2.1 The Rosen framework

Most studies in this literature, including this one, follow the framework in the seminal article by Rosen (Reference Rosen1974), which explores the pricing of multiattribute products in a competitive market. Rosen distinguishes between a household bid function for a product attribute and the market price function, also known as the hedonic. In our terms (which differ from Rosen’s), $P$ is a bid and $S$ is an amenity. A bid function is an isoutility curve for $P$ and $S$ for a given type of household, and the hedonic, labeled $P^{E}$ , is the envelope of bid functions across households. The allocation of households across values of $S$ is known as “sorting.” Figure 1, which is analogous to Figure 2 in Rosen, provides an example of this bidding–sorting framework.

Figure 1 A bid-function envelope for school quality.

When applied to housing, Rosen’s framework highlights the fact that housing prices reflect both the underlying demands for amenities, bidding, and the allocation of households across neighborhoods with different levels of the amenity, sorting. Rosen proposes a two-step empirical approach that makes it possible to separate these two factors. The first step is to estimate a hedonic regression, $P^{E}$ as a function of $S$ , and then to differentiate the results to find the implicit or hedonic price, $\unicode[STIX]{x2202}P^{E}\!/\unicode[STIX]{x2202}S\equiv P_{S}^{E}$ , at each value of $S$ . Because each bid function is tangent to the envelope, the derivative of the envelope at the value of $S$ a household receives equals the implicit price it faces. The second step is to estimate the demand for $S$ by regressing $S$ on $P_{S}^{E}$ and other demand factors. An equivalent procedure is to estimate the inverse demand function, that is, to regress $P_{S}^{E}$ on $S$ (and other demand factors).

As is well known, the main problem with this procedure is that the implicit price is endogenous in the second-step regression (or, equivalently, $S$ is endogenous in the second-step inverse demand regression); see Taylor (Reference Taylor, Baranzini, Ramirez, Schaerer and Thalmann2008). If the hedonic function is nonlinear, households “select” an implicit price when they select a level of $S$ . And if the hedonic is linear, it yields no variation in implicit prices with which to estimate demand. Households have different preferences, so the level of $S$ and hence the implicit price they select depend on their observed and unobserved traits. Regressions of $S$ on the implicit price (or vice versa) are therefore subject to endogeneity bias. Another challenge facing this procedure is bias in the first-step regression due to omitted neighborhood variables. As we will see, attempts to address these challenges sometimes introduce endogeneity into (or change the meaning of) the first-step regression.

The literature on local public finance (reviewed in Ross & Yinger, Reference Ross, Yinger, Cheshire and Mills1999) complements the Rosen framework in the case of public services and neighborhood amenities (both designated by $S$ ). This literature emphasizes the importance of household sorting across communities (or neighborhoods) with different values of  $S$ . Figure 1 illustrates the most fundamental sorting theorem, namely, that households with higher demand for $S$ and hence with steeper bid functions sort into locations with higher $S$ . Sorting plays a critical role in our federal system, and this sorting theorem has been explored by several scholars, notably Epple and Sieg (Reference Epple and Sieg1999), Epple, Romer and Sieg (Reference Epple, Romer and Sieg2001), and Epple, Peress and Sieg (Reference Epple, Peress and Sieg2010). In this context, a “steeper bid function” is one that is steeper at a given value of $S$ , which we call “relative” steepness. This relative steepness is not the same as Rosen’s implicit price, which is the slope of the envelope at the level of $S$ a household consumes, because this implicit price is affected by $S$ . The slope of a bid function at a given $S$ is a household type’s marginal willingness to pay (MWTP) for the amenity at that level of $S$ . As shown in Figure 1 and panel A of Figure 2, the high level of $S$ consumed by high-demand households may push them so far down their demand curve that the implicit price they face is below the implicit price faced by low-demand households – despite the higher relative slope that goes with high demand.Footnote ²

Figure 2 Sorting and the distribution of MWTP.

