Abstract
This paper has two objectives. First, we identify a problem with the ability of the discrete-continuous choice framework and conditional demand functions to fully describe consumer preferences in the presence of kinked budget constraints. Second, we propose and illustrate an alternative, preference-based method for estimating consumer responses to price changes under these conditions. This framework recognizes that commitments to commodities such as pools or outdoor landscaping influence how water consumption responds to price changes as part of the long-run consumption adjustments. With microdata on household choices, this approach could be used to design increasing block rate structures that recognize the differences in water consumption responses with these commitments. (JEL Q25)
I. Introduction
This paper has two objectives. First, we identify a general problem in using the discrete-continuous choice (DCC) framework and conditional demand functions for applied welfare analysis with changes in kinked budget constraints. To illustrate the relevance of our argument, we consider the estimation issues that arise for the case of models for residential water demand with increasing block rate structures. Second, we propose an alternative, preference-based method for estimating consumer responses under these conditions. By specifying a direct utility function and estimating its parameters, our approach can evaluate policies that alter all the attributes of a block pricing structure. For the case of residential water demand, in many areas of the United States, especially in the arid Southwest, this type of change is being discussed as a component of increased incentives for conservation.1 At a national level, it is also part of the U.S. Environmental Protection Agency’s (EPA’s) policy known as the “Four Pillars of Sustainable Infrastructure.”2
Conventional models for commodity demand (or labor supply) in the presence of piecewise linear budget constraints use a parametric specification for the conditional demand (labor supply) function to describe the selection of a “best facet” on the budget constraint and with it, an amount of the commodity demanded (or labor supplied). The choice is decomposed into the discrete and continuous components. This strategy has been widely adopted in describing labor supply responses including differences in participation status (i.e., working, retired, etc.), and the extent of participation, part- time versus full-time. It has also been an important component of modeling the strategies used to consistently estimate electricity and water demands in the presence of block rate pricing structures.
By contrast, our proposed method uses a different maintained assumption. It specifies household preferences using a direct utility function. As a result, there are also assumptions that underlie the model’s results. One of the important questions arises from identification. That is, given a preference specification, a key issue concerns whether we can identify and estimate the structural parameters using the first-order conditions from the constrained choice problem. The limited price variation usually encountered in applying the model to households in a single region can confound the ability to observe quantity adjustments.
We illustrate the logic for our method with an application to the residential water demand for the Phoenix, Arizona, metropolitan area. Our empirical model uses monthly consumption data for 2005 for the “average” household (in a single-family dwelling) in the service areas of each of 43 water service providers in the Phoenix Active Management Area (PAMA) of Arizona. By intersecting the GIS descriptions of the water providers’ service areas with the GIS records for the 2000 U. S. Census at the block group level and the Maricopa County (the county within the PAMA) parcel records, we construct a set of economic, residential property, and demographic variables to serve as instruments. Our empirical analysis is intended to offer a proof of concept. Ideally, the model would be estimated with household-level data and provide a basis for welfare analysis of alternative rate structure designs to promote long-run water conservation. Demand information with sufficient variation in price schedules and detail about pools, landscaping, and other water-intensive consumption choices is scarce. As a result, we evaluate the model in this paper using a difficult standard: average consumption responses to varied price schedules. These data have the variation in price schedule but provide only the average residential customer response. Thus, we lose the microlevel variation that might be expected to be important in identifying the key preference parameters. To gauge the relative performance of the model in this setting we compare our results with the conventional DCC framework applied to the same sample. These estimates yield insignificant income coefficients and smaller (in absolute magnitude) conditional price elasticity estimates.
The point estimate for price elasticity based on our model using a direct utility function is close to that in the literature that uses the DCC framework. This is at least partially due to the way the elasticity is computed.3 We use a conventional definition for the price elasticity implied by this preference function as if choices were made facing a constant marginal price. This strategy is adopted because our interest is in the comparability of our measures of local price responsiveness to those from the DCC framework. An important motivation for the utility function approach would be in considering the nonmarginal price changes associated with designing rate structures. With the preference-based approach, we estimate preference parameters directly and could measure welfare impacts of largescale changes in the price schedule. To be relevant, however, these analyses would require microdata on household responses and must await availability of these data sources.
Section II describes the main elements of our argument. It begins by describing the DCC framework. After that, we explain one of the issues motivating this paper—an apparent contradiction between the Hausman (1979) interpretation of what is possible within the DCC framework and the Bockstael-McConnell (1981) conclusion about the difficulties facing welfare measurement with nonlinear budget constraints. With that background, we then summarize how Reiss and White (2006) explain the appropriate strategy for welfare measurement with nonlinear budget constraints.4 Within that context, we describe the problems with using the DCC logic to evaluate large changes in a block pricing structure for resources whose demands are largely determined by the utilization of complementary goods. Section III outlines the estimation strategy. Section IV describes the data sources, and Section V presents our results. The last section comments on the potential role measures of the responsiveness of water demand to major revisions in the price schedule can play in the design of policies intended to be part of developing sustainable infrastructure.5
II. Background and Economic Consistency
Context
Water demand is one of the prominent applications for models explaining consumer choice in the presence of kinked budget constraints. Hewitt and Hanemann’s (1995) paper is the first, and arguably the most comprehensive, application of the DCC framework to water demand. Their analysis follows the general recommendations of Hausman (1979) and Moffit (1986) for these situations. A conditional demand is specified. In their case, it is a constant elasticity function. The price and virtual income for the conditional demand are defined by the first-stage selection of an optimal budget segment given the block pricing structure. Two errors are included in the specification. One represents optimization error and the second captures unobserved heterogeneity. Given assumed distributions for these errors (usually independent normality), observed water use, and knowledge of the specific features of the pricing structure faced by households, the framework defines the likelihood function for a sample of households’ water consumption. Variation in the pricing structures, the maintained distributions for the errors, and the parametric specification for conditional demand function allows statistically consistent estimates to be developed.
