The excess burden associated to characteristics of the goods: Application to housing demand by ** Amelia Bilbao ** Celia Bilbao *** José M. Labeaga DOCUMENTO DE TRABAJO 2005-09

April, 2005

* We are grateful to participants in a seminar at the Instituto de Estudios Fiscales, Madrid and Samuel Gil-Martín for some useful comments. José M. Labeaga acknowledges financial support from project BEC2002-04294- C02. The usual disclaimer applies.

** Universidad de Oviedo.

*** FEDEA, UNED, Madrid. Corresponding author: José M. Labeaga. FEDEA , C/ Jorge Juan 46, 28001 – Madrid. Phone: +34 914359020 Fax: +34 915779575. E-mail: jmlabeaga@fedea.es

Los Documentos de trabajo se distribuyen gratuitamente a las Universidades e Instituciones de Investigación que lo solicitan. No obstante están disponibles en texto completo a través de Internet: http://www.fedea.es/hojas/publicaciones.html#Documentos de Trabajo

ABSTRACT

This paper shows that conditional subsidies aimed at certain kinds of goods do not merely generate the traditional excess burden: there is an alternative welfare loss as the distortion affects good characteristics rather than prices. We provide empirical evidence of the welfare loss after estimating a linear hedonic price model and using predicted prices in a quadratic almost ideal demand system for housing characteristics. We identify the sources of the losses and we also provide a monetary measure that suggests that, under plausible parameter values, it can be quantitatively high.

Keywords: welfare loss, rationing, hedonic prices, virtual prices, quadratic demand model.

1. Introduction

Traditionally, the excess burden has been defined as the welfare loss that a tax or subsidy produces above the tax revenue in the former or on public spending in the latter case. The loss arises because both taxes and subsidies distort the relative prices faced by both producers and consumers. In the case of conditional subsidies, which are aimed at promoting certain types of goods or goods satisfying some specific requirements, another kind of loss apart from the excess burden can be generated due to distortions in the quantities supplied of the characteristics of the good.

Conditional subsidies for certain kind of goods only appear when the subsidized good is heterogeneous in the sense that there are many types of the same commodity in the market. Under these circumstances, a new welfare loss arises where the inefficiency emerges due to distortions in quantities of the characteristics of the good instead of distortions in prices. The reason is that the beneficiary of a conditional allowance has to consume a good with some given characteristics, which are not necessarily in line with his choice in the market. In some sense, the welfare loss analysed here has many important economic features in common with rationing. Examples of this phenomenon occur when governments make a direct provision for goods or services, such as public health, public education or housing. The beneficiaries of these policies have their choice limited to a fixed segment of the market where they have to choose their family doctor, the school of their children or their house, from a closed list.

The purpose of this paper is to show that the aforementioned welfare loss exists and can be quantified. To achieve this goal, it is first necessary to model the consumer’s problem in the face of constraints upon the characteristics of the good. The analysis is based on the proposal of Lancaster (1966). According to this theory, utility is not directly derived from the consumption of goods but from their characteristics. The most common application of this approach is the so-called hedonic prices model (Rosen, 1974). Second, it is necessary to quantify the welfare loss generated by these kinds of subsidies. The theory of rationing proposes instruments to calculate the welfare losses when consumers face certain restrictions on consumption goods (see Neary and Roberts, 1980 or Kooreman, 1990). In this paper, we compute the excess burden under standard assumptions from the theory of rationing. We modify the usual approach in that the consumer now faces constraints concerning the characteristics of the goods rather than the goods themselves.

¹ Pareto-optimal allocations are not altered by the introduction of lump-sum taxes, even in the back of movements of relative prices.

To provide empirical evidence of the welfare loss, we estimate a quadratic demand model for housing characteristics in Spain, where the prices used are those predicted in the hedonic equations. The results of this empirical exercise show that the welfare losses for subsidized housing in Spain are far from insignificant, reaching some 33 per cent of the market value. It would be ideal to provide a traditional measure of the excess burden in order to compare it with the measure we propose. It is worth emphasizing that in contrast to ordinary welfare instruments, the results obtained allows us to identify the sources of the losses according to the characteristics of the good, as well as providing a measure of the intensity of the losses.

The paper continues as follows. In section 2 we establish the theoretical basis for the existence of the welfare loss we propose. In section 3 we describe an empirical method that allow us to adjust hedonic prices equations and to estimate the demand for characteristics. We carry out a simulation exercise in section 4 in order to quantify our proposed measure of welfare. Section 5 concludes.

2. Theoretical framework

The analysis provided in this paper is based on the proposals of Lancaster (1966). In this framework, utility is not directly derived from goods, but from their intrinsic characteristics or attributes. This specific feature is considered in our analysis. The most common application of this approach is the so-called hedonic prices model (Rosen, 1974): the types of a heterogeneous good z, are described by an n-vector of characteristics . These components can be measured quantitatively and objectively so that the consumers’ perceptions of the quantities of characteristics are similar. Consumers may, of course, differ in their subjective valuations of alternative models. We also assume that there are a large number of differentiated products and that the choice among several combinations of z is continuous.

Every heterogeneous commodity has a quoted market price associated with a fixed value of the vector z, so that the implicit product market reveals a function relating prices and characteristics. This function is equivalent both for buyers and sellers. It is assumed that has continuous second derivatives. The consumer tries to maximize his utility where has the usual properties - in particular, it is assumed to be strictly concave - and x is a vector of the remaining goods. Utility is maximized subject to the following non-linear budget constraint:

² In order to simplify the model we assume that there is only one heterogeneous good in the economy and the consumer buys just one unit of such good, such that p(z) represents expenditure on this good.

\[\boldsymbol {q} \boldsymbol {x} + p (\boldsymbol {z}) = m\tag{1}\]

where m represents total expenditure and is the price vector associated with the goods x.