Rosen (Reference Rosen1974) recognized that his framework was “similar in spirit to Tiebout’s (Reference Tiebout1956) analysis of the implicit market for neighborhoods, local public goods being the ‘characteristics’ in this case” (p. 40). Nevertheless, the literature on local public finance has shifted the framework used by most studies of this topic in two ways. First, it assumes that people care about housing services, $H$ , which are a function of the structural characteristics of housing, and public services, $S$ , but it treats these two components differently. Housing services are determined by a function that is common across households, and households are able to alter the amount of $H$ their house delivers. In contrast, house buyers treat public services as fixed and households must compete for entry into places where $S$ is high, which leads to household sorting.Footnote ³ This approach alters the algebra of a hedonic equation but does not alter the basic principles. As we will see, for example, it leads to expressions of MWTP for $S$ per unit of housing services, labeled MWTP^H. Because MWTP = (MWTP^H)(H) and our assumptions imply that sorting is not influenced by $H$ , we can readily move between the two MWTP concepts. The slope of the hedonic in Figure 1, for example, could be interpreted as MWTP^H.

Second, this approach provides a way to simplify the supply side. Rosen’s framework applies to goods with multiple traits. Heterogeneous firms each have an offer function for each trait, and the market price function is a joint envelope of the bid and offer functions. With restrictive assumptions, it is possible to derive a form for this joint envelope (as in Epple, Reference Epple1987). In the case of housing, however, housing suppliers do not supply neighborhood amenities and the focus has been on the demand side. We think this is entirely appropriate.

2.2 Rouwendal’s quadratic envelope

An envelope is a mathematical concept and further insight into the Rosen framework can be gained by showing the assumptions about bidding and sorting that are necessary to derive alternative forms for the hedonic envelope. The hedonic vices we identify below arise even without a formal treatment of bidding and sorting, but the use of this mathematical link helps clarify the sources of these vices and why they lead to faulty inferences about WTP. We start with a quadratic envelope derived by Rouwendal (Reference Rouwendal1992) and then turn to an envelope based on constant-elasticity demand functions developed by Yinger (Reference Yinger2015b ).

An envelope is the solution to two equations: a family of crossing curves, written in implicit form as $f\{P,S,Y\}=0$ (where $P$ and $S$ are variables and each member of the family has a different value of the parameter, $Y$ ) and $f_{Y}\{P,S,Y\}=0$ (where $f_{Y}$ is the derivative of $f$ with respect to the parameter). In our case, $f$ is a bid function: $P$ is the bid (or its log) for a unit of housing services, $S$ is the amenity, and $Y$ is income. As we will see, the $f_{Y}$ equation describes the sorting equilibrium.

Rouwendal’s (Reference Rouwendal1992) hedonic envelope assumes that the bid function takes the form:

where $M$ indicates demand determinants (including income, $Y$ ) and $S$ indicates a public service or amenity.Footnote ⁴ Rouwendal then treats $M$ as the parameter that varies across households. To find the envelope, therefore, Rouwendal differentiates (1) with respect to $M$ to obtain

Setting this derivative equal to zero and solving for $M$ yields

where $\unicode[STIX]{x1D6FC}_{2}<0$ . Now substituting equation (3) into equation (1) yields the envelope:

where

Three points are worth emphasizing. First, Rouwendal (Reference Rouwendal1992) explains that equation (3) shows the value of $M$ that maximizes the bid at a given value of $S$ . This interpretation fits the intuition of sorting, which allocates housing at a given location (= given value of $S$ ) to the household with the demand factors ( $M$ ) that lead to the highest bid for $S$ at that location. Another way to make this point is that $M$ determines the slope of a bid function at a given $S$ ; it is therefore the parameter that determines sorting. Thus, equation (3) describes the sorting equilibrium.

Second, the slope of the envelope reflects both the slope of the underlying bid functions and the slope of the sorting equilibrium. Bid functions generally get flatter as the amenity increases because of the law of diminishing marginal rate of substitution (MRS), but sorting implies steeper bid-function slopes as $S$ increases. In the Rouwendal case, the bid functions (equation (1)) are linear in $S$ , which corresponds to linear utility functions, that is, to a situation in which the MRS does not diminish as $S$ increases. In this case, bidding has the same impact on the slope of the envelope at all values of $S$ . It follows that the slope of the envelope has to increase as $S$ increases because the sorting factor is the only one at work.