The DCC model characterizes consumer preferences with the specification of a conditional demand function. The same functional form and parameterization is used to describe all of the segments of an increasing block rate structure. This restriction assures that the parameters can be identified. To illustrate our argument here, we selected the simplest of the popular specifications, a linear conditional demand:
[1]Water is assumed to be priced with an increasing block rate structure with the nonlinear price for water defined as c(w) and the form of the relationship given as
[2]where Z is a vector of sociodemographic and household-specific attributes, pi is the marginal price of the ith block, and mi is the virtual income of the ith block. The virtual income of the second block, m2, is defined by equation [3]:
[3]Following Hausman (1979), the quasiindirect utility function implied by this conditional demand specification is derived by Roy’s identity. The first step links the specified conditional Marshallian demand from equation [1] to the relationship implied by Roy’s identity in equation [4]. This sequence describes how the differential equation given on the right-hand side of the equation is related to the empirical demand function:
[4]Solving this differential equation yields equation [5], where c is the constant of integration:
[5]As a rule, initial conditions help to relate c to the other factors that may influence the quantity demanded and also serve to provide the utility index. For our purposes, without loss of generality, we can write the expression for the indirect utility function implied by this demand using6
[6]A part of Hausman’s argument for the model noted that the log-likelihood function can be defined from the conditional demand functions. Thus, it is possible to recover parameter estimates for [6] using only the conditional demand given in [1]. This conclusion relies on the restriction of a common conditional demand specification for all facets of the budget constraint. Moreover, it restricts the demand parameters to be equal across all blocks. The estimator resembles one for interval-censored data, adjusted to reflect the effect of block choice on the implied virtual income (i.e., actual income adjusted by the difference between what would be implied if all units consumed were priced at the marginal price implied by the selected price block and what each household actually pays, as we have described in equation [3]). What is important about this logic is that it would appear that the estimated parameters of the conditional demand function contain sufficient information for welfare measurement when they are used in the scheme outlined by Reiss and White. However, this suggestion overlooks the importance of the assumption that conditional demand functions “look just like” unconditional demand functions aside from the adjustment to the income term. We can investigate the importance by examining the Reiss-White proposal in more detail.
Welfare Analysis with the DCC Model
A key conclusion of the earlier work discussing problems with applied welfare analysis in the presence of nonlinear budget constraints is that with a nonlinear budget constraint, the Marshallian “prices as parameters” demand function (or labor supply function) does not exist. Bockstael and McConnell (1983) made this point in discussing the implications of the household production function (HPF) framework for applied welfare analysis. Reiss and White (2006) might seem to have proposed a resolution to the problem. However, their derivation of welfare measures with nonlinear prices relies on the ability to assume the existence of what appears to be an ordinary Marshallian demand function that can be integrated back to quasi-expenditure function. The resulting compensated demand is then used with the marginal price function to develop the welfare measures. This marginal price function helps in describing the path of price change.
Reconciling the two disparate arguments requires understanding the role for the assumed properties of the Marshallian demands defined for each value of the marginal price. Marginal prices can be continuous in the household production case considered by Bockstael and McConnell, or they can change discretely for the kinked budget constraint due to block rate pricing. It is this relationship that is giving rise to the nonlinearity in the budget constraint.7 As Bockstael and McConnell demonstrate in their note 1 (p. 810), with a general nonlinear budget constraint, the features of preferences and nature of the constraint are scrambled. In their example, each of the marginal cost functions will imply a different slope for the Marshallian demand function. Reiss and White’s argument requires that there exists a single Marshallian demand that describes how quantity demand changes for each marginal price.8 This is essentially the same as what is being assumed in the DCC framework. We must assume preferences are consistent with an identical conditional demand for all price facets of the kinked budget constraint.
This requirement stands in contrast with conclusions from the literature on the econometrics of kinked budget constraints that appear much more general in their scope. For example, Hausman (1979) has argued that demand (or supply) functions can be estimated without instrumental variables or a weighted average of prices. Roy’s identity, together with the convexity of the budget set, assures that the choice of segment and the quantity demanded can be based solely on the “demand functions.”9
The DCC framework describes the unconditional demand in a two-step framework. A conditional demand function is used to describe the choice of the “best” price segment on the kinked budget constraint as well as the quantity demanded, given the marginal price and virtual income implied by the first selection. The framework uses the logic implied by analytical (or numerical) solution to the differential equation defined by Roy’s identity to compute the correct response in quantity demanded to a movement along one segment of the faceted budget constraint. However, as Reiss and White make clear, this result requires the assumption that a common (across facets) Marshallian conditional demand exists and describes demand responses correctly for all the marginal prices.
