Identifying network density and scale economies for Japanese water supply organizations

Transcription

1 Papers Reg. Sci. 80, (2001) c RSAI 2001 Identifying network density and scale economies for Japanese water supply organizations Fumitoshi Mizutani, Takuya Urakami Graduate School of Business Administration, Kobe University, 2-1 Rokkodai, Nada-ku, Kobe , Japan ( toshi@rokkodai.kobe-u.ac.jp) Received: 10 March 2000 / Accepted: 8 January 2001 Abstract. Within this analysis we determine the optimal size (that is, with the minimum average cost) of a water supply enterprise and reconsider the matter of scale economies, using sample data from Japanese water supply organizations. After surveying evidence from previous studies of scale economies in the water supply industry, we estimate cost functions with three different cost models: the log-linear, translog and translog with a hedonic function. We obtain the result that economies of network density do exist, but that there are slight diseconomies of scale at the sample mean point. The optimal size of a water supply organization would be one supplying a population of approximately 766,000 people. JEL classification: L95, L11 Key words: Economies of scale, water supply, optimal size, average cost, translog cost function 1 Introduction The worldwide trend toward deregulation and privatization has manifested itself in Japan with the privatization in the late 1980s of two nationally owned giants, the Nippon Telephone and Telegraph Corporation and the Japan National Railway, as well as in the increased use of competition policy within the electricity supply industry. The water supply industry, however, remains to this day relatively unmoved by the encouraging results of these public-turned-private organizations. While many utility industries such as electric power, gas, railway, and telecommunications rely almost exclusively on private companies to supply and deliver their goods and services, the water industry remains dependent on the public sector, mainly in the form of municipal governments. Parts of this article have been supported by the Ministry of Education s Science Research Fund.

2 212 F. Mizutani, T. Urakami There were 1,960 main water supply enterprises in Japan in Almost all of them were publicly owned, and yet almost all of them were different from each other, the largest serving a population of 11 million people and the smallest serving a mere 5,000 people. Such huge differences in size, type of water supply organization, management style of each, costs incurred, and prices charged, have all stimulated debate on the future of the water industry. Central to the debate is whether or not economies of scale exist in the water industry. While cost structures have been analyzed and evidence of scale economies examined (e.g., Kim 1987), different studies produce different results. By using data sets from Japanese water supply agencies, we will reconsider whether or not scale economies exist in this industry. If scale economies are strongly in evidence, it follows that it would be in the interest of the Japanese water supply industry to reorganize itself. Policy makers might agree with such an assessment, but they are likely to question the form of any change. Should oversized corporations be divided into many, while smaller ones join forces? From a policy maker s perspective, it would be most useful if we could determine the optimal population size served by a water supply organization, which is defined here as the size at which that organization achieves the minimum average cost. After this introduction, the structure of our article is as follows. We first briefly summarize the methodologies used in previous cost studies, pointing out how they suggest the existence, or absence, of economies of scale. We distinguish between the concept of economies of network density and economies of scale. Second, we explain our three cost models: log-linear, translog, and translog with a hedonic function. We also define measures of scale economies in this section. We summarize sample selection and define the variables. After explaining certain estimating procedures for obtaining unavailable variables, we show the estimation results using our cost models, and then measure scale economies and network density economies. In the final and most important section, we determine the optimal size of a water supply agency based on our preferred estimated cost model. 2 A current policy issue in the water supply industry Modern water supply systems first appeared in Japan in the middle of the Meiji Era in 1887, and were found to be desirable in their capacity to stem disease and alleviate fire damage. Local governments began supplying water legally in 1890, and today remain almost the sole suppliers of water. This is in contrast to the situation in other public utility industries such as the electricity and gas industries, the railways and bus transportation, some of which have always been supplied wholly or in part by the private sector and others that have only recently become private. The number and variety of water suppliers in Japan indicate a certain degree of disorder within the industry. As of March 1998, there were 1,962 direct suppliers, of which six were owned and managed by prefectural governments, 617

3 Identifying network density and scale economies 213 by cities, 1,160 by towns, 90 by villages, 78 by cooperative societies, and 11 by the private sector. In addition to these 1,962 larger general water suppliers, there is a myriad of smaller, vaguely defined water supply organizations, perhaps even too small to have names, supplying, for example, communities of 30 or fewer families. According to a report by the Ministry of Health and Welfare in 1998, if we include organizations of this latter type, then the total number of all types of water supply organizations has reached a staggering 15,784. Not only are the suppliers clearly too numerous, but their differences in size are remarkable (see Table 1), no doubt reflecting differences in resources, available capital, costs incurred, and prices charged. The largest organization, the Tokyo metropolitan government s water bureau, employs 6,228 people and provides water for a population of 10,829,000. Conversely, there are only two employees at the smallest water supply organization, which supplies water for 2,000 people. In fact, most water suppliers in Japan fit into the latter category, with only 10% of organizations supplying populations of more than 100,000 in The size distribution indicates no sign of change, with no mergers or takeovers imminent in this labyrinthine structure of disparate but somehow connected water systems. Price variation stems in part from size differences, but since the Japanese water supply industry bases its prices mainly on a self-supporting (cost-recovery) principle, there are price differences according to region. For example, according to the Economic Planning Agency (1996), the monthly charge for ten cubic meters in April 1994 ranged from 319 yen to 3,090 yen. This tenfold difference in price is likely to be noted and criticized by consumers at the higher end of the price scale. The water distribution industry, however, is a natural monopoly. Although consumers cannot fully boycott the supply of such an indispensable resource as water, consumers would likely protest if they knew that the high prices they pay result in part from higher average costs for poorly situated or undersized suppliers. Public interest and awareness have strengthened in recent calls for reorganization and proposals for subsidy programs to encourage consolidation. The issue of optimal size should therefore be particularly interesting to policy makers in their efforts to achieve greater efficiency in the water supply industry. 3 Previous studies of economies of scale in the water supply industry 3.1 Methodology of cost studies We have surveyed international economic journals from 1960 to the present and we summarize here the methodology of cost studies, as well as the results on scale economies in the water supply industry. Since Ford and Warford s study in 1969, there have been many cost studies of the water supply industry with common features. The first commonality refers to the theoretical background. Between Ford and Warford s study in 1969 and Bruggink s in 1982, no study except Crain and Zardkoohi (1978) had a clear theoretical foundation for the cost function. Studies tended instead to describe the relationship between cost

