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The demand for housing services
JOURNAL OF
Journal of Housing Economics 13 (2004) 16–35
HOUSING ECONOMICS www.elsevier.com/locate/jhe
The demand for housing services Jeffrey E. Zabel Department of Economics, Tufts University, Medford, MA 02155, USA Received 19 August 2003
Abstract Despite the voluminous literature devoted to it, the concept of housing demand is ambiguous. Reconciling this ambiguity is important since the approach to estimating housing demand and the policy questions that can be addressed depends crucially on what interpretation is used. In this paper, housing demand is modeled as a continuous quantity that represents the flow of housing services. Confusion can arise over whether these services include not only those that arise from the housing structure but also from the neighborhood in which the house is located. The demand for housing services is derived from standard utility maximization and it is decomposed into structure and neighborhood demand equations. Another key issue that is addressed in this paper is the construction of the price of housing. Allowing for a single price of housing or multiple prices in a given housing market results in different indices for housing services that can produce very different estimates of the price and income elasticities for housing. This latter point is borne out in an empirical example using data from National version of the American Housing Survey for 1993 and 2001. Ó 2004 Elsevier Inc. All rights reserved.
1. Introduction The topic of housing demand, particularly in the context of neighborhood choice, has received a lot of attention recently. Historically, housing demand has also been the focus of much research. Numerous survey articles have been written including those by Quigley (1979), Mayo (1981), Olsen (1987), Smith et al. (1988), and Whitehead (1999). Despite the voluminous literature devoted to it, the concept of housing demand is ambiguous. Reconciling this ambiguity is important since the approach to E-mail address: jeff[email protected]. 1051-1377/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jhe.2003.12.002
J.E. Zabel / Journal of Housing Economics 13 (2004) 16–35
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estimating housing demand and the policy questions that can be addressed depends crucially on what interpretation is used. Rothenberg et al. (1991) note that the literature on housing demand falls into four categories: (1) the demand for housing services, (2) the demand for individual housing attributes, (3) the demand for owner occupancy versus renting (tenure choice), and (4) the spatial allocation of households. Each of these categories requires very different modeling and estimation strategies. Category (1) is useful for estimating price and income elasticities of housing demand. Housing demand is modeled as a continuous quantity that represents the flow of housing services. Thus, it is similar to other continuous economic variables like money and electricity and it is generally modeled as a function of the price of housing, non-housing expenditures, and socioeconomic variables. Category (2) has generated at least two large literatures. Starting with Rosen (1974), researchers have debated the validity of procedures for estimating the demand for specific structure characteristics (e.g., number of bedrooms) and neighborhood attributes (e.g., air quality). Another literature concerns the estimation of the demand for public goods based on the median voter model. Goldstein and Pauly (1981) showed that inconsistent estimates arise because of neighborhood sorting (Tiebout bias). Category (3) involves the estimation of a binary choice equation with rent/own as the two categories. These results can play an informative role when devising government policies to stimulate home-ownership. Category (4) includes applications of the random utility model developed by McFadden. McFaddenÕs (1978) paper focuses on neighborhood choice and it is the seminal paper in this area. Quigley (1985) was one of the first researchers to implement many of the new ideas and strategies in this paper. Recent analyses include Bajari and Kahn (2003) and Bayer et al. (2002). Category (1) is the focus of this paper. Even after restricting the analysis to the demand for housing services, the concept of housing demand is still ambiguous. This is because what constitutes housing services is unclear. Invariably, it includes the services that result from the structure and the land on which it sits. Also, given that the location of the house is fixed, it is reasonable to include the services that arise from the local amenities (Quigley, 1979 and Smith et al., 1988). Together, stock plus amenities can be considered to provide housing services. These are the services that one obtains by residing in a specific house in a specific location. What is observed in the housing market is the selling prices of houses that incorporate different amounts of these characteristics. The sales price represents the discounted present value of the total services provided by the house that include the structure services as well as the services that flow from the local amenities. Housing services is total value divided by the price per unit of housing services, that is, the sales price of the house divided by the (unit) price of housing. Generally, the price of a unit of housing services is not observed. This is because housing is a differentiated product that is made up of a set of characteristics that include the structure characteristics and also the local amenities. Thus, obtaining the price of a unit of housing services requires special techniques such as the hedonic method. This technique can be used to estimate the marginal prices of the house characteristics. This allows for the value of a ‘‘constant quality’’ house to be calculated for
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different geographical areas. These values can be used to generate an index of relative house prices. What constitutes a ‘‘constant quality’’ house will determine the particular price index that is generated. The main goal of this paper is to reduce the ambiguity surrounding the concept of housing demand. In particular, the notion of continuous housing demand is clarified. In the process, the different measures of housing services are reconciled. Also addressed is the related issue of the ‘‘price’’ of housing and how to estimate this ‘‘price.’’ Another goal of this paper is to show that a coherent measure of housing services requires that there be one price of housing services per housing market. This is the case whether or not housing demand includes neighborhood services. A number of researchers assume multiple prices per housing market (e.g., Olsen (1987), Sieg et al. (2002), and Rapaport (1997)). In this case, the appropriate measure of housing demand is the demand for housing structure. The ‘‘price’’ of housing structure is shown to be the value of the neighborhood. But the calculation of the housing structure index results in a strange measure of structure services. Under this framework of multiple prices, even a coherent demand for structure equation includes a ‘‘price’’ variable that is the amount of neighborhood quality purchased. The sign of the coefficient for this ‘‘price’’ variable will depend on whether structure and neighborhood quality are substitutes or complements. The final goal of this paper is to estimate the different models of housing demand using one dataset so that the results can be realistically compared. The paper is organized as follows. Section 2 begins with a general utility function and a derivation of the (implicit) demand for housing equation. In Section 3, housing demand is decomposed into the demand for structure and neighborhood. This is done because a common conception of housing demand is the demand for the structure itself. Also, because structure and neighborhood are inherently different, estimating their demands can involve separate approaches. In Section 4, the issue of the price of housing is addressed. Identifying the model requires that there be exogenous variation in this price. One approach is to estimate a hedonic house price model for multiple housing markets. It is shown that other approaches that are based on a single housing market result in models of housing demand that are not coherent. In Section 5, an empirical example is provided to compare estimates from different models of housing demand. Both a model of housing demand that includes structure and neighborhood services and a model of structure demand only are estimated. When the price index based on a single price per MSA is used, estimated price elasticities are relatively small as compared to when the price of housing is allowed to vary within the MSA. This can explain some of the relatively large values that exist in the literature.
2. The demand for housing services Let individual iÕs utility function depend on non-housing composite consumption, Ci , housing services, Hi , and own demographic characteristics that might affect preferences, zi
J.E. Zabel / Journal of Housing Economics 13 (2004) 16–35
Ui ¼ U ðCi ; Hi ; zi Þ:
19
ð1Þ
Assuming a static setting, an individual chooses how to allocate her income, yi , to Ci and Hi subject to the budget constraint Ci þ p Hi ¼ yi ;
ð2Þ
where the price of non-housing consumption is normalized to one and where p is the price of a unit of housing services. Assume that the individual maximizes utility subject to Eq. (2). Solving the budget constraint for Ci and substituting into the utility function gives the indirect utility function Vi ¼ Max : U ðyi p Hi ; Hi ; zi Þ: Hi
ð3Þ
Solving (3) yields the (implicit) housing demand equation p¼
oV oHi oV oyi
:
ð4Þ
Providing a specific form for the utility function (1) will give rise to an explicit housing demand equation. While many utility functions result in non-linear demand equations, typically a log-linear housing demand equation is specified ln Hi ¼ b0 þ b1 ln p þ b2 ln yi þ b3 ln zi þ ei ;
ð5Þ
where i is a random approximation error. This equation can be assumed to be an approximation to the underlying (non-linear) housing demand equation. Using (5) will produce estimates of the price and income elasticities, b b 1 and b b 2 . It will also allow for the estimation of the benefits from non-marginal changes in H or p. Public policy implications that can be addressed using (5) include the effect of vouchers versus cash transfers as a part of low-income housing programs.
3. The demand for structure and neighborhood characteristics An alternative approach to modeling continuous housing demand comes from decomposing housing demand into two components, the services that arise from the structure and from the neighborhood in which the house is located. Define Hsi and Hni to be the level of services provided by the structure and neighborhood characteristics and let utility now be a function of these indexes1 Ui ¼ U ðCi ; Hsi ; Hni ; zi Þ:
ð6Þ
One reason for splitting up H into Hs and Hn is that one can view Hs as being exogenous and Hn as being endogenous in the following sense. Since Hs emanates 1 An alternative approach to housing demand comes from defining utility as a direct function of the structure and neighborhood characteristics that are consumed by the individual, si and ni . This leads to demand equations for each of these characteristics. This corresponds to Category 2 (see Section 1) and has been covered extensively elsewhere (see Follain and Jimenez, 1985 and Palmquist, 1991 for surveys).