Third, the distinction between bid functions and the envelope is analogous to the distinction between short-run average cost curves (SRACs) and long-run average cost curves (LRACs). An LRAC is the envelope of a set of SRACs. Each SRAC applies at a different value of the capital stock, and a set of SRACs is estimated with the capital stock as a variable.Footnote ⁵ Any cost regression that includes capital stock must be interpreted as estimating SRACs, although it is only correctly specified for this purpose if it also includes interactions between capital stock and output. The same lesson applies to the hedonic envelope. If demand variables are included, the regression examines bid functions, not a hedonic envelope, and a bid-function regression should include interactions between demand factors and amenities. Equation (1) has an interaction between $M$ and $S$ , for example. Without such interactions, all households have the same bid-function slope at a given $S$ and sorting is not possible. Moreover, a bid-function regression must treat the amenity variables as endogenous because unobserved demand traits affect both amenity choices and housing prices.

2.3 The constant-elasticity case

Yinger (Reference Yinger2015b ) provides a more general discussion of sorting and extends envelope derivations to the case of constant-elasticity demand functions for amenities and housing. He starts by deriving a general form for a household bid function. He assumes, like most of the literature, that household utility depends on a composite good, $Z$ , housing services, $H$ , and a public service or amenity, $S$ . Households maximize their utility subject to the constraint that their income equals spending on $Z$ , spending on housing, PH, and property taxes, which are levied at rate $\unicode[STIX]{x1D70F}$ on house value, $V=\mathit{PH}/r$ , where $r$ is a discount rate. $P=P\{S,\unicode[STIX]{x1D70F}\}$ is a bid per unit of $H$ that depends on $S$ and $\unicode[STIX]{x1D70F}$ . This well-known (Ross & Yinger, Reference Ross, Yinger, Cheshire and Mills1999) maximization problem leads to

where “ $\,\hat{\;}\,$ ” denotes a before-tax bid and $\mathit{MB}_{S}$ is the MWTP for $S$ . The role of property taxes is not important for our purposes and is therefore not considered.Footnote ⁶

Yinger then assumes that the demands for $S$ and $H$ take the following forms:

and

where $W$ equals tax price, $K_{S}$ and $K_{H}$ are constants, and $N$ and $M$ are vectors of preferences and other demand variables than income $Y$ , both observable and unobservable. LaFrance (Reference LaFrance1986) derives these forms from a model of “incomplete” demand, in which one set of commodities ( $Z$ ) is not observed but affects the observed commodities ( $S$ and $H$ ) through a price index, which appears in $N$ and $M$ . More specifically, LaFrance shows that these forms are consistent with the integrability requirements of a demand system.Footnote ⁷ This result is important here because it preserves the link between estimated price elasticities and household WTP.

Substituting the inverse of (7), which equals $\mathit{MB}_{S}$ , and (8) into (6) and assuming that $\unicode[STIX]{x1D708}=-1$ ,Footnote ⁸ yields the following differential equation:

where

The solution to this differential equation, that is, the bid function, is

where $C$ is a constant of integration. Note that the slope of a bid function at a given value of $S$ depends on $\unicode[STIX]{x1D713}$ ; that is, the role of $\unicode[STIX]{x1D713}$ in equation (11) is the same as the role of $M$ in equation (1).

Yinger’s next step is to find the envelope of equation (11) using the sorting theorem. He reverses the mathematical steps in Rouwendal. Instead of arbitrarily specifying the constant and then differentiating the bid function to find $\unicode[STIX]{x2202}f\{P,S,\unicode[STIX]{x1D713}\}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}$ , Yinger draws on the sorting theorem to specify $\unicode[STIX]{x2202}f\{P,S,\unicode[STIX]{x1D713}\}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}$ , which describes the sorting equilibrium, and then integrates to find the constant. This approach opens the door to a more general derivation of hedonic envelopes and ensures a link between the form of the envelope and the assumptions about sorting.

