and physical resource constraints each POTW has distinct objectives for managing water quality. Federal grants for POTW construction have declined since the construction grant program was phased out in 1990. The resulting financial constraint has prompted the POTW to pass on expenditures for waste treatment to the private sector by requiring pretreatment of industrial waste prior to discharge to the POTW. In addition, particularly in the western United States where water resources are scarce, some POTWs place a high priority on reclamation to augment water supply and reduce waste. These observations lead to important questions for research. How do POTWs balance objectives of cost minimization and pollution prevention? Do they rank one objective over another? Do POTWs perform according to more stringent water quality standards than those mandated by regulations? The essay provides answers to these questions by estimating the parameters in a POTW’s objective function. The parameter estimates offer a basis for ranking several objectives of the POTW. We estimate the implicit trade-off between water quality and cost. The estimates provide an avenue for examining how a POTW U

I thank Larry Karp, Amos Golan, George Judge, and two anonymous referees for useful comments. 293 0095-0696r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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might respond to changes in its economic environment. The analysis can be useful in generating information to aid POTWs and policymakers in the choice among alternative instruments for managing wastewater. The findings of this study can have implications for environmental policy since changes in financial and physical constraints affect the POTW’s treatment decisions. For example, without federal grants, future emphasis may shift from remediation to prevention in order to control water pollutants. The contribution this study makes is estimating a dynamic model of wastewater treatment. The paper is organized as follows. Section II provides more information about wastewater treatment. Section III reviews literature. The model is explained in Section IV. Section V describes the data. The estimation is summarized in Section VI. Results are presented in Section VII. We use the parameters in Section VIII to simulate the POTW’s responses to changes in the economic environment. Section IX contains concluding comments.

II. OBJECTIVES FOR PUBLIC WASTEWATER TREATMENT The POTWs operate with one or more objectives. They accomplish wastewater treatment to prevent the threat to human health from waste-related diseases. Additionally, they seek to maintain ecosystem habitat of the receiving waters in which they discharge. Optimal behavior for the POTW means striking a balance to maximize expected utility for the city it serves. By equilibrating marginal costs with marginal benefits of each increment of treatment, the POTW decides on the level of treatment. There may be a balance of efficiency and equity in terms of the trade-off between cost minimization and water quality benefits for the whole community that the POTW serves. In the realm of command and control environmental regulations, POTWs have flexibility in meeting water quality standards that are set for protecting public health and environment. There are two ways that POTWs can meet any water quality standard. The first is through fluid treatment at the POTW site, which involves chemical and biological processing to remove conventional pollutants found in residential and commercial waste. The second is through pretreatment, which involves the reduction or elimination of unconventional pollutants such as toxics and heavy metals by industrial and commercial businesses prior to or in lieu of discharging to the POTW. Fluid treatment requires expenditures to cover capital and operationsrmaintenance costs at the POTW. The POTW’s expenditures for pretreatment include enforcement and monitoring of industrial discharges, waste minimization studies to identify changes in processing technology to influence the behavior of industries in preventing or minimizing the waste. Biochemical oxygen demand ŽBOD. represents a standard index of pollutants at various stages of the wastewater treatment. Water quality impacts of BOD are odor, reduced clarity, and reduced dissolved oxygen ŽDO. w19x. The BOD measures the sewage effluent’s use of DO which precludes use by marine organisms that depend on DO for survival w1x. A low BOD protects aquatic habitat and recreational uses. With the recent growth of commercial mariculture operations in the waters receiving the waste, the POTW must strive to reduce BOD to levels compatible with food safety standards. The BOD effect on DO depends upon the level of treatment that the sewage receives and the existing pollution ŽBOD. level

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in the receiving waterway. The BOD has a cumulative effect over time since the level at each point in time depends on previous periods in a linear manner w19, 26x. Since the BOD measure is taken in the benthic sediment on the sea floor where the waste is discharged, it has a residual effect that is longer than in the water column above. The BOD measure can be directly related to the amount of aerobic metabolism and organic matter Žfood supply. in the sediments w7x. BOD is used as the measure for water quality in the POTW’s objective function.