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from the physical structure of the house, the amount of Hs consumed depends on this physical structure. Changes in Hs occur through changes in this structure. Depreciation aside, this is caused by a conscious decision on the part of the owner to maintain/alter/add to the house. Hn arises from the amenities that are associated from the location of the house; such as measures of accessibility, which are generally fixed, and other factors like local public goods, environmental amenities and possibly the characteristics of oneÕs neighbors that can vary. Locally provided public goods such as school quality are determined by the residents of the town. The presence of environmental (dis)amenities like incinerators and factories that cause pollution may be related to house values—firms tend to locate in poorer areas and hence the level of environmental factors can be affected by the firmÕs decision. Finally, the characteristics of oneÕs neighbors change if they move. Thus, the amount of Hn that is consumed is not under the ownerÕs control the way that Hs is. A related issue is the problem that arises when estimating housing demand in the presence of neighborhood choice. Tiebout (1956) noted that people ‘‘vote with their feet.’’ This type of sorting is the basis for the so called ‘‘Tiebout bias’’ that arises when estimating the demand for public goods. Rapaport (1997) estimates a model of housing demand and community choice. She finds that controlling for community choice significantly affects the estimates of the price and income elasticities. Ioannides and Zabel (2003) estimate a model of housing demand, social interactions, and neighborhood choice. They find that controlling for neighborhood choice in the standard model without social interactions actually reduces (in magnitude) the price elasticity. When social interactions are included in the demand equation, controlling for neighborhood choice has little impact on price and income elasticities. Given that housing demand has been decomposed into structure and neighborhood amenities, the budget constraint can be expressed as Ci þ ps Hsi þ pn Hni ¼ yi :
ð7Þ
where ps and pn are the prices for a unit of structure and neighborhood service, respectively. Assume that the individual maximizes utility subject to this budget constraint. Solving for Ci in (7) and substituting into (6) gives the indirect utility function Vi ¼ Max : U ðyi ps Hsi pn Hni ; Hsi ; Hni ; zi Þ: Hsi ;Hni
ð8Þ
Solving (8) yields the (implicit) housing demand equations ps ¼
oV oHsi oV oyi
and pn ¼
oV oHni oV oyi
:
ð9Þ
Again, these equations are likely to be non-linear. If a log-linear demand equation is used as an approximation, it should include both ps and pn . For example, the structure demand equation would be ln Hsi ¼ b0s þ b1s ln ps þ b1n ln pn þ b2 ln yi þ b3 ln zi þ i :
ð10Þ
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4. Estimating the price of housing services At this point, it is not possible to estimate the housing demand Eq. (5) since neither H nor p is individually observed (but only their product). A number of methods have been used to obtain the price of housing services. One common approach is to estimate the hedonic house price function. This allows for the value of a ‘‘constant quality’’ house to be calculated for different geographical areas that can be used to generate an index of relative housing prices. In this sense, estimating the housing demand equation is analogous to estimating the two-stage demand equations for particular housing attributes. The key is that one needs exogenous variation in the housing price variable, p, in order to be able to estimate the housing demand equation. To provide this link to the house price hedonic (implicitly) define the expenditure function Eðsi ; ni ; p; uÞ as V ðyi Eðsi ; ni ; p; uÞ; H ðsi ; ni Þ; zi Þ ¼ u;
ð11Þ
where E is the minimum expenditure necessary to obtain the level of housing services H ðsi ; ni Þ given a fixed level of utility, u. H ðsi ; ni Þ is now explicitly expressed as a function of different amounts of structure, si , and neighborhood characteristics, ni . EðÞ can be obtained by inverting Eq. (11). If H ðsi ; ni Þ is homogeneous of degree one in si and ni , then it is possible to express EðÞ as the product of p and H ðsi ; ni Þ (see Sieg et al., 2002). Let P ðsh ; nh Þ be the value of house h that is a function of sh and nh . The value or price of the house represents the present discounted value of the flow of services provided by the dwelling in a given location. In equilibrium (when utility is maximized and the market clears), the minimum expenditure for house h, H ðsh ; nh Þ; will be equal to the market price for that house Eðsh ; nh ; pÞ ¼ p H ðsh ; nh Þ ¼ r P ðsh ; nh Þ;
ð12Þ
where r is the interest rate. Solving for P ðsh ; nh Þ and taking logs gives the house price hedonic ln P ðsh ; nh Þ ¼ ln Eðsh ; nh ; pÞ ln r ¼ ln H ðsh ; nh Þ þ ln p ln r:
ð13Þ
Given this link between H ðsh ; nh Þ and P ðsh ; nh Þ, it is clear that if p can be estimated it will be possible to solve for H and one can then estimate the demand for housing services Eq. (5). Two approaches are now considered for obtaining exogenous variation in the price of housing services. 4.1. Multiple markets Assume that p is the single price of housing services that holds for a given housing market. The most common practice assumes that housing markets are MSA-wide. Then p will indicate the price of a unit of housing services in Boston versus, say, Cleveland. It is possible for housing markets to exist at levels below the MSA. There has been a lot of research on housing sub-markets and how to define
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them (Goodman, 1981; Maclennan and Tu, 1996; Rothenberg et al., 1991). The key assumption is that there is one price per market, no matter what is the geographic level of the market. The more variation in p the ‘‘better’’ identified is the housing demand equation in terms of the precision of the parameter estimates, in general, and the price elasticity, in particular. Ohsfeldt and Smith (1988) point out that the amount of the exogenous variation in the price variable will strongly impact the accuracy of the parameter estimates. Using housing data across MSAs is likely to produce more variation in the price of housing services than is data from sub-markets within a single MSA. Also, a problem arises if one creates submarkets artificially, by estimating separate hedonic equations for sub-areas within an MSA, when these are not really separate markets. Then there is likely to be even less variation in the price of housing which will lead to a poorly identified housing demand equation. In this section, it is assumed that the data include information from more than one housing market, say J > 1 markets. The hedonic equation can be specified as ln Phj ¼ a0j þ a1j ln shj þ a2j ln nhj þ ehj ;
j ¼ 1; . . . ; J ;
ð14Þ
where j indexes markets. Rosen (1974) showed that parameters a1j and a2j are the percent increase in house value due to a 1% increase in structure and neighborhood characteristics in market j, respectively, and hence are interpreted as ‘‘hedonic’’ prices for these characteristics. The intercept term (a0j ) is the component of value that is constant across all houses in market j. A constant quality house price index can be calculated by evaluating the hedonic equation for each market at a constant level of s and n. The ratio of (constant quality) house prices in market j to a fixed market (say j ¼ 1) provides the (relative) price of housing in market j pj ¼
expða0j þ a1j ln s þ a2j ln nÞ ; expða01 þ a11 ln s þ a21 ln nÞ
ð15Þ
where s and n are evaluated at constant levels s and n. Note that p1 ¼ 1. One can also multiply this index by 100 so that p1 ¼ 100. If the coefficients for structure and neighborhood attributes are constant across markets (the percent increase in house value due to a 1% increase in structure and neighborhood characteristics, respectively, is common across markets) the index reduces to pj ¼
expða0j Þ : expða01 Þ
ð16Þ
One can easily see that there is only one price per market under this scenario. Price differentials arise because of market-wide factors that influence the desirability of living in one market versus another and possibly because of differences in the supply and demand for housing. Note that a downward sloping demand equation means that, on average and holding income constant, more housing services will be purchased in markets with a lower price of housing.
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Under this scenario of constant structure and neighborhood coefficients, housing services equals (hj is set to zero in (14) to ease exposition, hence this can be interpreted as average housing services) r Phj ðsh ; nh Þ pj r expða0j þ a1 ln sh þ a2 ln nh Þ ¼ expða0j Þ= expða01 Þ
H ðsh ; nh Þ ¼
¼ r expða01 þ a1 ln sh þ a2 ln nh Þ
ð17Þ
which is independent of j. This index is only a function of sh and nh which is a true index of housing services. When the coefficients for sh and nh are allowed to vary across markets, housing services equals H ðsh ; nh Þ ¼ ¼
r Phj ðsh ; nh Þ pj r expða0j þ a1j ln sh þ a2j ln nÞ expða0j þa1j ln sþa2j ln nÞ expða01 þa11 ln sþa21 ln nÞ
¼ r expða01 þ a11 ln sh þ a21 ln nh þ ðða1j a11 Þ ðln sh ln sÞ þ ða2j a21 Þ ðln nh ln nÞÞ:
ð18Þ
The first component on the right-hand side of the last equality in Eq. (18) is the same as Eq. (17). The second component reflects the deviation of the coefficients for sh and nh in market j from those in market 1 and the difference in the amounts of sh and nh embodied in house h as compared to the standard package s and n. Why not just use the first component? One answer is that the choice of which market is market 1 is arbitrary. The use of any other market as the comparison market will result is a different set of weights a11 and a21 that would result in a different set of relative values of housing services. So the second component counteracts the arbitrary choice of which market is market 1. Note that individual elements of this second term will be zero if either the coefficient in market j is the same as that in market 1 (e.g., a1j ¼ a11 ) or if the amount of the characteristic is the same as is in the standard house (e.g., sh ¼ s). To estimate the demand for housing structure or neighborhood, first decompose the price for housing services as pj ¼
expða0j Þ expða1j ln sÞ expða2j ln nÞ ¼ p0j psj pnj ; expða01 Þ expða11 ln sÞ expða21 ln nÞ
ð19Þ
where psj is the component of price that corresponds to structure, pnj is the component that corresponds to neighborhood, and p0j is the component that corresponds to market-wide characteristics. p0j is the price for the same house (neighborhood and structure) in two different markets (see Zabel (1999) for a similar decomposition). Note that psj and pnj reflect the cost of providing a unit of structure and neighborhood
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services, respectively. These prices will differ across markets because of differential costs of providing these services that arise from differences in demand and supply. They will also reflect the differences in wages across markets (not already captured in p0j ). Clearly, psj and pnj cannot be estimated if the coefficients for the structure and neighborhood characteristics are held constant across markets. Two measures of the services provided by housing structure can be constructed (comparable measures can be constructed for neighborhood service) Hshj ¼
r expða1j ln shj Þ ¼ r expða11 ln sh þ ða1j a11 Þ ðln sh ln sÞÞ psj
Hshj ¼
r expða0j þ a1j ln sh Þ p0j psj
ð20Þ
and
¼ r expða01 þ a11 ln sh þ ða1j a11 Þ ðln sh ln sÞÞ:
ð21Þ
These two measures are essentially the same since a01 will be common to all observations. The difference is reflected in the price that is used. The first index corresponds to a price of psj while the price that corresponds to the second index is p0 psj . The difference is that the second price includes not only the the price of a unit of structure or neighborhood in a given market but also includes the market-wide component that gives the ‘‘price’’ of buying housing in a given market. 4.2. A single market When the data only include one housing market, the hedonic becomes ln Ph ¼ a0 þ a1 ln sh þ a2 ln nh þ eh
ð22Þ
and hence a0 , alone, cannot provide the needed variation in p. While it is technically possible to obtain exogenous variation in the price of housing services with data from a single housing market, this relies on tenuous functional form restrictions (Bartik, 1987; Epple, 1987; Mendelsohn, 1985). Thus, to generate a price of housing that varies within a single market requires a different concept of house price. This is consistent with Olsen (1987, p. 998)) The assumption of no price variation is often made when the data refer to a single urban area. Unfortunately, this assumption is contrary to an implication of the simplest model of the determination of housing prices within an urban area. This model implies that the price per unit of housing services will be lower farther from the central business district because land prices are lower and the prices of other inputs used to produce housing are the same.
