Public service costs and city size : the case of urban water supply and sewerage services in Colombia

URR-8113 PUBLIC SERVICE COSTS AND CITY SIZE THE CASE OF URBAN WATER SUPPLY AND SEWERAGE SERVICES IN COLOMBIA by Johannes F. Linn and Nelson A. Valverde March 1981 Urban and Regional Economics Division Development Economics Department The World Bank Paper presented at the Eastern Economic Association Meetings in Philadelphia, April 1981 FlILE U'SPY{   ABSTRACT The purpose of the paper is to explore the relationship between water supply and sewerage service costs and city size in Colombia by drawing on various cross-sectional and time-series data sets. The paper concludes that simple associations between historical unit service costs and city (or system) size variables are insufficient evidence for judging the impact of city size on service costs that is of relevance for spatial policy. Furthermore, while it appears--contrary to U.S. and U.K. evidence -- that in Colombia water and sewerage costs are not falling with increasing city size, the analysis does not permit an unequivocal conclusion as to whether unit costs are constant across cities of different size, or rising with city size.  March 1981 Public Service Costs and City Size: The Case of Urban Water Supply and Sewerage Services in Colombia* Johannes F. Linn and Nelson A. Valverde Urban and Regional Economics Division World Bank T. INTRODUCTION One of the important policy concerns in developing countries during the last two decades has been the rapid urbanization process experienced in these countries, and the tendency towards concentration of urban population in the large cities. Frequently, the view is expressed by policy makers and academics alike that cities are growing too rapidly and too large and that something must be done to stop or reverse the "imbalances" thought to occur in the urbanization process. One particularly common concern has been that the costs of urbanization The authors wish to thank Roy W. Bahl, John English, Claudio Fernandez, Orville Grimes,Jr., DeAnne Julius, Martin Katzman, Doug Keare, Bertrand Renaud, Jerome L. Rothenberg,Robert Saunders and Alfredo Sfeir-Younis for their comments on an earlier version of this paper. Thanks are due also to Willi Wagner for effective computional assistance. However, the views expressed in the paper are exclusively those of the authors, and do not necessarily reflect the views of the World Bank.  - 2 - and of large city growth are in some sense 'excessive. 'l/ Recent critiques have stressed the conceptual and empirical weaknesses of these views (Linn, forthcoming; Preston, 1979; Renaud, 1979; Richardson, 1977). One particular weakness has been the lack of reliable information regarding the association, if any, between city size and urbanization costs. For developing countries few studies are available on urbanization costs, and even fewer are based on careful empirical analysis.2-/ The purpose of this paper is to demonstrate for an important set of urban services, viz., water supply and sewerage, the difficulties which arise when attempting to analyze the association between service costs and city size for only one set of services in a particular country (Colombia), even where the available data are unu- sually good. More specifically, the paper aims to demonstrate that the usual simple associations between historical unit service costs and city size that are frequently adduced as basi5' for judgements regarding urbanization costs are far too simplistic to permit meaningful conclusions on this issue for particular urban services, let alone for all urban services combined. Judging from the analysis of this paper even a more refined analysis will not necessarily lead to satisfactory answers on that score.-/ 1/ See for example Lewis (1977) for a prominent statement of these views. 2/ See Linn (forthcoming) for a review of the scant evidence on urbani- zation costs in developing countries. 3/ In any case, for the purpose of urbanization policy it is not only necessary to know what are the costs of urbanization and large city growth, but also what are the benefits. This latter question is even more difficult to answer than the former.  - 3 - The remainder of the paper is organized as follows: Section II discusses briefly the components of water production, distribution and disposal : the purpose of considering the likely influences operating on service costs. Section III reviews existing studies of water supply and sewerage costs. Section IV presents the estimation of water and sewerage cost and production functions in Colombia. Section V concludes by summarizing the findings of the paper. II. WATER SUPPLY AND SEWERAGE COSTS: SYSTMS COMPONENTS This section reviews the major systems components of water supply and sewerage services in order to derive an understanding of the nature of the processes involved and of the effects which city size may have on the different service costs components. Where possible, examples will be cited drawing on the authors' experience in Colombia. The provision of potable water falls into three stages: production, storage and distribution. The cost of water production