Housing Demand and Expenditures: How Rising Rent Levels Affect Behavior and Costs-of-Living over Space and Time David Albouy

Gabriel Ehrlich

University of Illinois and NBER

Congressional Budget Office

Yingyi Liu ∗ University of Illinois November 2014

∗ Albouy:

[email protected]; Ehrlich: [email protected]; Liu: [email protected]. During work on this project, Albouy was supported by the NSF grant SES-0922340 and was a David C. Lincoln Fellow. Any mistakes are our own. The views in this paper are the authors’ and should not be interpreted as the views of the Congressional Budget Office.

Abstract The share of income spent on housing has grown since 1970, along with incomes and the relative price of housing. This rising share is consistent with housing being a necessity if demand is sufficiently price inelastic. We estimate housing demand parameters using compensated and uncompensated frameworks over space and time, testing restrictions imposed by demand theory and household mobility. The estimates are largely consistent across frameworks and suggest that housing demand is slightly price and income inelastic, and obeys the restrictions of demand theory. We construct a non-homothetic constant-elasticity-of-substitution cost-of-living index that reflects housing’s greater importance in the expenditure bundle when housing prices are high and incomes are low. Keywords: Housing demand, affordability, cost-of-living, inflation, non-homothetic preferences, consumer economics. JEL Numbers: D12, E31, R21

1

Introduction

The share of Americans’ household budgets devoted to housing has risen considerably over the last four decades. Figure 1 illustrates this trend using several sources of data: the Bureau of Economic Analysis (BEA) indicates that the share of national income devoted to housing rose by 2 percentage points from 1970 to 2012, while the American Housing Survey (AHS) and Consumer Expenditure Survey (CEX) indicate a 7 percentage point increase over similar time periods. At the same time, the AHS, CEX, and Census all indicate that the typical share of income spent on rent by renting households rose by 10 percentage points, despite similar home-ownership rates of 64 percent in both 1970 and 2012. These trends support the recent claim by the Secretary of Housing and Urban Development that, “We are in the midst of the worst rental affordability crisis that this country has known” (Olick 2013). Economically, these trends are surprising, as household incomes have risen over the past four decades and housing has historically been viewed as a necessity. This means that the income elasticity of housing demand is less than one, and the share of expenditures devoted to housing should fall as incomes rise. Yet, the rising expenditure share suggests that housing may in fact be a luxury good. One possible resolution to this apparent contradiction lies in the considerable increase in the price of housing relative to the prices of other goods in the economy, which Figure 1 illustrates with the price index for shelter (the main component of housing) relative to non-shelter goods. If housing demand is price inelastic, the expenditure share on housing will rise as housing becomes more expensive relative to other goods, due to weak substitution effects. Figure 2 displays this possibility graphically. The rising relative price of housing suggests that the production possibility frontier (PPF) has expanded further in the direction of non-housing goods than in the direction of housing. Many non-housing goods may be traded internationally, but the production of housing depends on scarcer local factors, such as urban land, and may be subject to slower technological improvements. In that case, households will optimally increase their consumption of non-housing goods far more than of housing goods because of both income effects (illustrated by the movement from point A to point B in the figure) and substitution effects 1

(illustrated by the movement from point B to point C). The income effect causes the expenditure share to fall, seen in the distance between points B and D. The rise in the relative price, determined by the slope of the PPF, causes housing’s share to rise due to the limited substitutability between housing and non-housing goods: at the new prices, consumption would need to be at point E, not C, for the share spent on housing to not rise. In this paper, we investigate both cross-sectional and time-series data to identify the principal features of housing demand using an intuitive, yet adequately rich, framework. We demonstrate that cross-sectional data lends itself to estimating both uncompensated (Marshallian) and compensated (Hicksian) housing demand functions. The latter is based on the common assumption that the mobility of some households across cities acts to equalize the utility households receive from living in different locations. Because that assumption is not plausible over time, the time-series data lends itself only to estimating uncompensated demand. In either case, understanding housing demand is useful for measuring changes in costs-of-living over space (e.g. across cities) and time (i.e. inflation) realistically through a price index that incorporates substitution and income effects. Our investigation employs a parsimonious framework featuring housing expenditures, housing prices, measures of income, and more originally, non-housing prices. Unlike previous authors, we use data on non-housing prices to test restrictions imposed by demand theory, which serve as a check on the validity of our empirical methodology. Under such restrictions, we integrate a demand equation into non-homothetic utility and expenditure functions with a constant elasticity of substitution. These functions should be useful to researchers interested in housing consumption behavior or in how changes in cost-of-living affect welfare. The analysis also provides an unconventional examination of demand theory by using spatial variation, rather than more conventional temporal variation (e.g. Deaton 1986, Blundell et al. 1993). Combining cross-sectional evidence from various sources, it appears that the uncompensated own-price, income, and substitution elasticities are all near 2/3 in absolute value. The time series patterns are consistent with the cross-sectional results, but exhibit an additional secular trend towards greater housing consumption that invites further investigation. We also discuss potential