More specifically, Yinger (Reference Yinger2015a ,Reference Yinger b ) explores the case of one-to-one matching in which each household type, identified by a value for $\unicode[STIX]{x1D713}$ , sorts into a unique value for $S$ . In this case, Yinger shows that the nature of the sorting equilibrium does not directly depend on the distributions of $\unicode[STIX]{x1D713}$ and $S$ ; instead, it depends on the function that transforms the $\unicode[STIX]{x1D713}$ distribution into the $S$ distribution (or vice versa). Suppose, for example, that both $\unicode[STIX]{x1D713}$ and $S$ have normal distributions. Any normal distribution can be transformed into any other normal distribution with a linear transformation. In this example, therefore, the sorting equilibrium can be described by $S$ as a linear function of $\unicode[STIX]{x1D713}$ . To provide for a wider range of possible transformations between the $\unicode[STIX]{x1D713}$ and $S$ distributions, Yinger assumes that the sorting equilibrium can be described by

where the $\unicode[STIX]{x1D70E}$ terms are parameters to be estimated. Assuming a linear form for this equilibrium, as in the above example, corresponds to assuming that $\unicode[STIX]{x1D70E}_{3}=1$ . The sorting theorem states that household types with steeper bid-function slopes sort into locations with higher values of $S$ , so this theorem can be tested by determining whether $\unicode[STIX]{x1D70E}_{2}$ and $\unicode[STIX]{x1D70E}_{3}$ are both positive. Finally, note that equation (12) is analogous to Rouwendal’s equation (3), except that (12) is based on theory.

One-to-one matching is unlikely to hold exactly. Some household types may live in locations with more than one value of $S$ , for example, or some locations may be home to more than one household type. A method to consider these cases in a hedonic regression has been developed by Epple et al. (Reference Epple, Peress and Sieg2010), but this method is complex and restricts the analysis to a single amenity.Footnote ⁹ Moreover, Yinger (Reference Yinger2015a ) shows that one-to-one matching is likely to provide a close approximation to the case of one household type in more than one location. With many household types in a single location, however, no method can observe variation in WTP for more than one amenity across household types, because the price of housing in that location varies with amenity levels, but not with household bids.

Based on equations (11) and (12), Yinger (Reference Yinger2015b ) derives the bid-function envelope:

where $\unicode[STIX]{x1D706}_{1}=1+\unicode[STIX]{x1D708}$ ; $\unicode[STIX]{x1D706}_{2}=(1+\unicode[STIX]{x1D707})/\unicode[STIX]{x1D707}$ ; $\unicode[STIX]{x1D706}_{3}=\unicode[STIX]{x1D706}_{2}+1/\unicode[STIX]{x1D70E}_{3}$ ; $X^{(\unicode[STIX]{x1D706})}=(X^{\unicode[STIX]{x1D706}}-1)/\unicode[STIX]{x1D706}$ if $\unicode[STIX]{x1D706}\neq 0$ ; and $X^{(\unicode[STIX]{x1D706})}=\ln \{X\}$ if $\unicode[STIX]{x1D706}=0$ . The function $X^{(\unicode[STIX]{x1D706})}$ is the Box–Cox form. Figure 1 gives an example of this envelope. With multiple amenities, the terms on the right side other than the constant become summations across amenities, with different parameter values for each amenity. With $\unicode[STIX]{x1D708}=-1$ , the left side of equation (13) equals $\log \{\hat{P}^{E}\}$ , and this form can be incorporated into a standard log–linear regression with logged house value, $V$ , as the dependent variable, and with controls for structural housing traits, which determine housing services, $H$ , and the property tax rate; see Yinger (Reference Yinger2015b ).

As shown in Table 1, most parametric forms used for hedonic estimation (including linear, semilog, log–linear, Box–Cox, and quadratic) are special cases of (13). Some forms implicitly assume, for example, that $\unicode[STIX]{x1D707}=-\infty$ , which indicates a horizontal demand curve; a linear form also assumes that $\unicode[STIX]{x1D70E}_{3}=\infty$ , whereas a quadratic form assumes a linear sorting equilibrium ( $\unicode[STIX]{x1D70E}_{3}=1$ ). A semilog specification implicitly assumes that $\unicode[STIX]{x1D708}=-1$ and $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D70E}_{3}=\infty$ . Thus, most forms in the literature are based on stronger assumptions than those behind (13).