III. LITERATURE REVIEW Several economic studies of public wastewater treatment focus on compliance with environmental regulations w10, 20, 23x. These studies use a cost minimization framework for static analysis. McConnell and Schwarz w20, 23x develop a static framework to characterize the choice of wastewater treatment plant design for setting pollution levels endogenously. Their study identifies capital costs as an important financial constraint in achieving the optimal treatment through plant design. Their static analysis does not allow for the examination of dynamic decision making needed to meet the uncertain changes in both environmental and financial constraints over time. The environmental constraint is that water quality ŽBOD. in the previous time period affects current water quality since there is accumulation of pollutants. Water quality provides a direct link between past treatment decisions and current treatment decisions. The level of accumulated pollutants in any period determines what will be spent to treat waste that adds to the pollutant level. The solution to the POTW’s dynamic problem is a decision rule for treatment as a function of the state of water quality over the relevant time horizon instead of only one decision for the static problem of choosing a plant design. The financial constraint, as McConnell and Schwarz w20x note, is that federal construction grants are no longer available for building plants. Another constraint which McConnell and Schwarz do not include is the lack of grants to upgrade the operation and maintenance of existing plants for augmenting water supply through reclamation. While many municipalities in the United States have built treatment plants, they are facing post-construction financial constraints that may limit abilities to provide long-term treatment. Over time, the POTWs face uncertainty about influent water quality and financial support for treatment. The present study will incorporate uncertainty into a dynamic model of a POTW’s optimal behavior through time. This study’s objective parallels that of Fulton and Karp w11x. Fulton and Karp use data for the linear state equation and control rule of a public firm’s optimal behavior to estimate parameters of the quadratic objective function under uncertainty. Instead of the two-stage least squares estimation procedure Fulton and Karp use, this study uses maximum entropy to estimate a dynamic discrete time model. This new econometric estimation procedure does not require the restrictions needed by two-stage least squares and other traditional methods to convert the problem to a well-posed one by imposing the value of zero on some parameters w13x. The restrictions are required to estimate the parameters since the problem is underidentified: there are no unique estimates of the utility function in the inverse

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control problem. Given any linear state equation and linear control rule, there exists a continuum of linear-quadratic objective functions for which the observed control rule constitutes the optimal solution to the linear]quadratic problem w15x. Examples of restrictions include setting cross terms equal to zero and assuming error terms are normally distributed to impose more structure on the model and make it tractable. Restrictions are also used if the problem is ill-posed, where the number of unknown parameters exceeds the number of observations. While additional data may offer more precise estimates of parameters in the state equation and control rule, it cannot overcome the underidentification problem of finding unique parameters for the objective function w13x. IV. MODEL The maintained assumption is that the POTW is solving a linear]quadratic optimal control problem with uncertainty in water quality. The POTW chooses treatment expenditures to maximize expected discounted utility. The model is described with one control variable of treatment expenditures. The empirical analysis will add a control variable for describing two different types of treatment expenditures. The POTW’s objectives are constrained by the state equation of ambient water quality, representing the environment’s capacity to absorb pollutants over time. The state equation defines the change in water quality from wastewater discharges. The notation used in the model can be found in the Appendix. Ut s B1 Q t q B2 Q t2 q B3 K t q B4 K t2 q B5 Q t K t J s max E K

½

T

Ý d t Ž Ut . ts1

5

Ž 1. Ž 2.

subject to Q t s r 0 q r 1 Q ty1 q r 2 K t q e t .

Ž 3.

The utility function Ut in Eq. Ž1. is assumed to be a strictly concave quadratic function over arguments of water quality and treatment expenditures. Both types of treatment expenditures are combined into one control variable. Equations Ž2. and Ž3. specify the POTW’s maximization problem. Concavity of the value function J in Ž2. on the state variable Q t , follows from the arguments in Bertsekas Žw2x, Proposition 8, p. 253.. The expected utility function summarizes preferences for water pollution control. The state Eq. Ž3. specifies the linear relationship between ambient water quality at t, water quality at t y 1, the level of treatment in t, and some noise e t w19x. The problem represented by Eqs. Ž1. ] Ž3. is restated in the form of a linear]quadratic stochastic programming model with vector notation, where the state vector contains both the state and the control variables. J s max E ut

½

T

Ý d t Ž bX x t q xXt Bx t . ts1

5

Ž 4.

subject to x t s Ax ty1 q C u t q w q e 1 t ,

Ž 5.

297

WASTEWATER TREATMENT OBJECTIVES

where ut s w Kt x

Ž 6.

x t s w Qt K t x .