We assume that preferences are weakly separable between characteristics of the heterogeneous commodity and the rest of goods. We also assume that the consumer can allocate total expenditure in a two-stage budgeting process (Deaton and Muellbauer, 1980): at the first stage, expenditure is allocated between the heterogeneous good and the rest of goods, and then group expenditure is allocated to characteristics of the good.³ Both weak separability and two-stage budgeting imply that if any good or group of goods is rationed and if some other group of goods is separable from the rationed commodity, then the only effect of the rationing on the goods in the separable group is through total group expenditure (Deaton and Muellbauer, 1980). Under both weak separability and two-stage budgeting, the optimization problem above becomes:

\[\begin{array}{c} \text {Max V(z)} \\ \text {subject to: p(z) = m ^ {\prime}} \end{array}\tag{2}\]

where is the sub-utility function for the characteristics of the heterogeneous good and is the amount of income devoted to it.

Let uss assume that some consumers can only afford the heterogeneous good (for example a house) if the government establishes an allowance for it . Thus, the government establishes a subsidy s in such a way that , with y representing total expenditure after s is received. However, its beneficiaries can only consume the good with certain characteristics; hence, they see their consumption rationed to some combination of characteristics. For these consumers, problem (2) becomes:

\[\begin{array}{c} \text {Max V(z)} \\ \text {subject to: p(z) = y} \\ \mathbf {z} _ {H} \leq \mathbf {c} _ {H} \\ \mathbf {z} _ {L} \geq \mathbf {c} _ {L} \end{array}\tag{3}\]

where denotes the set of characteristics restricted by an upper bound and is the vector of satiation points associated with the consumption of such characteristics. On the other hand, represents the set of characteristics constrained by a lower bound and is the vector of minimum quantities to be consumed from such characteristics.

³ Separability of preferences is intimately related to two-stage budgeting but they are by no means equivalent: neither implies the other. What is true, however, is that weak separability is both necessary and sufficient for the second stage of two-stage budgeting (Deaton and Muellbauer, 1980).

If we denote the solution to problem (3) as and the value of the function at the optimum as , then the preference set associated with level , can be written as:

\[S _ {V ^ {*}} = \{(z _ {1}, \dots \dots . z _ {n}) / V (z _ {1}, \dots \dots . z _ {n}) \geq V ^ {*} \}\tag{4}\]

This set is convex due to the concavity of the utility function, . The convexity of and the differentiability of are sufficient conditions for the existence of a hyperplane tangent through the point on . This hyperplane is defined as the set of pairs such that points z belonging to the hyperplane verify:

\[\overline {{p}} z = \overline {{y}}\tag{5}\]

We can now establish that any vector z whose utility level exceeds will satisfy . Moreover, since this hyperplane is tangent to the indifference set , the solution to the following problem without restrictions over quantities of characteristics

\[\begin{array}{c} \text {Max V(z)} \\ \text {subject to:} \overline {{p}} z = \overline {{y}} \end{array}\tag{6}\]

is the vector , which is also the solution to problem (3) because it satisfies the first order conditions of problem (6). Notice also that the budget constraint corresponding to problem (6) is linear.

The components of the price vector are known as virtual prices and y is virtual income (Neary and Roberts, 1980). Virtual prices are defined as those that induce an unrationed consumer to consume the constrained quantities, i.e. he behaves as if he were facing the rationed situation. If the virtual price exceeds the market price, the household is (locally) constrained to consume a smaller quantity than he wishes and viceversa. Virtual or rationed income, , provides the amount of money needed to achieve the maximum level of utility that would be achieved when the consumer faces no restrictions on quantities. The difference between total expenditure at market prices and total expenditure at virtual prices provides a measure of the welfare loss in so far as it represents the change in expenditure that makes up for the utility loss brought about by the change in the rationed quantity. We can solve for the virtual prices, , by applying the Kuhn-Tucker conditions to the constrained problem (3) and the Lagrange conditions to the unconstrained problem (6). We can write the Lagrangian function associated with (3) as:

\[L _ {I} \left(\mathbf {z}, \lambda , \phi_ {i}, \gamma_ {i}\right) = V (\mathbf {z}) + \lambda (y - p (\mathbf {z})) + \sum_ {i \in H} \phi_ {i} \left(c _ {i} - z _ {i}\right) + \sum_ {k \in L} \gamma_ {k} \left(z _ {k} - c _ {k}\right),\tag{7}\]

and the first order conditions are:

\[\begin{array}{l l} \frac {\partial L _ {1}}{\partial z _ {i}} = \frac {\partial V}{\partial z _ {i}} - \lambda \frac {\partial p (z)}{\partial z _ {i}} - \phi_ {i} = 0 & i \in H \quad i) \\ \frac {\partial L _ {1}}{\partial z _ {k}} = \frac {\partial V}{\partial z _ {k}} - \lambda \frac {\partial p (z)}{\partial z _ {i}} + \gamma_ {k} = 0 & k \in L \quad i i) \\ \phi_ {i} (c _ {i} - z _ {i}) = 0 & i \in H \quad i i i) \\ \phi_ {i} \geq 0 & i v) \\ \gamma_ {k} (z _ {k} - c _ {k}) = 0 & k \in L \quad v) \\ \gamma_ {k} \geq 0 & v i) \\ z _ {H} \leq c _ {H} & v i i) \\ z _ {L} \geq c _ {L} & v i i i) \end{array}\tag{8}\]

The Lagrangian function associated with problem (6) is:

\[L _ {2} \left(z _ {1}, \dots .., z _ {n}, \lambda\right) = V (\mathbf {z}) + \lambda (\bar {y} - \bar {p} \mathbf {z})\tag{9}\]

and the corresponding first-order conditions are:

\[\frac {\partial L _ {2}}{\partial z _ {i}} = \frac {\partial V}{\partial z _ {i}} - \lambda \overline {{p}} _ {i} = 0\tag{10}\]