A key insight from (13) is that specifications consistent with sorting generally require two terms on the right side. The second term drops out if $\unicode[STIX]{x1D70E}_{3}=\infty$ , which, according to (12), implies that $S$ is not a function of $\unicode[STIX]{x1D713}$ , or, in other words, that sorting does not take place.Footnote ¹⁰ As the Rosen framework makes clear, the concept of sorting is essential for isolating WTP with heterogeneous households, so one-term hedonics should be regarded with great skepticism. A one-term Box–Cox specification consistent with sorting arises when $\unicode[STIX]{x1D70E}_{1}=0$ , that is, when $S$ is proportional to $\unicode[STIX]{x1D713}$ in equilibrium, so the first term in (13) drops out.Footnote ¹¹ This is Case 4C in Table 1.

2.4 Summary

In summary, the hedonic function is mathematically related to the bid-function envelopes. As we will see, the hedonic vices discussed in this paper are all related to this mathematical connection. Some of the vices reflect extreme assumptions about the bid functions or the envelope, others reflect an inconsistency between the forms assumed for these two functions, and still others arise because scholars confuse the bid-function and envelope concepts.

3.1 Functional form vices

3.2 Control variable vices

3.3 Interpretation vices

4.1 Evidence from Cleveland

This section explores hedonic vices with the Cleveland data set used in Yinger (Reference Yinger2015b ).Footnote ²⁷ Our analysis focuses on the price of housing in a CBG per unit of housing services, $H$ . We use a two-step procedure that appears in Epple et al. (Reference Epple, Peress and Sieg2010) and Yinger (Reference Yinger2015b ).Footnote ²⁸ The first step, which applies to a sample of 23,000 house sales, is a regression of the log of sales price on 17 structural characteristics of a house (including house age, house area, and lot area), 18 variables to measure within-block-group variation in amenities (including commuting distance and distance to a private school), and block-group fixed effects. These fixed effects summarize the impact of all neighborhood traits on sales prices. Complete results can be found in Yinger (Reference Yinger2015b , Table 1).

The second step, which applies to the sample of 1,665 block groups, is a regression of the coefficients of the block-group fixed effects (plus the first-stage constant term) on public services and neighborhood amenities. This dependent variable can be interpreted as the log of the after-tax price per unit of housing services, $P^{E}\{S\}$ . As shown in Tables 2–4, the data set for estimating this second step includes public services, neighborhood amenities, and neighborhood-level demand variables.Footnote ²⁹

We focus on the results for two measures. The first is the share of students in a school district who enter the twelfth grade and subsequently pass all five mandated state tests. Students are recorded as not passing if they either drop out before taking the tests or if they fail to pass all five before they graduate. The second is the share of a neighborhood’s population that is Black.Footnote ³⁰ In every case, these two amenity variables (and many others) are entered in quadratic form, so sorting is not ruled out. According to Table 1, this specification corresponds to an infinite demand elasticity and a linear approximation to the sorting equilibrium.Footnote ³¹

We divide the other public service and amenity variables into two sets. The set in Table 2, which we use in our “parsimonious” specification, consists of the variables that are, in our judgment, basic public service and amenity variables, representing the types of variables included in many previous studies. The set in Table 3, which we add for our complete specification, consists of additional public services and amenities that households seem likely to care about but which are rarely included in hedonic regressions. The regression results for the second step are presented in Table 5 and summarized in Figures 3 and 4. The results in this table use a quadratic specification for the continuous amenities; a linear alternative can be rejected at a high rate of confidence.

Figure 3 Comparison of alternative specifications, high-school passing rate.

Figure 4 Comparison of alternative specifications, neighborhood ethnicity.