This logic has been applied in modeling water and electricity demands. In these cases, the demand is also conditional to another set of choices that is usually assumed to be made prior to the utilization decisions giving rise to water (or, in a similar context, electricity) usage patterns. As a rule, these cases involve equipment or durables associated with residence. Once we acknowledge that large changes in water price structures would influence these types of decisions then we must consider whether the envelope condition associated with reevaluating commitments contains sufficient information to derive the associated long-run demand response. In general, we cannot be assured it will.
To the extent the nature of the price schedule affects decisions to install pools or adopt different types of residential landscapes, those effects cannot be detected with the estimates derived from the DCC framework. The DCC logic assumes the responses to changes in the implied marginal price along with the inframarginal expenditure adjustment is all that is needed to “predict” the optimal facet for water (or electricity). One realization of this assumption arises with the model requiring that the form of the conditional demand function and the values for its unknown parameters are constant across all facets of the budget constraint.
When we wish to redesign the price schedule in a general setting that assumes commitments to complementary goods might change, we do not have enough information to compute the marginal price and expenditure adjustments. These effects are important because it is often the case that policy makers are selecting higher-priced blocks to discourage investments in high-consumption complementary goods and, therefore, the associated higher water consumption.
Reiss and White (2006) demonstrate that exact welfare measures can be derived in nonlinear pricing situations. They propose a four-step logic:
Estimate ordinary demand functions relating quantity to the relevant marginal prices at the observe levels of the quantity demanded.10
Use these demand functions from Step 1 to recover the quasi-expenditure function analytically (Hausman 1978) or numerically (Vartia 1983), and derive the corresponding Hicksian demand (as our example illustrated).
At the observed quantity demanded, this expression for the Hicksian demand evaluated at an unknown constant marginal price and utility level should equal the inverse of the marginal price schedule also evaluated at this price.
The initial level of utility realized at the initial nonlinear price schedule should be consistent with the total expenditures implied by the price schedule and the virtual income (i.e.. Hicksian expenditures evaluated at the initial marginal price and utility plus the income adjustment defined earlier).11
The equations defined by Steps 3 and 4 provide two relationships and two unknowns: the initial marginal price consistent with the utility level realized and the level for the initial utility. With these two variables, they demonstrate compensated demand defined in terms of an artificial constant marginal price can be used. A price schedule change is then recovered with the initial price and utility as well as the new price. Exact welfare is the integral under the compensated demand between the two marginal prices corrected by the change in the adjustment to income.
Reiss and White observe a big advantage of their logic is that “a researcher can begin with an empirically satisfactory model for the ordinary demand function . . . and then compute exact consumer surplus under nonlinear prices without having to recover the direct utility function” (Reiss and White 2006, 12). The problem with this logic is that it ignores the effects of changes in these conditional demands when the modifications in the price schedule are large enough to adjust quasi-fixed goods, as we illustrate in the next section.
Complementary Goods
Suppose that Z in equation [4] is a measure of the swimming pool or landscape structure that a household has committed to. It is reasonable to assume there are complementarities between either of these goods and water demand. The compensated demand response implied by equation [5] will consistently describe household responses to changes in the marginal price only if the amount of Z is consistent with equality of the Marshallian virtual price for Z and the market price. Thus, we should expect
[7]with VZ and Vm the partial derivatives of equation [6]. Using the quasi-indirect utility function (equation [6]) this envelope condition means γ=dpZ. Substituting, we can write Roy’s identity for long-run adjustments as
[8]Suppose, however, the virtual price and market price cannot be assumed to be equal. In this case, the conditional demand specification for water is not enough information to describe how water demand would adjust in the long run, as there are opportunities to modify these quasi-fixed commitments. We need some basis for describing how Z would adjust to changes in p and pZ.12 That is, to recover the full adjustment to changes in p through the conditional demand, we are implicitly assuming the response implied by the envelope condition fully characterizes Z’s adjustment. Solving Roy’s identity uses changes in the water demand and p as well as initial conditions. It tells us nothing about how Z would adjust to nonlocal changes in p.
To illustrate, suppose we assumed the demand for Z was also linear and adjusted in the long run.13 Yet the demand given in equation [1] is a short-run demand for water. Assume equation [9] describes the long-run demand for Z. Using the envelope condition [7] and [9] we can illustrate why the Reiss-White procedure will not reflect the complementarities between w and Z:
[9]Substituting into equation [8] for Z and rearranging terms, we have
[10]This result illustrates how the long-run demand for w would imply that the marginal effect of p on water demand depends on pZ (i.e., the first term in equation [10]).