Equation Ž5. has A as a 2 = 2 matrix, C as a 2 = 1 vector, w as a 2 = 1 intercept vector, and e 1 t as a 2 = 1 error vector. The error term is included since we assume that the dependent variable x t evolves stochastically. The objective function in Eq. Ž4. includes B as an unknown 2 = 2 matrix and b as an unknown 2 = 1 vector. The POTW is solving for the optimal linear control rule for wastewater treatment expenditures. The optimal value of the control variable at each point in time is a linear function of the lagged state variable u t s Gx ty1 q s q e 2 t ,

Ž 7.

where G is a 2 = 2 unknown matrix of control parameters and s is a 2 = 1 dimensional intercept vector for the feedback rule. The 2 = 1 dimensional error vector e 2 t , is included for the stochastic state variable. The time invariant feedback rule G and the intercept vector s can be calculated according to the following Riccati equations from Chow w5, 6x, where B, A, and C are time invariant matrices. A q CG s R G s y Ž CX HC .

y1

Ž 8.

CHA

Ž 9.

B s H y d RX HR X y1

yCX w I y d R x

Ž 10 . X y1

br2 s CX HCs q CX I y d Ž R .

RX H w.

Ž 11 .

The goal of this study is to recover the unknown parameters of the state Eq. Ž5., the control rule Ž7., and the objective function Ž4. using observations of the state of the system x t and the control variable u t . The recovery of these parameters is referred to as an inverse control problem and proves challenging due to the underidentification of the parameters in the objective function. The parameters of Ž5. and Ž7. are initially estimated and, using the necessary conditions that result from the optimization of Ž4., the parameters B and b are recovered w13x. V. DATA The data consists of time series observations covering 11 years from 1983 to 1993. We obtain observations of the state variable Žwater quality. and both control variables Žexpenditures for treatment at the POTW and pretreatment off-site. from the El Estero Wastewater Treatment Plant in Santa Barbara, California w22x. We access data for the measure of water quality from annual ocean samples required under the National Pollutant Discharge Elimination System Permit w4x. The samples of BOD in ocean water originate from a sampling station located close to the POTW’s outfall. The disinfected wastewater flows through an outfall Žpipe. which extends 8750 feet into the Santa Barbara Channel at an ocean depth of 70 feet w22x. The treated wastewater is dispersed over a large area so as not to damage any one area of the ocean floor w22x.

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The on-site expenditures include capital costs for construction and equipment and operating costs of labor, insurance, maintenance and repair, power, consumable materials and tools, and administration. The pretreatment expenditures cover labor and consumable materials for enforcement and monitoring as well as laboratory analysis of industrial water samples. For the first 5 years of the data, the POTW made expenditures only for on-site treatment. Pretreatment expenditures were made only in the final 6 years. In order to use all of the data for 11 years in the estimation, we sum both types of treatment expenditures for the last 6 years into one control variable of total treatment expenditures. Subsequently, we separate the controls for a second estimation of the model using the last 6 years of the data when pretreatment is used. The second estimation allows for change in the objectives of the POTW over the time period of the data. During this period, Santa Barbara experienced severe water supply shortages and faced additional federal regulations for pretreatment. Water reclamation efforts began at the treatment plant in 1991 and expenditures for reclamation are included in the time series for treatment expenditures. A potentially important event that took place during the study’s data horizon was the introduction of mariculture production in the Santa Barbara channel Žin the late 1980s.. To meet the food safety standards of the shellfish produced in the mariculture operations, substantially more intensive treatment is required. The POTW may have anticipated the regulatory changes introduced in 1994 Žafter the data horizon..