4 214 F. Mizutani, T. Urakami Table 1. Size distribution of water supply enterprises by delivery population Delivery population More than 1,000 thousand to 1,000 thousand to 500 thousand to 250 thousand to 100 thousand to 50 thousand to 30 thousand to 20 thousand to10 thousand Less than 5 thousand Under construction Total 1,947 1,957 1,964 1,969 1,971 1,969 1,962 1,952 1,960 Source: Ministry of Health and Welfare (1998) and several factors using regression techniques. However, since Feigenbaum and Teeples s 1983 study (Crain and Zardkoohi 1978 should also be mentioned), cost studies have been rooted in economic theory, with cost functions based on the theory of cost minimization under the constraint of a production function. Second, in so far as the functional form is concerned, the translog cost function is prevalent in current cost studies. In the 1970s, the linear and log-linear cost functions were more common. These functional forms are relatively easier to deal with because they have fewer explanatory variables. But they have more constraints. For example, the log-linear cost function, which was used in Crain and Zardkoohi (1978) and elsewhere, assumes that the elasticity of substitution is one, and that the scale measure is constant, regardless of output level. In particular, this log-linear function s assumption that the scale measure is constant, regardless of output level differences, is crucial in the examination of economies of scale. In fact, it is more natural to assume that the scale measure varies with the output level. The translog cost function is therefore more advantageous than other functions and is considered to be a more flexible functional form, because the scale measure varies with output level, the elasticity of substitution is not one, and the function is not homothetic. Since Feigenbaum and Teeples (1983), the translog cost function has been widely used in cost studies. Third, it is important to control network factors and output quality factors. Although the cost function, based on textbook economic theory, is composed of

5 Identifying network density and scale economies 215 output and input factor prices, there are several important factors that should be controlled for in the public utility industry. Especially in a network industry such as the water supply industry, the factor of pipe length is important. For example, even if the size of output is the same, the cost structure could be different based on the network size. If region A has a denser population than region B but both have the same output, region A might be more cost advantageous with regard to capital than region B, because region B might need a longer line length than region A. The outcome means higher facility installment and maintenance costs. In an extremely densely populated Japanese city, the ratio of water output to network size will differ enormously from that of the same network size in a small community, a fact that makes it critically necessary to control the estimation bias due to network differences in the cost function. Although previous cost studies have not very well distinguished the output scale effect from the network scale effect, Kim (1987) and Kim and Clark (1988) did examine the existence of economies of scale based on the translog cost function with network characteristics. Water quality as well as network characteristics is also a crucial factor in the cost structure. 1 Even if the quality level of the final consumed water is the same among regions, the differences in the processes of water treatment certainly vary the cost structure. Obviously, if the water supply agency extracts the natural water from less populated mountain sites, then the quality of water could be higher than that in a populated urban river site, thus considerably lowering water treatment costs. Consumer differences also affect the cost structure. While the water supply agency provides a smaller water supply pipe to each individual general household consumer, for large-scale commercial or industrial consumers, larger caliber water supply pipes are required, increasing installment and maintenance costs. Quality measures and output measures are controlled in current cost studies by using the hedonic function (see for example, the studies by Feigenbaum and Teeples 1983 and Bhattacharyya et al. 1995a). These studies integrate output and quality measures as hedonic output measures. The integrated output measure is used for the cost function. The treatment of output quality measures might also be necessary in estimating a more accurate cost function for the water supply industry. 3.2 Evidence of economies of scale In our examination of previous cost studies, which did not aim to measure scale economies, we found evidence of economies of scale. In the summary shown in Table 2, we distinguish the concept of economies of scale from economies of network density. Although the definition is given in a later section, the essence 1 In addition to network characteristics and the quality of water, there are many other factors that affect cost structure, for example, the depth of the water table and the topography of the city. For example, large elevation differences require more storage tanks and booster pumps. In these cost functions, variables showing the topographical differences in a city or a town or water-intake conditions such as ground/surface water, are usually included as dummy variables.