We could, as sometimes proposed with another situation involving a nonlinear budget constraint, assume that the nonlinearity of the conditional demand is sufficient to identify a role for Z’s adjustment.14 However, what is estimated will be conditional to the relationship between Z and pZ in the envelope condition.
The DCC model decomposes water demand into two steps, the choice of a best segment and then, conditional on that selection, a demand response to changes in that segment’s marginal price, given the adjustment to income required by the increasing block rate structure. In principle, goods that are complements for water, such as swimming pools or landscape systems, have demands that depend on the full price schedule for the water that is utilized with them. Once these commitments are made, they are quasi-fixed goods that influence the parameters of the Marshallian demand for water, as our simple example illustrates. Large changes in the price schedule are likely to change these commitments. They cannot be identified from the short-run responses in the water usage alone. We illustrated this point in very simple terms by using the envelope condition together with an assumed demand function for one complementary good. The bottom line is we do not escape making additional assumptions. The single conditional demand function is not sufficient.
Arrufat and Zabalza (1986) identify a related issue in describing advantages of their use of a Consumer Expenditure Survey (CES) direct utility function to describe labor supply choices. They note:
There are two reasons why, even when dealing with convex budget sets, it may be convenient to estimate a specific utility index. First, we may want to use our estimates to predict the effects of introducing non-convexities in the budget constraint . . . unless we have at our disposal an explicit utility index, in either direct or indirect form, we will not be able to predict behavioral responses to these types of changes in the opportunity set. Second, and less obvious, if there exist optimization errors . . . the conventional measures of welfare change cannot be inferred from areas under market supply curves (even if compensated), since neither initial nor final positions will correspond to tangency positions. (p. 48, emphasis added)
This discussion recognizes that the estimates from the demand model approach do not permit analysis of large-scale reform in pricing (or tax) structures.
III. Model
Our alternative model begins with a specification for the direct utility function. We believe that this strategy offers a more transparent basis for characterizing relationships between water and the complementary goods that influence how water is used. This approach relies on a composite of structural restrictions as well as the need to observe sufficient variation in the price schedules for water along with household conditions to identify and estimate the parameters of that preference function.
As a practical matter, commodities such as water, electricity, and other utilities are associated with a small fraction of a budget. Small price changes in any one of them are unlikely to induce large reallocation of income among all goods. This feature of demand can compound the difficulties in estimating household demand. The effects of small price changes may primarily induce reallocation of the expenditures on household utilities. Recognizing this prospect for estimation of the preference parameters using data with limited variation in the price structures, we assume utilities enter preferences in a separable subfunction and treat their expenditures as a fraction of income that varies with income. Our estimates for this share are based on the CES average for utilities (including water). In 2006, the CES identified 13 different income classes (see Bureau of Labor Statistics 2007).15
The preference specification describing the choice of utilities corresponds to the von Haefen, Phaneuf, and Parsons (2004) direct utility function and is given by16
[11]If we assume the expenditure on this component of a household’s budget is m = sy (with s equal to a value based on household income), then we can embed the budget constraint into the utility function. As in
[12]where m−c(w) corresponds to a household’s virtual expenditures on the other components of utilities and will depend on the block rate structure faced by each household (see Hewitt and Hanemann 1995 for a detailed discussion). Z is a matrix of sociodemographic variables, q is a matrix of water-specific attributes such as lot size and presence of a pool, θ is a demand shift parameter (e.g., similar to a threshold consumption in a Stone-Geary specification) that also contains the identification of summer versus winter time span, and ρω and ρx are substitution parameters.17 The complementary goods (treated as quasifixed commitments) enter the utility function as part of a scaling function. This is consistent with the utilization logic described earlier. By specifying the direct utility function, we include the equivalent of the information that would be needed in the set of long-run demands for complementary goods. The expression, exp (qβ)w+θ, can be considered as a quality adjusted, or a quantity augmentation based on prior commitments, measure for the use of water. The first of these interpretations is similar to that of von Haefen, Phaneuf, and Parsons (2004) in a recreation application.
Further intuition for these parameters can be offered by recognizing that this function is a variation on Mukerji’s (1963) constant ratio of elasticity of substitution function. The Hanemann–Von Haefen et al. function can be derived using a monotonic transformation of the Mukerji generalization of the CES function. This recasting of the model allows Smith’s (1974) derivation of the price and income elasticities implied by this function to be used to describe local responsiveness within budget segments and thus to compare the estimates implied by the function with what has been derived using conditional demand functions within a DCC framework.18
The estimating equation for the model’s parameters uses the first-order condition for the local optimization component of the household’s budget:
[13]It is able to include the implications of the kinked budget constraint by defining mi as the virtual available income for utilities in block i and pi the relevant marginal price. The expression is conditional on the block selected. Water demand, price, and expenditures on remaining utilities (i.e., mi−piw) are all endogenous variables. Taking the natural logarithms of [13] and rearranging terms yields
[14]where ε can be assumed to correspond to either optimization error or unobserved household heterogeneity.