2

problems from aggregation bias, as well as how the changing income distribution should affect overall housing demand.

2

Motivation and Related Literature

Mirroring our own findings, the Joint Center for Housing Studies of Harvard University (JCHS, 2013) documents that from 2000 to 2012, the median share of renters’ incomes devoted to contract rent rose nearly five percentage points to 27.4 percent and that 28 percent of renting households now spend more than half of their incomes on rent. Concerns about housing affordability are most acute when housing is a necessity and demand is price inelastic. This is particularly true for low income households, who have experienced few income gains in America’s largest, most expensive cities (Baum-Snow and Pavan, 2013). Low-skilled households’ inability to substitute away from expensive housing appears to account for their choosing to live in cheaper cities (Moretti 2013), while those who remain in expensive cities must earn higher wages relative to other skill groups (Black, Kolesnikova, and Taylor 2009).1 Households’ ability to substitute between housing and non-housing goods is a key factor affecting house prices, tax incidence, and population density within and across metropolitan areas. When substitutes for housing are limited, housing demand is price inelastic, and increases in demand, for instance due to rising incomes, will lead to large local price increases in places where housing supply is price inelastic. Regardless of supply, housing will constitute a larger share of household budgets in areas where, or eras when, it is more expensive. If instead, demand is unit elastic, housing expenditures for similar households should not change across space or time. Accurately measuring the price elasticity of housing demand is important to understanding whether the recent resurgence of housing prices in some markets is sustainable or a potential signal of a new housing bubble. Recent price increases are likely be more sustainable if housing demand is 1

Using grocery data, Handbury (2013) estimates a non-homothetic log-logit utility function with a CES superstructure to argue that high-income households may find large cities to be more “affordable” because they contain a greater range of groceries suited to their tastes. We reinforce this conclusion by finding the large cities are more affordable for high-income households as they spend less on housing.

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inelastic. In that case, rents in high-priced markets may continue to take a larger share of income, and housing policy may need to prioritize problems in housing affordability and production over issues related to lending and credit. Economists’ interest in housing demand and expenditures has a long and distinguished history, featuring a wide range of estimates of the price and income elasticity of housing demand. Articles reviewed in Mayo (1981) find uncompensated price elasticities that range from slightly positive to less than minus one. Popular estimates in the middle include Pollinsky and Ellwood’s (1979) estimate of -0.7 and Hanushek and Quigley’s (1980) estimates of -0.64 in Pittsburgh and -0.45 in Phoenix.2 More recently, Davis and Ortalo-Magne (2009) argue that the median expenditure on housing among rentals is roughly constant across metro areas, implying a price elasticity of negative 1. Few articles estimate elastic price demand, with elasticities greater than one.3 While some studies use non-housing price data to deflate their numbers, none actually use it to test the validity of the housing demand specification, as we do here. The income elasticity of demand for housing, which measures households’ propensity to consume more housing services as their incomes grow, also has important implications for house prices and quantities. Classical studies, such as Engel (1857) and Schwabe (1868), tended to find an income (or more precisely, an expenditure) elasticity for housing demand of less than one, which became known as “Schwabe’s Law of Rent”.4 As summarized by DeLeeuw (1971), Mayo (1981) and later Harmon (1988), most studies have been consistent with Schwabe’s Law, with some important exceptions. For instance, Muth (1960) estimated an income elasticity well over one. One major source of differences among studies is on how to measure income income: most researchers suggest using a measure of “permanent income” – of which various varieties have been proposed – to correct for attenuation bias caused by transitory income shocks. For owner-occupiers, it also 2 There is a large literature on this topic, including Muth (1960), Reid (1962), Rosen (1985), Goodman and Kawai (1986), Goodman (1988) Ermisch et al. (1996), Goodman (2002), and Ionnides and Zabel (2003). Most estimate uncompensated price elasticities ranging from -1 to -0.3 and income elasticities from 0.4 to 1. 3 Kau and Sirmans (1979) estimated price elasticity shifting from -2.25 to -1 from year 1876 to 1970 using historical data from Chicago. However, these are based off land-price gradients and are not robust to expected sorting behaviors or changes in commuting costs. 4 Some confusion regarding Engel’s findings stems from Wright’s (1875) confused statement of the results in English; see Stigler (1954) for a discussion.