In the parsimonious regression, the impact of high-school quality is large (the first two columns of row 1 in Table 5 and Figure 3). However, the results from the parsimonious regression are potentially biased from omitted amenities. Using both sets of amenity controls (row 2) enhances the statistical significance of key variables. Using this preferred model, a house in a district with a 0.77 passing rate sells for about 34% more than one with a 0.13 passing rate.Footnote ³² Adding the demand variables instead (row 3), has a much more dramatic effect, namely, to substantially lower the magnitude and significance of the school-quality variable. If taken at face value, these estimates imply that house value goes up only 12.2% when the district passing rate goes from 0.13% to 0.77%. As explained earlier, however, these estimates should be interpreted as a biased bid-function regression based on extreme sorting assumptions. Adding interactions with income (but, to keep the estimation manageable, not the other demand variables) leads to bid functions that vary with income (row 4, which gives results for the median block-group income). Along this bid function, plotted in Figure 3, the WTP for housing increases 18.8% from the district with the lowest passing rate to the district with the highest passing rate; as discussed earlier, this estimate is subject to endogeneity bias. As shown by the second two columns of Table 5 and by Figure 4, the results for neighborhood ethnicity also depend heavily on specification. The parsimonious regression (row 1 of Table 5 and Figure 4) indicates that house values drop 24.6% as one moves from 0% Black to 74% Black, but also drop 3.6% as one moves from 100% to 74% Black. Because largely Black neighborhoods tend to have poorer amenities, the impact of ethnicity is smaller when additional controls for amenities are added; to be specific, the second row of Table 5 indicates that house values are 20.3% lower at 74% Black than at 0% Black and 2.6% lower at 74% than at 100% Black. These highly significant results are consistent with the survey evidence, which indicates that some people are willing to pay more to live in a largely Black neighborhood than in an integrated or White neighborhood (Yinger, Reference Yinger2014).

As in the case of the high-school passing rate, adding income leads to an understatement of the amenity effect. The third row of Table 5 indicates that going from 0% to 63% Black (the minimum has shifted) lowers house values by 17.7%, whereas going from 100% to 63% Black lowers them by 6.7% (see Figure 4). Adding interactions between income and both the school and ethnicity variables yields (possibly biased) bid-function estimates. The bid function for the median homeowner income is tangent to the envelope in the second row at 10% Black and declines continuously as the percentage of Blacks increases. Interpreting this estimate as an envelope is obviously a serious error.

Overall, these results are consistent with the view that bid functions, which include demand variables, have different shapes than the envelope. With demand variables (but no interactions) included, the regression is a biased, no-sorting bid-function regression, not a hedonic. More reasonable results can be obtained for a bid-function regression when the required interactions with income are added, but this regression still is subject to endogeneity bias. We know of no instruments to solve this problem.

Under certain assumptions, these results are also consistent with the hypothesis that home buyers are able to determine income and other demand traits at the neighborhood level, consider these traits to be indicators of neighborhood quality, and adjust their housing bids (and hence the envelope) accordingly. These assumptions are that (a) neighborhood and household demand traits are uncorrelated and (b) neighborhood demand traits capture important dimensions of the home buyer’s perceptions of neighborhood quality beyond the neighborhood amenities in Table 3. We find both assumptions to be implausible: Sorting leads to relatively homogeneous neighborhoods, and the amenities in Table 3 cover a wide range of observable, statistically significant neighborhood traits. Scholars who include neighborhood-level demand variables in a hedonic regression should explain why they are willing to make these assumptions.

We also use the Cleveland data to examine how neighborhood fixed effects alter estimated results. In the case of the high-school variable, adding census-tract fixed effects to the parsimonious specification results in an envelope with a negative (but statistically insignificant) slope.Footnote ³³ Turning to neighborhood ethnicity, the envelope with tract fixed effects starts and ends at the same place as the parsimonious specification in Figure 4 but is essentially linear. These results do not prove, of course, that these changes in the slopes of the bid functions associated with tract fixed effects reflect the impact of tract-level demand factors. Instead, these results could signify that the tract fixed effects eliminate bias caused by the omission of nondemand tract traits. Nevertheless, these results are consistent with the possibility that the inclusion of tract or other small-area fixed effects introduces inappropriate controls for demand variables. These controls may be measured with error, because they ignore within-tract variation in demand, and they may be accompanied by legitimate controls for nondemand factors, but neither of these possibilities can justify their use. If the coefficients of the tract effects reflect demand factors to a significant degree, a possibility that cannot be ruled out a priori or tested, then the regression is a hybrid somewhere between a hedonic envelope and a set of bid functions – not a true hedonic.