VI. ESTIMATION We use the generalized maximum entropy ŽGME. method of estimation ŽJudge and Golan w18x, Golan and Judge w12x, and Golan et al. w14x. to recover the unknown and unobservable parameters in our dynamic model. The advantages of GME are Ži. it is nonparametric in errors, Žii. it can easily be applied to the estimation of nonlinear systems, Žiii. it is designed to work with small, incomplete data sets, Živ. it does not require restrictions on parameters to estimate ill-posed problems Že.g., collinearity in observations. or underidentified problems such as the one in this study. The inverse control problem of this study is underidentified since there exists a continuum of linear]quadratic objective functions for which the observed control rule constitutes the optimal solution w13x. In terms of parameters of the model, underidentification means there is a family of matrices B ŽEq. Ž3.., for which a particular matrix G ŽEq. Ž5.. is an optimal solution w15x. All matrices in that family are observationally equivalent. Additional data cannot overcome this problem, but can enhance the recovery of the estimates of the parameters in the state equation and control rule. Therefore, GME is a useful method to estimate this problem. The GME recovers probability distributions. The procedure involves maximizing an entropy measure subject to a statistical model for the inverse control problem and recovering unknown and unobservable parameters through a reparameterization defining the parameters with probabilities. We follow Judge and Golan w18x in reparameterizing the unknown parameters. The GME reparameterization reduces the relevant parameter space to a finite set of support values, Z, that span the possible range of values for each unknown parameter. Recovery of the entropy-

WASTEWATER TREATMENT OBJECTIVES

299

maximizing probabilities for these support values w12x is accomplished with the reparameterization. Thus, instead of estimating the parameter directly, we estimate a probability distribution. The parameter is then expressed as the product of a support space Z and corresponding probabilities p in the reparameterized form. The support values may incorporate nonsample information if there is some basis in economic theory or empirical knowledge for restricting the magnitude or sign of the unknown parameters. We use the state equation and control rule ŽEq. Ž5. and Ž7.. for the statistical model and then use GME to recover the parameters. The moments of the data for the problem are defined by Eqs. Ž5. and Ž7.. The unknown parameters appear in the following equations in vector and matrix form. xt s

Qt A11 w s q Kt 0 0

ut s

G11 s q 0 0

0 0

0 0

Q ty1 C1 e q u t q 1t K ty1 1 0

Q ty1 e q 2t . K ty1 0

Ž 16 . Ž 17 .

These equations lead to estimates for the first element of each of the parameter matrices and vectors A, C, G, w, s, e 1 t , and e 2 t through reparameterization. Using matrix A from equation Ž5. as an example, reparameterization entails defining a set of two or more discrete points Z A s w z1A , z 2A , . . . , z gA xX in the support, each with a corresponding probability vector p A s w p1A , p 2A , . . . , pgA xX to carry out the following reparameterization X

Z Ap A s

Ý z gA pgA s A.

Ž 18 .

g

Reparameterizing the unknown parameters in the state equation Ž5. and the control rule Ž7. yields the statistical model: State equation. Z A p A x ty1 q Z C p C u t q Z w p wt q Z e 1 vte 1 s x t .

Ž 19 .

Z G p G x ty1 q Z s p st q Z e 2 vte 2 s u t ,

Ž 20 .

Control rule.

where Z i, p i are the support space and probabilities for parameter i, Ž i s A, C, G, w, s. and Z j, v j are support space and probabilities for parameter j, Ž j s e 1 t , e 2 t .. To recover the parameters we maximize entropy ŽEq. Ž21.. subject to the statistical model. max p, v

X

Ý  yp i , log e p i y v j

X

log e v j 4

Ž 21 .

ij

for i s A, G, C, H, w, s t , and j s e 1 t , e 2 t subject to the statistical model defined in Eqs. Ž8. ] Ž10., Ž19., and Ž20.. The entropy maximization is also subject to the condition that the probabilities within the probability vector for each parameter

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LINDA FERNANDEZ

sum to one and the parameter definitions as a product of the probability vector and the support space. Further details of the estimation are included in Fernandez w9x. We use GAMS to solve the numerical optimization problem. VII. ESTIMATION RESULTS The GME estimation results of model parameters appear in Table I. The table also includes results obtained from estimating the problem by Chow’s two-stage least squares w5x. Chow’s two-stage least-squares method is used to compare the GME results with those of a traditional estimation technique. The method involves running OLS on the state and control equations and then using the Riccati equation to compute the estimation of the objective function. Standard errors are obtained for the 2SLS estimates in the state and control equations by OLS but not for the objective function since these estimates are found analytically through the Riccati equation w5x. In order to make the problem well posed, and therefore amenable to estimation by two-stage least squares, some restrictions were imposed to reduce the number of parameters in the objective function for identification. Specifically, the coefficient of QK equals zero and the error terms in the state equation Ž5. and control rule Žequation 7. are assumed to be normally distributed. The utility function is restated for ease of identifying the estimated parameters in Table I. Utility function. Ut s B1 Q t q B2 Q t2 q B3 K t q B4 K t2 q B5 Q t K t . The two estimation techniques yield somewhat different estimated values of the parameters in the state equation and control rule. The control rule resulting from both estimation procedures gives positive values of treatment for levels of pollutant below the regulated limit of 30 mgrliter BOD w8x. From the GME estimation, the intercept term of y7.40 implies that the level of 26 mgrliter BOD acts as a target level for the POTW to make expenditures for wastewater treatment. This is the level of water pollution which triggers treatment. The target is an indication of the POTW’s emphasis on the water quality objective in the utility function, since the target is more stringent than water quality standards. This result is consistent with the fact that Santa Barbara does pursue reclamation of wastewater to augment its potable water supply. The standards of potable water are higher than for that of ambient ocean water which receives the effluent discharges from the POTW.