4

makes sense to include implicit rental income from home equity.5 With such disparate findings, theoretical models of housing demand have taken great latitude in modeling housing demand. Many models in urban and macro-economics assume a fixed demand for housing, perfectly inelastic to price and income. This provides a simple derivation of the mono-centric city model, seen in Mills (1967), and used for urban welfare accounting in Desmet and Rossi-Hansberg (2013). Other models, such as the search and matching one of Piazzesi and Schneider (2010), assume housing demand is responsive to price but not income, as with quasilinear preferences. Another common approach is to specify preferences as Cobb-Douglas, implying price and income elasticities of one. Examples include Eeckhout (2004) and Davis and OrtaloMagne (2011), Michaels, Rauch and Redding (2012), and Guerreiri, Hartley and Hurst (2013). While a certain level of abstraction is a necessary component of a useful and tractable economic model, we must remain aware of the limitations such assumptions have on the conclusions we make about housing and its interaction with other markets. The secular rise in housing expenditures appears to be understudied by economists. Piketty (2014) finds that the value of residential capital relative to economic output has increased substantially over the last hundred years.6 Gyourko, Sinai, and Mayer (2013) find that the difference in housing values between the typical and highest-price locations has widened considerably over the last five decades. In addition, Davis and Heathcote (2007) present evidence of persistent real growth in land values, which accounts for an increasing share of housing costs over recent history. These findings are consistent with limited substitution possibilities between land and non-land inputs in housing production, as found in Albouy and Ehrlich (2012). Those authors, as well as Davis and Palumbo (2007), document that land values are extremely heterogeneous across time and space. Theoretically, as population and incomes rise, an inelastic derived demand for land can 5 See Hansen et al. (1998) and references therein for estimates less than one, Larsen (2002) for an estimate of approximately one, and Cheshire and Sheppard (1998) for an estimate greater than one, noting that the latter study estimates elasticities for housing attributes rather than for a unified bundle. 6 We note that housing is a capital asset that provides a flow consumption services to its owner. This asset is a composite of land and structure, the latter of which typically depreciates over time. We follow the bulk of the literature in estimating demand for a composite housing good, but the shape of the housing demand function can have important implications for land values separately from housing values. For more on the production of housing services from local land and construction inputs, we recommend Albouy and Ehrlich (2012).

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lead to land values taking up an ever increasing share of the economy.7

3

Household Demand for Housing in a System of Cities.

3.1

Household Budgets and Preferences

We use a standard static model of housing demand, and embed it in a richer equilibrium framework with local household amenities, similar to Rosen (1979) and Albouy (2008). The national economy contains many cities, indexed by j, which share a population of potentially mobile households. Households supply labor in their city of residence; they consume a housing good y with price pj , and a non-housing good x with price cj .8 Households earn total income mj = I + (1 − τ )wj , determined by a constant unearned income, I, and which varies due to local wage levels wj , after taxes, τ . In this static setting, household expenditure equals household income. Household preferences over the consumption good, housing, and location are modeled by a utility function U (x, y; Qj ), where Qj represents a city-specific amenity conceptualized as “quality-oflife”. The indirect utility function for a household in city j is then given by V (pj , cj , mj ; Qj ) = maxx,y (U (x, y; Qj )|cj x + pj y = (1 − τ )wj + I). The expenditure function for a household in city j is likewise given by e(pj , cj , u; Qj ) = minx,y (cj x + pj y|U (x, y; Qj ) ≥ u).