Rearranging (8i, ii) and (10) leads to:

\[\begin{array}{l} \overline {{p}} _ {i} = \frac {\partial p}{\partial z _ {i}} + \frac {\phi_ {i}}{\lambda} \\ \overline {{p}} _ {k} = \frac {\partial p}{\partial z _ {k}} - \frac {\gamma_ {k}}{\lambda} \end{array}\]

\[\begin{array}{c} i \in H \\ k \in L \end{array}\tag{11}\]

Taking into account that and the marginal utility of money, are non negative, we can establish from (11) that at the optimum:

\[\nabla_ {H} p \leq \overline {{p}} _ {H} \quad \text { and } \quad \nabla_ {L} p \geq \overline {{p}} _ {L}\tag{12}\]

where and are the gradients of the price function, , with respect to variables belonging to sets H and respectively. That is, the market prices of the characteristics are lower than or equal to virtual prices when the consumer is constrained by the lower bound on the quantity of characteristics, and viceversa when the consumer is restricted by the upper bound. From (11), (8iii) and (8v), all evaluated at , we get:

\[\left(\bar {p} _ {H} - \nabla_ {H} p\right) \left(c _ {H} - z _ {H}\right) = 0 \text { and } \quad \left(\nabla_ {L} p - \bar {p} _ {L}\right) \left(z _ {L} - c _ {L}\right) = 0\tag{13}\]

which has the usual interpretation associated with the complementary slackness theorem in that if at the optimum, then the virtual price of the ithcharacteristic equals its hedonic market price. The same applies when

It is important to determine or at least establish a threshold for the virtual income, y , using the dual interpretation of (6):

\[\begin{array}{c} \text {Min} \overline {{p}} \mathbf {z} \\ \text {subject to V(z) \geq V^{*}} \end{array}\tag{14}\]

whose solutions are and y . Then, under certain conditions (see a formal derivation in Appendix 1), we have:

\[\bar {y} \geq y + (\bar {p} _ {H} - \nabla_ {H} p) c _ {H} + (\nabla_ {L} p - \bar {p} _ {L}) c _ {L}\tag{15}\]

Moreover, if the hedonic price function is linearly homogeneous, an application of Euler’s theorem to implies that (15) is satisfied with equality. This allows a money metric measure of the utility loss to be calculated. Since the Lagrange multiplier, λ, of problem (6) is the marginal utility of income, the loss can be obtained as . This justifies the equivalence between the constrained and the unconstrained problems and provides a relationship between real and virtual income and market and virtual prices.

3. Empirical model: hedonic prices and housing demand

The empirical evaluation of the utility loss is similar to the measure of a standard excess burden except for the information required. The first step consists of accounting for consumer behaviour when facing the demand for characteristics of the heterogeneous good. In order to obtain parameter estimates, we need to know the demand for and price of each characteristic, information that does not appear explicitly in the market. We solve this lack of information by estimating the hedonic price model proposed by Rosen (1974) as well as the demand system for characteristics.⁴ In this context, it is not necessary to model the supply side of the market because our interest consists of the price of the characteristics faced by consumers (for a simultaneous model of supply and demand, see for instance Salo, 1994).

We assume that a given unit of a heterogeneous good is represented by whose components are the measurable characteristics of the good (size, shape, colour, etc.). The market price, , is a function associated with that vector of characteristics. We assume that the market is competitive. Consumers and producers are assumed to be price takers, although the implications of this assumption when the hedonic price function is non-linear are non-trivial.

The empirical methodology has two stages. In the first stage, we estimate the hedonic price function that relates the prices of the heterogeneous good to its characteristics, from which the implicit marginal price for each characteristic can be computed. These prices, , are obtained as:

\[\mathrm {p_ {i} (\mathbf {z}) = \frac {\partial \mathrm{p} (\mathbf {z})}{\partial \mathrm {z_ {i}}}}\tag{16}\]

At the second stage, we estimate the demand system for characteristics with those prices computed at the first stage (for recent applications using a linear expenditure demand system see Cheshire and Sheppard, 1998 and Wilhelmsson, 2002) In addition to prices, this function includes total expenditure, as a proxy for income, and some demographic variables,

\[z _ {i} = D (p _ {1} (z _ {1}), p _ {2} (z _ {2}), \dots \dots , p _ {n} (z _ {n}), y, d) \quad i = 1, \dots , n\tag{17}\]

Certain difficulties must be dealt with when it comes to applying the above method. The first problem arises from the non-linearity of the hedonic price function, which makes the budget constraint non-linear. The second problem is the simultaneous determination of prices and quantities. Even though consumers cannot alter the price structure, they can affect marginal prices with the consequence that there is correlation between prices and the error terms. In this situation, Ordinary Least Squares (OLS) produces inconsistent results (see Bartik, 1987; Epple, 1987; Meldenson, 1984; Cheshire and Sheppard 1998 or Wilhelmsson, 2002). The third issue which arises is the impossibility of separately identifying the price function and the demand equation unless identification restrictions are proposed.⁵ Both equations are related in such a way that the second stage of the process may only reproduce the information already obtained in the first stage of the estimation method (see Brown and Rosen, 1982 or McConnell and Phipps, 1987).

⁴ In fact, Rosen (1974) proposes a model to estimate marginal bid functions.He considers these to be the compensated inverse demands. In empirical studies, however, the demand functions are directly estimated.

Several solutions have been proposed in the literature in order to solve the above problems. The first consists of assuming that the household budget constraint is linear.⁶ The solution to the second problem is the use of an instrumental variable procedure (Cheshire and Sheppard, 1998), while the third problem can be dealt with by introducing an exogenous factor into the marginal price in the specification of the price function, thereby making it possible to identify the demand functions. Empirical studies do this by using data from several markets, so that a hedonic function is calculated for each market even though a common demand equation for each characteristic is imposed in all markets. Thus, the demand parameters are identical in all markets, while prices in each market differ. Actualy, theoretical progress has been made on the nonparametric estimation and identification of scalar additive hedonic models (Ekeland, Heckman and Nesheim, 2002) within a single market.