OlsenÕs notion of housing services corresponds to that of the housing structure. The ‘‘price’’ of these services reflects the value of living near the central business district. One can view this as a component of neighborhood quality so the ‘‘price’’ of a unit of structure service is the value of locating this unit of service in a given neighborhood. This ‘‘price’’ for housing structure can be constructed as the house price
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index that holds the structure constant but does not account for neighborhood characteristics. As shown in Zabel (1999) such an index can be constructed as psh ¼ expða0 þ a2 ln nh Þ ¼ expða0 Þ expða2 ln nh Þ:
ð23Þ
This ‘‘price’’ varies across neighborhoods since nh varies. As is apparent from (19) this is just the index of neighborhood quality weighted by the market-wide index. This latter term will be constant since the data are from one housing market. Thus psh is a measure (weighted sum) of the amount of neighborhood quality. This ‘‘price’’ for housing services is the value of the neighborhood in which the house is located and can be considered to be the cost of locating a house in a particular neighborhood. Using this definition of ‘‘price,’’ it follows that the price one pays for additional structure service (say adding a room to the house) will be different across neighborhoods. The price is not related to the cost of building the addition but to the quality of the neighborhood. It is clear that the ability to estimate psh requires that neighborhood variables be included in the house price hedonic. Since, in this context, housing services refers to the services from the housing structure, the numerator should be the same as the structure services equation in the previous section (Eq. (21)). The result is Hsh ¼
r expða0 þ a1 ln sh Þ ¼ r expða1 ln sh a2 ln nh Þ: expða0 þ a2 ln nh Þ
ð24Þ
This is a strange measure of structure services since it depends on neighborhood services, nh . If, instead, the numerator of the (total) housing services Eq. (17) is used the result is Hsh ¼
r expða0 þ a1 ln sh þ a2 ln nh Þ ¼ r expða1 ln sh Þ: expða0 þ a2 ln nh Þ
ð25Þ
This is a measure of the services from housing structure that does not depend on nh . Note, though, that the value of these services is now r P ðsh ; nh Þ. This includes the value of housing structure and neighborhood quality. Given that the price of housing services varies across neighborhoods, one might ask ‘‘why would one pay more for the same service in different neighborhoods?’’ The answer is now clear: the value of these services includes the value of the neighborhood. Given that the dependent variable is a measure of the services obtained from the housing structure, the housing demand equation will be similar to that in Eq. (10). It is important to note that this equation includes both the price of the services provided by housing structure and neighborhood quality, ps and pn , respectively. Recall from Eq. (23) that ps equals expða0 þ a2 ln nh Þ which is a measure of neighborhood quality. So the dependent variable is the amount of housing structure purchased and the price is the amount of neighborhood quality that is purchased. Is this a demand equation? The sign on the price variable will depend on whether s and n are substitutes or complements. Also, given that ps is a measure of neighborhood quality, it is not clear what pn would then be. An example where the price of housing services is a measure of neighborhood quality is in Rapaport (1997) who looks as six counties in Tampa and calculates a
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different price of housing for each county. In this case, each county is also a school district. She generates these prices by estimating a house price hedonic for each county and taking the ratio of predicted values using a constant quality house pj ¼
expða0j þ a1j ln sÞ : expða01 þ a11 ln sÞ
ð26Þ
Rapaport only includes house characteristics in her model so the differences in prices across the counties reflect the different levels of public goods provision which is captured in the county constants (a0j Õs). This is comparable to Eq. (23) where, now, the ‘‘price’’ for housing services is the value of each county as determined by its level of public goods provision. RapaportÕs measure of housing services is H ðsh ; nh Þ ¼ ¼
r Phj ðsh Þ pj r expða0j þ a1j ln sh Þ expða0j þa1j ln sÞ expða01 þa11 ln sÞ
¼ r expða01 þ a11 ln sh þ ða1j a11 Þ ðln sh ln sÞÞ:
ð27Þ
This is the same measure of structure services as Eq. (21). The big difference between the two demand equations is that the price variable in RapaportÕs model (Eq. (27)) reflects the differences in county services while the price in the structure service Eq. (21) reflects differences in the cost of an additional unit of structure service. Again, the sign of the ‘‘price’’ variable will depend on whether structure and neighborhood quality are substitutes or complements. Rapaport estimates the price and income elasticities to be )0.786 and )0.224 (Table 3—extended specification estimated using OLS).2 Another approach to calculating the price of housing services was developed by Polinsky and Ellwood (1979). This requires information on the price per unit of land and a construction cost index. This method allows for the generation of a house price index that is based on the input prices associated with each dwelling. Thus this index is comparable to the hedonic price index that does not account for neighborhood quality (ps in Eq. (23)). Polinsky and Ellwood derive the price per unit of housing by setting it equal to the cost per unit of housing (assuming equilibrium) and by specifying a CES unit cost function (corresponding to a CES production function for housing services as a function of land and housing capital) 1 1r 1r pH ¼ ar pS1r þ dr pK ; ð28Þ where pS is the unit price of land, pK is the unit price of capital, and r is the elasticity of substitution. Polinsky and Ellwood use Federal Housing Administration data that includes pS . The authors note that that the data do not include location information below the city level and hence the impact of location on the price of housing will be 2 Rapaport (1997) also controls for community choice. In this case the price and income elasticities are )1.628 and 0.181.
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27
incorporated through its effect on pS : Thus, like Rapaport, the price of housing, pH , will reflect the value of neighborhood services. Polinsky and EllwoodÕs housing demand equation specifies the log of sales price normalized by the Bureau of Labor Statistics (BLS) price index, p0 , as the dependent variable and pH divided by p0 as the price of housing. Their estimates of the price and income elasticities are )0.67 and 0.39.
5. Empirical example In this section, an empirical example is undertaken to estimate the different models of housing demand. The data are taken from the 1993 and 2001 National versions of the American Housing Survey (NAHS). Using two years allows for a check on the generality of the results. The analysis is restricted to MSAs with at least 100 houses in the sample. There are 36 such MSAs in 1993 with a total of 7276 observations. In 2001, there are 38 qualifying MSAs with a total of 7694 observations. The number of observations per year for each MSA is given in the table of Appendix A. The first step in the empirical analysis is to estimate house price hedonic equations to obtain the price for housing services. A separate equation is estimated for each of the qualifying MSAs. Structure variables include the number of bedrooms, full bathrooms, and total rooms, the age of house (and its square), whether a garage is present, whether water leaked into the home from outside in the last 12 months, whether there are open cracks or holes in the walls or ceiling and, whether there is broken plaster or peeling paint over 1 square foot. Neighborhood variables include whether or not the house is in the central city, and owner characteristics (the log of permanent income, age, grade, and dummy variables for married, male, and white).3 The dependent variable is the natural log of the ownerÕs valuation of the house and property.4 After estimating the house price hedonics for each MSA, overall price indices and their components (Eq. (19)) were constructed for 1993 and 2001; MSA 360 (Anaheim) is the comparison MSA with a price index value that is normalized to 100. Next, a series of log-linear demand equations are estimated. Regressors include price, (permanent) income, household size, and dummy variables that indicate if the individual is married, graduated from high school, moved in last five years, 3 The ownerÕs characteristics are used to proxy for objective neighborhood characteristics since these do no exist in the public version of the AHS. Zabel and Kiel (1998) provide a rationale for using these variables. Two variables, the ownerÕs ranking of the quality of the structure and the neighborhood (1–10 scale) were also included but the results did not change significantly. These variables are not exogenous (Zabel and Kiel, 1998). A number of other structure characteristics were tried but were not significant and hence are not included in this analysis. Note that access to the confidential data in the AHS would allow one to merge in variables at the census tract level from the decennial censuses. These can be used as proxies for neighborhood characteristics (e.g., Ioannides and Zabel, 2003). 4 Goodman and Ittner (1992) and Kiel and Zabel (1999) have analyzed the accuracy of ownersÕ valuations and they find that the average owner over-estimates the value of their house by 5%. Kiel and Zabel find that this over-valuation is greatest for new owners and declines with length of tenure. They recommend that tenure be included in the house price hedonic when ownerÕs valuation is the dependent variable. Other than tenure, though, the over-valuation is not related to the observed characteristics of the owner, house, or neighborhood.
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Table 1 Summary statistics: 1993 and 2001 Variable
1993 Mean
Log of housing demand Log of price Log of income Log of permanent income High school Changed hands ( 5 years White Number of persons Married Number of observations
7.301 4.363 10.562 10.562 0.864 0.317 0.852 2.784 0.638 7276
2001 SD 0.606 0.337 0.955 0.507 0.343 0.465 0.355 1.497 0.481
Mean 7.842 4.212 10.931 10.931 0.866 0.347 0.797 2.776 0.615
SD 0.678 0.396 1.151 0.598 0.340 0.476 0.402 1.546 0.487
7694
and is white. Means and standard deviations for theses variables are given in Table 1. When estimating the demand for housing services, the dependent variable is the natural log of the ratio of house value to the overall price index (pj ) (Eq. (18)). Model 1 includes income and Model 2 includes permanent income.5 Results are given in Table 2. As expected, the permanent income elasticity is larger than the income elasticity (0.394 versus 0.150 for 1993 and 0.362 versus 0.166 for 2001). The price elasticity for Model 2 is significantly negative but small, )0.091 for 1993 and )0.052 for 2001. Previous estimates of the price and income elasticities have tended to vary considerably because of different levels of aggregation, varying measures of income and the price of housing, and separate model specifications. An often cited paper on the review of empirical analyses of housing demand is Mayo (1981). He notes that results depend on which of four procedures are used to obtain the price of housing. Those studies that use the BLS ‘‘family workersÕ budgets’’ have a range of price elasticities of ()0.3, )0.9). Those studies that obtain the price variable from housing production functions have a range of price elasticities of ()0.67, )0.76). The one study that uses a hedonic price index obtains a price elasticity of )0.53. Finally, the study that obtains the price of housing from the Housing Allowance Demand Experiment estimates the price elasticity to be )0.2. A recent review of the literature by Ermisch et al. (1996) finds a range of values for the price elasticity of ()0.5, )0.8). Estimates of the income elasticity have shown even more variation than the price elasticity. MayoÕs review of the literature shows that when permanent income is used with disaggregated data, the range of estimates for owners is (0.36, 0.87) with most in the (0.5, 0.7) range. Harmon (1988) reviews the literature and carries out an empirical analysis. His best estimate of the income elasticity is 0.7. 5 Permanent income is obtained by regressing the log of income on a cubic polynomial in age and years of education, dummy variables that indicate if the individual is married, male, black, or Hispanic and whether or not the unit lies in the central city of the MSA. MSA dummy variables are also included to capture differences in average income levels across MSAs.
Model 1 1993 Log of price Log of income Log of permanent income Log of price log of permanent income Log of permanent income squared High school Changed hands White Number of persons Married Constant Observations R2
)0.043 (0.019) 0.150 (0.008)
**
Significant at 1%.
2001 0.004 (0.020) 0.166 (0.007)
1993 )0.091 (0.020) 0.394 (0.020)
0.238 (0.020) 0.026 (0.014) 0.278 (0.019) 0.009 (0.005) 0.159 (0.016) 5.325 (0.109) 7276 0.17
Standard errors in parentheses. Significant at 5%.
