It is the policy of Cornell University actively to support equality .... between total treatment cost on an annualized basis to some measure of system size (e.g. plant .... on delivery cost of a proportional change in population served is now seen to be (A-11-ro) ..... systems are financed over a 20-year period at an 8% interest rate.
WP 96-15 November 1996
Working Paper Department of Agricultural, Resource, and Managerial Economics Cornell University, Ithaca, New York 14853-7801 USA
Economies of Size in Water Treatment vs. Diseconomies of Dispersion for Small Public Water Systems
by
Richard N. Boisvert and Todd M. Schmit
It is the policy of Cornell University actively to support equality of educational and employment opportunity. No person shall be denied admission to any educational program or activity or be denied employment on the basis of any legally prohibited dis crimination involving, but not limited to, such factors as race, color, creed, religion, national or ethnic origin, sex, age or handicap. The University is committed to the maintenance of affirmative action programs which will assure the continuation of such equality of opportunity.
Economies of Size in Water Treatment vs. Diseconomies of Dispersion for Small Public Water Systems
By Richard N. Boisvert and Todd M. Schmit
Abstract
This paper outlines a method to determine the tradeoff between economies of size in water treatment and diseconomies of distribution. Empirical results for New York are used to identify the implications for the rehabilitation and consolidation of rural water systems.
-
Economies of Size in Water Treatment vs. Diseconomies of Dispersion for Small Public Water Systems By Richard N. Boisvert and Todd M. Schmit·
Introduction We know that the financial burden facing small public water systems in complying with the 1986 and subsequent amendments to the Safe Drinking Water Act can be substantial, in large measure because they are unable to take full advantage of the economies of size in water treatment (EPA, 1993a). To capture the benefits of these economies of size, it is often suggested that costs of water supply can be minimized through the formation of regional water systems consisting of a group of small systems or one or more systems hooked to a larger system (Clark and Stevie, 1981). In this way, the costs to all users can be reduced. To perform their function, however, water utilities also must be physically connected to their customers, and for purposes of economic analysis, we must define two separate components to a water supply system: the treatment plant and the delivery system. The cost functions for the components differ substantially. While the unit costs of treatment generally decline with the quantity of service, the cost of delivery (transmission and distribution) is affected by the nature of the service area (Clark and Stevie, 1981). The delivery cost may very well rise as the service territory increases in size and spatial complexity, and the economies of size in treatment may well be offset by the diseconomies of water transmission and distribution.
Strictly from an
economic perspective, a water system's optimal size must be determined by the tradeoff between the economies of size for water production and treatment and any diseconomies of delivery to the point of use (Dajani and Gemmell, 1973). The purpose of this paper is to identify a method by which to determine the size for small water systems in New York that will minimize the combined cost treatment and delivery for commonly used treatment options and representative differences in the characteristics of rural service areas. To our knowledge, the tradeoffs between economies of size and the diseconomies of delivery have only been examined for much larger systems than those examined here, and this work is nearly 20 years old (Clark and Stevie, 1981).
•
• The authors are Professor and Research Support Specialist, respectively, in the Department of Agricultural, Resource, and Managerial Economics, Cornell University.
2
The treatment options that are examined include chlorination, slow sand, direct, and other types of filtration suitable for small systems, and aeration. The differences in the service territories are captured for the most part by population density. Data from which the cost equations for treatment and delivery are estimated come from loan files for community drinking water improvements from the Rural Development offices across New York.
Rural
Development's Water and Waste Disposal Loan and Grant Program, along with the data for these water systems and a theoretical discussion of indirect cost functions and appropriate measures of economies of size, are described in other reports prepared for EPA in support of this ongoing research (Boisvert et ai. 1996; Schmit and Boisvert, 1996). This paper begins with a discussion of the nature of the cost functions for treatment and delivery.
Next, there is a discussion of the appropriate way to combine the costs of these
components. After the estimated cost functions are described, the empirical results are presented. The paper concludes with a statement of the important policy implications.
The Components of the Cost of Water Supply As stated above, the costs of a community water system can be separated into two major components: those related to water treatment and other activities at the water plant and those related to water transportation or distribution to point of use. We begin with a discussion of treatment costs.
Treatment Cost Functions For purposes of discussion and empirical estimation below, treatment costs can be divided into two main elements: capital costs for construction, which can be put on an annual basis, and annual operation and maintenance (O&M) costs. In the literature, the relationship between total treatment cost on an annualized basis to some measure of system size (e.g. plant design capacity, average daily flow, or population served) is generally represented by an exponential function, which is linear in logarithms:
where TCt is the total annualized cost of treatment and P is a measure of output. If we define economies of size (SCE) by the proportional increase in cost for a small proportional increase in output, then, following Christensen and Green (1976),
(2) SCE = I-a InTCt fa InP,
3 which is equal to I-a. for the cost function given in equation (1). Economies of size exist if SCE = I-a. > 0, and diseconomies exist if the SCE is negative. It is also true that economies of size in
this case are invariant regardless of the level of output (Boisvert et ai., 1996; and Christensen and Greene, 1976). The practical implication of this specification is that if economies of size are estimated to exist, average costs will continue to fall regardless of how large the system becomes. This may not, however, be a reasonable assumption, because for a given treatment technology, economies of size may be exhausted at a certain point, implying that average costs should begin to rise as the size of the system expands beyond this point. To deal with this potential difficulty, we can re-specify the cost function as: (3)
TC,
= ~pa+51nP.