The hedonic model allows us to calculate the implicit valuation of the characteristics of which the heterogeneous good is composed. The model supplies the information necessary to estimate demand functions for characteristics. As the welfare losses or gains are defined through the cost function, both stages are needed to obtain the parameters of such functions. When the direct utility function is known, the loss is easily computed by solving the Lagrangian function. If we wish to obtain a money measure of the welfare loss, we should calculate the differences between the costs associated with each alternative. When the direct utility function is not known, it is necessary to apply the assumptions of the theory of rationing through the dual approach (see Deaton and Muellbauer, 1980; Kooreman, 1990 or Neary and Roberts, 1980).

The dual approach uses the unconditional demand system,⁷ where the rationed quantities are introduced, with prices and income (or total expenditure) as the unknown variables. Prices and income obtained in this way are virtual prices and virtual income respectively. As already pointed out, the difference between total expenditure at market prices and the rationed expenditure provides a measure of the welfare loss (gain), representing the change in expenditure which would compensate the family for the change in utility due to the purchasing restrictions. Thus, the welfare loss can be expressed as:

⁵ For a revision of the methodological problems in applying the hedonic method, see Ohsfeldt (1988).

⁶ For an empirical exercise considering nonlinear budget constraints, see Seko (2002).

⁷ In this paper, we refer to the unconditional demand system as the one in which only the budget constraint is imposed, with no restrictions imposed on quantities.

\[L o s s = e (p, u _ {l}) - e (\bar {p}, u _ {l}) = y _ {l} - \bar {y}\tag{18}\]

where is the initial price vector, is the vector of virtual prices and is the cost function.

In order to evaluate the welfare loss defined in (18), we choose a highly heterogeneous good, housing, where specific tax reforms affecting conditional subsidies can be modelled. Rationing is motivated by the fact that the recipient is constrained with regard to the type of house he can buy, named officially protected housing (viviendas de protección oficial). Thus, our objective is to evaluate consumers’ welfare losses given that they can only buy from a part of the housing market spectrum. We apply the hedonic model of Rosen (1974) in order to estimate prices of the characteristics. Second, we estimate demand functions for characteristics using the Quadratic Almost Ideal Demand System (QUAIDS) proposed by Banks, Blundell and Lewbel (1997).⁸ Once consumer behaviour is known, we evaluate the inefficiency caused by rationing. We use the dual approach because the system used lacks direct utility.

The sample used contains information corresponding to the five biggest cities of the region of Asturias for 1996: Oviedo, Gijón, Avilés, Mieres and Langreo. In order to estimate the hedonic price functions, we need to impose additional restrictions, as in Palmquist (1984) and Parsons i) the parameters of the hedonic price function vary across but not within cities. This implies that housing markets are only segmented across cities, in a way such that price changes can emerge because of supply conditions or variations in consumers’ preferences. This guarantees that price variations are exogenous to consumer decisions; ii) producers or consumers cannot individually alter prices because the hedonic method accounts for individual but not aggregate behaviour; iii) demand parameters are assumed to be invariant between cities. As a result, we estimate a hedonic price function for each market and a common demand function for the five housing markets; iv) we assume weak separability in preferences between housing characteristics and other goods in the consumer basket (see Gorman, 1981); v) we assume that the hedonic price function is linear in order to have a linear budget constraint.

⁸ This system is a rank three integrable extension to the rank two Almost Ideal Model of Deaton and Muellbauer (1980). The distinction with regard to the correct rank of a demand model is an empirical one as we will check below. Recently, Lewbel (2003) has proposed a rational rank four demand system.

⁹ See also Can (1992) for methodological issues in hedonic housing prices models estimated using spatial dependence, heterogeneity and heteroskedasticity. Knight, Carter, Hill and Sirmans (1993) present Monte Carlo results to evaluate the predictive performance of different estimation methods applied to the hedonic model. For a recent application of a hedonic price model for islands, see Bonnetain (2003).

The last hypothesis has non-trivial implications when it comes to estimating and interpreting both hedonic prices and demand functions. As identification problems are assumed away, cost and indirect utility functions for carrying out welfare analysis are available and the results of the QUAIDS are easy to interpret. Non-linearity of the hedonic price function can improve the estimation and precision of price variation within each market (see Knight, Carter Hill and Sirmans, 1993). However, our main interest lies in price variation across markets so that linearity could be a problem whenever the housing market is homogeneous (see Parson, 1986). The reason is that the marginal price of each housing characteristic would be constant and demand functions cannot be properly estimated without price variation. This is the main argument for considering segmented housing markets (see Brown and Rosen 1982; Can, 1992; Diamond and Smith, 1985; Ohsfeldt and Smith, 1990; Palmquist, 1984; or Parsons, 1986).

3.1. Estimation of hedonic price equations

In the first stage, we estimate the hedonic price equations. We need data both for prices and housing characteristics and we use two sources. First, public agencies and estate agents provide us with prices and characteristics related to real transactions of free market houses sold during 1996. Among the housing characteristics they provide the size of the house in square metres, number of bathrooms, existence of a heating system, the floor on which the flat is located, existence of a garage, whether the house is new or second hand and the street where the house is located. Second, the Environment Office of the Principality of Asturias provides information about the environmental conditions of the area where the house is located, in particular (sulfur dioxide) emissions in the different areas of the five cities, measured in normalized micrograms per cubic metre of air. We are aware that some infomation on environmental quality such as noise of the area or suspended particles in the air could have been exploited, but unfortunately this kind of information is not available for all the cities (see Department of Environmental Quality Air Quality Division Database for explanations on air quality measures).