*
Model 2
0.256 (0.022) )0.009 (0.015) 0.235 (0.018) )0.005 (0.005) 0.221 (0.017) 5.331 (0.108) 7694 0.18
0.091 (0.022) )0.014 (0.015) 0.255 (0.019) 0.002 (0.005) 0.030 (0.018) 3.223 (0.191) 7276 0.17
Model 3 2001 )0.052 (0.021) 0.362 (0.017)
0.131 (0.024) )0.048 (0.016) 0.235 (0.019) )0.004 (0.006) 0.094 (0.019) 3.637 (0.174) 7694 0.17
1993
Model 4 2001
1993
1.621 (0.424)
3.373 (0.390)
)0.505 (0.016)
)3.372 (0.452)
)3.101 (0.351)
0.431 (0.021)
)0.163 (0.040)
)0.312 (0.035)
0.214 (0.022)
0.224 (0.017)
0.127 (0.023) )0.009 (0.014) 0.255 (0.019) 0.001 (0.005) 0.042 (0.018) 19.078 (2.636) 7276 0.18
0.154 (0.024) )0.032 (0.016) 0.218 (0.018) )0.007 (0.005) 0.093 (0.019) 14.608 (2.182) 7694 0.19
2001 )0.353 (0.014) 0.270 (0.018)
0.023 (0.024) 0.040 (0.024) )0.008 (0.016) )0.046 (0.016) 0.103 (0.020) 0.095 (0.018) )0.001 (0.005) 0.007 (0.006) )0.041 (0.019) 0.046 (0.019) 4.379 (0.200) 5.425 (0.166) 7276 7694 0.17 0.10
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Table 2 Results for housing demand model: 1993 and 2001
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Table 3 Price and income elasticities for Model 3 Percentile
Price elasticity 1993
10 25 50 75 90
0.0209 )0.0434 )0.1101 )0.1601 )0.1972
Income elasticity 2001
(0.0347) (0.0234) (0.0199) (0.0252) (0.0318)
0.2268 0.0724 )0.0696 )0.1783 )0.2674
1993 (0.0394) (0.0262) (0.0206) (0.0240) (0.0304)
0.1269 0.2959 0.4710 0.6025 0.7000
2001 (0.0341) (0.0222) (0.0212) (0.0291) (0.0372)
0.1630 0.3554 0.4817 0.5523 0.6426
(0.0289) (0.0199) (0.0193) (0.0260) (0.0336)
The study that is closest to this empirical example is Goodman (1988). He estimates a model of tenure choice and housing demand using the 1978 NAHS. He estimates the price of housing using a hedonic price model. Goodman only includes four regional variables, and dummies for California, Washington, DC, and New York City. This only yields seven prices. The housing demand equation is a linear function of permanent and transitory income, price, value-to-rent ratio, age, household size, race, and sex. The dependent variable is the house value divided by the house price index. The estimated price and income elasticities are approximately )0.5 and 0.25, respectively. The comparable model from the above results is Model 2 which has a smaller price elasticity ()0.091 or )0.052) but a slightly higher income elasticity (0.394 or 0.362).6 Mayo (1981) notes that the log-linear model implies constant elasticities which is not necessarily consistent with reality. He derives price and income elasticities that vary with income using the Stone–Geary utility function. Note that it is possible to allow for price and income elasticities to vary with income in the log-linear demand model by interacting income with itself and price. When these two terms are added to the housing services demand equation (call it Model 3), both are significant (Table 2). Table 3 gives the price and income elasticities evaluated at different points of the distribution for price and income. Thus, the price and income elasticities do appear to vary with income. The price elasticity is particularly variable in 2001; it is more than three times as large (in magnitude) at the 90th percentile as compared to the 50th percentile. (The income elasticity is evaluated at the 50th percentile for price.) The demand for housing services is then estimated assuming that the data consist of only one housing market (Model 4). That is, the ‘‘price’’ of housing services reflects the neighborhood quality (Eq. (23)). This provides the variation in the price variable that is necessary to estimate the housing demand equation.7 Compared to the results for Model 2 (assuming one price per MSA), the ‘‘price’’ elasticities for 1993 and 2001 are both much larger in magnitude, )0.505 and )0.353 (Table 2). This model is similar to the ones in Rapaport (1997) and Polinsky and Ellwood (1979) since the price varies within the MSA. Recall that RapaportÕs estimate of the price elasticity is )0.786 while the one in Polinsky and Ellwood is )0.67. This makes it clear that the relatively higher 6 In a later paper, Goodman (2002) estimates the income elasticity to be between 0.4 and 0.45 which is very similar to the results in this paper. 7 Note that since there are actually multiple MSAs in the dataset, the product of this price and the MSA-wide price is used.
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Table 4 Results for demand for structure model: 1993 and 2001 Variable
Model 5 1993
Model 6 2001
1993
2001
)0.004 (0.001) )0.011 (0.001) Log of structure price )0.001 (0.001) )0.011 (0.001) Log of neighborhood )0.128 (0.008) )0.008 (0.008) price Log of permanent 0.137 (0.010) 0.104 (0.009) 0.147 (0.010) 0.105 (0.009) income High school 0.081 (0.012) 0.083 (0.012) 0.076 (0.011) 0.082 (0.012) Change hands )0.007 (0.007) )0.014 (0.008) )0.019 (0.007) )0.015 (0.008) 0.051 (0.009) 0.034 (0.009) 0.050 (0.009) White 0.044 (0.010) Number of persons 0.030 (0.003) 0.033 (0.003) 0.031 (0.003) 0.033 (0.003) Married 0.028 (0.009) 0.065 (0.010) 0.020 (0.009) 0.065 (0.010) Constant 5.571 (0.094) 5.454 (0.084) 5.575 (0.093) 5.436 (0.085) Observations 7276 7694 7276 7694 R2 0.15 0.14 0.18 0.14
Standard errors in parentheses. * Significant at 5%. ** Significant at 1%.
estimates obtained by Rapaport and Polinsky and Ellwood are due, at least in part, to the use of a different concept of the price of housing (see Section 4). Next, the demand for structure services is estimated. To the authorsÕs knowledge, this is the first time such an equation has been estimated. The dependent variable is the log of the ratio of the value of the structure (including the constant) divided by the product of the structure price (ps ) and the MSA-wide price (p0 ) (Eq. (21)). Model 5 includes just the price of structure and Model 6 includes both the prices of structure and neighborhood as regressors. The results are presented in Table 4. The price elasticity in Model 5 is negative but small (significant for 2001), )0.001 and )0.011 for 1993 and 2001, respectively and is lower in magnitude that the price elasticity of demand for housing services (Model 2). The (permanent) income elasticity has fallen by more than 50% when compared to Model 2. The addition of the neighborhood price has little effect on the own price elasticity (though it is now significant in both years). The cross-price elasticity is larger in magnitude than the own price elasticity and is significant for both years. It appears that housing structure and neighborhood are complements. Finally, a version of Model 5 is estimated where terms that interact income with itself and price are added to allow the price and income elasticities to vary with income (similar to Model 3). In this case, neither the price elasticity nor the income elasticity varies much across the distributions of price and income.
6. Conclusion This paper helps to reconcile the ambiguity surrounding housing demand. The ambiguity is understandable since there are, at least, four distinct categories into which housing demand falls. In this analysis, housing demand is modeled as a
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continuous quantity that represents the flow of housing services. The demand for housing services equation is derived from underlying utility maximization. It is also decomposed into structure and neighborhood demand equations. A key result is that the derivation of coherent housing services and housing structure indices requires a single price per market. When the price of housing is allowed to vary within a market, the appropriate measure of housing demand is the demand for housing structure and the ‘‘price’’ of housing structure is the value of the neighborhood. But the calculation of the housing structure index results in a strange measure of structure services. A coherent index is based on a value of housing structure that includes the value of the neighborhood. The demand for structure equation thus includes a price variable that is the amount of neighborhood quality purchased. The sign of the coefficient for this price variable will depend on whether structure and neighborhood quality are substitutes or complements. A empirical exercise was carried out to estimate the various models using one dataset. Generally, the price elasticity, while significantly negative, is small in magnitude. This is true for both housing services and housing structure. The (permanent) income elasticity for housing services is between 0.35 and 0.40. The income estimate is similar to that in Goodman (1988) (possibly the most comparable study) but the price elasticity is smaller in magnitude. The model that allows price and income elasticities to vary by income produces estimates of the price elasticity (evaluated at mean income) that do vary around the median but still do not approach the magnitudes of some values in the literature. When the price variable is allowed to vary within the market, the estimated price elasticities are much larger in magnitude than in the case of one price per market. This helps to explain some of the relatively large estimates of the price elasticity in the literature. One interesting result is the similarity of the estimates based on the 1993 and 2001 data sets. While these are both from the American Housing Survey, it does make it clear that the results are not particular to a given year and provides some support for their generality. Acknowledgments I am grateful to Yannis Ioannides, Jerry Rothenberg, two referees, and the editor for useful comments on an earlier draft of this paper.
Appendix A Observation counts by year and MSA MSA Anaheim-Santa Ana, CA Atlanta, GA
Number of observations Code
1993
2001
360 520
184 165
185 165
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Appendix A. (continued) MSA Baltimore, MD Bergen-Passaic, NJ Boston, MA Chicago, IL Cincinnati, OH-KY-IN Cleveland, OH Columbus, OH Dallas, TX Detroit, MI Fort Lauderdale-Hollywood, FL Fort Worth-Arlington, TX Houston, TX Kansas City, MO-KSa Los Angeles-Long Beach, CA Miami-Hialeah, FL Milwaukee, IL Minneapolis-Saint Paul, MN Nassau-Suffolk, NY New York City, NY Newark, NJ Norfolk-Virginia Beach, VA-NC Oakland, CA Philadelphia, PA-NJ Phoenix, AZ Pittsburg, PA Riverside-San Bernadino, CA Sacramento, CA Saint Louis, MO-IL Salt Lake City-Ogden, UT San Diego, CA San Francisco, CA San Jose, CA Seattle, WA Tampa-Saint Petersburg-Clearwater, FL Washington, DC-MD-VA Chicago Areas, IL Northern New Jersey Areas, NJ
Number of observations Code
1993
2001
720 875 1120 1600 1640 1680 1840 1920 2160 2680 2800 3360 3760 4480 5000 5080 5120 5380 5600 5640 5720 5775 6160 6200 6280 6780 6920 7040 7160 7320 7360 7400 7600 8280
178 117 212 545 107 185
153 101 219 502 111 168 94 156 500 128 101 218 116 518 169 122 222 261 428 145 113 195 480 237 175 126 100 140
8840 9991 9993
154 388 119 102 218 99 531 139 129 230 276 451 134 110 172 390 203 197 139 100 171 100 165 114 133 161 178 280
175 135 123 161 193 273 134 152
a The cutoff of 100 observations per MSA was made prior to dropping observations with missing variables.
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References Bajari, P., Kahn, M.E., 2003. Estimating Housing Demand with an Application to Explaining Racial Segregation in Cities, National Bureau of Economic Research, Working Paper 9891. Bartik, T.J. Estimating hedonic demand parameters with single market data: the problems caused by unobserved tastes. The Review of Economics and Statistics 69, 178–180. Bayer, P., McMillian, R., Ruben, K., 2002. An Equilibrium Model of Sorting in an Urban Housing Market: a Study of the Causes and Consequences of Residential Segregation, Mimeo, Yale University. Epple, D. Hedonic prices and implicit markets: estimating demand and supply functions for differentiated products. Journal of Political Economy 95, 59–80. Ermisch, J.F., Findlay, J., Gibb, K. The price elasticity of housing demand in Britain: issues of sample selection. Journal of Housing Economics 5, 64–86. Follain, J.R., Jimenez, E. Estimating the demand for housing characteristics: a survey and critique. Regional Science and Urban Economics 15, 77–107. Goldstein, G.S., Pauly, M.V. Tiebout bias on the demand for local public goods. Journal of Public Economics 16, 131–144. Goodman, A.C. Housing submarkets within urban areas: definitions and evidence. Journal of Regional Science 21, 175–185. Goodman, A.C. An econometric model of housing price, permanent income, tenure choice, and housing demand. Journal of Urban Economics 23, 237–353. Goodman, A.C. Estimating equilibrium housing demand for ‘‘Stayers’’. Journal of Urban Economics 51, 1–24. Goodman, J.L., Ittner, J.B. The accuracy of home ownersÕ estimates of house value. Journal of Housing Economics 2, 339–357. Harmon, O.R. The income elasticity of demand for single-family owner-occupied housing: a empirical reconciliation. Journal of Urban Economics 24, 173–185. Ioannides, Y.M., Zabel, J.E., 2003. Interactions, Neighborhood Selection and Housing Demand, mimeo, Tufts University. Kiel, K.A., Zabel, J.E. The accuracy of owner provided house values: the 1978–1991 American housing survey. Real Estate Economics 27, 263–298. Maclennan, D., Tu, Y. Economics perspectives on the structure of local housing systems. Housing Studies 11, 387–406. McFadden, D. Modelling the choice of residential location. In: Karlquist, A. et al. (Eds.), Spatial Interaction Theory and Planning Models. North-Holland, Amsterdam, pp. 75–96. Mayo, S.K. Theory and estimation in the economics of housing demand. Journal of Urban Economics 10, 95–116. Mendelsohn, R. Identifying structural equations with single market data. The Review of Economics and Statistics 67, 525–529. Ohsfeldt, R.L., Smith, B.A., 1988. Assessing the accuracy of structural parameter estimates in analysis of implicit markets. Land Economics 64, 135–146. Olsen, E.O. The demand and supply of housing service: a critical survey of the empirical literature. In: Mills, E.S. (Ed.), Handbook of Regional and Urban Economics, vol. 2. North-Holland, Amsterdam. Palmquist, R.B. Hedonic methods. In: Braden, J.B., Kolstad, C.D. (Eds.), Measuring the Demand for Environmental Quality. Elsevier Science Publishers, The Netherlands. Polinsky, A.M., Ellwood, D.T. An empirical reconciliation of micro and grouped estimates of the demand for housing. Review of Economics and Statistics 61, 199–205. Quigley, J.M. What have we learned about urban housing markets? In: Mieszkowski, P., Straszheim, M. (Eds.), Current Issues in Urban Economics. John Hopkins University Press, Baltimore. Quigley, J.M. Consumer choice of dwelling, neighborhood and public services. Regional Science and Urban Economics 15, 41–63. Rapaport, C. Housing demand and community choices. Journal of Urban Economics 42, 243–260. Rosen, S. Hedonic prices and implicit markets: product differentiation in pure competition. Journal of Political Economy 82, 34–55.