3.2

The Housing Expenditure Share and Uncompensated Demand

In order to take the model to the data, we approximate the relationships described above around their national average values. Denote the fraction of household expenditures on housing in city j 7

This may happen if land-saving technological improvements are weak or stifled by regulation. Rising demand may reverse earlier declines in land values engendered by transportation improvements. Even in a modernizing economy, if the demand for housing is price inelastic, Ricardo’s (1817) and George’s (1879) concerns that land may accrue an outsize share of national income may be more relevant than they have seemed for much of the past century. This reasoning is based on insights from standard two-sector models, although it deserves a full formalization. 8 For simplicity, the exposition of the theoretical model will refer to a system of cities and call individual geographical units as such. However, the empirical work using the Consumer Expenditure Survey data will be at the state level, and the empirical work using the Census data will also use the non-metropolitan portions of states. Therefore, the geographies considered in this model are more properly conceived as ‘areas’, with the term ‘city’ used for concreteness rather than precision.

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as sjy ≡ (pj y j )/mj . Log-linearizing this equation produces the identity ˆ j. sˆjy = pˆj + yˆj − m

(1)

A hat over a variable represents its log deviation from the (geometric) national average, i.e., zˆj = d ln z j = dz j /¯ z . We take local price and income levels as parametric, so that the only behavioral variable in the share is housing consumption, y. Consumption is determined by the uncompensated (Marshallian) demand function y j = y(pj , cj , mj ; Qj ). Log-linearizing this function produces ˆj . yˆj = y,p pˆj + y,c cˆj + y,m m ˆ j + y,Q Q

(2)

The parameter y,p is the uncompensated own-price elasticity of housing demand, y,c is the uncompensated elasticity of housing demand with respect to non-housing prices (or cross-price elasticity), y,m is the income elasticity, and y,Q is the elasticity with respect to quality of life. Equation 2 is an identity for infinitesimal changes, and an approximation for larger changes. If housing is a normal good, then y,m > 0, and housing obeys the law of demand, y,p < 0. It is unclear whether housing is a gross substitute for non-housing goods, i.e., whether y,c > 0, because the cross-price elasticity will exhibit positive substitution effects and negative income effects of unknown magnitudes. Housing could also be a gross complement or substitute for non-market quality of life, i.e. y,Q ≷ 0, if amenities somehow alter the marginal rate of substitution between housing and non-housing goods.9 We combine equations 1 and 2 to demonstrate how expenditure shares depend on behavioral responses to local attributes: ˆj pj + y,c cˆj + (y,m − 1)m ˆ j + y,Q Q sˆjy = (1 + y,p )ˆ 9

(3)

We have not modeled how households with low tastes for housing may be inclined to seek out more amenable areas (see Black et al. 2002). Albouy and Lue (2014) present evidence that household sizes, age, and marital status vary little across metropolitan areas (they vary more within), suggesting such selection issues are not of first-order importance.

7

Unrestricted, equation (3) is merely definitional. Rationality of preferences requires that the demand function be homogenous of degree zero in prices and income (p, c, m), so that y,p + y,c + y,m = 0. This restriction requires that there be “no money illusion,” so that proportional increases in all prices and income do not lead to changes in behavior. Adding a constant to equation 3 motivates the following regression equation:

ln sjy = α0 + α1 ln pj + α2 ln cj + α3 ln mj + α4 q j + ej = α0 + α1 (ln pj − ln cj ) + α3 (ln mj − ln cj ) + α4 q j + ej

(4) (5)