Our final sample corresponds to 364 real transactions made in 1996 in the five most important cities in Asturias, with 80 corresponding to Oviedo, 98 to Gijón, 68 to Avilés, 54 to Mieres and 64 to Langreo. All these data correspond to urban areas of the five municipalities, so rural areas are excluded. The second step is to define the variables which enter the hedonic prices equations. Since we are interested in a parsimonious specification, we assume that the characteristics we have included provide an accurate picture of the house. We try to approximate the size (“quantity”) and quality of the house, the quality of the environment and the location of the house. Detailed explanations of the how these variables were constructed are provided in Appendix 2. Since we assume linearity of the hedonic prices equations, the empirical specification is:

\[\begin{array}{c} p _ {j m} = \lambda_ {0 m} + \lambda_ {1 m} s q m _ {j m} + \lambda_ {2 m} d b s _ {j m} + \lambda_ {3 m} d h s _ {j m} + \lambda_ {4 m} d h _ {j m} + \lambda_ {5 m} d g _ {j m} + \lambda_ {6 m} d a g e _ {j m} + \lambda_ {7 m} d i s t _ {j m} + \\ \lambda_ {8 m} S O _ {2 j m} + e _ {j m} \end{array}\tag{19}\]

where j stands for transactions, m is the market (i.e. the five cities we consider) and e is the error term, which satisfies the usual assumptions.

Table 1: Hedonic price equations

Notes. 1. Columns 1, 2, 3, 4 and 5 correspond to Oviedo, Gijón, Avilés, Mieres and Langreo, respectively. 2. Standard errors are in parenthesis. 3. F-test in a test of joint significance.

The coefficient of each characteristic is its implicit marginal price, which is equivalent to the mean price when the price function is linear. We estimate the equations for the five cities using OLS and the results are presented in Table 1. They seem to be as expected both in statistical and economic terms. From a statistical point of view, each set of regressors explains a high percent of the variance of the housing price and their coefficients are jointly significant. From an economic viewpoint, these results seem to confirm the adequacy of the hedonic model and as such can be considered to constitute an important source of information for agents participating in this market, such as buyers, suppliers or the public sector.

All the cofficients are positive, thus classifying characteristics as goods, as opposed to bads. The intercept is negative, indicating that total expenditure devoted to housing is higher than its market price. The intercept is highest for Oviedo, followed by Gijón, Avilés, Mieres and Langreo. It should be noted that Oviedo is the capital of the region and Gijón the biggest city. Some implications can be drawn from these results. The most expensive city in terms of housing quantity is Oviedo (869.54 euro/squared meter), followed by Gijón (794.12), Mieres (587.20), Avilés (553.90) and Langreo (483.85), which reflects the relative importance of the five cities of Asturias. Regarding the prices of the remaining characteristics, the most expensive housing market is Oviedo, with the exception of number of bathrooms (Gijón) and garage (Mieres). In order to get some insight into the latter result, it should be taken into account that Mieres is the only city where most of the garages are individual and closed while in the other cities they are normally opened garages.

3.2. Demand system estimation for housing characteristics

Once the implicit prices of each of the housing characteristics are obtained, the second step of our empirical exercise consists of estimating a demand system for characteristics. We use a QUAIDS model, imposing common coefficients for the five markets (see for instance King, Ohsfeldt, 1988; Ohsfeldt y Smith, 1990; Palmquist, 1984; Parsons, 1986 or Wilhemsson 2002). Marshallian demands for the QUAIDS can be expressed as:

\[w _ {i} = \alpha_ {i} + \sum_ {j} \gamma_ {i j} \ln p _ {j} + \beta_ {\mathrm{i}} \ln \left(\frac {x}{a (p)}\right) + \frac {d _ {i}}{b (p)} \left(\ln \left(\frac {x}{a (p)}\right)\right) ^ {2}\tag{20}\]

where is the budget share of characteristic and are parameters, prices, and price indexes and x is total expenditure. Omitting the subindex corresponding to transaction for ease of notation, and are defined as):

\[\begin{array}{l} \log a (p) = \alpha_ {0} + \sum_ {k} \alpha_ {k} \ln p _ {k} + \frac {1}{2} \sum_ {k} \sum_ {j} \gamma_ {k j} \ln p _ {k} \ln p _ {j} \\ b (\mathrm{p}) = \prod_ {i = 1} ^ {n} p _ {\mathrm{i}} ^ {\beta_ {i}} \end{array}\tag{21}\]

On differentiating the share equation (20) with respect to lnx and to respectively we obtain

\[\mu_ {i} = \frac {\partial w _ {i}}{\partial \ln x} = \beta_ {i} + \frac {2 d _ {i}}{b (p , z)} \left[ \ln \left(\frac {x}{a (p , z)}\right) \right]\tag{22}\]

\[\mu_ {i j} = \frac {\partial w _ {i}}{\partial \ln p _ {j}} = \gamma_ {i j} - \mu \left(\alpha_ {i} + \sum_ {k = 1} ^ {j} \gamma_ {k j} \ln p _ {k}\right) - \frac {d _ {i} \beta_ {i}}{b (p , z)} \left[ \ln \left(\frac {x}{a (p , z)}\right) \right] ^ {2}\tag{23}\]

yielding total expenditure elasticities . The uncompensated price elasticities are given by where is the Kronecker delta. The compensated price elasticities, , are calculated from the Slutsky equation,

In order to estimate (20), we need prices of housing characteristics, the quantities of these characteristics, and total housing expenditure, since we use two stage budgeting or weak separability among housing and other goods. We use prices obtained in the hedonic price model. In this sense, prices are exogenous under the assumptions that they are obtained by interaction of supply and demand and that the hedonic price functions are linear. Since we have prices and quantities of characteristics, we construct total expenditure on housing and budget shares for all characteristics, i.e., quantity, quality, location, and environmental quality. Notice first that to construct the budget share of quality, we follow King (1976) in defining quality expenditure as the sum of the nominal expenditures on each characteristic composing quality (baths, heating system, height of the flat, garage and age). Second, to construct the price of quality, we take a standard house to be one that is new, with more than one bath, heating system, garage and located in a third floor.