TABLE I Results from First Estimation Method

A11

C1

G11

w

s

B1

B2

B3

B4

B5

GME 0.45 y1.69 0.25 19.75 y7.40 47.54 y2.35 3.31 y49.4 y8.12 Chow 0.57 y0.85 0.23 17.69 y6.47 y4.32 y0.89 y5.47 y0.01 0 2SLS Standard 0.35 1.35 0.04 11.78 2.04 error

WASTEWATER TREATMENT OBJECTIVES

301

Large differences exist between the two sets of parameter estimates in the objective function. The GME method results in negative coefficients B2 and B4 for the squared terms and a coefficient B5 for the cross term that is small enough in absolute value to be consistent with a negative definite Hessian matrix of the objective function. The results imply a concave function for the maximization problem. Two-stage least squares yields the same result although the coefficient B5 has been restricted to equal zero. The lack of any cross term in the estimated utility function by two-stage least squares is due to the restrictions necessary to achieve identification of the parameters. This restriction is arbitrary and is not based on a strong justification for excluding cross effects of water quality and treatment expenditures on utility. When viewed separately, terms of the objective function such as B1 , may be significantly different in terms of sign and magnitude. The reason for a sign difference between the two methods might be due to the lack of an intercept term. However, the sign and magnitude of the parameter do not provide complete knowledge of the relative importance of the arguments and the effect they have on the level of the control variables. This arises primarily because the parameters estimated are not unit free, since they apply to dissimilar items such as water pollutant levels and treatment expenditures. The B1 term is incorporated in the calculation of utility elasticities in order to compare different units of measure. Using elasticities overcomes the problem of different units of measure Ždollars and mgrliter. between the control and state variables, respectively. The disparity in elasticity measures from the two estimation methods gives rise to different interpretations of the POTW’s objectives. The elasticity measures are y0.77 for water quality and y0.02 for treatment expenditures according to the parameters obtained from GME. All elasticities are evaluated for the sample year of 1992. These elasticities suggest the POTW’s utility is influenced much more by water quality than treatment expenditures. According to the two-stage least-squares method, the elasticity measures are y0.99 for water quality and y0.30 for treatment expenditures. The hypothesis that the estimates of the state equation from the two methods are the same is rejected through an F test since the value of 66.60 is above the F statistic for three restrictions and eight degrees of freedom. The F test is formed by setting the OLS estimates equal to the GME values. The hypothesis that the estimates of the control rule from both methods are equal cannot be rejected. The GME proves to be a feasible alternative to traditional estimation ŽChow’s 2SLS. allowing the recovery of meaningful information without imposing restrictions. One restriction that is applied in the 2SLS method pertains to the B5 parameter on the cross-product term between treatment expenditures and water quality illustrates how important the control objective is for the POTW. This restriction is arbitrary and applied in order to reduce the number of unknown parameters to be estimated. It is not a restriction based on economic theory and yet, it does influence and distort the economic interpretations of the balancing of the POTW’s objectives measured in the utility elasticities. A useful experiment for comparing the estimation techniques is to apply the restriction on the cross term for GME that is used for Chow’s 2SLS. This experiment results in estimates for a utility function that does not have a negative definite Hessian matrix which would guarantee a maximum for the expected utility