6 216 F. Mizutani, T. Urakami Table 2. Previous cost studies and results on economies of scale Researcher Area Data Year Model Cost RTD RTS Ford and Warford (1969) U.K (cs) log-linear AVC Mann and Mikesell (1976) U.S (cs) linear AVC Morgan (1977) U.S (cs) linear VC Crain and Zardkoohi (1978) U.S (cs) log-linear VC Bruggink (1982) U.S (cs) log-linear VC Feigenbaum and Teeples (1983) U.S (cs) translog VC Kim (1987, 1995) Kim and Clark (1988) U.S (cs) translog TC Bhattacharyya et al. (1994) U.S (cs) translog AVC Bhattacharyya et al. (1995a) U.S (cs) translog VC Bhattacharyya et al. (1995b) U.S (cs) translog VC Takada and Shigeno (1998) Japan (pl) translog TC Kuwahara (1998) Japan n.a. (pl) translog TC This study Japan 1994 (cs) translog TC Note: (1) cs: cross-sectional data, ts: time series data, pl: pooling data; (2) AVC: average variable cost, VC: total variable cost, TC: total cost; (3) RTS: return to scale, RTD: return to network density of these concepts is summarized as follows. Economies of network density are measured by the return to network density, defined in terms of the relative increase in output resulting from a proportionate increase in all inputs, but holding the network condition constant. The return to scale is defined as the relative increase in output from a proportionate increase in all inputs. These conceptual differences have been widely accepted in other network industries such as the railway industry (see for example, Caves et al. 1985). In the water supply industry, Bhattacharyya et al. (1995b) examined economies of network density for various cost functions. Furthermore, Takada and Shigeno (1998) stated the conceptual differences clearly and examined both economies of scale and network density. These empirical results show the possible existence of economies of network density in the variable costs. However, as for total cost, there is no clear evidence for either economies of scale or economies of network density. For example, the results of Kim s several studies suggest the existence of economies of network density, but weak diseconomies of scale, while Takada and Shigeno (1998) and Kuwabara (1998) suggest strong economies of network density and weak economies of scale. These varying results hinder current policy making efforts in the Japanese water supply industry, which is now focusing on the issue of whether or not water supply organizations should be consolidated to attain scale benefits. These results also suggest the necessity for considering network

7 Identifying network density and scale economies 217 size as well as output size when deciding the optimal size of a water supply organization, a process we think is useful in providing policy information relevant to the Japanese water supply industry. 4 Cost model and measure of scale economies 4.1 Cost model We choose a translog cost function as the basic cost model of water supply. The main reason is that the functional form is more flexible than those of log-linear and linear cost functions. Because the main purpose here is to examine whether or not there exist economies of scale, it is necessary to choose the more flexible form for the cost model. However, in order to avoid differences in the results according to functional form differences, we also estimate the cost functions by using three different methods. The selected models are the simple log-linear cost function, the translog cost function and the translog cost function with a hedonic specification for output measure. The special feature of this study is the inclusion of network factors and output quality factors. The cost measure used here is total cost (C ), which includes both variable components such as labor, energy and material costs, and fixed components such as capital costs. The total cost is a function of output (Q), input factor prices (P i ) and network characteristics (N k ). In the hedonic-type cost function, the output measure is specified as a hedonic output measure (Y ), which is a function of output (Q) and output quality measures (Z h ). The first cost model is a simple log-linear cost function (Model 1). In this model, second-power and cross-terms of variables in the translog cost function are excluded. This most basic form of cost function model is a Cobb-Douglas type function as follows: LnC = α 0 + α Q lnq + Σ i β i lnp i + Σ k γ k lnn k, (1) where C is total cost, Q is service output, P i is input factor price (i = l (labor), e (energy), m (material), k (capital)), N k is network characteristics (k = n (network density), c (utilization rate)). We further impose the restriction on input factor prices that Σ i β i =1. Second, the translog cost model (Model 2) is specified as follows: LnC = α 0 + α Q lnq + Σ i β i lnp i + Σ k γ k lnn k +1/2α QQ (lnq) 2 + 1/2Σ i Σ j ε ij (lnp i )(lnp j )+1/2Σ k Σ l µ kl (lnn k )(lnn l )+ Σ ı δ Qi (lnq)(lnp i )+Σ k δ Qk (lnq)(lnn k )+Σ i Σ k ζ ik (lnp i )(lnn k ). (2) In this model, we also impose restrictions on input factor prices such that Σ i β i = 1, Σ i δ Qi =0,Σ j ε ij =0,Σ i ζ ik =0,ε ij = ε ji,ζ ik = ζ ki,µ kl = µ lk. The i-th input s cost share equation is defined as follows:

8 218 F. Mizutani, T. Urakami S i = P i X i /C (3) where S i is the i-th input s cost share, and X i is the i-th input s demand. Furthermore, we apply Shephard s Lemma to this model and the cost share equation can be written as follows: S i = β i + Σ j ε ij (lnp j )+δ Qi (lnq)+σ k ζ ik (lnn k ). (4) The third kind of cost model is a translog cost function with a hedonic specification of output (Model 3). The basic structure of the cost model is shown as follows: LnC = α 0 + α Q lny + Σ i β i lnp i + Σ k γ k lnn k +1/2α QQ (lny ) 2 + 1/2Σ i Σ j ε ij (lnp i )(lnp j )+1/2Σ k Σ l µ kl (lnn k )(lnn l )+ Σ i δ Qi (lny )(lnp i )+Σ k δ Qk (lny )(lnn k )+Σ i Σ k ζ ik (lnp i )(lnn k ). (5) The restrictions on input factor prices are again Σ i β i =1,Σ i δ Qi =0,Σ j ε ij = 0,Σ i ζ ik =0,ε ij = ε ji,ζ ik = ζ ki,µ kl = µ lk. The hedonic specification for the output measure is shown as follows: lny =lnq + Σ h η h lnz h (6) where Z h is output quality measures, (h = tr (treatment level), hr (household ratio), ndam (non-dam water acquisition index), nund (non-underground water index)). We also impose a restriction among the quality measures, namely, Σ h η h =1. Furthermore, we use Shephard s Lemma with respect to Equation (5) and the i-th input s cost share equation is obtained as follows: S i = β i + Σ j ε ij (lnp j )+δ Qi (lny )+Σ k ζ ik (lnn k ). (7) 4.2 Measures of scale economies With regard to the measures of economies of scale and density, we choose the following, which were used by Caves et al. (1984, 1985). The return to density, RTD, a measure of economies of density, is the inverse of the output elasticity of cost when other variables in the cost function are fixed. The return to scale, RTS, a measure of economies of scale, is the inverse of the sum of output elasticity and network length elasticity of cost. From this definition, we can obtain the following results for these models. For each cost model, the scale and density measures are summarized in Table 3. RTD = 1/[ lnc / lnq] (8) RTS = 1/[( lnc / lnq)+( lnc / lnn n )]. (9)

9 Identifying network density and scale economies 219 Table 3. Measures of economies of scale and network density Model RTD: return to network density RTS: return to scale Model 1: log-linear 1 / α Q 1/(α Q + γ n ) Model 2: translog 1/[α Q + α QQ (lnq)+σ i δ Qi (lnp i )+ 1/[[α Q + α QQ (lnq)+σ i δ Qi (lnp i )+ Σ k δ Qk (lnn k )] Σ k δ Qk (lnn k )]+[γ n + Σ l µ nl (lnn l )+ δ Qn (lnq)+σ i ζ in (lnp i )]] Model 3: hedonic 1/[α Q + α QQ (lnq + Σ h η h Z h )+ 1/[[α Q + α QQ (lnq + Σ h η h Z h )+ Σ i δ Qi (lnp i )+Σ k δ Qk (lnn k )] Σ i δ Qi (lnp i )+Σ k δ Qk (lnn k )]+ [γ n + Σ l µ nl (lnn l )+ δ Qn (lnq + Σ h η h Z h )+Σ i ζ in (lnp i )]] 5 Estimation of cost functions 5.1 Sample selection and definition of variables The observations used in this study are from 112 water supply organizations for the fiscal year 1994 (FY1994). These observations are selected from 1,891 water supply organizations appearing in the FY1994 Annual Statistics of Local Public Corporations (Chiho Koei Kigyo Nenkan FY1994) issued by the Ministry of Local Government. To avoid bias regarding size differences among organizations, we chose water supply organizations from different sized categories: 15 observations from very large cities and prefecture-owned organizations, 20 observations each from cities with 30 to 50 thousand populations, 50 to 100 thousand, 100 to 150 thousand, 150 to 300 thousand, and cities with more than 300 thousand people. After eliminating three organizations with incomplete data sets, we finally chose 112 water supply organizations for this analysis. These observations are geographically well distributed throughout Japan and almost all of the water supply organizations are municipally-owned (e.g., city or town) public sector organizations. The main data source is the FY1994 Annual Statistics of Local Public Corporations. The variables used here are defined as follows. First, total cost (C ) consists of variable and fixed costs, which are the sum of labor, energy, materials, and capital costs. Labor costs include some allowances as well as basic salaries. Energy costs consist of electricity expenditure and other energy expenditures for the operation of machines and pumps. Material cost is categorized as repair expenditures in the Annual Statistics. Capital costs are depreciation and interest payments for facilities and other items. Depreciation expenditures apply to both tangible and intangible assets in the supply facilities. Interest payments consist of payments of corporate bonds and debt payments. Output (Q) is measured as annual total volume of water delivered. As for input factor prices, wages (P l ) are obtained by dividing annual labor expenditures by the number of employees in the water supply division. The energy price (P e ) is obtained by dividing energy expenditures by energy consumption. Although energy consumption is monitored by the government, not all organizations release data to the public. Therefore, we estimate energy