In the demand equations we include some demographics through translation, i.e., , and scaling, and . Thus, the empirical specification for equation (20) is:

\[w _ {i} = \alpha \left(Z _ {i}\right) + \sum_ {j} \gamma_ {i j} \ln p _ {j} + \beta \left(Z _ {i}\right) \ln \left(\frac {x}{a (p)}\right) + \frac {d \left(Z _ {i}\right)}{b (p)} \left(\ln \left(\frac {x}{a (p)}\right)\right) ^ {2} + u _ {i}\tag{24}\]

where , with being the percentage of young people (aged less than 26 years) in each city, being the percentage of people with the highest educational level (university degree) in each city, and measuring the composition of the household, i.e., the average number of household members in each city. We estimate the demand system (24) by iterative non-linear least squares (see Blundell and Robin, 1999). We begin by using a Stone price index approximation for . Since the common intercept, in (21), is not identified, we assign it the value 3.0. Results are independent of the value for this parameter provided it is below the minimum of total expenditure in household characteristics. We then estimate the model with no theoretical restrictions whatsoever, and next we impose homogeneity. With the homogeneous parameter estimates at hand, we use minimum distance to impose symmetry because it implies cross-equation restrictions. We use the restricted parameters because compliance with the theoretical restrictions is required in order to carry out the welfare exercise. All the variables are jointly significant. The total expenditure parameters are significant in all equations, as are their squared terms, thus confirming the rank three model as opposed to the rank two model. The price parameters are also significant and most of the sociodemographic variables are significant at standard levels. Young people demand less quantity and more quality of housing than old people, although the interaction between age and total expenditure shows that young people demand more quantity once their income reaches a threshold. Education shows a similar pattern to that of age: the more education, the more demand for quality. Educated people demand more quantity when their income increases up to a critical threshold. Finally, and as expected, the larger the household size, the larger the house they need, and they achieve it at the expense of quality. The demand for location follows the same pattern as that for quantity, while the demand for environmental quality follows the same pattern as that for quality.¹⁰ Concerning the fulfilment of the theoretical restrictions, the quantity and quality equations satisfy homogeneity values of 0.11 and 0.55, which has to be compared with a theoretical value of a with 1 degree of freedom) while this restriction is rejected in the location equation figure of 8.93). A global test of homogeneity and symmetry takes a value of 31.85 which has to be compared with a with 6 degrees of freedom. It clearly rejects the null, thus indicating a violation of the symmetry assumptions. Notwithstanding, we are going to use the homogeneity and symmetry-restricted parameters in order to conduct the welfare simulations in the next section.

¹⁰ The coefficients of the environmental quality demand equation can be derived by additivity.

Table 2. Demand system estimates

Notes. 1. w1, w2 and w3 are, repectively the shares of quantity, quality and location. 2. Standard errors robust to heteroskedasticity are in parenthesis. 3. c. price elasticity is the compensated price elasticity. 4. u. price elasticity is the uncompensated price elasticity.

We also report in Table 2 the total expenditure elasticity and the uncompensated and compensated price elasticities. As shown in equations (22) and (23), the demand model we use allows us to calculate a distribution for the elasticities. We only report the figures at mean values, although these distributions are available upon request. All these figures have been calculated using the homogeneity and symmetry-restricted parameters. Quality and location are price–inelastic characteristics while quantity is very elastic to price. Total expenditure elasticities show that quantity is a necessity at mean values but is a luxury at low values of income, whereas the reverse is true for both kinds of quality and location. We should bear in mind when comparing these figures that we measure some characteristics in a rather different way and we use a demand system which allows much more flexibility both in price and total expenditure responses. In particular, income elasticities for size and own-price elastiticies for location are in the range provided by Wilhelssom (2002). Income elasticities for size and quality are in line with those provided by Follain and Jiménez (1985). Our figure for income elasticity of location is close to that of Palmquist (1992) for quietness. Environmental quality is price inelastic, being a luxury at low values and a necessity at high values of income.

4. An evaluation of welfare measures

We are going to evaluate welfare losses (gains) produced by direct housing policies established in the Spanish Housing Plan (Plan de Vivienda y Suelo 1996-1999). Once the direct utility function behind the demand system is known, the welfare loss simply consists of calculating the difference in the utility levels between the housing allowance and a house with the same characteristics bought in the market at the same price. Unfortunately, the QUAIDS has no direct utility function, and hence we are compelled to use the dual approach. To do this, we need first to calculate virtual prices, solving the demand functions when the consumer purchases the restricted quantities of characteristics. We have to solve the system of equations

\[w _ {i} = \frac {\bar {p} _ {i} \bar {z} _ {i}}{\bar {x}} = \alpha (Z _ {i}) + \sum_ {j = 1} ^ {K} \gamma_ {i j} \ln \bar {p} _ {j} + \beta (Z _ {i}) \ln \left(\frac {\bar {x}}{a (p)}\right) + \frac {d (Z _ {i})}{b (p)} \left(\ln \left(\frac {\bar {x}}{a (p)}\right)\right) ^ {2}\tag{25}\]

where are virtual prices, are rationed quantities and x total rationed housing expenditure. The unrestricted total housing expenditure x, corresponds to the value that the characteristics of the rationed housing will acquire in the market and we can calculate it by means of the hedonic price function of the corresponding market using hedonic or market prices . System (25) can be solved by an iterative Newton method (see for instance Kooreman,

1990), where virtual prices and total rationed housing expenditure are the unknown variables of the system.

We need to have rationed quantities in order to solve (25) and we gather information provided by the regional government of Asturias (Principado de Asturias) in order to get them. We have data on housing subsidies corresponding to transactions taking place from January 1992 to December 1996 with all the relevant information (squared meters, whether the house is new or second hand, and location). Unfortunately, the latter variable is only available for purchases made before April 1995. Accordingly, we use data from January 1995 to April 1995 because the data for 1996 are not complete. We believe that the use of information for 1995 instead of 1996 will not substantially affect the analysis because subsidized houses according to the general plans of the regional governments have similar characteristics for long time periods. Then, we calculate the characteristics of a standard house in each protection regime and for each of the five markets. The standard house is obtained by taking average sample values given that since subsidized houses in each regime have similar characteristics in such a way that mean values adequately represent all houses.