Equation 5 follows 4 from imposing the homogeneity assumption as α1 + α2 + α3 = 0. If we demean the right-hand side variables, the regression coefficients α0 = ln s¯y , α1 = 1 + y,p , α2 = y,c , and α3 = y,m − 1. s¯y = eα0 is the geometric mean of expenditure shares. The own-price uncompensated elasticity is simply the coefficient on housing prices minus one, y,p = α1 − 1, while the income elasticity is the coefficient on income plus one, y,m = α3 + 1. Quality of life cannot be observed directly but only proxied by observable amenities, qj , meaning y,Q cannot be identified in a fully cardinal sense without additional assumptions. The same holds true of other demand shifters. Consistent estimation of this equation requires that nonhousing goods are properly accounted for by the index cj , that preferences across cities are the same, that preferences can be aggregated across households, and that we have an appropriate (arguably permanent) measure of income mj .

3.3

Compensated Demand with Household Mobility and Heterogeneity

The uncompensated demand function may be converted into a compensated (Hicksian) demand function by substituting in the expenditure function, i.e. y H (p, c, m; Q) = y(p, c, e(p, c, u; Q); Q). Alternatively, we may log-linearize the expenditure function directly, yielding ˆ j + m,u uˆj m ˆ j = s¯y pˆj + (1 − s¯y )ˆ cj + m,Q Q

8

(6)

where m,u is the elasticity of expenditures with respect to utility, and m,Q is the elasticity of expenditures with respect to quality of life. Substituting equation 6 into equation 3 yields a relatively complicated equation that may be simplified using the Slutsky equations. These give the relationships among the uncompensated, or Marshallian, price elasticities, and the compensated, or Hicksian, price elasticities: y,p = H y,p − H H s¯y y,m and y,c = H y,c − s¯x y,m . Here y,p and y,c represent the compensated elasticities of housing

demand with respect to housing prices and consumption prices, respectively.10 Rationality requires that compensated demand functions are homogenous of degree zero in prices. This means that the H own and cross-price elasticities of compensated housing demand should sum to zero, H y,p +y,c = 0.

Combining these insights yields the following equation for differences in the expenditure share in terms of relative prices, quality of life, and utility: ˆj sˆjy = (H pj − cˆj ) + (H uj + (H y,p + 1 − s¯y )(ˆ y,u − m,u )ˆ y,Q − m,Q )Q

(7)

H Here H y,Q is the compensated elasticity of housing demand with respect to quality of life and y,u

is a similar elasticity for income. To simplify further, we impose the restriction households are equally well-off across cities. When households are sufficiently mobile, then prices and wages equilibriate so that households are indifferent across inhabited locations. In that case, utility for a particular type of household should not vary across cities. Rather, utility differences only represent inherent differences across households, such as different earnings potentials. We parameterize income in city j as mj = ζ j wj , where ζ j is the skill level and wj is the wage level that compensates these household for living in that city.11 To interpret the coefficient, we posit that our utility function is money metric around national The first substitution yields sˆjy = (1 + y,p − s¯y + s¯y y,m )ˆ pj + [y,c − (1 − y,m )(1 − s¯y )]ˆ cj + (y,Q − (1 − ˆ j − (1 − y,m )m,u u y,m )m,Q )Q ˆj . Besides the Slutsky equations we also substitute in the identities H y,Q = y,Q + H y,m m,Q and y,u = y,m m,u to get the resulting equation. 11 When household types vary within city, the compensating wage differences will vary according to their tastes for housing, quality of life, and taxes. 10

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averages: u(x, y; Q) = e(¯ p, c¯, u˜(x, y; Q), Q). This added simplification allows us to write utility differences in terms of differences in the skill index uˆj = ζˆj , and impose m,u = 1 and H y,u = y,m . Note that we implicitly impose the restriction that the skill index affects housing consumption through income, and not through differences in tastes.12 These simplifications yield ˆj sˆjy = (H pj − cˆj ) + (y,m − 1)ζˆj + (H y,p + 1 − s¯y )(ˆ y,Q − m,Q )Q

(8)

Equation 8 then motivates the following empirical specification using data across cities: ln sjy = β0 + β1 pˆj + β2 cˆj + β3 ζˆj + β4 q j + ej

(9)