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maximization problem. Since B2 equals zero, there is an eigenvalue equal to zero. The concavity of the utility function is not guaranteed for maximization in this case. Hence, the restriction is not reasonable in this model. Estimation Results with Two Control Variables This subsection examines in greater detail the estimation using six years of the data after pretreatment and water reclamation activities were introduced by the POTW in 1987 which means the model contains two control variables. It is possible to examine three objectives namely water quality enhancement, pollution prevention and cost effectiveness with the final years of the data. The state variable gauges the water quality enhancement objective, consistent with the model described previously. The variable for pretreatment expenditures can provide information about the pollution prevention objective. Cost effectiveness can be analyzed through the trade-off between pretreatment and on-site treatment expenditures. Hence, the model is estimated with two controls. One control variable Ž O . is expenditures for on-site wastewater treatment at the treatment facility and the other control variable Ž P . is expenditures for pretreatment activities of enforcement and monitoring at industries that discharge to the treatment plant. Equation Ž5. is modified to include two control variables shown in Eq. Ž13.. Qt wt A11 O xt s q 0 t s 0 Pt 0 0

0 0 0

0 0 0

Q ty1 C1 Oty1 q 1 Pty1 0

C2 0 1

et Ot q 0 . Ž 13 . Pt 0

The off-site pretreatment expenditures have been part of the program for the last six years of the data. The problem is ill-posed due to scant data and underidentification. There are fewer observations Ž6 in total. than unknown parameters to be estimated Ž16 in total.. Scant data is a separate problem Žill posed. from that of lack of identification, which means there is a continuum of objective functions for which the observed control rule constitutes the optimal solution to the linear]quadratic problem. Due to both problems, we cannot use a traditional estimation technique since we have no basis for justifying zero restrictions on many parameters. Therefore, maximum entropy is most useful here. Tables II and III contain the results of the estimation with two control variables. The utility function is provided in order to identify the estimates from Table III. U s 79.4Q y 3.77Q 2 q 2.43O y 49.83O 2 y 4.73 P y 14.32 P 2 y 7.04QO y 10.08QP y 30.54OP. The C1 and C2 parameters indicate that pretreatment is more effective dollar for dollar than on-site treatment in reducing the level of pollutant: C1 is estimated at only y1.75 for on-site treatment against C2 at y1.92 for pretreatment. TABLE II Estimates of the State Equation and Control Rule w

A11

C1

C2

G1

G2

s1

s2

22.5

0.79

y1.75

y1.92

0.19

2

y3.24

y7.51

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WASTEWATER TREATMENT OBJECTIVES TABLE III Estimates of the Objective Function B1

B2

B3

B4

B5

B6

B7

B8

B9

79.4

y3.77

2.43

y49.83

y4.73

y14.32

y7.04

y10.08

y30.54

In order to compare the relative importance of three arguments in the POTW’s objective function, the utility elasticities with respect to these arguments are again computed. Table IV indicates the marginal utility measures and the utility elasticities. We use 1992 data for the values in Table IV. The utility function is decreasing in each argument Žlevel of pollutant, expenditures for on-site treatment, and expenditures for pretreatment.. The utility is higher from having less of the pollutant, and there is more utility from savings than for expenditures made for treatment of wastewater. The utility elasticity with respect to Q t Žthe level of pollutant affecting water quality. outweighs that of the two controls. This larger negative number indicates a greater importance of this variable in the objective function relative to the other variables, Ot and Pt . Similar to the previous estimation, the POTW’s utility is influenced much more by water quality than treatment expenditures. This emphasis on water quality is consistent with the intercept term that signals the POTW to treat wastewater according to a more stringent level of water quality than is mandated by regulations. During the last six years of the data set, this more stringent level is 24 mgrliter, which is below the regulated level of 30 mgrliter. Although pretreatment appears more effective than on-site treatment in changing water quality, the POTW is still emphasizing on-site treatment, as indicated by the larger negative number for utility elasticity. The emphasis may be justified since approximately 79% of the waste volume at the POTW comes from domestic sewage and domestic sewage can presently be handled only by on-site treatment w22x. There is no ability to substitute one treatment for another. VIII. IMPACT OF CHANGES IN THE EFFICACY OF THE TREATMENT INSTRUMENTS With estimates of the objective function we can simulate the POTW’s behavior under hypothetical changes in its economic environment. The specific changes investigated here are increases in the efficacy of on-site treatment and pretreatment. We assume technical equipment or process changes made at the treatment plant and off-site at industries allow for greater impact on water quality. Through-