Our analysis does not confront the issues associated with choice at the kinks of the budget constraint versus tangencies to a facet. As we discuss below, the use of consumption for an “average” household implies that all choices can be interpreted as tangencies with optimization errors that are on average zero. Our framework uses GIS methods to intersect water service provider service areas with U. S. Census block areas for Maricopa County as well as with the parcel records for residential properties. These variables characterize water providers’ customers. They are used to define instruments for the quantity of water demanded and virtual expenditures.
IV. Data
Background
Ideally, our model (and the DCC specification) would be applied with household-level data. These data are generally not in the public domain. When studies have had access to microdata, they often span wide geographic domains with quite different patterns of water use or they have very limited price variation. Olmstead, Hanemann, and Stavins’s (2007) work is an example of the first situation, and Pint’s (1999) the second.
We illustrate our model and compare it to the DCC framework using data that are consistently in the public domain. These data are based on average household water usage for residential customers in each of a set of water providers. We exploit the variation in price schedules and construct the bill that would have resulted for a household with this water usage for each provider. Thus, our example considers a potentially important issue that is secondary to the main objective of the paper. That is, we investigate whether differences in modeling assumptions between a preference- based formulation and the DCC model can be detected with relatively coarse aggregate data.
Data Sources
The Phoenix Active Management Area is shown in Figure 1 and corresponds to approximately the northern two-thirds of Maricopa County. It encompasses all of what would be considered the Phoenix metropolitan area. The data for our application were collected from a variety of sources. The 2005 price schedule for each water service provider is reported by the Arizona Water Infrastructure Financing Authority. Table 1 summarizes the features of these price schedules. The first panel highlights the variation in the price schedule across providers. Each element in the table displays the range of marginal prices (per thousand gallons) for each block. The second panel provides the average width of the blocks across the uniform two-, three-, and four-block schedules. Each is measured in gallons.
Water Service Providers in the Phoenix Active Management Area
Summary of Price Schedules in the Phoenix Active Management Area
Information on aggregate single-family consumption in 2005 and number of single-family customers was obtained from imaged records of the Arizona Department of Water Resources’ Schedule F, available for monthly water consumption by residence type. These data allow construction of water used by an “average” single-family residential consumer. Based on that monthly water consumption and each provider’s price schedule we can recover measures for their marginal price and expenditures on water. These two datasets provide 516 provider/month observations for water consumption and price schedule information.
Census data for 2000 were adapted to conform to the service areas of each of the providers to construct measures of expenditures on utilities as well as other economic and demographic variables characterizing their customers. This was accomplished by intersecting the census block groups with the water service provider areas.19 Average household income together with the CES schedule of the proportions of income spent on utilities by income class was used to construct the total expenditures on utilities for each area. The parcel records from the Maricopa County Assessor were used to develop summary statistics for housing measures at the census block group level and ultimately intersected with water service provider areas. This allows measurement of the percentage of households with pools located in a given service provider area. To control for climatic conditions, monthly data on cooling degree days with a 65 °F baseline and total precipitation during the month were obtained from the National Climatic Data Center of NOAA for the Phoenix Sky Harbor Airport station.
There are approximately 60 water service providers that serve the Phoenix metropolitan area. Of these, 43 have complete records for residential customers in single-family dwellings. The first panel of Table 2 provides some summary statistics for the demographic characteristics for the residential customers across providers. The second panel in the table summarizes the variation in temperature and precipitation across the months in our sample. The majority of the water service providers that we eliminated are simply irrigation districts that may provide water to a few households, usually at a fixed cost.20
Summary Statistics for Sociodemographic and Weather Variables
As with most water studies, such as those by Hewitt and Hanemann (1995) and Olmstead, Hanemann, and Stavins (2007), we are exploiting structural restrictions inherent in our model as well as the variation across water service providers in price schedules, sociodemographic, and housing characteristics to identify the model’s parameters. All of these water service providers are in the Phoenix metropolitan area. There is no variation in linked weather conditions across water service providers, because they are derived from a single weather station. Weather-related variables vary with the month of consumption.
V. Results
We present the estimates for the DCC model as well as our proposed alternative framework. The findings for our model are presented first. Equation [14] implies that our model explains variation in marginal prices based on water consumption and the net expenditures for other utilities excluding spending on water as well as other exogenous variables describing variations in households’ circumstances and weather conditions. Water consumption and the expenditures measure are jointly determined with choices that lead to the marginal prices by virtue of water’s block rate pricing structure. As a result, we need to consider selecting instruments for these variables. Fortunately our application uses summaries of household-level information based on the geographic scale of the service areas of water providers. As a result it offers some advantages in resolving this issue. We construct summaries of the households’ characteristics in each water provider’s service area using the shape files for these areas together with census data. These summary statistics are most likely to satisfy the criteria for instruments in that we would expect them to be correlated with the summaries of water consumption and the income measure that underlies our utilities expenditure variable at the water provider level, but uncorrelated with the marginal price measure.21 This price is defined once a representative household’s water consumption in each provider area is placed in the relevant price schedule for that provider.