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Rothenberg, J., Galster, G.C., Butler, R.V., Pitkin, J. The Maze of Urban Housing Markets. The University of Chicago Press, Chicago. Sieg, H., Smith, K., Banzhaf, S., Walsh, R. Interjurisdictional housing prices in locational equilibrium. Journal of Urban Economics 52, 131–153. Smith, L.B., Rosen, K.T., Fallis, G. Recent developments in economic models of housing markets. Journal of Economic Literature 26, 29–64. Tiebout, C.M., 1956. A pure theory of local public expenditures. Journal of Political Economy 64, 416– 424. Whitehead, C.M.E. Urban housing markets: theory and policy. In: Cheshire, P., Mills, E.S. (Eds.), Handbook of Regional and Urban Economics, vol. 3. North-Holland, Amsterdam. Zabel, J.E. Controlling for quality in house price indices. Journal of Real Estate Finance and Economics 19, 223–241. Zabel, J.E., Kiel, K.A., 1998. The Impact of Neighborhood Characteristics on House Prices: What Geographic Area Constitutes a Neighborhood? Wellesley College Working Paper 98-04.
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HOUSING ECONOMICS www.elsevier.com/locate/jhe
The demand for housing services Jeffrey E. Zabel Department of Economics, Tufts University, Medford, MA 02155, USA Received 19 August 2003
Abstract Despite the voluminous literature devoted to it, the concept of housing demand is ambiguous. Reconciling this ambiguity is important since the approach to estimating housing demand and the policy questions that can be addressed depends crucially on what interpretation is used. In this paper, housing demand is modeled as a continuous quantity that represents the flow of housing services. Confusion can arise over whether these services include not only those that arise from the housing structure but also from the neighborhood in which the house is located. The demand for housing services is derived from standard utility maximization and it is decomposed into structure and neighborhood demand equations. Another key issue that is addressed in this paper is the construction of the price of housing. Allowing for a single price of housing or multiple prices in a given housing market results in different indices for housing services that can produce very different estimates of the price and income elasticities for housing. This latter point is borne out in an empirical example using data from National version of the American Housing Survey for 1993 and 2001. Ó 2004 Elsevier Inc. All rights reserved.
1. Introduction The topic of housing demand, particularly in the context of neighborhood choice, has received a lot of attention recently. Historically, housing demand has also been the focus of much research. Numerous survey articles have been written including those by Quigley (1979), Mayo (1981), Olsen (1987), Smith et al. (1988), and Whitehead (1999). Despite the voluminous literature devoted to it, the concept of housing demand is ambiguous. Reconciling this ambiguity is important since the approach to E-mail address: jeff[email protected]. 1051-1377/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jhe.2003.12.002
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17
estimating housing demand and the policy questions that can be addressed depends crucially on what interpretation is used. Rothenberg et al. (1991) note that the literature on housing demand falls into four categories: (1) the demand for housing services, (2) the demand for individual housing attributes, (3) the demand for owner occupancy versus renting (tenure choice), and (4) the spatial allocation of households. Each of these categories requires very different modeling and estimation strategies. Category (1) is useful for estimating price and income elasticities of housing demand. Housing demand is modeled as a continuous quantity that represents the flow of housing services. Thus, it is similar to other continuous economic variables like money and electricity and it is generally modeled as a function of the price of housing, non-housing expenditures, and socioeconomic variables. Category (2) has generated at least two large literatures. Starting with Rosen (1974), researchers have debated the validity of procedures for estimating the demand for specific structure characteristics (e.g., number of bedrooms) and neighborhood attributes (e.g., air quality). Another literature concerns the estimation of the demand for public goods based on the median voter model. Goldstein and Pauly (1981) showed that inconsistent estimates arise because of neighborhood sorting (Tiebout bias). Category (3) involves the estimation of a binary choice equation with rent/own as the two categories. These results can play an informative role when devising government policies to stimulate home-ownership. Category (4) includes applications of the random utility model developed by McFadden. McFaddenÕs (1978) paper focuses on neighborhood choice and it is the seminal paper in this area. Quigley (1985) was one of the first researchers to implement many of the new ideas and strategies in this paper. Recent analyses include Bajari and Kahn (2003) and Bayer et al. (2002). Category (1) is the focus of this paper. Even after restricting the analysis to the demand for housing services, the concept of housing demand is still ambiguous. This is because what constitutes housing services is unclear. Invariably, it includes the services that result from the structure and the land on which it sits. Also, given that the location of the house is fixed, it is reasonable to include the services that arise from the local amenities (Quigley, 1979 and Smith et al., 1988). Together, stock plus amenities can be considered to provide housing services. These are the services that one obtains by residing in a specific house in a specific location. What is observed in the housing market is the selling prices of houses that incorporate different amounts of these characteristics. The sales price represents the discounted present value of the total services provided by the house that include the structure services as well as the services that flow from the local amenities. Housing services is total value divided by the price per unit of housing services, that is, the sales price of the house divided by the (unit) price of housing. Generally, the price of a unit of housing services is not observed. This is because housing is a differentiated product that is made up of a set of characteristics that include the structure characteristics and also the local amenities. Thus, obtaining the price of a unit of housing services requires special techniques such as the hedonic method. This technique can be used to estimate the marginal prices of the house characteristics. This allows for the value of a ‘‘constant quality’’ house to be calculated for
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J.E. Zabel / Journal of Housing Economics 13 (2004) 16–35
different geographical areas. These values can be used to generate an index of relative house prices. What constitutes a ‘‘constant quality’’ house will determine the particular price index that is generated. The main goal of this paper is to reduce the ambiguity surrounding the concept of housing demand. In particular, the notion of continuous housing demand is clarified. In the process, the different measures of housing services are reconciled. Also addressed is the related issue of the ‘‘price’’ of housing and how to estimate this ‘‘price.’’ Another goal of this paper is to show that a coherent measure of housing services requires that there be one price of housing services per housing market. This is the case whether or not housing demand includes neighborhood services. A number of researchers assume multiple prices per housing market (e.g., Olsen (1987), Sieg et al. (2002), and Rapaport (1997)). In this case, the appropriate measure of housing demand is the demand for housing structure. The ‘‘price’’ of housing structure is shown to be the value of the neighborhood. But the calculation of the housing structure index results in a strange measure of structure services. Under this framework of multiple prices, even a coherent demand for structure equation includes a ‘‘price’’ variable that is the amount of neighborhood quality purchased. The sign of the coefficient for this ‘‘price’’ variable will depend on whether structure and neighborhood quality are substitutes or complements. The final goal of this paper is to estimate the different models of housing demand using one dataset so that the results can be realistically compared. The paper is organized as follows. Section 2 begins with a general utility function and a derivation of the (implicit) demand for housing equation. In Section 3, housing demand is decomposed into the demand for structure and neighborhood. This is done because a common conception of housing demand is the demand for the structure itself. Also, because structure and neighborhood are inherently different, estimating their demands can involve separate approaches. In Section 4, the issue of the price of housing is addressed. Identifying the model requires that there be exogenous variation in this price. One approach is to estimate a hedonic house price model for multiple housing markets. It is shown that other approaches that are based on a single housing market result in models of housing demand that are not coherent. In Section 5, an empirical example is provided to compare estimates from different models of housing demand. Both a model of housing demand that includes structure and neighborhood services and a model of structure demand only are estimated. When the price index based on a single price per MSA is used, estimated price elasticities are relatively small as compared to when the price of housing is allowed to vary within the MSA. This can explain some of the relatively large values that exist in the literature.
2. The demand for housing services Let individual iÕs utility function depend on non-housing composite consumption, Ci , housing services, Hi , and own demographic characteristics that might affect preferences, zi
J.E. Zabel / Journal of Housing Economics 13 (2004) 16–35
Ui ¼ U ðCi ; Hi ; zi Þ:
19
ð1Þ
Assuming a static setting, an individual chooses how to allocate her income, yi , to Ci and Hi subject to the budget constraint Ci þ p Hi ¼ yi ;
ð2Þ
where the price of non-housing consumption is normalized to one and where p is the price of a unit of housing services. Assume that the individual maximizes utility subject to Eq. (2). Solving the budget constraint for Ci and substituting into the utility function gives the indirect utility function Vi ¼ Max : U ðyi p Hi ; Hi ; zi Þ: Hi
ð3Þ
Solving (3) yields the (implicit) housing demand equation p¼
oV oHi oV oyi
:
ð4Þ
Providing a specific form for the utility function (1) will give rise to an explicit housing demand equation. While many utility functions result in non-linear demand equations, typically a log-linear housing demand equation is specified ln Hi ¼ b0 þ b1 ln p þ b2 ln yi þ b3 ln zi þ ei ;
ð5Þ
where i is a random approximation error. This equation can be assumed to be an approximation to the underlying (non-linear) housing demand equation. Using (5) will produce estimates of the price and income elasticities, b b 1 and b b 2 . It will also allow for the estimation of the benefits from non-marginal changes in H or p. Public policy implications that can be addressed using (5) include the effect of vouchers versus cash transfers as a part of low-income housing programs.
3. The demand for structure and neighborhood characteristics An alternative approach to modeling continuous housing demand comes from decomposing housing demand into two components, the services that arise from the structure and from the neighborhood in which the house is located. Define Hsi and Hni to be the level of services provided by the structure and neighborhood characteristics and let utility now be a function of these indexes1 Ui ¼ U ðCi ; Hsi ; Hni ; zi Þ:
ð6Þ
One reason for splitting up H into Hs and Hn is that one can view Hs as being exogenous and Hn as being endogenous in the following sense. Since Hs emanates 1 An alternative approach to housing demand comes from defining utility as a direct function of the structure and neighborhood characteristics that are consumed by the individual, si and ni . This leads to demand equations for each of these characteristics. This corresponds to Category 2 (see Section 1) and has been covered extensively elsewhere (see Follain and Jimenez, 1985 and Palmquist, 1991 for surveys).