Additional assumptions are needed to carry out the analysis. First of all, the program of subsidies is restricted so that it cannot affect market prices. However, we have carried out sensitivity analysis when the prices of the characteristics are affected by the protection system, and since the hedonic price functions are linear the effect on prices is also linear. Second, we feel that the results are not affected by endogenous sample selection when carrying out the simulations because households have the same preferences for housing characteristics whether they benefit from the program or not. Third, there are no restrictions on the quantity of characteristics in the competitive market.

We carry out simulations for the two existing protection regimes for the adquisition of housing as outlined in the Spanish Housing Plan 1996-1999. One of these is referred as Official Protection Housing (VPO) and the other one is known as fixed price housing (VPT). The latter distinguishes between new houses at fixed price (VPTN) and second hand houses at fixed price (VPTV). We conduct the exercise for the five different markets, namely Oviedo, Gijón, Avilés, Mieres and Langreo, in 1996. Although we only present results for Oviedo in Table 3, those corresponding to the other markets are available upon request. The first column in the table reports housing characteristics; the second column presents the quantities of characteristics according to the different regimes (see Bilbao, 1998, for details); the third column collects hedonic (market) prices for each of the characteristics. The total expenditure at market prices is provided in column four. Column five reports virtual prices obtained by solving (25), while column six gives rationed total expenditure on each characteristic. We show the difference between rationed and unrationed total expenditure on each characteristic in the last column and the total difference (for all characteristics) in the final row of each panel.

Table 3. Simulation results (Oviedo)

Notes. 1. All figures in the table are in euros. 2. Quantity: squared meters. Quality: index defined using number of bathrooms, presence of heating system, height of the flat, presence of garage and age of the house. Location: distance to the city centre in meters. Environmental quality: SO₂ micrograms per cubed meter. 3. Losses (gains) in percentage are in parenthesis. 4. Virtual prices are calculated solving equation (25).

The difference between rationed and unrationed total expenditure is an exact measure of welfare, as defined in the theoretical section. Whenever this difference is negative, the consumer is worse off because he is forced to purchase a given house instead of choosing it in the market according to his preferences. The monetary measure that would compensate the consumer for this loss is provided in the last row of each panel. In parenthesis we provide the relative (to total expenditure in housing) loss so as to be able to compare the three regimes. The differences in welfare losses among the different regimes are significant and they deserve further analysis.

The biggest inneficiency loss corresponds to the VPO regime. If we are willing to assume that housing purchased in the market does not suffer from inefficiencies produced by policy measures, the welfare loss represents 33 per cent of the market value. The second relative loss corresponds to houses under the VPTN regime and amounts to 24.54 per cent of its market price. Houses purchased under VPTV generate a loss which represents 14.55 per cent of its market value. The main difference among regimes is that houses at VPO and VPTN are all new buildings and they must satisfy certain requirements on quality and quantity, whereas houses under the VPTV regime only suffer size restrictions. As a consequence, welfare losses brought about by rationing quantity are high, while the excess of quality is larger in VPO and VPTN houses. VPTV houses are near the choice of the consumer according to quality, so the loss is smaller. Another feature of VPTV houses is that they are typically close to the city centre.

We observe bigger losses in quantity than in the other housing characteristics. Quantity is a necessity at low income levels and people who benefit from housing policies are relatively poor. Quality generates welfare gains (except in houses belonging to the VPTV regime). Quality is a luxury at low and mean income levels so that a low-income consumer would be better off by substituting quality by quantity. Location produces some welfare losses except for houses in the VPTV regime. The reason is that houses in the VPO and VPTN regimes are relatively far away from the city centers in comparison with VPTV houses. Finally, environmental quality is lowly valued by consumers. Spanish consumers are probably still not conscious enough about this issue.

5. Conclusions

We have developed a new concept of welfare loss which differs from the traditional excess burden. This new loss relies on the setup of conditional subsidies aimed at promoting the consumption of heterogeneous goods. The inefficiency emerges because the beneficiaries of these kinds of policies have to consume a good with certain characteristics which may not be the same those they would have chosen had they been able to choose freely in the market. In the second part of the paper we have modelled consumers’ behaviour when facing restrictions upon the characteristics of the goods. The modelling has succeeded in showing that with specific conditioned subsidies, consumers may experience welfare losses and these are probably different from the traditional excess burden. Unfortunately, we cannot compute both measures in our dataset. We estimate the proposed measure using a sample of transactions from five cities of a Spanish region (Asturias). First, we adjust linear hedonic prices equations; second, we estimate a very flexible demand system on housing characteristics; finally, we compute the welfare losses under three different regimes of purchasing. We show the importance of computing these measures as well as their heterogeneity, both across characteristics and across purchasing regimes. We believe that these figures should help design appropriate housing policies in the future.

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Appendix 1. Derivation of the virtual income equation

We have defined virtual income as:

\[\begin{array}{l} \overline {{y}} = \sum_ {i \varepsilon H} \overline {{p}} _ {i} z _ {i} ^ {*} + \sum_ {k \varepsilon L} \overline {{p}} _ {k} z _ {k} ^ {*} = \sum_ {i \varepsilon H} \left(\frac {\partial p (z)}{\partial z _ {i}} + \frac {\phi_ {i}}{\lambda}\right) z _ {i} ^ {*} + \sum_ {k \varepsilon L} \left(\frac {\partial p (z)}{\partial z _ {k}} - \frac {\gamma_ {k}}{\lambda}\right) z _ {k} ^ {*} = \\ \sum_ {i \varepsilon H} \left(\frac {\partial p (z)}{\partial z _ {i}}\right) z _ {i} ^ {*} + \sum_ {i \varepsilon H} \left(\overline {{p}} _ {i} - \frac {\partial p (z)}{\partial z _ {i}}\right) z _ {i} ^ {*} + \sum_ {k \varepsilon L} \left(\frac {\partial p (z)}{\partial z _ {k}}\right) z _ {k} ^ {*} + \sum_ {k \varepsilon L} \left(\frac {\partial p (z)}{\partial z _ {k}} - \overline {{p}} _ {k}\right) z _ {k} ^ {*} \end{array}\tag{A.1}\]

or equivalenty,

\[\bar {y} \geq \nabla p (z ^ {*}) z ^ {*} + (\bar {p} _ {H} - \nabla_ {H} p) c _ {H} + (\nabla_ {L} p - \bar {p} _ {L}) c _ {L}\tag{A.2}\]