= β0 + β1 (ˆ pj − cˆj ) + β3 ζˆj + β4 q j + ej

(10)

where β0 = ln s¯y , β1 = H y,p + 1 − sy = −β2 and β3 = y,m − 1. Similar to the uncompensated β0 ˆj case, s¯y = eβ0 and y,m = β3 + 1, but H y,p = β1 − 1 + e . In practice, ζ is an index estimated

from the average log wages households would earn in a typical city based on their human capital and other location-invariant characteristics. The main testable restriction here is that the coefficients on the price of housing and nonhousing goods should be of opposite signs and equal magnitudes, i.e., β1 + β2 = 0. This may be seen as a joint test of both demand theory and mobility.13 When this restriction holds we use the elasticity of substitution between housing and non-housing goods, σD ≡ −(ˆ y j − xˆj )/(ˆ pj − cˆj ) = −H y,p /(1 − s¯y ), so that β1 = (1 − s¯y )(1 − σD ). When the elasticity of substitution is less (greater) than one, housing demand is said to be price inelastic (elastic), and the expenditure share of housing rises (falls) with the relative price of housing, p/c. One advantage of the compensated specification is that it estimates the elasticity of substitution without directly using information on income. Quality of life amenities may affect the income share of housing if H y,Q 6= m,Q , which means 12 If households with more skills like housing less (more) than those with fewer skills, the income elasticity estimate will be biased downwards (upwards). 13 If mobility does not hold, then the coefficients would not be equal. Income effects in the uncompensated elasticities would likely push coefficients on both housing and non-housing prices downwards.

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amenities and housing are either are net complements or substitutes. Such relations are not obvious a priori. If 0 > H y,Q > m,Q , compensated improvements in quality of life reduce housing consumption less than other consumption. Households could spend more on their properties to enjoy a nice climate. The opposite could also be true: nice weather may make people spend time away from their houses, while cold weather could cause them to consume more housing.

3.4

Potential Biases

Most examinations of housing demand do not consider variation in non-housing prices. Yet, these are potentially important as non-housing prices co-vary positively with housing prices. In places where housing is expensive, labor-intensive services, such as haircuts and restaurant meals, also tend to be expensive. This covariance arises because the workers who provide those services must purchase local housing. Such services may also contain a land component in their cost. We model this covariance with the projection cˆj = ρˆ pj + v j

(11)

where ρ > 0 and v j is white noise. Substituting this projection into equation 8, together with the elasticity of substitution, gives ˆj sˆjy = (1 − s¯y )(1 − σD )(1 − ρ)ˆ pj + (y,m − 1)ζˆj + (H y,Q − m,Q )Q

(12)

We may write the estimated elasticity of substitution as a function of the parameters σ ˆD = 1 − β1 /[(1 − eβ0 )(1 − ρ)] The higher is ρ, the more ignoring non-housing prices will bias βˆ1 towards zero and σ ˆD towards one.14 An upward bias in the estimated price elasticity of housing is likely to occur if worker skill levels are omitted from the regression. As seen in Moretti (2013), higher skilled individuals tend 14

Technically, Davis and Ortalo-Magne’s (2011) data support an elasticity of substitution of 0.85. However, their index of rental costs differs from ours by controlling for commuting costs, and thus exaggerating the actual price differences faced by households (e.g. that suburban dwellers in the New York suburbs face Manhattan prices), biasing their results towards one. Their study also suffers from the other potential biases we correct for.

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to locate in areas with more expensive housing. Formally, if ζ j is positively correlated with pj , and housing is a normal good, i.e., y,m > 0, then excluding a measure of skill will bias β1 downwards. For instance, a negative relationship between prices and expenditure shares induced by households in more skilled cities spending a smaller fraction of their incomes on housing could lead to the inference that demand is highly price elastic even if the true elasticity is unity or lower. Another potential bias stems from using renters. It could be the case that households are less likely to own, and more likely to rent, in more expensive areas. In that case, only the poor rent in cheap cities, while more affluent households also rent in expensive cities. If housing is a necessity, and our controls for utility are incomplete, then selection bias of this kind would make the estimated expenditure-rent gradient more negative. One may attempt to deal with this problem by controlling for the home-ownership rate, but this rate is also endogenous. Finally, there is the issue of unobserved taste in housing. Households that care more for housing will prefer to locate in areas where housing is less expensive. This selection effect would also make the expenditure-rent gradient more negative, as those who wish to consume the least amount of housing locate in the most expensive areas.