TABLE IV Marginal Utilities and Elasticities Argument Qt Ot Pt

Marginal Utility

Utility Elasticity

y180.20 y725.39 y772.61

y0.82 y0.38 y0.02

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out, we assume the POTW’s objective function remains unchanged. The simulations provide more evidence of tradeoffs between the POTW’s objectives. The impact of efficacy changes on treatment expenditures and water quality can be estimated by solving for the level of treatment using the estimated objective function along with a new state equation that reflects the hypothetical efficacy changes through changes in the parameters on each control variable. Increase in the efficacy of on-site treatment is estimated by a 25% increase in the parameter C1 from 1.75 to 2.19. Likewise, increase in the efficacy of pretreatment is simulated by a 25% increase in the parameter C2 from 1.92 to 2.4. The resulting levels of pretreatment, on-site treatment, and water quality from solving the maximization problem are presented in graphical form in Figs. 1]3 for three different simulations, each ranging over a hypothetical 10-year period. First, the GME parameters estimated in the previous section in the model with two control variables are used in the simulation labeled ‘‘benchmark.’’ The benchmark simulation serves as a basis for comparison with the cases of changing the parameters on the control variables in the state equation to parallel increased efficacy measures of each type of treatment expenditure. ‘‘Revised pretreatment’’ and ‘‘revised on-site treatment’’ refer to cases of efficacy changes of 25% improvement in the removal of BOD at the industry level and POTW, respectively. Comparing the benchmark case with the other two cases indicates that the POTW is particularly concerned about water quality. This finding is consistent with the utility elasticities in the previous section. More is spent for on-site treatment under the benchmark case than under both ‘‘revised’’ scenarios as shown in Fig. 1. The least is spent when the efficacy of on-site treatment increases, making BOD removal more cost effective. We can explore the price and income effects due to shifts in the indifference curve, while keeping the same utility function. A change in the efficacy of the controls is equivalent to a price change. Because the price of on-site treatment has fallen, the POTW has greater ‘‘real’’ income. Both the price effect and income effect result in a trade-off for the POTW between savings and achieving a lower level of pollutant in the water. The income effect of increasing utility from savings may dominate since neither control can directly substitute for the other if the efficacy of one changes. Pretreatment is rather a complementary good, which enhances on-site treatment by removing other pollutants from the 20% of industrial discharges to the POTW which on-site treatment cannot handle. Since aggregate expenditures for pretreatment include a combination of money spent on enforcement and monitoring and technology transfer through advice on

FIG. 1. On-site treatment expenditures.

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305

FIG. 2. Water quality.

pollution prevention strategies in each industry, it is plausible for any of these activities to simultaneously influence changes in expenditures. The graphs depict a hypothetical time horizon and not an actual one so the interpretations of the control rules depicted are suggested from sample and nonsample information used in the estimation of parameters. A sharp decline in expenditures during the third and fourth years of the simulation may be the result of some time lag adjustment in pretreatment expenditures. Since these expenditures include enforcement and monitoring and technology transfers, there is a chance that previous pretreatment efforts Žin the first and second year. have triggered a decline in wastewater flows from industries targetted through pretreatment. Figure 2 illustrates the better level of water quality under the revised efficacy of on-site treatment. This result is consistent with a higher utility elasticity for on-site treatment than for pretreatment. The level of pretreatment under all three scenarios appears in Fig. 3. The level of pretreatment needed when its efficacy increases is slightly less than the benchmark case. The level of expenditures on pretreatment is lowest when the efficacy of on-site treatment increases. This trend is explained by a higher percentage of residential effluent exists over industrial effluent which is treated with on-site treatment only. The peak occurring in years 3]4 and steady rise in years 7]10 for the benchmark case parallels the other graph in that the activities covered by pretreatment expenditures Ženforcement, monitoring, technology transfer . can

FIG. 3. Pretreatment expenditures.