4.2 A hedonic simulation

We conducted simulations to shed further light on the biases that may be caused by a hedonic regression that is misspecified and/or that includes neighborhood-level demand variables. These simulations are based on the forms in equations (10), (11), and (13). As discussed in Yinger (Reference Yinger2015a ), the assumption of one-to-one matching, which is used to derive equation (13), may not be a good approximation to an actual market equilibrium under some circumstances. These simulations therefore indicate possible biases but do not reveal whether these biases arise in any particular market.

Our simulation consists of four steps. The first step is to fill in a table of neighborhoods, each with a unique income and a unique combination of two neighborhood traits, school quality ( $S$ ) and an index of other amenities ( $A$ ). The second step is to create a table of household types, each with a unique combination of relative MWTP^H for the two neighborhood traits. The third step is to allocate household types to neighborhoods on the basis of their relative MWTP^Hs and then to use the winning household’s bids and a random error to define the market price, that is, the hedonic, in each location. The final step is to run several different hedonic regressions using the simulated data and to determine the extent to which the estimated average MWTP^H for a given regression provides a biased estimate of the “true” average MWTP^H from the simulation.Footnote ³⁴

This approach is similar in spirit to the elaborate, iterative simulation procedures devised by Kuminoff et al. (Reference Kuminoff, Parmeter and Pope2010) and Kuminoff and Jarrah (Reference Kuminoff and Jarrah2010), who build on Cropper, Deck and McConnell (Reference Cropper, Deck and McConnell1988). Our procedures are simpler because we assign a single household type to each neighborhood and because we eliminate the need for iteration by creating neighborhoods with every possible combination of the two amenities. With this approach, households can be allocated according to the standard sorting theorem, which says that households sort across the values of any given amenity according to the relative steepness of their bid function. As discussed earlier, this relative steepness (or, equivalently, relative MWTP^H) can be measured by the $\unicode[STIX]{x1D713}$ term in equation (10). Without a complete table of the two amenities, a household type with a very high $\unicode[STIX]{x1D713}_{A}$ might outbid a household type with a high $\unicode[STIX]{x1D713}_{S}$ in a high- $S$ , high- $A$ neighborhood. In this case, iteration is needed to determine the equilibrium household allocation. With our approach, however, every possible combination of the two amenities is available. Thus, a household type with a high $\unicode[STIX]{x1D713}_{A}$ can find a neighborhood with both a high level of $A$ and with the level of $S$ that is consistent with its $\unicode[STIX]{x1D713}_{S}$ . In other words, this household can obtain its preferred level of $A$ without having to “overrule” the bid of a household with a higher $\unicode[STIX]{x1D713}_{S}$ .

Our approach also differs from Kuminoff et al. (Reference Kuminoff, Parmeter and Pope2010) and Kuminoff and Jarrah (Reference Kuminoff and Jarrah2010) because we focus on housing bids per unit of $H$ , not overall bids. As discussed earlier, the standard model in the literature is one in which households sort based on public services and amenities, but not on the basis of the structural characteristics of housing. In the long run, housing services adjust to match the sorting equilibrium, not the other way around. This approach matches the empirical results in the previous section, which look at housing prices per unit of $H$ . Measures of WTP per unit of $H$ must, of course, be multiplied by $H$ before they can be used in benefit-cost analysis. Our simulations would add no insight to this step, however, because they are designed to shed light on an analysis of sorting, not on the determinants of $H$ .Footnote ³⁵

Although we switch to a per-unit-of- $H$ measure, we follow Kuminoff et al. (Reference Kuminoff, Parmeter and Pope2010) by focussing on average MWTP. This approach is not an endorsement of average MWTP as a welfare measure. Instead, it is based on the idea that average MWTP provides a reasonable way to summarize the extent to which a given empirical approach leads to a pattern of MWTP estimates that differ from the “true” underlying pattern.