To illustrate the potential importance of our instruments for the model, we report two alternative sets of instruments with our model. The first, and our preferred approach, develops instruments for the log of the average household’s expenditures on other utilities (taking account of the implications of the block pricing structure for the virtual income available) and the average water consumption by month using the predictions from first-stage regression models that include temperature degree days, total monthly precipitation, the percentage of houses with pools, the average house value (as assessed by homeowners in the census), total number of lots in the service area, and number of rental units. A simple model with fixed effects for each water service provider together with the weather variables is the second. Unfortunately, the later cannot distinguish the effects of the variation in the socioeconomic characteristics of residential customers by providers’ service areas and these providers’ price schedules. The first-stage estimates for the first of these estimators are given in Table 3.
First-Stage Regression of Expenditures and Water Consumption
The estimates for the preference model parameters using each of the two sets of instruments are given in Table 4. Both models specify the Z matrix as composed of a vector of ones (for an intercept), our temperature measure, and the number of customers the water service provider serves. q contains the percentage of pools in the service provider area; and θ contains a constant and dummy variables identifying whether the monthly consumption was during the summer and winter periods.22 The shoulder period between summer and winter is the omitted category. We estimate equation [14] using nonlinear least squares with each set of instruments.
Estimates of Utilities Subfunction
The estimates based on the demographic, economic, and weather variable instruments yield more precise results for the model. With instruments based on the provider fixed effects we find the model attributes all the price variation to the provider effects. This is important because differences in the price schedules along with the variation in the average household’s water consumption in each provider’s residential customer class provide the variation in marginal prices. With provider fixed effects used to define instruments, it appears that the variation in the price schedule is being captured completely by these variables. As a result, it is not surprising that the parameter estimates are not significant. A Hausman (1978) specification test decisively rejects the fixed-effects specification (χ2 = 115.83 with p = 0.00).23
To evaluate these estimates further, we computed the price and “utilities expenditures” elasticities at the sample mean for the price of water and for the virtual utilities expenditures. These results are given in Panel A of Table 5. Below each estimate we report the Z-statistic based on the asymptotic standard errors. In addition, we computed the elasticity for each month and water provider. A summary of these results is given in Panel B, with the mean, median, and the minimum and maximum values across the providers in the summer months.
Estimated Price and “Income” Elasticities
There are several caveats in interpreting these estimates. These measures are not the estimates that would be implied with a change in the structure of the increasing block rate structure. They apply to local changes. Our approach recovers estimates of all the preference parameters that would be required to derive a response to a large change in the price structure. Indeed, this is a key motivation for the model. We report these local price and expenditure elasticities because they are comparable to most of the literature and are more comparable to the elasticities implied by the conditional demands estimated with the DCC framework.24 In our case, measures of the demand responses to large changes require computation of the solutions to the constrained optimization problems associated with a baseline price structure and the proposed change.
Our estimates are also conditioned by the maintained separability assumption we discussed at the outset. This restriction influences how the estimates can be compared with findings from the existing literature. For the income elasticity, the relationship is straightforward, provided we interpret the response to an income change as a marginal change to a preexisting optimal water consumption choice. σwy=σwsσsy, with σwy the conventional income elasticity of demand; σws the utilities expenditure elasticity (i.e., what we estimated); and σsy the elasticity of the composite of utilities expenditures with respect to household income. Using estimates for σsy from the literature, we can adapt the results in Table 4 to develop a measure of the income elasticity implied by our findings. For example, Blanciforti and Green (1983) report an estimate of σsy = 0.62.25 This implies our estimate for the income elasticity of demand for water using the measures for σws of 0.63 (i.e., 0.62 * 1.02).
The price elasticities also require some adjustment (σwp=ηwp−σwsσmp), where ηwp is our estimate for the price elasticity, holding the utilities expenditures constant, and σmp is the elasticity of these expenditures with respect to the price of water, holding income constant. Since we are expressing the price elasticities in absolute magnitudes, these expressions imply the comparable unconditional price elasticity would be smaller than our estimates.26
As noted at the outset, we estimated the DCC model using a constant elasticity specification of the conditional demand function to compare our results. We follow Hewitt and Hanemann (1995), Waldman (2000), and Olmstead, Hanemann, and Stavins (2007) in our structuring of the model. Table 6 shows two sets of parameter estimates for DCC model, using the expenditure on utilities as the income measure in the first column and average income in the second. Several aspects of the results should be noted. First, applications of the DCC model to date have used microdata. Our finding that the variance of the optimization error is larger than the unobserved heterogeneity (ση>σε) are likely due to the structure of our data. Second, the price elasticity for the conditional demand elasticity is comparable in magnitude to our estimate of the local elasticity using the preference model. However, there is a striking difference in the income elasticity estimates. Under the DCC model, there is an insignificant income effect. By contrast, our estimates of the income elasticity using the preference function are close to unity.