For this specification, we have: (4)
SCE=I-(a.+28lnP),
and the economies of size can vary with the level of output. l Besides its flexibility, this function can be used to test the hypothesis that returns to size are invariant with respect to output through a simple t-test on the parameter
o.
Further, the parameters in equation (3) can be estimated by
ordinary least squares by transforming it into logarithmic form: (5) In TC,
=In ~
+a. In P +0 (In p)2.
From an empirical point of view, there are two issues that must be addressed in using this formulation to estimate water system treatment costs. First, to estimate treatment cost functions empirically for various water treatment technologies, we need a single measure of output that is related both to capital and O&M costs. In the engineering equations accompanying the BAT document prepared for EPA for small water systems (Malcolm Pirnie, Inc., 1993) and elsewhere, system design capacity is used as the measure of system size in order to estimate certain components of capital cost, and average daily flow is the measure of size used to estimate the variable costs of operation and maintenance. Neither is an ideal measure for both, nor for a measure of size for delivery costs. Therefore, as a compromise, and to be as meaningful as possible from a policy perspective, system size or output, for much of the work in this larger The cost function in equation (1) and (3) are most often written in logarithmic form--taking the natural logarithms of both sides (Boisvert et al., 1996). It is in the logarithmic form that the expressions for the
economies of size are most easily derived. It is also in this way that the parameters of the two functions
are estimated econometrically. .
1
,.
4
research project for EPA, is specified in terms of the population served. This, after all, is the measure used by EPA and others to classify systems by size. This measure of output is also directly related to the number of service connections. The fact that this measure of output works well for many analytical purposes is well documented by Boisvert et aI. (1996), where they estimate the relationship between population served and both average daily flow and design capacity nationally and regionally based on data from the FRDS-II data system (EPA, 1993b). Although this first issue would have to be addressed regardless of the nature of the cost functions being estimated or the nature of the data, the second issue relates to the fact that in the sample of Rural Development water systems, there were generally insufficient observations for a particular type of treatment to allow for the estimation of separate equations for each type of treatment. To deal with this difficulty, equation (3) was re-specified as:
In this specification, differences in costs by treatment are reflected by coefficients associated with
the zero-one variables di • That is, the variable di takes on a value of 1 if the observation in the data is associated with treatment i, and is zero otherwise. In logarithmic form, this equation becomes:
For any treatment i, the measure of economies of size becomes: (8)
SCE
= 1- (Ui +20
InP).
Delivery System Cost Function In some respects, estimating the costs of water delivery is more complex than treatment.
Water is transported to point of use through a system of transmission pipelines and distribution mains. The transmission pipelines are the major trunk lines that transport large volumes of water and connect the treatment plant to the pumping station and ultimately to the distribution system. Thus, the major components of distribution system costs include pipelines, pumping stations and water towers, and energy to move the water through the system. Costs of supplying water to customers also rises with the distance from the water source. To capture much of this complexity in a cost function for water distribution, Clark and Stevie (1981) assume that capital costs are determined by pipe length alone, while the energy costs are a functioI) of both flow and distance.
5
Because of the availability of data, and the small-size of the systems studied here, we rely on a different specification. More is said about this below and in Schmit and Boisvert (1996), but it was impossible to disentangle energy costs from other O&M costs in the grant and loan files. Thus, while energy costs for distribution are embodied in the entire analysis and are related directly to output (as measured by population served), the effect of distance on energy costs of distribution is reflected only indirectly through a measure of population density. The cost function for water system delivery is also specified in exponential form as:
where TCd is total cost of delivery, P is population served, L is linear feet of pipe, and H is the number of water hydrants. Thus, according to this specification, the total cost of delivery is a function of population served, as well as the linear feet of pipe and the number of hydrants per person served. These latter two variables reflect the density of population in the service territory, and combined with the population itself in the equation account for the increasing size of the service territory as population served rises and population density falls.
Although quite
dissimilar algebraically from the cost functions for delivery specified by Clark and Stevie (1981) and by Ford and Warford (1969), this specification is consistent with what they believed were the primary determinants of water delivery cost. This cost function is also much more convenient to work with analytically. As with the cost for treatment, equation (9) is also linear in logarithms, and in that form its parameters can be estimated by ordinary least squares. The function can be written as: (10)
InTCd =In''( +AlnP+11[InL-lnP]+ro[InH-lnP],
In this form In P appears three times in the equation, and estimating it in this fonn (with the
differences in logarithms of L and P and H and P being specified as separate variables) is equivalent to estimating the function: (11)
InTCd =In''( +(A-11-ro)lnP+111nL+rolnH,
In this form, we are able to estimate the parameters In"(, 11, and ro, but the proportional net effect
on delivery cost of a proportional change in population served is now seen to be (A-11-ro). The total cost of water delivery is now specified as a function of population served, the total length of water pipe and the number of hydrants. To estimate total system costs below, we use this
•
6
equation, but manipulate it in such a way so that the population density of the service territory is specified directly.