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from the physical structure of the house, the amount of Hs consumed depends on this physical structure. Changes in Hs occur through changes in this structure. Depreciation aside, this is caused by a conscious decision on the part of the owner to maintain/alter/add to the house. Hn arises from the amenities that are associated from the location of the house; such as measures of accessibility, which are generally fixed, and other factors like local public goods, environmental amenities and possibly the characteristics of oneÕs neighbors that can vary. Locally provided public goods such as school quality are determined by the residents of the town. The presence of environmental (dis)amenities like incinerators and factories that cause pollution may be related to house values—firms tend to locate in poorer areas and hence the level of environmental factors can be affected by the firmÕs decision. Finally, the characteristics of oneÕs neighbors change if they move. Thus, the amount of Hn that is consumed is not under the ownerÕs control the way that Hs is. A related issue is the problem that arises when estimating housing demand in the presence of neighborhood choice. Tiebout (1956) noted that people ‘‘vote with their feet.’’ This type of sorting is the basis for the so called ‘‘Tiebout bias’’ that arises when estimating the demand for public goods. Rapaport (1997) estimates a model of housing demand and community choice. She finds that controlling for community choice significantly affects the estimates of the price and income elasticities. Ioannides and Zabel (2003) estimate a model of housing demand, social interactions, and neighborhood choice. They find that controlling for neighborhood choice in the standard model without social interactions actually reduces (in magnitude) the price elasticity. When social interactions are included in the demand equation, controlling for neighborhood choice has little impact on price and income elasticities. Given that housing demand has been decomposed into structure and neighborhood amenities, the budget constraint can be expressed as Ci þ ps Hsi þ pn Hni ¼ yi :
ð7Þ
where ps and pn are the prices for a unit of structure and neighborhood service, respectively. Assume that the individual maximizes utility subject to this budget constraint. Solving for Ci in (7) and substituting into (6) gives the indirect utility function Vi ¼ Max : U ðyi ps Hsi pn Hni ; Hsi ; Hni ; zi Þ: Hsi ;Hni
ð8Þ
Solving (8) yields the (implicit) housing demand equations ps ¼
oV oHsi oV oyi
and pn ¼
oV oHni oV oyi
:
ð9Þ
Again, these equations are likely to be non-linear. If a log-linear demand equation is used as an approximation, it should include both ps and pn . For example, the structure demand equation would be ln Hsi ¼ b0s þ b1s ln ps þ b1n ln pn þ b2 ln yi þ b3 ln zi þ i :
ð10Þ
J.E. Zabel / Journal of Housing Economics 13 (2004) 16–35
21
4. Estimating the price of housing services At this point, it is not possible to estimate the housing demand Eq. (5) since neither H nor p is individually observed (but only their product). A number of methods have been used to obtain the price of housing services. One common approach is to estimate the hedonic house price function. This allows for the value of a ‘‘constant quality’’ house to be calculated for different geographical areas that can be used to generate an index of relative housing prices. In this sense, estimating the housing demand equation is analogous to estimating the two-stage demand equations for particular housing attributes. The key is that one needs exogenous variation in the housing price variable, p, in order to be able to estimate the housing demand equation. To provide this link to the house price hedonic (implicitly) define the expenditure function Eðsi ; ni ; p; uÞ as V ðyi Eðsi ; ni ; p; uÞ; H ðsi ; ni Þ; zi Þ ¼ u;
ð11Þ
where E is the minimum expenditure necessary to obtain the level of housing services H ðsi ; ni Þ given a fixed level of utility, u. H ðsi ; ni Þ is now explicitly expressed as a function of different amounts of structure, si , and neighborhood characteristics, ni . EðÞ can be obtained by inverting Eq. (11). If H ðsi ; ni Þ is homogeneous of degree one in si and ni , then it is possible to express EðÞ as the product of p and H ðsi ; ni Þ (see Sieg et al., 2002). Let P ðsh ; nh Þ be the value of house h that is a function of sh and nh . The value or price of the house represents the present discounted value of the flow of services provided by the dwelling in a given location. In equilibrium (when utility is maximized and the market clears), the minimum expenditure for house h, H ðsh ; nh Þ; will be equal to the market price for that house Eðsh ; nh ; pÞ ¼ p H ðsh ; nh Þ ¼ r P ðsh ; nh Þ;
ð12Þ
where r is the interest rate. Solving for P ðsh ; nh Þ and taking logs gives the house price hedonic ln P ðsh ; nh Þ ¼ ln Eðsh ; nh ; pÞ ln r ¼ ln H ðsh ; nh Þ þ ln p ln r:
ð13Þ
Given this link between H ðsh ; nh Þ and P ðsh ; nh Þ, it is clear that if p can be estimated it will be possible to solve for H and one can then estimate the demand for housing services Eq. (5). Two approaches are now considered for obtaining exogenous variation in the price of housing services. 4.1. Multiple markets Assume that p is the single price of housing services that holds for a given housing market. The most common practice assumes that housing markets are MSA-wide. Then p will indicate the price of a unit of housing services in Boston versus, say, Cleveland. It is possible for housing markets to exist at levels below the MSA. There has been a lot of research on housing sub-markets and how to define
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them (Goodman, 1981; Maclennan and Tu, 1996; Rothenberg et al., 1991). The key assumption is that there is one price per market, no matter what is the geographic level of the market. The more variation in p the ‘‘better’’ identified is the housing demand equation in terms of the precision of the parameter estimates, in general, and the price elasticity, in particular. Ohsfeldt and Smith (1988) point out that the amount of the exogenous variation in the price variable will strongly impact the accuracy of the parameter estimates. Using housing data across MSAs is likely to produce more variation in the price of housing services than is data from sub-markets within a single MSA. Also, a problem arises if one creates submarkets artificially, by estimating separate hedonic equations for sub-areas within an MSA, when these are not really separate markets. Then there is likely to be even less variation in the price of housing which will lead to a poorly identified housing demand equation. In this section, it is assumed that the data include information from more than one housing market, say J > 1 markets. The hedonic equation can be specified as ln Phj ¼ a0j þ a1j ln shj þ a2j ln nhj þ ehj ;
j ¼ 1; . . . ; J ;
ð14Þ
where j indexes markets. Rosen (1974) showed that parameters a1j and a2j are the percent increase in house value due to a 1% increase in structure and neighborhood characteristics in market j, respectively, and hence are interpreted as ‘‘hedonic’’ prices for these characteristics. The intercept term (a0j ) is the component of value that is constant across all houses in market j. A constant quality house price index can be calculated by evaluating the hedonic equation for each market at a constant level of s and n. The ratio of (constant quality) house prices in market j to a fixed market (say j ¼ 1) provides the (relative) price of housing in market j pj ¼
expða0j þ a1j ln s þ a2j ln nÞ ; expða01 þ a11 ln s þ a21 ln nÞ
ð15Þ
where s and n are evaluated at constant levels s and n. Note that p1 ¼ 1. One can also multiply this index by 100 so that p1 ¼ 100. If the coefficients for structure and neighborhood attributes are constant across markets (the percent increase in house value due to a 1% increase in structure and neighborhood characteristics, respectively, is common across markets) the index reduces to pj ¼
expða0j Þ : expða01 Þ
ð16Þ
One can easily see that there is only one price per market under this scenario. Price differentials arise because of market-wide factors that influence the desirability of living in one market versus another and possibly because of differences in the supply and demand for housing. Note that a downward sloping demand equation means that, on average and holding income constant, more housing services will be purchased in markets with a lower price of housing.
J.E. Zabel / Journal of Housing Economics 13 (2004) 16–35
23
Under this scenario of constant structure and neighborhood coefficients, housing services equals (hj is set to zero in (14) to ease exposition, hence this can be interpreted as average housing services) r Phj ðsh ; nh Þ pj r expða0j þ a1 ln sh þ a2 ln nh Þ ¼ expða0j Þ= expða01 Þ
H ðsh ; nh Þ ¼
¼ r expða01 þ a1 ln sh þ a2 ln nh Þ
ð17Þ
which is independent of j. This index is only a function of sh and nh which is a true index of housing services. When the coefficients for sh and nh are allowed to vary across markets, housing services equals H ðsh ; nh Þ ¼ ¼
r Phj ðsh ; nh Þ pj r expða0j þ a1j ln sh þ a2j ln nÞ expða0j þa1j ln sþa2j ln nÞ expða01 þa11 ln sþa21 ln nÞ
¼ r expða01 þ a11 ln sh þ a21 ln nh þ ðða1j a11 Þ ðln sh ln sÞ þ ða2j a21 Þ ðln nh ln nÞÞ:
ð18Þ
The first component on the right-hand side of the last equality in Eq. (18) is the same as Eq. (17). The second component reflects the deviation of the coefficients for sh and nh in market j from those in market 1 and the difference in the amounts of sh and nh embodied in house h as compared to the standard package s and n. Why not just use the first component? One answer is that the choice of which market is market 1 is arbitrary. The use of any other market as the comparison market will result is a different set of weights a11 and a21 that would result in a different set of relative values of housing services. So the second component counteracts the arbitrary choice of which market is market 1. Note that individual elements of this second term will be zero if either the coefficient in market j is the same as that in market 1 (e.g., a1j ¼ a11 ) or if the amount of the characteristic is the same as is in the standard house (e.g., sh ¼ s). To estimate the demand for housing structure or neighborhood, first decompose the price for housing services as pj ¼
expða0j Þ expða1j ln sÞ expða2j ln nÞ ¼ p0j psj pnj ; expða01 Þ expða11 ln sÞ expða21 ln nÞ
ð19Þ
where psj is the component of price that corresponds to structure, pnj is the component that corresponds to neighborhood, and p0j is the component that corresponds to market-wide characteristics. p0j is the price for the same house (neighborhood and structure) in two different markets (see Zabel (1999) for a similar decomposition). Note that psj and pnj reflect the cost of providing a unit of structure and neighborhood
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J.E. Zabel / Journal of Housing Economics 13 (2004) 16–35
services, respectively. These prices will differ across markets because of differential costs of providing these services that arise from differences in demand and supply. They will also reflect the differences in wages across markets (not already captured in p0j ). Clearly, psj and pnj cannot be estimated if the coefficients for the structure and neighborhood characteristics are held constant across markets. Two measures of the services provided by housing structure can be constructed (comparable measures can be constructed for neighborhood service) Hshj ¼
r expða1j ln shj Þ ¼ r expða11 ln sh þ ða1j a11 Þ ðln sh ln sÞÞ psj
Hshj ¼
r expða0j þ a1j ln sh Þ p0j psj
ð20Þ
and
¼ r expða01 þ a11 ln sh þ ða1j a11 Þ ðln sh ln sÞÞ:
ð21Þ
These two measures are essentially the same since a01 will be common to all observations. The difference is reflected in the price that is used. The first index corresponds to a price of psj while the price that corresponds to the second index is p0 psj . The difference is that the second price includes not only the the price of a unit of structure or neighborhood in a given market but also includes the market-wide component that gives the ‘‘price’’ of buying housing in a given market. 4.2. A single market When the data only include one housing market, the hedonic becomes ln Ph ¼ a0 þ a1 ln sh þ a2 ln nh þ eh
ð22Þ
and hence a0 , alone, cannot provide the needed variation in p. While it is technically possible to obtain exogenous variation in the price of housing services with data from a single housing market, this relies on tenuous functional form restrictions (Bartik, 1987; Epple, 1987; Mendelsohn, 1985). Thus, to generate a price of housing that varies within a single market requires a different concept of house price. This is consistent with Olsen (1987, p. 998)) The assumption of no price variation is often made when the data refer to a single urban area. Unfortunately, this assumption is contrary to an implication of the simplest model of the determination of housing prices within an urban area. This model implies that the price per unit of housing services will be lower farther from the central business district because land prices are lower and the prices of other inputs used to produce housing are the same.