Discrete-Continuous Choice Model Estimates
Comparisons of the local price elasticity measures (in Table 5) from our preference model compared to the conditional elasticity estimates with the DCC model indicate that they are quite consistent. While a statistical test for the difference between these estimates is not possible, their relative proximity suggest, for practical purposes, they would not imply differences in local responsiveness to small changes in price. This is not surprising since the two models are describing average responsiveness around the same marginal prices. However this comparison misses the point of the use a preference-based model. The effects of a large change in the rate structure for water could not be evaluated using the DCC estimates for price elasticity. This model can describe movements within a preexisting price schedule. It does not permit analysis of changes that would call into question the equality of conditional demand across the price facets of the budget constraint. Our model would allow evaluation of such changes because it identifies estimates of the preference parameters. With microdata and greater detail on swimming pools, landscape infrastructure, and variables describing other large water-using capital decisions, we could relax the assumption that the share of expenditures on utilities varies with income and expand the range of structural parameters estimated with the framework.
VI. Summary and Implications
This paper argues that the conventional approach for analyzing commodity demand in the presence of kinked budget constraints cannot evaluate the implications of large changes in the pricing structure for water. Moreover, it relies on the assumption of a constant conditional demand function across different marginal price segments in an increasing block rate structure. We proposed an alternative that uses a preference specification as the primitive in estimation. It allows us to consider welfare changes to large-scale changes in the price structure. The data available for our application, while comparable to the most common information available for characterizing water demand, describes the average response rather than the individual household response. Aggregate data for all residential customers served by 43 water service providers in the Phoenix area are used to estimate our preference-based model and a version of the DCC model.
An application to the average residential customer based on aggregate data is of interest because these data are readily available for most water demand applications. To date, the DCC model has been exclusively associated with applications with access to extensive household-level water consumption data as well as sufficient price variation to recover demand responsiveness. When these data do not exist, policy analysts are forced to adapt existing (and often old) estimates to describe how households’ water demand responds to changes in the block rate pricing structure.
Our strategy exploits the economic and demographic diversity of households along with GIS techniques to characterize the demographic and economic features of the spatially delineated service areas for the water providers included in our sample. In addition, rather than derive an estimator that uses the preference function to predict the facet of the budget constraint selected and then the amount of the commodity demanded, we focus on developing instruments for the choice variables and estimating parameters of the separable subfunction of preferences. With detailed micro-records, our framework could be extended using the two-error framework and logic comparable to Zabalza’s work to develop a maximum likelihood estimator. These are the types of estimates needed to address the challenge of designing price structures that encourage water conservation by reflecting the full costs of complementary, water-using, capital goods.
The next step in evaluating this strategy for modeling water demand calls for applying the method to a detailed household-level data set. Our example provides motivation for taking this step by demonstrating its feasibility and consistency in measuring the local responsiveness water demand to small price changes. To our knowledge, there do not appear to have been welfare analyses of reforms in water pricing that would be consistent with promoting conservation while maintaining minimum levels of assured access to all. It seems reasonable to expect that large changes in water pricing would promote different types of landscaping decisions as well as other long-run household choices for other capital goods such as pools and, in arid areas, gray water systems for recapture of wastewater. A preference-based model is a more appropriate basis for integrating decisions that alter the amount and mix of water-using capital goods along with the monthly rates of use of water they would require.
Footnotes
The authors are, respectively, assistant professor, Department of Economics and Finance, University of Wyoming; and W. P. Carey Professor of Economics, Arizona State University. Partial support for this research was provided by ASU’s Decision Center for a Desert City. Thanks are due to Josh Abbott, Rob Williams, Chris Goemans, Jay Shimshack, and the participants of the University of Colorado Workshop in Environmental and Resource Economics; to two anonymous referees for constructive comments; to Eric Moore and Jonathan Eyer for excellent research assistance; and to Michael Tschudi for developing the GIS linkages between census and PAMA water provider service areas.
↵1 In June 2007, U.S. News and World Report profiled the global water problem. There continue to be professional and popular assessments raising concern about water management and availability (i.e., The Economist, December 8, 2007, and National Geographic, February 2008), as well as renewed interest in economic research (see Olmstead, Hanemann, and Stavins 2007; Mansur and Olmstead 2006; Hanak and Chen 2007).
↵2 The EPA Assistant Administrator for Water, Benjamin Grumbles, in a 2006 briefing on EPA’s water policies, said that current water prices signal it is available and cheap. This statement has been echoed in most of the popular assessments.
↵3 The initial area of application of the DCC framework was in modeling labor supply. It is therefore natural to consider whether preference-based and conditional supply models have yielded similar results as with our demand models. This is a more difficult judgment to make than might initially be expected, because to our knowledge no one has applied the two to the same data set. There is wide variation in the estimates for labor supply elasticities. For example, in another context the second author reported estimates for the Marshallian supply elasticity varying from 0.002 to 0.21, without considering preference-based versus DCC comparisons (see Smith, Pattanayak, and Van Houtven 2003). For comparison, the original Burtless and Hausman (1978) study presenting the DCC framework found a labor supply elasticity of essentially zero (i.e., 0.00003). Their estimated income elasticity was −0.0477. Chetty (2006) reports a range of Hicksian labor supply elasticities in the literature from 0.088 to 1.040. Combining these with the corresponding income elasticities (assuming nonwage income is a third of total income) yields a wide range of estimates for the Marshallian supply response from a variety of labor supply models. Finally Zabalza (1983), using a preference-based approach with a CES utility function for women’s labor supply, reports estimates comparability in his labor supply estimates to conventional estimates (see his discussion on pp. 326–27).