depends primarily on the accessibility of the water source, the level of treatment required given a desired quality of water, and the need for pumping. In principle one would expect that as more and more water needs to be produced for a growing settlement, the more readily and cheaply accessible sources of water are first utilized, while the more remote or deeper, and therefore more costly, sources are tapped sub- sequently. In Bogota (Colombia), for example, the recent Chingaza  - 4 - project required the construction of a dam and transmission of water over a considerable distance through mountainous terrain, whereas previously sources much closer to the city provided sufficient water. In water treatment technological economies of scale exist with large- size treatment plants showing lower unit costs than plants of smaller size. However, eeee increasing density and city size result in higher pollution of ground water and river sources, more treatment ig bs required, thus tending to off-set the cost rom technological economies of scale. In Cali (Colombia), for example, the relatively clean water of the Cali River was exhausted some years ago, and there- fore water had to be drawn from the more polluted Cauca River, requiring much more extensive and costly treatment. Costs also tend to rise if gravity flow water has to be substituted for by pumped water, or if water is to be pumped from deeper wells. On the other hand, technolo- gical economies of scale in pumping may reduce costs as larger pumps are installed. The prevalent pattern in water production therefore appears to be that technological economies of scale tend to lower produc- tion costs as system size increases, while the natural limitations on the input of hydraulic resources tend to raise costs as the quantity produced increases with increasing settlement size. In water storage, technological economies of scale again exist because the tank surface (and thus the need for construction inputs) increases less than in proportion with tank volume, and thus larger storage tanks tend to have smaller unit costs.-/ However, the 4/ Stanford Research Institute (1968), p. 87, gives engineering cost estimates in support of this contention.  - 5 - actual size of storage tanks may not increase as cities grow, because new tanks of similar size as the existing ones may be added to meet , 11< the increasing demand the least-cost geographical disper- sion of storage facilities. Furthermore, storage tanks may have to be located in increasingly inaccessible or costly land, thus off-setting any economies of scale. The costs of distribution tend to increase with declining density, because more piping per household is required to serve the lower density settlements as compared with the more densely populated areas.-/ There exists some evidence that average densitites have declined historically in Bogota and Cali (Ingram, 1980). More relevant for the present concern, however, is the question whether incremental residential growth in larger cities on balance tends to be of greater or lesser density than in the smaller cities. Little is known on this score.-/ What is more, as settlements expand in area, the length of transmission lines from the source may increase, and economies of scale, which could result from using larger pipes, may not be realized when new pipes are laid parallel to existing pipes rather than replacing the existing pipes by larger new pipes. Recent water system expansions in 5/ The inverse relationship between density and water distribution costs per connection is demonstrated, for example, for the case of Cali in Linn (1976). 6/ In Cali, average densities in recent years were above those of Bogota (Ingram, 1980). For the sample of Colombian cities discussed in Section IV.B. below, there exists a very weak negative correlation between residential density and population size.  - 6 - Cartagena have involved precisely this type of investment in parallel transmission pipes (Linn, 1975). Moreover, distribution costs may rise as a result of the inaccessibility of the terrain in which expansion of the cities takes place. For instance, the Colombian cities of Bogota, Cali and Medellin are surrounded by hillsides where capital costs of distribution networks tend to be higher and secondary pumping is frequently required.-/ In Cartagena (Colombia) a number of low income neighborhoods are located in swampy marshlands making access extremely difficult and costly. Thus, offsetting cost factors are again at work, and although technological economies of scale may exist in distribution, these may not be in fact utilized, or may be more than outweighed by increasing difficulty of access. Similar arguments apply to sewerage. Sewerage service involves collection, treatment, and disposal. As with water distribution, higher density will tend to lower collection costs on technological grounds, Xut in a rapidly growing town or city, collection pipes of similar size may have to be laid sequentially, and increasing difficulty of access and terrain may raise unit costs. For treatment and disposal, decreasing unit costs would result on technological grounds for increased size of the treatment plant. However, plants are generally built sequentially and therefore may not increase much in size. As settlements grow the 7/ For Cali, it has been estimated that the average incremental cost of water supply for the hill-side communities is some 30 percent above that for the plane areas of the city (Linn, 1976).  - 7 - need for sewerage treatment is likely to increase. For example, some of the large Colombian cities (e.g., Bogota, Cali, Cartagena) have now reached the threshold where treatment of sewerage has become necessary and therefore a major cost increase is expected for sewerage services. An important additional element for sewerage costs is the choice of technology. Low cost sanitation technologies such as pit latrines and low cost septic tanks have generally not been favored by urban sanitary engineers, who have preferred the much more costly waterborne sewerage systems. As a result, urban sewerage systems have tended to involve much higher unit costs than rural or village systems, where low-cost technologies have generally been applied. A recent World Bank research study, however, has shown that low-cost technologies provide acceptable levels of sanitary waste disposal even in urban areas of developing countries, except for the very densely built-up central business districts, where waterborne sewerage systems are required (Kalbermatten et al., 1980). Summarizing these largely a priori arguments, one may conclude that water supply and sewerage systems typically involve technological economies of scale for given technologies, but the potential cost savings from increased size are counteracted by a number of factors: First, human settlements do not grow in discrete jumps, but continuously over time. Thus, capacity must be added sequentially involving frequently units of constant, rather than increasing size. Second, as cities grow they may expand into areas which are more difficult and costly to service. Third, increased congestion and pollution, and reduced carrying capacity  - 8 - of the environment associated with increased city size result in the need for more treatment at the source of water and more costly tech- nologies of sewage disposal to ensure comparable levels of environmental quality. Finally, declining densities may be associated with increases in distribution costs. In any case, there are many components to a water supply and sewerage system for each of which mutually offsetting impacts of city growth may be at work, and what is more, there are many other factors at work in determining water supply and sewerage costs which may not have anything to do at all with city size and growth, in par- ticular climatic and geological conditions and availability of hydraulic resources. III. REVIEW OF EXISTING STUDIES Given the complexity of the cost structure of water supply and sewerage systems and the many different factors influencing these costs, it is not surprising that comparative estimates of unit costs for settlements of different sizes are difficult to carry out. Two types of approaches have in the past been taken in estimating the relationship between water and sewerage service costs and system (or city) size. First, engineering cost studies have investigated the technological relationships between the size of system components and the costs of installing these components, including transmission pipes, pumping and storage facilities, and treatment plants. These studies, such as the one carried out -S ur Stanford Research Institute (1968) for India, tend to show declining costs, because they capture  -9- the impact of economies of scale and density, but fail to allow for differences in accessibility of water resources, for differential needs for water and sewerage treatment, for poor accessibility of newly serviced neighborhoods, and for the fact that systems are built sequentially. A second approach has been based on a cross-section analysis of service cost data of actually operating systems in urban areas of different size in specified regions or countries. The methodology used has typically involved regression analysis using average historical cost per capita, per connection, or per quantity supplied as the depen- dent variable, and measures of system or city size, of density, and population growth as explanatory variables. Table 1 summarizes the major results. For the U.S. and the U.K., studies have shown mostly average costs declining with city size. In France, Japan, Brazil and Colombia average water and sewerage costs were observed to be constant or increas- ing with city size for cities beyond 50,000 - 100,000 inhabitants. $~number of lcese cross- sectional studies . First, there are the missing variables: Hydraulic resource availability will vary from city to city. If larger cities in a particular country on balance are more poorly endowed with water resources than smaller cities, then the cross- sectional cost function will show rising costs although they may be falling over time, given the underlying cost structure of the service (see Figure 1). Another reason for average cost differences may be found in differential service standards applied in different  - 10 - Table 1: Results of Cross-Section Studies of Water and Sewerage Costs Authors(s) Service Country/Region Results Andrews (1971) Water U.S. (New Hampshire and - "modest economies of size" New England) Hines (1969) Water U.S. (Wisconsin) - constant or decreasing unit costs of water production Isard and Coughlin Sewerage U.S. (Massachusetts) - decreasing unit costs (1957) Saunders and Warford Water U.S. - decreasing unit costs (1976) Ford and Warford Water U.K. (England and Wales) - economies of scale with respect (1969) to quantity of water supplied Prud'homme (1973) Water and Sewerage France - per capita expenditure constant or rising with city size Terao (1975) Water Japan - falling unit costs with city size up to medium size cities (50,000-100,000 inhabitants), rising thereafter. Rizzieri (1979)3 Water and Sewerage Brazil - investment costs per capita constant when adjusting for differential quantity per capita supplied in cities of different size Water Brazil - distribution network costs increase with city size at constant levels of density Villamizar (1977) Water Colombia - per capita investment cost constant with city size Water and Sewerage Colombia - per capita cost increasing with city size Republic of Colombia Water Colombia - per capita cost declining for (1977) cities between 10,000 and 70,000 inhabitants Sewerage Colombia - per capita cost declining for cities between 10,000 and 70,000 inhabitants  - 11 - Figure 1: Hypothetical Example of Missing Variable: Hydraulic Resources Avv Wgo j C -A AtL r t4e~4-k (A eu sr*P < G Figure 2: Hypothetical Example of Missing Variable: Excess Capacity C*4 Lttw -L La Se L __ ~ ~ ~ JL~ . _ _ ; - 4 r,i,l 64,~ 6~  - 12 - size cities. For example, water quality standards may be higher or more effectively implemented in larger than in smaller cities reflecting either different levels of local demand or more effective enforcement by governmental agencies. Other reasons for higher costs which are not strictly caused by larger city size, may be found in topological limitations experienced by specific cities, where city growth encoun- ters physical barriers such as swamps or mountain sides-/ Yet another missing variable relates to capacity utilization. Average historical costs tend to be lowest when there is little or no excess capacity, i.e., shortly before new investment in system expansion; and they tend to be highest when there is a high degree of excess capacity, i.e., shortly after system expansion. Thus, if larger cities tend to have more excess capacity than smaller cities, then a cross-sectional regression may result in an estimate of rising costs, when in fact the underlying cost structure shows declining long-run costs (Figure 2). On the other hand, if newer plants embody more recent levels of (lower cost) tech- nologies then an off-setting effect may enter into the picture, which again, however, is not as such causally associated with city size. Finally, there is the question of efficiency in operations of water systems in different cities. If water agencies in larger cities Qf a particular country tend to be operated more efficiently than in 8/ Villamizar (1977) attempted to allow for this factor by introducing a dummy variable (approximately) reflecting differences in topolo- gical conditions between cities in Colombia.  - 13 - smaller cities, then average water costs may well be lower there than in smaller cities, even if the underlying cost structure for equally effective management may involve rising costs. In all these cases of missing variables, the important point is that although at a par- ticular -t.hr- 4 time there may exist a statistical association between city size and average historical cost, the relationships may be quite spurious, inKe-scse*s?'that the underlying causal relation- ship is not with city size per se, but with other variables such as hydraulic resources, topology, capacity utilization or the extent to which technical progress is embodied in new capacity. .0wh.ie ~~~~~~~~~~~~~~~~~~~~~r other. Lwt and othtr places is axtremel, limitLdf A second type of problem relates to the definition of the cost concept. Historical cost of past systems expansions is not likely to be the relevant concept for purposes of spatial policy. More appropriate would be the incremental cost of future systems expansions in different cities as required by expected growth in the cities' populations. Finally, there are ubiquitous data problems which are almost always encountered. g these the difficulty of estimating reliable capital costs, ks-4A highes-( Moreover, system components can often not be considered separately, since available cross-city data often show combined water and sewerage costs, and rarely distinguish between production, transmission, and distribution costs, or between recurrent and capital costs.  - 14 - The upshot of these considerations is that given the usually available data and approaches used in engineering or cross-sectional analysis of water costs across cities of different sizes, the results commonly encountered cannot readily be taken at face value. The next section reports on some attempts by the authors to improve the analysis in a number of ways drawing on what is an unusually rich set of data for selected Colombian cities regarding water and sewerage service costs and other systems attributes. IV. ESTIMATION OF WATER COST AND PRODUCTION FUNCTIONS FOR SELECTED COLOMBIAN CITIES Two types of cost data for water and sewerage services are available for selected large and intermediate size cities in Colombia: First, for 1973, 1975, and 1977 average historical costs have been estimated for up to 22 cities. Second, for 1978 average incremental cost estimates of projected systems expansions are available for twenty large, medium and small cities. In addition, for 1975 the historical cost data have been complemented by information on residential density, capacity utilization, water loss, labor and capital stock estimates and a number of other relevant variables. Finally, for the case of Bogota, time series data on service costs are available for the years 1961 through 1974.2/ Given this available data set, this section reports estimation results, first, for the simple association between 9/ Appendix 1 discusses the data and their sources in some detail. Appendix 2 lists the cities that are covered in the empirical analysis below.  - 15 - historical service costs and systems size in 1973, 1975 and 1977; second, for an in-depth analysis of cost functions in 1975; third, for an attempt to estimate water and sewerage service production functions for 1975; fourth, for an estimation of the association of average incremental water and sewerage costs with system size variables; and, finally, for the intertemporal association between unit costs and system size in Bogota. A. Simple Association of Historical Costs with System Size, 1973, 1975 and 1977. Table 2 summarizes the regression results for 1973. Two cost concepts, total operational costs and total costs of water supply and sewerage,- are related alternatively to water consumption, number of subscribes and population in bivariate log-linear regressions. Equations 2.1 through 2.6 indicate that total operational costs exhibit an elasticity with respect to system size which is slightly greater than, but never significantly different from unity, whether city popu-: lation size, number of subscribers or water consumption is used as the dependent variable. Total costs in contrast show an elasticity signi- ficantly greater than unity for two of the size variables, viz. number of subscribers and population. It thus appears that operational costs 10/ Total operational cost are defined to include all operational costs but not depreciation. Total cost is defined as total operational cost plus depreciation plus eight per cent of net fixed assets. The last conpoment is included as an estimate of annuitized histo- rical capital costs. The regressions cover up to 16 cities with population ranging from 3 million down to 50,000. See Appendix 1 for definition of variables for this and all subsequent tables.  Table 2: Log-linear Regression Equations Relating Service Costs to Annual Water Consumption, Number of Subscribers, Population and Water Consumption per Capita, 1973 Independent Variables 1/ Water 2 Equation Dependent Constant Water Number of Consumption R Number of No. Variable Term Consumption Subscribers Population Per Connection (F-ratio) Observations 2.1 ln TOC -.3266 1.064 .8891 13 (.113) (88.l7) 2.2 ln TC -.7215 1.112 .9039 13 (.109) (103.41) 2.3 In TOC 8.493 1.120 .9258 13 (.096) (137.25) 2.4 In TC 8.502 1.169* .9394 13 (.090) (170.52) 2.5 In TOC 2.298 1.192 .8711 16 (.123) (94.60) 2.6 ln TC 2.094 1.240* .f810 16 (.122) (103.67) 2.7 ln TOC -1.481 1.030 1.460 .8907 13 (.147) (1.028) (40.76) 2.8 ln TC -2.208 1.068 1.622 .9604 13 (.141) (.987) (48.41) 1/ Numbers in parenthese represent the standard error of the regression coefficient. * Significantly different from unity at the 5% significance level. Legend: TOC = total operational cost; TC = total cost.  - 17 - tend to change in proportion with city size while, total costs (and, by infergnce, capital costs) tend to increase more than in proportion with system size. Interestingly, if water consumption per connection is entered as an independent variable into the equa- tion jointly with number of subscribers, the size of the regression coefficient for the connection variable declines and is no longer significantly different from unity in the case of total cost (Equations 2.7 and 2.8). Table 3 shows the results of applying the format of Equations 2.7 and 2.8 to the 1975 data base, where 21 cities ranging in population from 3.3 million inhabitants to 20,000. In this case, operational costs again show unitary elasticities with respect to system size; the same holds true for the components of operational costs, i.e., labor and other operational costs. However, capital and total costs increase significantly more than in proportion with system size, even when allowing for differences in consumption per connection. For 1977 total operational cost data for water supply in a sample of 22 cities ranging from 220,000 to 20,000 inhabitants is available. Bivariate linear regressions were run to analyze the rela- tionship between operational cost per cubic