Then, by applying the Wickells-Johnson theorem:

\[\bar {y} = \varepsilon_ {c} (z ^ {*}, t = 1) p (z ^ {*}) + (\bar {p} _ {H} - \nabla_ {H} p) c _ {H} + (\nabla_ {L} p - \bar {p} _ {L}) c _ {L}\tag{A.3}\]

where is the elasticity of return at the optimum. If , then:

\[\bar {y} \geq p (z ^ {*}) + (\bar {p} _ {H} - \nabla_ {H} p) c _ {H} + (\nabla_ {L} p - \bar {p} _ {L}) c _ {L}\tag{A.4}\]

Since , we have the result in equation (15).

Appendix 2. Definition of variables

The variables used in the hedonic price equations are the following:

Dependent variable: price paid for the house (excluding taxes). It is measured in millions of pesetas. (6000 euro).

Among the independent variables we try to approximate:

Quantity:

- squared meters of the house (sqm)

Quality:

- bathrooms (dbs): dummy variable taking value 1 whenever the house has more that 1 bathroom and 0 otherwise.

- heating system (dhs): dummy variable taking value 1 whenever the house has a heating system and 0 otherwise.

- height of the flat (dh): floor on which the flat is located. We can also construct dummies, but there seems to be a linear relationship between height and price such that the results are not affected.

- garage (dg): dummy variable taking value 1 when the house has a garage.

- age of the house (dage): dummy variable taking value 1 when the house is new.

Location:

- distance to the city centre (dist): distance to centre in straight line measured in metres. We redefine it to get positive sign for its coefficient: with being the maximum distance and the distance to the centre from house i.

Environmental quality:

- quantity of : quantity of in the area where the house is located. In order to get a positive sign for its coefficient, we define a new variable , with being the maximum level of and the level corresponding to house i.

DOCUMENTOS DE TRABAJO

References

1. 2005-09: “The excess burden associated to characteristics of the goods: application to housing demand”, Amelia Bilbao, Celia Bilbao y José M. Labeaga.

References

1. 2005-08: “La situación laboral de los inmigrantes en España: Un análisis descriptivo”, Ana Carolina Ortega Masagué

References

1. 2005-07: “Demographic Uncertainty and Health Care Expenditure in Spain”, Namkee Ahn, Juan Ramón García y José A. Herce.

References

1. 2005-06: “EL NO-MAGREB. Implicaciones económicas para (y más allá de) la región”, José A. Herce y Simón Sosvilla Rivero.

References

1. 2005-05: “The real picture: Industry specific exchange rates for the euro area”, Simón Sosvilla-Rivero y Sonia Pangusión.

References

1. 2005-04: “A Residential Energy Demand System for Spain”, Xavier Labandeira, José M. Labeaga y Miguel Rodríguez.

References

1. 2005-03: “The Evolution of Retirement”, J. Ignacio Conde-Ruiz., Vincenzo Galasso y Paola Profeta.

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1. 2005-02: “Housing deprivation and health status: Evidence from Spain” Luis Ayala, José M. Labeaga y Carolina Navarro.

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1. 2005-01: “¿Qué determina el éxito en unas Oposiciones?”, Manuel F. Bagüés.

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1. 2004-29: “Demografía y empleo de los trabajadores próximos a la jubilación en Cataluña”, J. Ignacio Conde-Ruiz y Emma García

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1. 2004-28: “The border effect in Spain”, Salvador Gil-Pareja, Rafael Llorca-Vivero, José A. Martínez-Serrano y Josep Oliver-Alonso.

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1. 2004-27: “Economic Consequences of Widowhood in Europe: Cross-country and Gender Differences” Namkee Ahn.

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1. 2004-26: “Cross-skill Redistribution and the Tradeoff between Unemployment Benefits and Employment Protection”, Tito Boeri, J. Ignacio Conde-Ruiz y Vincenzo Galasso.

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1. 2004-25: “La Antigüedad en el Empleo y los Efectos del Ciclo Económico en los Salarios. El Caso Argentino”, Ana Carolina Ortega Masagué.

References

1. 2004-24: “Economic Inequality in Spain: The European Union Household Panel Dataset”, Santiago Budría y Javier Díaz-Giménez.

References

1. 2004-23: “Linkages in international stock markets: Evidence from a classification procedure”, Simon Sosvilla-Rivero y Pedro N. Rodríguez.

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1. 2004-22: “Structural Breaks in Volatility: Evidence from the OECD Real Exchange Rates”, Amalia Morales-Zumaquero y Simon Sosvilla-Rivero.

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1. 2004-21: “Endogenous Growth, Capital Utilization and Depreciation”, J. Aznar-Márquez y J. R. Ruiz-Tamarit.

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1. 2004-20: “La política de cohesión europea y la economía española. Evaluación y prospectiva”, Simón Sosvilla-Rivero y José A. Herce.

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1. 2004-19: “Social interactions and the contemporaneous determinants of individuals’ weight”, Joan Costa-Font y Joan Gil.

References

1. 2004-18: “Demographic change, immigration, and the labour market: A European perspective”, Juan F. Jimeno.

References

1. 2004-17: “The Effect of Immigration on the Employment Opportunities of Native-Born Workers: Some Evidence for Spain”, Raquel Carrasco, Juan F. Jimeno y Ana Carolina Ortega.

TEXTOS EXPRESS

References

1. 2004-02: “¿Cuán diferentes son las economías europea y americana?”, José A. Herce.

References

1. 2004-01: “The Spanish economy through the recent slowdown. Current situation and issues for the immediate future”, José A. Herce y Juan F. Jimeno.