↵4 To our knowledge, none of the water demand models have attempted to measure the consumer surplus associated with price changes. Most of their focus has been on correctly measuring unconditional price elasticities in the presence of kinked budget constraints.
↵5 For example, see Hanak and Chen (2007).
↵6 The direct utility function implied by the equation is
↵7 It also influences the relationship between the sum of the quasi-expenditure function and the inframarginal expenditure adjustment to derive the unobserved arguments needed for the Hicksian welfare measure in the Reiss and White (2006) framework. See notes 10 and 11 for a further discussion.
↵8 Following Edlefsen (1981) it is possible to define a relationship (correspondence) between the marginal price/cost implied by the nonlinear constraint for each value of this partial derivative and in principle numerically implement the Reiss-White logic. At each of the steps of the process, define the Marshallian relationship for that marginal price, apply a Vartian approximation, and solve the implied extension to Reiss and White’s twoequation framework for a specific marginal price and utility level. A comparable issue arises with hedonic models and the definition of “demand” functions for the characteristics of heterogeneous goods. See Brown and Rosen (1982), Mendelsohn (1985), and Palmquist (2005) for discussion.
↵9 In the context of a labor supply framework, Hausman (1980) develops this conclusion directly, citing his earlier result, and concludes, “An econometric model of both labor force participation and labor supply can be based solely on a labor supply equation specification. Neither the direct not indirect utility function is needed” (p. 165). This same argument has been applied to consumer demand models where decisions about budget segment and amount demanded parallel the choice of participation and extent of work in the labor supply literature.
↵10 This first step is the practical constraint that avoids the numerical approach proposed by Edelfsen and described in note 3. To begin the process, the analyst must have an estimate of the Marshallian demand function. Without locally constant prices and the assumption of constant parameters across all price segments, we do not have the requisite demand function Reiss and White assume as a starting point.
↵11 This process is consistent with Epstein’s (1981) discussion of consistent application of duality with nonlinear constraints. See his Theorem 5 and the example of household production functions.
↵12 This is not a new point. Hausman (1978) noted that we only learn about variations in the own price of the good with the solution of the differential equation implied by Roy’s identity. As La France and Hanemann (1989) and von Haefen (2003) document, we need additional assumptions and a line integral for more price changes.
↵13 Chetty and Szeidl (2007) used this argument in a dynamic setting to explain discrepancies between some types of behavior and measures of the coefficient of relative risk aversion. Assuming two classes of goods, food and committed consumption, they demonstrate that the income elasticity of food consumption is larger, implying greater food consumption response to income shocks with commitments than would be present without them.
↵14 This was Brown and Rosen’s (1982) argument. See Palmquist (2005) for further explanation and Chattopadhyay (1999) for an example.
↵15 Our sample is composed of average consumption levels, and there are no area specific measures of these expenditure shares at the level of census data. To account for differences with income we use the census measures of household income for each water provider’s service area along with the CES to assign one of 13 different values for the expenditure share ranging from about 0.068 to 0.108.
↵16 This is based on Hanemann’s (1984) earlier proposed model for a generalized corner solution model.
↵17 These two parameters implicitly allow for complementary or substitute relationships between the quantity of water consumed and these other goods.
↵18 An analysis of the responses to changes in the full price schedule is also possible and would require simulating the constrained optimization for different price schedules. This is outside the scope of this paper.
↵19 The construction of the demographic information required intersecting the service provider areas with the census block groups. Two statistics were defined: (1) the percentage of each block group contained in the service provider area, and (2) the percentage of the service provider area contained in each of the block groups. The first was used to aggregate census variables describing distributions (e.g., income, race, etc.) and the second for summary statistics such as means or medians.
↵20 The only other reason a provider was deleted from consideration stemmed from the fact that it was not required to report monthly consumption. The size of the water service provider’s service area determines these reporting requirements.
↵21 See Angrist and Krueger (2001) for an accessible summary of the conditions for instruments and the advantages of natural experiments as one source for them. Our strategy is similar to the logic proposed by Bayer, Ferreira, and McMillan (2007) in that we exploit spatially delineated measures that should be correlated with our endogenous variables but cannot by construction be related to our measures for the marginal prices.
↵22 Summer is defined as June, July, August, and September, whereas winter is defined as December, January, February, and March.
↵23 We conducted this test with and without the intercept, and the results in both cases favor the model using the GIS-constructed instruments over that based on provider fixed effects.
↵24 In their appendix, Olmstead et al. document the simulation methods used to develop measures for elasticities for the unconditional demand.
↵25 More recently Taylor (2005) has estimated income elasticities using the CES. His results range from 0.31 to 0.42 for 1999 cross-specifications for a category he designates as utilities. Water is not separately identified in the ACCRA price indexes he uses with the expenditure survey. Water expenditures are included in the CES. So strictly speaking, his price index fails to reflect the covariation in their prices.
↵26 We were unable to locate estimates of the responsiveness of utility expenditures to the price of water; these do not appear to have been developed.