10 220 F. Mizutani, T. Urakami consumption using the available data sets. The estimation method of energy consumption is explained in the next section. The material price (P m ) is obtained by dividing repair expenditures by fixed assets. The capital price (P k ) is defined as the sum of the depreciation rate and the interest rate on short-term bonds held by governments. The depreciation rate is obtained by dividing depreciation by fixed assets at the beginning of the fiscal year. For the network characteristic variables, the network length (N n ) is defined as the total main distributing pipe length after subtracting small pipes connected to each house, a figure also estimated by the available data set (see the next section). The utilization rate (N c ) is defined as the ratio of daily water delivery volume to designated daily volume of waterintake. Finally, there are two kinds of quality variables for output measures. The first measure is the purifier level (Z t ). The purifier level is obtained by dividing annual clear water volume by annual designated volume of water-intake. The second measure is the household ratio (Z f ), which is defined as the ratio of the residential water delivery to total water consumption. In addition to these variables, we set up two other variables such as non-dam ratio (Z ndm ) and nonunderground water ratio (Z nug ). The non-dam ratio is defined by dividing total water-intake, less water-intake by dam by total designated water-intake volume. The non-underground water ratio is defined by dividing total water-intake, less underground water-intake by total designated water-intake volume. The statistics of variables used here are summarized in Table Estimation of several missing explanatory variables Because not all variables are available from observations in the public domain, we must obtain the missing ones from other sources. In this study the main solution is to use regression results based on different data sets. The energy consumption needed for the energy price and the network length are not available in the original data set. Therefore, both the energy consumption and the network length are estimated before we estimate the cost models. The estimation results for these missing variables are summarized in Table 5. An energy price is not available for all observations so we must estimate this based upon available data sets. As we explained in the previous section, the energy price used here is obtained by dividing electricity expenditure by electricity consumption. Data for electricity expenditure are available for all observations, but electricity consumption is not, so we estimate electricity consumption by using different data sets. The methodology for the estimation of electricity consumption (E e ) is as follows. First, we estimate the electricity consumption model by using prefecturally aggregated data sets, which are for 47 prefectures, then estimate electricity consumption by substituting explanatory variables for each observation. After calculating the regressions, we find that electricity consumption is best explained by the capacity of the distributing reservoirs and water-towers (W c ). However, another problem arises, as 42 of 112 water supply organizations have

11 Identifying network density and scale economies 221 Table 4. Statistics on used variables a Variable Unit Mean Standard Mini- Maxi- Definition deviation mum mum C 1 million 9, , ,239.0 Sum of labor, (total yen energy, materials cost) and capital costs Q 1,000 m 3 66, ,132 2,954 1,746,210 Annual total (output) water delivery P l 1000 yen 7, , , ,718.1 Annual labor (wage) /employee expenditure per employee P e yen / kwh Annual energy (energy expenditure per price) unit of energy consumption P m , Annual repair (material expenditure per price) fixed assets P k Sum of the (capital price) depreciation rate and the interest rate of short-term bonds issued by government N n km ,229.9 Total main (network distributing pipe length) length after subtracting small pipes N c Ratio of daily (utilization water delivery ratio) volume to designated daily volume of water-intake Z tr Ratio of annual (purifier clear water volume level) to designated volume of water-intake Z hr Ratio of (household residential water ratio) delivery to total water consumption Z ndm Ratio of total (non-dam water-intake, index) less water-intake by dam, to total designated water-intake volume Z nug Ratio of total (non-under- water-intake, ground) less underground water-intake to designated water -intake volume a Energy consumption, used in order to obtain the energy price (P e), and the network length (N n), are estimated from a different data set. The details of the estimation procedures are explained in the next section.

12 222 F. Mizutani, T. Urakami Table 5. Estimation results of selected variables a,b Dependent variable Electricity Capacity of distributing Network length consumption reservoir and water-tower lne e W c lnn n Constant 6.459*** 11, *** (0.763) (8,955) (0.733) lnw c 0.914*** (0.059) Q d 1.434*** - (0.036) lnsa 0.834*** (0.100) Adjusted R Number of observations Type of observation Prefecturally Original water Prefecturally aggregated supply organization aggregated Note: a b The equations for estimation are as follows: (a) lne e = θ 0 + θ 1 lnw c (b) W c = θ 2 + θ 3 Q d (c) lnn n = θ 4 + θ 5 lns a where E e is electricity consumption, W c is capacity of distributing reservoir and water-tower, Q d is water distribution, N n is network (pipe) length, S a is supplied area. *** 1% significance, ** 5% significance, * 10% significance. no information on the capacity of the distributing reservoirs and water-towers. We estimate the capacity of the distributing reservoirs and water-towers for these 42 water supply organizations based on the regression results from the available data set (i.e., the other 70 water supply organizations).we find that the capacity of the distributing reservoirs and water-towers have a linear relationship with water distribution (Q d ). Finally, regarding the network length (N n ), defined as the main distributing pipe length, the methodology for the estimation is similar to the method for electricity consumption. We estimate the regression model of network length by using prefectural data sets, then estimate the network length by substituting explanatory variables for each water supply organization. Thus we find that the network length, defined as the main distributing pipe length, is well explained by the supplied area. As Table 5 shows, the goodness-of-fit in these models is rather high and most variables are significant at the 1% level. However, it could be argued that these estimated explanatory variables are incorrect. To test whether the estimated results are reasonable, we compare the estimation results with the actual values of available water supply organizations. If our estimation results are not much different from the actual values, then the methods used here are thought to be acceptable. Although the test results are very limited in number, we conclude that our methods are acceptable. For example, as for electricity consumption, in