OlsenÕs notion of housing services corresponds to that of the housing structure. The ‘‘price’’ of these services reflects the value of living near the central business district. One can view this as a component of neighborhood quality so the ‘‘price’’ of a unit of structure service is the value of locating this unit of service in a given neighborhood. This ‘‘price’’ for housing structure can be constructed as the house price
J.E. Zabel / Journal of Housing Economics 13 (2004) 16–35
25
index that holds the structure constant but does not account for neighborhood characteristics. As shown in Zabel (1999) such an index can be constructed as psh ¼ expða0 þ a2 ln nh Þ ¼ expða0 Þ expða2 ln nh Þ:
ð23Þ
This ‘‘price’’ varies across neighborhoods since nh varies. As is apparent from (19) this is just the index of neighborhood quality weighted by the market-wide index. This latter term will be constant since the data are from one housing market. Thus psh is a measure (weighted sum) of the amount of neighborhood quality. This ‘‘price’’ for housing services is the value of the neighborhood in which the house is located and can be considered to be the cost of locating a house in a particular neighborhood. Using this definition of ‘‘price,’’ it follows that the price one pays for additional structure service (say adding a room to the house) will be different across neighborhoods. The price is not related to the cost of building the addition but to the quality of the neighborhood. It is clear that the ability to estimate psh requires that neighborhood variables be included in the house price hedonic. Since, in this context, housing services refers to the services from the housing structure, the numerator should be the same as the structure services equation in the previous section (Eq. (21)). The result is Hsh ¼
r expða0 þ a1 ln sh Þ ¼ r expða1 ln sh a2 ln nh Þ: expða0 þ a2 ln nh Þ
ð24Þ
This is a strange measure of structure services since it depends on neighborhood services, nh . If, instead, the numerator of the (total) housing services Eq. (17) is used the result is Hsh ¼
r expða0 þ a1 ln sh þ a2 ln nh Þ ¼ r expða1 ln sh Þ: expða0 þ a2 ln nh Þ
ð25Þ
This is a measure of the services from housing structure that does not depend on nh . Note, though, that the value of these services is now r P ðsh ; nh Þ. This includes the value of housing structure and neighborhood quality. Given that the price of housing services varies across neighborhoods, one might ask ‘‘why would one pay more for the same service in different neighborhoods?’’ The answer is now clear: the value of these services includes the value of the neighborhood. Given that the dependent variable is a measure of the services obtained from the housing structure, the housing demand equation will be similar to that in Eq. (10). It is important to note that this equation includes both the price of the services provided by housing structure and neighborhood quality, ps and pn , respectively. Recall from Eq. (23) that ps equals expða0 þ a2 ln nh Þ which is a measure of neighborhood quality. So the dependent variable is the amount of housing structure purchased and the price is the amount of neighborhood quality that is purchased. Is this a demand equation? The sign on the price variable will depend on whether s and n are substitutes or complements. Also, given that ps is a measure of neighborhood quality, it is not clear what pn would then be. An example where the price of housing services is a measure of neighborhood quality is in Rapaport (1997) who looks as six counties in Tampa and calculates a
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J.E. Zabel / Journal of Housing Economics 13 (2004) 16–35
different price of housing for each county. In this case, each county is also a school district. She generates these prices by estimating a house price hedonic for each county and taking the ratio of predicted values using a constant quality house pj ¼
expða0j þ a1j ln sÞ : expða01 þ a11 ln sÞ
ð26Þ
Rapaport only includes house characteristics in her model so the differences in prices across the counties reflect the different levels of public goods provision which is captured in the county constants (a0j Õs). This is comparable to Eq. (23) where, now, the ‘‘price’’ for housing services is the value of each county as determined by its level of public goods provision. RapaportÕs measure of housing services is H ðsh ; nh Þ ¼ ¼
r Phj ðsh Þ pj r expða0j þ a1j ln sh Þ expða0j þa1j ln sÞ expða01 þa11 ln sÞ
¼ r expða01 þ a11 ln sh þ ða1j a11 Þ ðln sh ln sÞÞ:
ð27Þ
This is the same measure of structure services as Eq. (21). The big difference between the two demand equations is that the price variable in RapaportÕs model (Eq. (27)) reflects the differences in county services while the price in the structure service Eq. (21) reflects differences in the cost of an additional unit of structure service. Again, the sign of the ‘‘price’’ variable will depend on whether structure and neighborhood quality are substitutes or complements. Rapaport estimates the price and income elasticities to be )0.786 and )0.224 (Table 3—extended specification estimated using OLS).2 Another approach to calculating the price of housing services was developed by Polinsky and Ellwood (1979). This requires information on the price per unit of land and a construction cost index. This method allows for the generation of a house price index that is based on the input prices associated with each dwelling. Thus this index is comparable to the hedonic price index that does not account for neighborhood quality (ps in Eq. (23)). Polinsky and Ellwood derive the price per unit of housing by setting it equal to the cost per unit of housing (assuming equilibrium) and by specifying a CES unit cost function (corresponding to a CES production function for housing services as a function of land and housing capital) 1 1r 1r pH ¼ ar pS1r þ dr pK ; ð28Þ where pS is the unit price of land, pK is the unit price of capital, and r is the elasticity of substitution. Polinsky and Ellwood use Federal Housing Administration data that includes pS . The authors note that that the data do not include location information below the city level and hence the impact of location on the price of housing will be 2 Rapaport (1997) also controls for community choice. In this case the price and income elasticities are )1.628 and 0.181.
J.E. Zabel / Journal of Housing Economics 13 (2004) 16–35
27
incorporated through its effect on pS : Thus, like Rapaport, the price of housing, pH , will reflect the value of neighborhood services. Polinsky and EllwoodÕs housing demand equation specifies the log of sales price normalized by the Bureau of Labor Statistics (BLS) price index, p0 , as the dependent variable and pH divided by p0 as the price of housing. Their estimates of the price and income elasticities are )0.67 and 0.39.
5. Empirical example In this section, an empirical example is undertaken to estimate the different models of housing demand. The data are taken from the 1993 and 2001 National versions of the American Housing Survey (NAHS). Using two years allows for a check on the generality of the results. The analysis is restricted to MSAs with at least 100 houses in the sample. There are 36 such MSAs in 1993 with a total of 7276 observations. In 2001, there are 38 qualifying MSAs with a total of 7694 observations. The number of observations per year for each MSA is given in the table of Appendix A. The first step in the empirical analysis is to estimate house price hedonic equations to obtain the price for housing services. A separate equation is estimated for each of the qualifying MSAs. Structure variables include the number of bedrooms, full bathrooms, and total rooms, the age of house (and its square), whether a garage is present, whether water leaked into the home from outside in the last 12 months, whether there are open cracks or holes in the walls or ceiling and, whether there is broken plaster or peeling paint over 1 square foot. Neighborhood variables include whether or not the house is in the central city, and owner characteristics (the log of permanent income, age, grade, and dummy variables for married, male, and white).3 The dependent variable is the natural log of the ownerÕs valuation of the house and property.4 After estimating the house price hedonics for each MSA, overall price indices and their components (Eq. (19)) were constructed for 1993 and 2001; MSA 360 (Anaheim) is the comparison MSA with a price index value that is normalized to 100. Next, a series of log-linear demand equations are estimated. Regressors include price, (permanent) income, household size, and dummy variables that indicate if the individual is married, graduated from high school, moved in last five years, 3 The ownerÕs characteristics are used to proxy for objective neighborhood characteristics since these do no exist in the public version of the AHS. Zabel and Kiel (1998) provide a rationale for using these variables. Two variables, the ownerÕs ranking of the quality of the structure and the neighborhood (1–10 scale) were also included but the results did not change significantly. These variables are not exogenous (Zabel and Kiel, 1998). A number of other structure characteristics were tried but were not significant and hence are not included in this analysis. Note that access to the confidential data in the AHS would allow one to merge in variables at the census tract level from the decennial censuses. These can be used as proxies for neighborhood characteristics (e.g., Ioannides and Zabel, 2003). 4 Goodman and Ittner (1992) and Kiel and Zabel (1999) have analyzed the accuracy of ownersÕ valuations and they find that the average owner over-estimates the value of their house by 5%. Kiel and Zabel find that this over-valuation is greatest for new owners and declines with length of tenure. They recommend that tenure be included in the house price hedonic when ownerÕs valuation is the dependent variable. Other than tenure, though, the over-valuation is not related to the observed characteristics of the owner, house, or neighborhood.
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J.E. Zabel / Journal of Housing Economics 13 (2004) 16–35
Table 1 Summary statistics: 1993 and 2001 Variable
1993 Mean
Log of housing demand Log of price Log of income Log of permanent income High school Changed hands ( 5 years White Number of persons Married Number of observations
7.301 4.363 10.562 10.562 0.864 0.317 0.852 2.784 0.638 7276
2001 SD 0.606 0.337 0.955 0.507 0.343 0.465 0.355 1.497 0.481
Mean 7.842 4.212 10.931 10.931 0.866 0.347 0.797 2.776 0.615
SD 0.678 0.396 1.151 0.598 0.340 0.476 0.402 1.546 0.487
7694
and is white. Means and standard deviations for theses variables are given in Table 1. When estimating the demand for housing services, the dependent variable is the natural log of the ratio of house value to the overall price index (pj ) (Eq. (18)). Model 1 includes income and Model 2 includes permanent income.5 Results are given in Table 2. As expected, the permanent income elasticity is larger than the income elasticity (0.394 versus 0.150 for 1993 and 0.362 versus 0.166 for 2001). The price elasticity for Model 2 is significantly negative but small, )0.091 for 1993 and )0.052 for 2001. Previous estimates of the price and income elasticities have tended to vary considerably because of different levels of aggregation, varying measures of income and the price of housing, and separate model specifications. An often cited paper on the review of empirical analyses of housing demand is Mayo (1981). He notes that results depend on which of four procedures are used to obtain the price of housing. Those studies that use the BLS ‘‘family workersÕ budgets’’ have a range of price elasticities of ()0.3, )0.9). Those studies that obtain the price variable from housing production functions have a range of price elasticities of ()0.67, )0.76). The one study that uses a hedonic price index obtains a price elasticity of )0.53. Finally, the study that obtains the price of housing from the Housing Allowance Demand Experiment estimates the price elasticity to be )0.2. A recent review of the literature by Ermisch et al. (1996) finds a range of values for the price elasticity of ()0.5, )0.8). Estimates of the income elasticity have shown even more variation than the price elasticity. MayoÕs review of the literature shows that when permanent income is used with disaggregated data, the range of estimates for owners is (0.36, 0.87) with most in the (0.5, 0.7) range. Harmon (1988) reviews the literature and carries out an empirical analysis. His best estimate of the income elasticity is 0.7. 5 Permanent income is obtained by regressing the log of income on a cubic polynomial in age and years of education, dummy variables that indicate if the individual is married, male, black, or Hispanic and whether or not the unit lies in the central city of the MSA. MSA dummy variables are also included to capture differences in average income levels across MSAs.
Model 1 1993 Log of price Log of income Log of permanent income Log of price log of permanent income Log of permanent income squared High school Changed hands White Number of persons Married Constant Observations R2
)0.043 (0.019) 0.150 (0.008)
**
Significant at 1%.