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have lagged effects, where some effort by the treatment plant results in reduced wastewater and pollutant concentration. The decline in year 5 could be due to compliance with industry improvements in pollution control technology as a result of prior enforcement, monitoring, and technology transfer efforts in years 3 and 4. IX. CONCLUSION This study has moved beyond static analysis of wastewater treatment to analyze dynamic behavior with available data. The dynamic behavior incorporates various objectives over time. We have shown how the dynamic estimation problem can be formulated and solved using the GME method when there are more unknown parameters than data points. The problem required recovering the parameters of the objective function and state equation for a linear]quadratic control problem. We obtain estimates of the implicit trade-off between cost minimization and water quality in the POTW’s objective function. Empirical estimates of the objective function indicate that the POTW attaches more weight to water quality than to cost minimization. Based on an intercept term in the control rule, the POTW was found to emphasize water quality with a more stringent water quality standard than is federally regulated. This result may be explained by Santa Barbara’s particular interest in reclaiming water Žgiven its recurring water shortages. and maintaining ambient water quality Žgiven the intensive use of the ocean for commercial shellfish production and recreational use.. Augmenting the utility function to define the separate controls of on-site treatment and pretreatment enables further investigation of the POTW’s objectives. We have found through the utility elasticity measures and through simulation of responses to changes in the efficacy of treatment instruments, that the POTW emphasizes on-site treatment. This is based on the larger amount of domestic sewage versus industrial waste entering the plant that can only be treated on-site. As more unconventional pollutants are generated from increased industrial and commercial activity, pretreatment may receive greater emphasis since it has greater impact on changing water quality than on-site treatment when the efficacy changes. The POTW in fact plans to increase enforcement activity for more pollution prevention at the industrial sources where these unconventional pollutants are generated w3x. The aim is to reduce pollutants which currently constrain efforts to increase water reclamation. For example, more investment by industries for catchment screens may reduce the industrial loads of grease and oil. In order to study pretreatment in more detail we need another pollutant index to obtain a better measure of unconventional pollutants Že.g., the concentration of heavy metals.. Adding another question that measures influent concentration of wastewater as well as total flow that enters the treatment plant from industrial and residential sources can serve as a clear measure of success of pretreatment efforts aimed at particular sectors discharging to the POTW. In order to add this detail, we need more data. The present pretreatment program targets only toxics from industrial sources. On-site treatment is able to handle the conventional pollutants from residential toilets, sinks, and showers. However, toxic substances used in the home, including crankcase oil, paint, household cleaners, and pesticides also make their way to the treatment plant. The same principle behind pretreatment of reducing the volume and concentration of waste at the source applies to households as well as industries. Since water quality is clearly important to the POTW to augment its water

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supply, it may resort to expanding pretreatment to the residential sector to ensure that water can be reclaimed from all effluent. Efforts to handle toxic wastes from households currently consist of public education for voluntary participation to minimize the dumping of toxic products in the wastewater stream. Further analysis of treatment alternatives can be done with this model by including the POTW’s expenditures for voluntary pretreatment efforts. APPENDIX: NOMENCLATURE J E Ut d Ot Pt Kt Qt Q ty1 et , ¨ t t T B1 B2 B3 B4 B5

r0 r1 r2 bX xt B A C ut w e1 t G s e2t H A11 G11 Zi pi Zj pj

Value of the objective function Expected value operator Utility function Discount rate On-site treatment expenditures Pretreatment Žoff-site. expenditures Ot q Pt , the sum of both types of treatment expenditures Ambient water quality Žunits of BOD in mgrliter. in time t Ambient water quality in time t y 1 Žlagged. Error terms Time period Final time period Coefficient on linear variable of water quality in the utility function Coefficient on quadratic variable of water quality in the utility function Coefficient on linear treatment variable in the utility function Coefficient on quadratic treatment variable in the utility function Coefficient on cross term of treatment and water quality variables in the utility function Parameter for scalar intercept in the state equation Coefficient for lagged water quality variable in the state equation Coefficient for treatment variable in the state equation Vector of coefficients for variables in the utility function Vector of variables Matrix of coefficients for quadratic variables in the utility function Matrix of coefficients for lagged variables Vector of coefficients for control variables Vector of control variables in the state equation Intercept vector in the state equation Error vector in the state equation Matrix of coefficients for lagged variables in the control rule Intercept vector in the control rule Error vector in the control rule Matrix in Riccati equation First element in A matrix of coefficients on lagged variables in the state equation First element in the G matrix of coefficients on lagged variables in the control rule Vector of support space for each parameter indexed by i s A, C, G, w, s Vector of probabilities for each parameter indexed by i s A, C, G, w, s Vector of support space for each error term indexed by j s e 1 t , e 2 t Vector of probabilities for each error term indexed by j s e 1 t , e 2 t

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