13 Identifying network density and scale economies 223 Kobe city s case, the estimation results are rather good (i.e., 68.4% accurate) so that we decided to use this model to estimate electricity consumption to obtain the energy price. It is noteworthy that these results are best from both a rational and explanatory perspective, although we have performed many other regression analyses in order to supplement the explanatory variables. 5.3 Estimation results Estimation results of three different models are shown in Table 6. The estimation methods are the SUR (Seemingly Unrelated Regression) for all cost models with input share equations. The goodness-of-fit in the regression is acceptably high for all three models. The estimated models meet almost all of the required properties. Because we impose restrictions on the cost models, symmetry and homogeneity in input factor prices are satisfied. However, as for both the monotonicity and the concavity conditions (Diewert and Wales 1987), Model 1 satisfies the condition globally while Models 2 and 3 do so only locally. However, the first-order coefficients show the correct sign. Although we also estimated coefficients for many different kinds of cost models by changing variables, these are the best results. Some key first-order coefficients in these models such as output, input prices and network length show quite similar values among the three models. We also performed statistical tests for heteroskedasticity for these three models. Both results of the LM test and Ramsey s RESET test (e.g., Maddala 1988) show that we would not reject the hypothesis of homoskedasticity. 2 A likelihood ratio (LR) test 3 is also performed to test whether all results from these three models are different. The values of the log of likelihood function are individually for Model 1, for Model 2, and for Model 3. First, the null hypothesis that all the parameters of second-power and crossproduct variables are jointly equal to zero is tested. The estimated LR test statistic 2 First, an LM statistic and p-value (in parentheses) for these three models are obtained as follows: Model 1 LM = (0.851) Model 2 LM = (0.717) Model 3 LM = (0.715) Second, the results of Ramsey s RESET are as follows: Model 1 û = y y 3 adj. R 2 = ( ) ( ) ( ) Model 2 û = y y 3 adj. R 2 = ( ) ( ) ( ) Model 3 û = y y 3 adj. R 2 = ( ) ( ) ( ) where û : estimated residuals y : explained variable. Numbers in parentheses are estimated standard errors. 3 We performed an LR test based on Christensen and Greene (1976) and Greene (1993).

14 224 F. Mizutani, T. Urakami is and the value of the chi-squared statistic with K = 19 restrictions at the 1% level of significance is The null hypothesis is therefore rejected and Model 2 is preferred to Model 1. Second, we test the null hypothesis that all the parameters on the quality variables in the hedonic cost model are jointly equal to zero. The estimated LR test statistic is and the value of the chi-squared statistic with K = 4 at the 1% level of significance is Therefore, the null hypothesis that all parameters on the quality variables in the hedonic model are jointly equal to zero is rejected. As a result, Model 3 is preferable to Model 2 and we subsequently use Model 3 to investigate the size of the minimum average cost. 6 Scale economies and optimum size of firm 6.1 Measures of economies of scale and network density Based on the estimation results from using three kinds of cost models, we calculated the RTD and RTS according to the equations shown in Table 3. Table 7 reports the estimates of the degree of RTD and RTS for five different groups of water supply organizations by the size of water supply volumes. First, a large organization represents the largest 15 utilities in the sample, which provide more than 100 million m 3 of water supply. A medium large organization represents the largest 35 utilities in the sample, which provide more than 20 million m 3 of water supply. A medium small represents 29 utilities with more than 10 million m 3 of water supply and a small represents the other 33 utilities. Finally, an average water supply organization represents the sample mean. The results show that, first, there are economies of network density for all plant sizes. From our calculations, the RTD is about to for an average organization. Compared with both Takada and Shigeno (1998) and Kuwahara (1998), our results show much smaller values and are closer to Kim (1987) and Kim and Clark (1988). Presumably, the main reason why previous studies in Japanese water supply organizations show higher values for RTDs is because both Takada and Shigeno, and Kuwahara limited their samples to smaller organizations. Certainly, our result for a small organization is close to that of Kuwahara. We chose observations from different size organizations so that the RTD shows relatively smaller values. As an overall conclusion we believe that there are economies of network density. In so far as for the evaluation of economies of scale is concerned, the RTS measures the long-term effects of changes in output and network, assuming full capacity utilization. In our analysis here, full capacity utilization means that we assume the utilization ratio (N c ) should be 100%. Based on our calculation for an average organization, the values of the RTS are between and 0.921, indicating diseconomies of scale. Compared with the results of previous studies, our results (i.e., the existence of diseconomies of scale) resemble those of Kim (1987) and Kim and Clark (1987), but differ from Takada and Shigeno (1998). However, the degree of diseconomies of scale in our study is larger than