2001 0.004 (0.020) 0.166 (0.007)
1993 )0.091 (0.020) 0.394 (0.020)
0.238 (0.020) 0.026 (0.014) 0.278 (0.019) 0.009 (0.005) 0.159 (0.016) 5.325 (0.109) 7276 0.17
Standard errors in parentheses. Significant at 5%.
*
Model 2
0.256 (0.022) )0.009 (0.015) 0.235 (0.018) )0.005 (0.005) 0.221 (0.017) 5.331 (0.108) 7694 0.18
0.091 (0.022) )0.014 (0.015) 0.255 (0.019) 0.002 (0.005) 0.030 (0.018) 3.223 (0.191) 7276 0.17
Model 3 2001 )0.052 (0.021) 0.362 (0.017)
0.131 (0.024) )0.048 (0.016) 0.235 (0.019) )0.004 (0.006) 0.094 (0.019) 3.637 (0.174) 7694 0.17
1993
Model 4 2001
1993
1.621 (0.424)
3.373 (0.390)
)0.505 (0.016)
)3.372 (0.452)
)3.101 (0.351)
0.431 (0.021)
)0.163 (0.040)
)0.312 (0.035)
0.214 (0.022)
0.224 (0.017)
0.127 (0.023) )0.009 (0.014) 0.255 (0.019) 0.001 (0.005) 0.042 (0.018) 19.078 (2.636) 7276 0.18
0.154 (0.024) )0.032 (0.016) 0.218 (0.018) )0.007 (0.005) 0.093 (0.019) 14.608 (2.182) 7694 0.19
2001 )0.353 (0.014) 0.270 (0.018)
0.023 (0.024) 0.040 (0.024) )0.008 (0.016) )0.046 (0.016) 0.103 (0.020) 0.095 (0.018) )0.001 (0.005) 0.007 (0.006) )0.041 (0.019) 0.046 (0.019) 4.379 (0.200) 5.425 (0.166) 7276 7694 0.17 0.10
J.E. Zabel / Journal of Housing Economics 13 (2004) 16–35
Table 2 Results for housing demand model: 1993 and 2001
29
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Table 3 Price and income elasticities for Model 3 Percentile
Price elasticity 1993
10 25 50 75 90
0.0209 )0.0434 )0.1101 )0.1601 )0.1972
Income elasticity 2001
(0.0347) (0.0234) (0.0199) (0.0252) (0.0318)
0.2268 0.0724 )0.0696 )0.1783 )0.2674
1993 (0.0394) (0.0262) (0.0206) (0.0240) (0.0304)
0.1269 0.2959 0.4710 0.6025 0.7000
2001 (0.0341) (0.0222) (0.0212) (0.0291) (0.0372)
0.1630 0.3554 0.4817 0.5523 0.6426
(0.0289) (0.0199) (0.0193) (0.0260) (0.0336)
The study that is closest to this empirical example is Goodman (1988). He estimates a model of tenure choice and housing demand using the 1978 NAHS. He estimates the price of housing using a hedonic price model. Goodman only includes four regional variables, and dummies for California, Washington, DC, and New York City. This only yields seven prices. The housing demand equation is a linear function of permanent and transitory income, price, value-to-rent ratio, age, household size, race, and sex. The dependent variable is the house value divided by the house price index. The estimated price and income elasticities are approximately )0.5 and 0.25, respectively. The comparable model from the above results is Model 2 which has a smaller price elasticity ()0.091 or )0.052) but a slightly higher income elasticity (0.394 or 0.362).6 Mayo (1981) notes that the log-linear model implies constant elasticities which is not necessarily consistent with reality. He derives price and income elasticities that vary with income using the Stone–Geary utility function. Note that it is possible to allow for price and income elasticities to vary with income in the log-linear demand model by interacting income with itself and price. When these two terms are added to the housing services demand equation (call it Model 3), both are significant (Table 2). Table 3 gives the price and income elasticities evaluated at different points of the distribution for price and income. Thus, the price and income elasticities do appear to vary with income. The price elasticity is particularly variable in 2001; it is more than three times as large (in magnitude) at the 90th percentile as compared to the 50th percentile. (The income elasticity is evaluated at the 50th percentile for price.) The demand for housing services is then estimated assuming that the data consist of only one housing market (Model 4). That is, the ‘‘price’’ of housing services reflects the neighborhood quality (Eq. (23)). This provides the variation in the price variable that is necessary to estimate the housing demand equation.7 Compared to the results for Model 2 (assuming one price per MSA), the ‘‘price’’ elasticities for 1993 and 2001 are both much larger in magnitude, )0.505 and )0.353 (Table 2). This model is similar to the ones in Rapaport (1997) and Polinsky and Ellwood (1979) since the price varies within the MSA. Recall that RapaportÕs estimate of the price elasticity is )0.786 while the one in Polinsky and Ellwood is )0.67. This makes it clear that the relatively higher 6 In a later paper, Goodman (2002) estimates the income elasticity to be between 0.4 and 0.45 which is very similar to the results in this paper. 7 Note that since there are actually multiple MSAs in the dataset, the product of this price and the MSA-wide price is used.
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Table 4 Results for demand for structure model: 1993 and 2001 Variable
Model 5 1993
Model 6 2001
1993
2001
)0.004 (0.001) )0.011 (0.001) Log of structure price )0.001 (0.001) )0.011 (0.001) Log of neighborhood )0.128 (0.008) )0.008 (0.008) price Log of permanent 0.137 (0.010) 0.104 (0.009) 0.147 (0.010) 0.105 (0.009) income High school 0.081 (0.012) 0.083 (0.012) 0.076 (0.011) 0.082 (0.012) Change hands )0.007 (0.007) )0.014 (0.008) )0.019 (0.007) )0.015 (0.008) 0.051 (0.009) 0.034 (0.009) 0.050 (0.009) White 0.044 (0.010) Number of persons 0.030 (0.003) 0.033 (0.003) 0.031 (0.003) 0.033 (0.003) Married 0.028 (0.009) 0.065 (0.010) 0.020 (0.009) 0.065 (0.010) Constant 5.571 (0.094) 5.454 (0.084) 5.575 (0.093) 5.436 (0.085) Observations 7276 7694 7276 7694 R2 0.15 0.14 0.18 0.14
Standard errors in parentheses. * Significant at 5%. ** Significant at 1%.
estimates obtained by Rapaport and Polinsky and Ellwood are due, at least in part, to the use of a different concept of the price of housing (see Section 4). Next, the demand for structure services is estimated. To the authorsÕs knowledge, this is the first time such an equation has been estimated. The dependent variable is the log of the ratio of the value of the structure (including the constant) divided by the product of the structure price (ps ) and the MSA-wide price (p0 ) (Eq. (21)). Model 5 includes just the price of structure and Model 6 includes both the prices of structure and neighborhood as regressors. The results are presented in Table 4. The price elasticity in Model 5 is negative but small (significant for 2001), )0.001 and )0.011 for 1993 and 2001, respectively and is lower in magnitude that the price elasticity of demand for housing services (Model 2). The (permanent) income elasticity has fallen by more than 50% when compared to Model 2. The addition of the neighborhood price has little effect on the own price elasticity (though it is now significant in both years). The cross-price elasticity is larger in magnitude than the own price elasticity and is significant for both years. It appears that housing structure and neighborhood are complements. Finally, a version of Model 5 is estimated where terms that interact income with itself and price are added to allow the price and income elasticities to vary with income (similar to Model 3). In this case, neither the price elasticity nor the income elasticity varies much across the distributions of price and income.
6. Conclusion This paper helps to reconcile the ambiguity surrounding housing demand. The ambiguity is understandable since there are, at least, four distinct categories into which housing demand falls. In this analysis, housing demand is modeled as a
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continuous quantity that represents the flow of housing services. The demand for housing services equation is derived from underlying utility maximization. It is also decomposed into structure and neighborhood demand equations. A key result is that the derivation of coherent housing services and housing structure indices requires a single price per market. When the price of housing is allowed to vary within a market, the appropriate measure of housing demand is the demand for housing structure and the ‘‘price’’ of housing structure is the value of the neighborhood. But the calculation of the housing structure index results in a strange measure of structure services. A coherent index is based on a value of housing structure that includes the value of the neighborhood. The demand for structure equation thus includes a price variable that is the amount of neighborhood quality purchased. The sign of the coefficient for this price variable will depend on whether structure and neighborhood quality are substitutes or complements. A empirical exercise was carried out to estimate the various models using one dataset. Generally, the price elasticity, while significantly negative, is small in magnitude. This is true for both housing services and housing structure. The (permanent) income elasticity for housing services is between 0.35 and 0.40. The income estimate is similar to that in Goodman (1988) (possibly the most comparable study) but the price elasticity is smaller in magnitude. The model that allows price and income elasticities to vary by income produces estimates of the price elasticity (evaluated at mean income) that do vary around the median but still do not approach the magnitudes of some values in the literature. When the price variable is allowed to vary within the market, the estimated price elasticities are much larger in magnitude than in the case of one price per market. This helps to explain some of the relatively large estimates of the price elasticity in the literature. One interesting result is the similarity of the estimates based on the 1993 and 2001 data sets. While these are both from the American Housing Survey, it does make it clear that the results are not particular to a given year and provides some support for their generality. Acknowledgments I am grateful to Yannis Ioannides, Jerry Rothenberg, two referees, and the editor for useful comments on an earlier draft of this paper.
Appendix A Observation counts by year and MSA MSA Anaheim-Santa Ana, CA Atlanta, GA
Number of observations Code
1993
2001
360 520
184 165
185 165
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Appendix A. (continued) MSA Baltimore, MD Bergen-Passaic, NJ Boston, MA Chicago, IL Cincinnati, OH-KY-IN Cleveland, OH Columbus, OH Dallas, TX Detroit, MI Fort Lauderdale-Hollywood, FL Fort Worth-Arlington, TX Houston, TX Kansas City, MO-KSa Los Angeles-Long Beach, CA Miami-Hialeah, FL Milwaukee, IL Minneapolis-Saint Paul, MN Nassau-Suffolk, NY New York City, NY Newark, NJ Norfolk-Virginia Beach, VA-NC Oakland, CA Philadelphia, PA-NJ Phoenix, AZ Pittsburg, PA Riverside-San Bernadino, CA Sacramento, CA Saint Louis, MO-IL Salt Lake City-Ogden, UT San Diego, CA San Francisco, CA San Jose, CA Seattle, WA Tampa-Saint Petersburg-Clearwater, FL Washington, DC-MD-VA Chicago Areas, IL Northern New Jersey Areas, NJ
Number of observations Code
1993
2001
720 875 1120 1600 1640 1680 1840 1920 2160 2680 2800 3360 3760 4480 5000 5080 5120 5380 5600 5640 5720 5775 6160 6200 6280 6780 6920 7040 7160 7320 7360 7400 7600 8280
178 117 212 545 107 185
153 101 219 502 111 168 94 156 500 128 101 218 116 518 169 122 222 261 428 145 113 195 480 237 175 126 100 140
8840 9991 9993
154 388 119 102 218 99 531 139 129 230 276 451 134 110 172 390 203 197 139 100 171 100 165 114 133 161 178 280
175 135 123 161 193 273 134 152
a The cutoff of 100 observations per MSA was made prior to dropping observations with missing variables.
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