meter of water sold as dependent variable and city population size, number of water connections and quantity of water consumed respectively as independent variables. In no case was there a significant correlation between the unit cost and the system size variables. Even a quadratic formulation of the regression equation, designed to indicate a non-linear relationship  Table 3: Log-Linear Regressions Relating Service Costs to Number of Subscribers and Water Consumption per Capita, 1975 Independent Variables - 2 Dependent Constant Water Consumption Adj. R Number of Equation No. Variable Term Subscribers Per Connection (ln WQ/N) (F-Ratio) Observations (ln N) 3.1 ln LC 3.1731 1.1032 .5865 .9319 21 (.0902) (.4574) (137.9) 3.2 In OC 1.6446 1.1700 .8412 .8186 21 (.1693) (.8586) (46.1) 3.3 ln TOC 3.4699 1.1102 .6303 .9252 21 (.0959) (.4864) (124.6) 3.4 lnKTC -.0579 1.3679 -.0857 .9156 21 (.1166) (.5915) (109.5) 3.5 In TC 2.7552 1.2174 .3773 .9416 21 (.0892) (.4526) (162.3) 1/ Numbers in brackets are standard errors * Significantly different from unity at the 5% significance level. Legend: LC = labor cost; OC = other operating costs; TOC = total operating costs; KTC capital costs; TC = total costs.  - 19 - between system size and unit cost did not yield significant regression coeficients at the 5% confidence level. The best fitting regression equation was as follows: TOC = .0321 + .9043 QW77 - 5.5234 QW2 77 77 ~~~~~~Q77 (.5822) (3.4314) -2 R = 0.0276 (F Value: 1.2975) Where TOC7 is total operational cost in 1977, QW77 is the quantity of water sold in the same year, and figures in brackets refer to standard errors. If anything this equation points to an inverted U-shaped unit cost function which runs counter to the ususal hypo- thesis of a U-shaped relationship between system size and unit cost. -/ Overall, one would thus conclude that for water and sewerage costs in 1973, 1975 and 1977, the samples of Colombian cities show constant operational unit costs, but also indicate increasing capital and thus total unit costs in relation to city size. Interestingly, the inclusion of a number of smaller cities in 1975 sample, when compared with the 1973 sample3appears to have strengthened rather than weakened the evidence of increasing capital and total unit costs. 11/ It also runs counterAevidence in Republic of Colombia (1977) where a declining cost funtion was estimated for cities in the 10-70,000 size bracket.  - 20 - B. Multivariate Regression Analysis of Historical Costs, 1975 For 1975 multivariate analysis was possible for historical service costs of water supply and sewerage in 15 large and medium- size Colombian cities. Tables 4 and 5 summarize the results of stepwise regression for average labor costs, other operating costs, total operating costs, capital costs and total costs. Beside the two system size variables QW and N (water consumption and number of connec- tions), which are used alternatively, the following variables were tested for their significance in step-wise linear regression: a dummy variable reflecting whether or not the fixed capital stock had been revalued (R); the proportion of water connections also serviced by the sewer sytem (S/W); the rate of population growth of the city (PG); the quality of water as reflected by the level of treatment (QU); excess capacity as a proportion of total water production (EC); average residential density of the city (D); and the percentage of water lost (unaccounted for) in distribution (WL). In considering Tables 4 and 5, the following general obser- vations may be made: First, the density (D) and water loss (WL) variables do not appear as significant explanatory variables in any of the regres- sions. On theoretical grounds one would have expected the former variable to be negatively related with average cost, the latter pos- sitively.2.- Second, the significance of the other independent variables 12/ There is no evidence of strong multicollinearity between these two variables and other independent variables, except that D is quite highly and inversely correlated with population growth.  - 21 - Table 4: Stepwise Linear Regression of Service Costs per Quantity of Water Consumed, 1975 Equation No. 4.1 4.2 4.3 Dependent Variable OC/QW KTC/QW TC/QW Constant Term 1.0850 1.0994 4.3918 Independent Variables QW .0051 .0114+ (.0021) (.0036) R .7527+ (.1934) S/W -.0103+ -.0222 (.0035) (.0087) PG .1358 (.0757) * QU -.1908 (.1345) EC D WL Adj. R2 .0675 .8244 .4910 F-Value 2.0133 17.4294 7.7515 No. of Observations 15 15 15 Note: (LC/QW) and (TOC/QW) showed no significant relationship with any of the independent variables at 10% level of confidence. Legend: * significantly different from zero at 10% confidence level. ** = significantly different from zero at 5% confidence level. + = significantly different from zero at 1% confidence level. OC = operational cost; KTC = capital cost; TC = total cost; QW = water consumption; R = dummy variable for revaluation of fixed capital stock (1 = yes; 0 - no); S/W = proportion of water connections also connected to sewerage; PG - population growth rate; QU = quality as measured by water treatment level; EC = percent of excess capacity; D = residential density; WL = percentage water loss. LC = labor cost; TOC = total operational cost.  - 22 - Table 5: Stepwise Linear Regression of Service Costs per Subscriber, 1975 Equation No. 5.1 5.2 5.3 5.4 Dependent Variable LC/N TOC/N KTC/N TC/N Constant Term 104.0286 150.3942 80.9710 317.2731 Independent Variable N .2713 .4800+ (.0589) (.1415) R 23.0722 (10.1217) S/W -.8821 -.6203 -1.3223 (.5566) (.2089) (.5695) PG QU EC -.3991 -.7939 (.2278) (.3835) D WL Adj. R2 .1287 .0974 .7825 .5551 F-Value 3.0686 (2.5112) (17.7839) (6.8231) No. of Observations 15 15 15 15 Legend: * = significantly different from zero at 10%; ** = significant at 5%; + = significant at 1% confidence level. N - number of connections; other variables defined as in Table 4. Note: (OC/N) showed no significant variation with any of the independent variables.  - 23 - depends considerably on which cost components are considered, and also on whether average cost is calculated per quantity of water consumed or per number of water connections. Considering specifically each of the independent variables, one finds that the system size variables QW and N are never signifi- cantly different from zero for any of the operating cost components, but that they are significantly positive for capital and total unit cost, irrespective of which size variable is selected (Equations 4.2, 4.3, 5.3, and 5.4 in Tables 4 and 5). Despite the inclusion of all other independent variables the hypothesis that average historical capital and total service costs increase with city size in Colombia cannot be rejected. The dummy variable reflecting fixed capital stock revaluations is significant only for the capital cost equations 4.2 and 5.3. The sign is positive as expected, since capital stock revaluation is desig- ned to bring the book value of fixed capital costs in line with actual estimated replacement cost, and thus tends to raise estimated histori- cal capital costs.l3/ The significant negative coefficient for the variable (S/W) (proportion of water connections hooked up to the sewerage system) is surprising. One might have expected that combined water and sewerage costs per water subscriber or per quantity of water consumed increases with the proportion of water users connected to the sewer system. The 13/ As mentioned earlier capital cost is estimated as 8 percent of book value of fixed capital stock.  - 24 - fact that a strong reverse tendency is observed may possibly be due to the fact that (S/W) is in fact a proxy for efficiency in the management of the utility agency, in the sense that those agencies which are more efficiently managed are able to produce a higher proportion of sewerage connections at lower costs than their inef- ficiently managed counterparts. Alternatively, the causation may run from unit costs to proportion of sewerage connections, in the sense that where unit costs are low (e.g., because of favorable topological or hydraulic conditions), the water company may be more readily able to connect its users to the sewerage system. 14/ Population growth (PG) is a (marginally) significant inde- pendent variable only in equation 4.2, with a positive sign indicating that capital costs per unit of water consumed tend to be higher in cities experiencing rapid population growth. This provides some, albeit weak, support for the frequently voiced hypothesis that rapid growth, rather than large size, results in higher costs of service provision due to the difficulties encountered in adjusting to rapid change. The quality variable (QU), which actually reflects the level of water treatment, rather than the degree of quality of the water produced, is generally not significantly related to cost, except in equation 4.1, where it is negatively related to total operating cost. 14/ There is no evidence of strong multicollinearity with any of the other independent variables.  - 25 - This is again a surprising result since one might have expected that higher levels of treatment would go hand in hand with higher costs. Differential efficiency in system management or reverse causation between systems costs and treatment may again explain the estimation result. The proportion of excess capacity (EC) is negatively related with historical labor and total costs per connection (equations 5.1 and 5.4). This is as 4&h expected, since lower levels of excess capa- city mean that historical costs can be more widely distributed among consumers. What is surprising is that this conclusion does not hold equally for the equations in Table 4 or for the capital cost equations. - / In summary, the inclusion of various relevant variables which might determine average historical water and sewerage costs has left the hypothesis intact that these costs tend to increase with system size in Colombia, particularly as regards capital costs. C. Multivariate Regression Analysis of Production Functions, 1975 The availability of employment and capital stock data for water and sewerage services in selected Colombian cities makes it possible to estimate cross-sectional production functions. The functional expression chosen draws on the specification of Segal (1976) 15/ There may be a problem of multicollinearity among the variable EC with the variables QU, PG and D, for which the simple corre- lation coefficients are between 0.4 and 0.6.  - 26 - who explored the existence of returns to scale in city size for 58 SMSAs in the U.S. In somewhat simplified terms Segal used the following production function: Q. AS CS Ka L. Where Q is output of city i, A a shift parameter, S a city size variable, Ci a variable reflecting specific climatic or geographic conditions of the city, and K and L capital stock and employment respectively. This equation permits to distinguish between economies or diseconomies of scale in the use of factors of production K and L on the one hand, and the existence of "size effects", or net agglome- ration economies or diseconomies, on the other hand. Scale economies (diseconomies) are indicated if (a+ S) is greater (smaller) than unity; agglomeration or size economies (diseconomies) are indicated if Y is greater (smaller) than zero. For statistical estimation the production function is trans- formed by first dividing both sides of the equation by Li and then taking natural logarithms, such that ln K1Iig = ln A + yln S + 6ln Ci + a ln L-i + (a+ 3 -l)ln Li This formulation avoids the problem of multicollinearity between the F4e'e"4(A I. variables Ki and L Segal further a _ the problem multicollinear- ity between S and Li by using a dummy variable for S, setting it equal to e for large cities, and equal to 1 for medium size and smaller cities.  - 27 - For the case of water and sewerage services in Colombia Segal's approach was utilized without significant modifications, except that variable Ci was replaced by a number of city specific variables reflecting such factors as density, water loss, capital stock revaluation, water quality, etc. Table 6 reports the results of various alternative specifications of the regression equation using ln (QW/L) as the dependent variable. -/ Throughout, the coefficient for (ln L), which is defined as (a +$- 1), is found to be negative and significantly different from zero (with the exception of equation 6.5). This result implies that there are diseconomies to scale in the use of factor inputs for water supply sewerage. However, when considering the coefficients of the size- $effect variables S1 and S2, it appears that economies of agglomera- tion are prevalent, since the coefficients are positive and signi- ficantly different from zero when S2 is used (equation 6.6). The vairables Sl and S2 are dummy variables set so that ln(Sl) is unity for cities over 500,000 inhabitants, and zero for all others; ln(S2) is unity for cities with more than 150,000 inhabitants, and zero for all others. As mentioned earlier, this formulation avoids the serious collinearity problem encountered when using the population or connection 16/ Regressions were also run for ln(N/L) as the dependent variable. Since the results were qualitatively comparable to those for ln(QW/L), they are not reported separately here.  - 28 - Table 6: Production Function: Multivariate Regression Analysis with Quantity of Water as Output Variable Equation No. 6.1 6.2 6.3 6.4 6.5 6.6 Dependent Variable ln(QW/L) ln(QW/L) ln(QW/L) ln(QW/L) ln(QW/L) ln(QW/L) Constant Term -1.8124 -1.6819 -2.9916 -1.6243 -1.7794 -1.3817 Independent Variables ln(K/L) .4974+ .6545 .5353 .5019 .3257 .2886 (.1767) (.2440) (.1733) (.1867) (.1414) (.1004) lnL -.1139* -.1093* -.1217* -.1170* -.1148 -.2617 + (.0780) (.0786) (.0758) (.0840) (.0917) (.0702) In Sl .0013 (.3957) + ln S2 .5590 (.2285) ln S/W .2815 (.2097) ln WL -.0510 (.3368) ln R -.2563 (.2733) Adj. R2 .2975 .2904 .3415 .2352 .2031 .4988 F -Value 3.9643 2.9097 3.4202. 2.4352 2.5288 6.9712 No. of Observations 15 15 15 15 19 19 Legend: * significantly different from zero at 10%, ** at 5%, + at 1% confidence level. L = labor employed; K = fixed capital stock; P = population. S = dummy variable (e for cities greater than 500,000 inhabitants, 1 1 for all others). S = dummy variable (e for cities greater than 150,000 inhabitants, 1 for all others) R = dummy variable (e if fixed capital stocks revalued, 1 if not) All other variables defined as in Table 4.  - 29 - variables(P and N, respectively) in conjunction with the employment variable. Judging from a comparison of equations 6.5 and 6.6 it appears that S2 is the better explanatory variable for measuring a size-effect, since its coefficient is highly significant, and its inclusion in the equation in place of Si increases the adjusted R from 0.20 to 0.50. This result would appear to indicate that there is a significant size effect at a threshold of about 150,000 to 200,000 inhabitants such that water and sewerage production per employee tends to be significantly larger in cities above that size, than in cities below that size, keeping the capitalvrlabor ratio and labor inputs constant. Introducing separately variables reflecting the proportion of water subscribers also connected to the sewerage system [ln(S/W)], the percentage of water losses tln(W/C)I, and the dummy variable for capital stock revaluation (ln R) shows that these variables are not significantly associated with ln(QW/L), in the absence of a size-effect variable.