15 Identifying network density and scale economies 225 Table 6. Estimation results of cost models: coefficients and standard errors Parameter Model 1 Model 2 Model 3 Parameter Model 1 Model 2 Model 3 α ε mm (0.062) (0.053) (0.049) (0.008) (0.007) α Q ε mk (0.043) (0.054) (0.048) (0.008) (0.008) β l ε kk (0.009) (0.020) (0.019) (0.043) (0.047) β e ξ ln (0.005) (0.010) (0.017) (0.013) (0.012) β m ξ lc (0.014) (0.019) (0.017) (0.048) (0.047) β k ξ en (0.011) (0.019) (0.020) (0.007) (0.006) γ n ξ ec (0.064) (0.080) (0.072) (0.025) (0.024) γ c ξ mn (0.230) (0.261) (0.235) (0.014) (0.013) α QQ ξ mc (0.049) (0.044) (0.053) (0.050) δ Ql ξ kn (0.010) (0.009) (0.014) (0.014) δ Qe ξ kc (0.005) (0.004) (0.052) (0.052) δ Qm µ nn (0.010) (0.009) (0.099) (0.088) δ Qk µ nc (0.010) (0.010) (0.236) (0.224) δ Qn µ cc (0.061) (0.054) (1.113) (1.008) δ Qc η tr (0.171) (0.152) (0.055) ε ll η hr (0.042) (0.044) (0.075) ε le η ndam (0.011) (0.011) (0.059) ε lm η nund (0.008) (0.008) (0.011) ε lk Pseudo R (0.041) (0.044) ε ee Log of (0.006) (0.006) likelihood ε em function (0.004) (0.004) ε ek Number of (0.012) (0.012) observations

16 226 F. Mizutani, T. Urakami Table 7. Estimated results of return to network density and scale a Size of water Output Network Utilization Type of RTD RTS supply (1000 m 3 ) length (km) ratio (%) models Model Large 355,550 2, Model Model Model Average 66, Model Model Model Medium large 42, Model Model Model Medium small 15, Model Model Model Small 6, Model Model a (1) When the RTS is calculated, we assume that the utilization ratio should be 100%. in Kim (1987), and Kim and Clark (1987). One reason why our results show larger diseconomies of scale is the assumption of a utilization ratio of 100%, which causes the diseconomies to become larger. However, even after relaxing this assumption, our results still show diseconomies of scale; the diseconomies still hold for different sizes of organization. We conclude therefore that there are economies of network density, but no scale economies in Japanese water supply organizations. 6.2 Estimation of optimum size of firm In this section we try to estimate the optimal size of a water supply organization based on the estimated cost function. Optimal size means an organization size with minimum average cost. Since we specify a single output measure in the cost function, the optimal size in the cost structure is at the point with the least average cost (see for example, Baumol et al. 1982). We here use the hedonic cost model shown in Equation (5). The main aim in this section is to determine the size with minimum cost. Among several factors affecting cost structure, we focus on output (Q) and network length (N n ); that is, we find the minimum average cost point in terms of output and network length. From Equation (5) the average cost function can be obtained as follows: AC = C /Q = (1/Q)EXP[α 0 + α Q (lnq + Σ h η h lnz h )+Σ i β i lnp i + Σ k γ k lnn k + 1/2α QQ (lnq + Σ h η h lnz h ) 2 +1/2Σ i Σ j ε ij (lnp i )(lnp j )+

17 Identifying network density and scale economies 227 1/2Σ k Σ l µ kl (lnn k )(lnn l )+Σ i δ Qi (lnq + Σ h η h lnz h )(lnp i )+ Σ k δ Qk (lnq + Σ h η h lnz h )(lnn k )+Σ i Σ k ζ ik (lnp i )(lnn k )]. (10) To calculate the minimum average cost, we assume that other factors such as input factor prices and quality factors such as treatment level, are fixed at the sample mean point. From the first-order conditions for the minimum average cost, we can obtain the following results (see Appendix 1): α Q + α QQ lnq + δ Qn lnn n 1 = 0 (11) γ n + µ nn lnn n + δ Qn lnq =0. (12) By solving these equations and substituting values of these parameters from the results shown in Table 6, we obtain the result that the minimum average cost is at the point of 261,084 thousands m 3 of water supply volume (Q) and 1,221 km of network length (N n ). This number is the optimal size of a water supply organization in terms of average cost. However, from the policy maker s point of view, it is more useful to know the optimal size of the water consumption population, as this constitutes the most typical measure of the optimal organization s size. We therefore translate from our results and find that, in fact, according to our data set of water supply enterprises, there is a log-linear relationship between the water consumption population (V p ), water supply volume (Q), and network length (N n ). The regression result for this relationship shows the following reasonable outcome: lnv p = lnQ d lnN n (13) (0.249)(0.018) (0.026) adj.r 2 =0.981 where V p is water consumption population, Q d is water distribution (Q = Q d ), and N n is network (pipe) length. The numbers in parentheses are the estimated standard errors. We decide to obtain the optimal size of a water supply organization using Equation (13). Substituting Q = 261,084 thousands m 3 and N n = 1,221 km into Equation (13), we obtain the result that V p = 766,000. The optimal size (with regard to minimum average cost) in terms of the water-supplied population would thus be 766, Conclusion The primary purpose of this study has been to calculate the optimal size (with respect to minimum average cost) of a water supply organization and to reexamine whether there are economies of scale and economies of network density in the water supply industry. This matter is highly relevant in Japan, where many think there are too many water supply organizations and that they should be consolidated to attain scale advantages. We employed total cost models for the water supply industry. Our conclusions are summarized as follows:

18 228 F. Mizutani, T. Urakami (1) There are economies of network density at the sample mean. However, the magnitude of these economies is not large. (2) There are diseconomies of scale at the sample mean, but the scale disadvantage is not large. (3) The optimal size of the water supply agency, which attains the minimum average cost, is a size with output Q = 261,084 thousands m 3 and a network length of N n = 1,221 km. (4) For this output and network size, the optimal size of a water-supplied population is about 766,000. In March 1998 there were 1,962 main direct water suppliers. If our calculation is correct that the optimal size of a water delivery population is 766,000 then simple mathematics tells us that Japan, with a national population of 120 million, should have only about 157 water supply organizations. We can see from Table 1 that the delivery populations of 1,887 of Japan s water supply enterprises are smaller than 250,000. From a purely economic point of view, we may assert that these enterprises should be merged and reorganized. Justifiable human concerns often delay the realization of proper economic choices, but at some point in the future, a more efficient, more uniform, less complicated water supply system will evolve in Japan. We are hopeful that the normal time-consuming evolutionary process will not be prolonged by any inertia on the part of those who find acceptance of the status quo less strenuous than positive change. Appendix 1 The minimum average cost The total cost function of water supply organization is expressed as follows: LnC = α 0 + α Q lny + Σ i β i lnp i + Σ k γ k lnn k +1/2α QQ (lny ) 2 + (A.1) 1/2Σ i Σ j ε ij (lnp i )(lnp j )+1/2Σ k Σ l µ kl (lnn k )(lnn l )+ Σ i δ Qi (lny )(lnp i )+Σ k δ Qk (lny )(lnn k )+Σ i Σ k ζ ik (lnp i )(lnn k ) The service output measure is also expressed as follows: LnY =lnq + Σ h η h lnz h. (A.2) Substituting Equation (A.2) into Equation (A.1), the following equation is obtained: LnC = α 0 + α Q (lnq + Σ h η h lnz h )+Σ i β i lnp i + Σ k γ k lnn k + 1/2α QQ (lnq + Σ h η h lnz h ) 2 +1/2Σ i Σ j ε ij (lnp i )(lnp j )+ 1/2Σ k Σ l µ kl (lnn k )(lnn l )+Σ i δ Qi (lnq + Σ h η h lnz h )(lnp i )+ Σ k δ Qk (lnq + Σ h η h lnz h )(lnn k )+Σ i Σ k ζ ik (lnp i )(lnn k ). (A.3) By taking the antilogarithm of both sides of (A.3) and dividing the result by Q, we get the average cost function AC :

19 Identifying network density and scale economies 229 AC = (1/Q)EXP[α 0 + α Q (lnq + Σ h η h lnz h )+Σ i β i lnp i + Σ k γ k lnn k + 1/2α QQ (lnq + Σ h η h lnz h ) 2 +1/2Σ i Σ j ε ij (lnp i )(lnp j )+ 1/2Σ k Σ l µ kl (lnn k )(lnn l )+Σ i δ Qi (lnq + Σ h η h lnz h )(lnp i )+ Σ k δ Qk (lnq + Σ h η h lnz h )(lnn k )+Σ i Σ k ζ ik (lnp i )(lnn k )]. (A.4) From the average cost function, we find the point with the minimum average cost. The main focus of this analysis is to find the minimum average cost in terms of output measure, Q, and network length, N n. Differentiating the average cost function by both service output and network length, the following results can be obtained from the first order conditions for the minimum average cost. (AC )/ Q = (C /Q)/ Q = (1/Q 2 )[EXP(G)[α Q + α QQ (lnq + Σ h η h lnz h )+Σ i δ Qi lnp i + Σ k δ Qk lnn k ] EXP(G)] = 0 (A.5) (AC )/ N n = (C /Q)/ N n (A.6) = [1/(N n Q)]EXP(G)[γ n + Σ k µ kn lnn k + δ Qn lnq + Σ i ζ in lnp i ]=0 where G = α 0 + α Q (lnq + Σ h η h lnz h )+Σ i β i lnp i + Σ k γ k lnn k + 1/2α QQ (lnq + Σ h η h lnz h ) 2 +1/2Σ i Σ j ε ij (lnp i )(lnp j )+ 1/2Σ k Σ l µ kl (lnn k )(lnn l )+Σ i δ Qi (lnq + Σ h η h lnz h )(lnp i )+ Σ k δ Qk (lnq + Σ h η h lnz h )(lnn k )+Σ i Σ k ζ ik (lnp i )(lnn k ). Because EXP(G)/= 0, Q /= 0, N n /=0, we can obtain the following results from both Equations (A.5) and (A.6): α Q + α QQ (lnq + Σ h η h lnz h )+Σ i δ Qi lnp i + Σ k δ Qk lnn k 1 = 0 (A.7) γ n + Σ k µ kn lnn k + δ Qn lnq + Σ i ζ in lnp i =0. (A.8) Among several factors that affect cost structure, in this study we find the minimum average cost point in terms of output (Q) and network length (N n ). In order to calculate the minimum average cost, we assume that other factors, except for output and network length, are fixed at the sample mean point. Because each variable is divided by the sample mean, we can obtain the following results from Equations (A.7) and (A.8): α Q + α QQ lnq + δ Qn lnn n 1 = 0 (A.9) γ n + µ nn lnn n + δ Qn lnq =0. (A.10)

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