One key feature of previous simulations and of ours is that they draw a new set of amenity values for each run. Kuminoff et al. (Reference Kuminoff, Parmeter and Pope2010) implement this step by drawing a new sample of houses including the amenities in their neighborhoods; our approach is to draw neighborhood incomes and associated amenity levels. To obtain the “true” values for MWTP in each run, Kuminoff et al. then average household MWTP across the amenity values associated with that run.Footnote ³⁶ In these simulations, household bids are nonlinear functions of amenity levels, so the average MWTP varies across runs not only because households and bids differ, but also because the calculations are conducted for different sets of amenity values. A better approach, in our view, is to find average MWTP (or, in our case, average MWTP^H) across runs for the same set of amenity values. Our objective is to determine bias in estimates of average MWTP, not to determine how estimates of average MWTP vary as amenity levels vary. Kuminoff et al. cannot follow this approach because they do not observe households or MWTP at amenity values other than those observed for a given run. We can follow this alternative approach by making use of the hedonic envelope in equation (13), which indicates the MWTP^H at all values of an amenity and therefore can be used to find the average MWTP^H for a selected set of amenity values.

In these simulations, bids and hence the hedonic do not depend on neighborhood income. As in an actual housing market, however, neighborhood income is correlated with the level of the amenities. This correlation reflects past sorting and may also reflect amenity choices by voters, as in Epple et al. (Reference Epple, Romer and Sieg2001). Moreover, as shown by equation (10), the $\unicode[STIX]{x1D713}$ terms are functions of household income, $Y$ , among other things. Because high- $\unicode[STIX]{x1D713}$ households sort into high-amenity locations, these simulations (and actual housing markets) are characterized by a correlation between household income (as represented by the $\unicode[STIX]{x1D713}$ terms) and neighborhood income. These simulations are designed to determine whether this correlation results in biased coefficients for the amenity variables when neighborhood income is included in a hedonic regression. These simulations also show whether and by how much estimated average MWTP^H is biased when the hedonic regression is misspecified. Key parameters for the simulations are based on the analysis of Cleveland area data in Yinger (Reference Yinger2015b ).

Illustrative results from our simulation for the school-quality amenity, $S$ , are presented in Table 6. These results describe cases with different price elasticities of demand for $S$ ; different values for the exponent in equation (12), which characterizes the sorting equilibrium; and different correlations between $S$ and neighborhood income, NI.Footnote ³⁷ In all these cases, a quadratic specification provides a good approximation to the underlying hedonic and to the “true” average MWTP^H – which is derived using equation (14). Adding NI to the quadratic regressions has little impact on the estimated average MWTP^H, and the coefficient of NI is, as theoretically predicted and discussed earlier, usually small and insignificant. This table also shows, however, that a linear specification may yield a severe underestimation of average MWTP^H, and that adding NI to a linear regression may boost this underestimation by 40% or more.Footnote ³⁸

These results do not prove that a quadratic form always provides a good approximation to the true average MWTP^H, that a linear form always understates average MWTP^H, or that adding NI always makes this understatement worse. Nevertheless, these results reinforce two key conclusions of Section 3: (a) linear forms should be met with great skepticism and (b) adding small-neighborhood demand measures to a hedonic, particularly one with a simple specification, may lead to downward bias in estimates of average MWTP^H. These results also suggest a procedure for minimizing these problems, namely, to estimate hedonic regressions with quadratic forms, which allow for sorting and appear at least to minimize the underestimate of average MWTP that can arise when neighborhood-level demand variables are included.Footnote ³⁹

In summary, neighborhood-level household demand variables should not be significant when they are added to a hedonic regression that is specified correctly with no omitted variables. In practice, however, these variables may prove to be significant for two reasons. First, they may be either neighborhood traits themselves or proxies for unobserved neighborhood traits. Second, they may help predict prices when added to a misspecified hedonic regression, particularly one that assumes a constant MWTP (or a constant MWTP^H – these lessons apply to both concepts). This situation poses a challenge for scholars, because these two possibilities cannot be formally distinguished. Estimating a regression without neighborhood-level demand traits may lead to omitted-variable bias and an associated overstatement of average MWTP, whereas including these variables may result in a severe underestimate of this average. The vast majority of existing studies include neighborhood-level demand variables or small-neighborhood fixed effects while assuming that MWTP is constant. The authors of these studies interpret these results to mean that the demand variables eliminate omitted-variable bias. The analysis in this paper indicates that a more likely explanation is that these demand variables are significant only because the functional form for MWTP is not correct.