Total Costs for the System Having specified these two cost components as functions of population served, total system costs can be obtained by adding equations (6) and (9).
Having done this, it is
conceptually possible to define expressions for the system's average and marginal costs, as well as for the measure of economies of size given in equation (2). These expressions, however, are complicated algebraically, and are not very enlightening. Therefore, they are not reproduced here. It is also possible using the total cost relationship to identify the optimal size for a water system (the point at which average cost per person served is a minimum) once the treatment technology and population density (as measured by UP and HIP) are known. To understand the relative importance of the economies of size for treatment and the diseconomies of size associated with transmission and distribution, one can easily compare the optimal size here with the optimal size considering only the cost of treatment. If the hypothesis by Clark and Stevie (1981) and others is true, then the optimal system size based on total cost should be substantially below that when only treatment costs are considered. This situation is pictured in Figure 1. In Figure 1, treatment costs, CT, rise first at a decreasing rate, and then at an increasing rate. Thus, average costs per person served initially falls and then rises as system size increases. The minimum average cost size is at PT, where a ray out of the origin is tangent to the CT curve. Distribution costs, CD, rise at an increasing rate throughout; thus, average costs always increase with system size. For this reason, when the two components of cost are added together, average total costs increase initially at a decreasing rate, but begin to increase at an increasing rate at a system size below that when treatment costs are considered in isolation. This means that the minimum average cost system size when both cost components are considered will be below that when only treatment costs are considered (i.e., at Pm rather than PT). Some empirical evidence supporting the hypothesis represented in this figure is presented in the empirical section below. Once the optimal system sizes are determined in this way, water systems around the country in places with similar treatment needs and population densities would in theory all construct systems of this size. Unfortunately, this country's rural populations are not scattered so neatly across the landscape so as to accommodate replication of these optimal size water systems organized around well defined service territories. Rather, rural population centers would rarely
7
Figure 1. Total and Average Cost Curves for Water Treatment (T), Water Distribution (D), and Combined Treatment and Distribution (TD) CID
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.. ...•....••.....••......
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Table C4. Minimum Average Cost Population Levels for Alternative Treatments and Transmission and Distribution Costs Treatment Plus Trans. and Distribution (c) Extension Beyond Minimum Cost (d) Average Cost Per Capita Average Cost Per Capita Population Treatment Only (b) Trans. & Total Trans. &
AC Population Treatment Dist'n. Density Population Total Population Treatment Dist'n. Total
Slow Sand Filtration
16,800 $130 8,400 $124 $192 $316 12,700 $123 $194 5.0 $317 6,100 $126 $328 9,300 2.0 16,800 $130 $454 $124 $331 $455 5,000 $128 $414 16,800 $130 $543 7,600 $125 1.3 $419 $544 $130 3,300 $134 $617 16,800 $751 5,100 $128 0.7 $624 $752 2,600 16,800 $130 $138 $749 $887 4,000 $131 0.5 $757 $888 5.0 2.0 1.3 0.7 0.5
31,800 31,800 31,800 31,800 31,800
$76 $76 $76 $76 $76
11,300 7,000 5,300 3,100 2,300
Direct Filtration $78 $194 $81 $329 $415 $84 $91 $616 $746 $96
5.0 2.0 1.3 0.7 0.5
22,300 22,300 22,300 22,300 22,300
$103 $103 $103 $103 $103
9,800 6,700 5,300 3,300 2,500
Other Filtration $100 $193 $103 $329 $415 $105 $111 $617 $116 $748
5.0 2.0 1.3 0.7 0.5
57,000 57,000 57,000 57,000 57,000
$35 $35 $35 $35 $35
9,400 4,500 3,100 1,600 1,200
$40 $45 $49 $57 $62
Aeration $193 $325 $409 $606 $734
$272 $410 $499 $707 $843
17,000 10,600 8,100 4,700 3,500
$77 $78 $80 $85 $89
$196 $332 $420 $623 $754
$272 $411 $500 $708 $844
N
VI
$293 $431 $520 $729 $864
14,800 10,100 8,000 5,000 3,800
$99 $100 $101 $106 $109
$195 $332 $419 $624 $756
$294 $432 $521 $730 $865
$233 $370 $458 $663 $796
14,300 6,900 4,800 2,500 1,800
$39 $42 $45 $51 $56
$195 $329 $414 $613 $742
$233 $371 $459 $664 $797
(a) Population density is defined as people per hundred If of transmission and distribution pipe, evaluated over the range in the data. People per hydrant is adjusted proportionately to the changes in the density levels. (b) Annualized cost functions assume a 20 year time period and 6% discount rate. (c) Transmission and distribution costs include storage and pump station components. (d) Extension limit refers to the maximum population extension for consolidation, after which lower costs result from constructing a separate treatment and transmission/distribution system for the extension considered.
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...