3.5

Non-Homothetic Utility and Expenditure Functions for Housing

In modeling housing demand, economists almost invariably use utility functions that are homothetic or quasilinear, imposing income elasticities of one or zero. Since housing is a major consumption item, in many applications we believe it may be worth using a more general utility function that allows for non-homotheticity. Here, we propose to use a non-homothetic separable family CES function from Sato (1977), with Qj = 1. To minimize on new notation, we write it as " U (x, y; 1) =

γσδx

σ−1 σ

+ 1 − σ − γδ

γσ(δ − 1)y

σ−1 σ

+ 1 − σ − γ(δ − 1)

σ # γ(σ−1)

(13)

This function contains three paramters: a distribution parameter δ, a substitution paramter, σ, and a non-homotheticity parameter, γ. In the limit, as γ → 0 this function becomes a standard

12

CES function. Our restricted log-linear model provides three parameters that map well to this utility function. We demonstrate in the appendix that when all of the variables are demeaned that β0 = σ ∗ln(1−δ) = ln s¯y , β1 = (1− s¯y )(1−σ), β3 = −γ(1− s¯y )(1−σ). Inverting, the parameters can be expressed recursively as σ = 1 − β1 /(1 − eβ0 ), δ = 1 − eβ0 /σ , γ = −β3 /β1 . We may construct an ideal cost-of-living index using the expenditure function

e(p, c, u; 1) =

 

1−σ σ γ(1−σ)

1−σ

σ

 1  1−σ

c δ u + p (1 − δ)  iσ  1 γ + (1 − σ − γδ)(1 − u γ(1−σ)  σ ) γσ h

(14)

This expenditure function differs from a standard CES function: when γ(1 − σ) > 0, then at higher levels of utility, households place greater importance on non-housing costs, c. By choosing a base level of utility and prices, we may then construct a cost-of-living index, which we detail below.

4

Data

The main variables in our inter-metropolitan analysis are the income or expenditure share devoted to housing, a price index of for housing, a price index for non-housing goods, and intermetropolitan income and potential earnings differentials. We also consider metropolitan amenity levels. The primary data we use in the analysis comes from the 2000 Census 5 percent microdata samples from IPUMS, which we use to calculate the expenditure share devoted to housing across different metropolitan areas, a housing price index across areas, and wage and income differentials across areas. We measure the prices of non-housing goods across metro areas using the panel of inter-area price differences for the years 1982-2012 constructed by Carrillo et al. (2013). The analysis uses several auxiliary data sources, which are cited throughout the paper. Although the theoretical relationships in section 3.2 are straightforward, in practice several decisions are necessary to take the model to the data. First is the decision on whether to include homeowners in the sample. Ideally both renters and homeowners would be included in order to have a more representative sample. However, the inclusion of homeowners can be problematic

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because homeowners’ monthly payments depend heavily on factors such as when they bought their homes and the terms of their mortgages, rather than on the inherent characteristics of the home. These factors can cause homeowners’ monthly payments to diverge from the theoretical quantity of interest, the value of the service flow that homeowners receive from their housing. This paper reports results both including and excluding homeowners from the estimation. In specifications that include homeowners, they are assigned an imputed annual rent of 6.2 percent of the reported value of the home, following Albouy and Hanson (2014). Second, in the static model of the theory, expenditure is equal to income, but this equality does not hold in dynamic reality. The difference between the two quantities leads to an ambiguity when calculating the expenditure share devoted to housing. In the main results, we calculate the expenditure share on housing using wage and salary income as the denominator. Third, it is necessary to decide whether to include expenditures on utilities such as electricity, fuel, and water in housing expenditures. The main analysis uses the inclusive, or ‘gross rent’ concept.

4.1

Rental Indices

To operationalize the concept of relative housing prices across cities, we construct inter-metropolitan price indices using hedonic regression methods. We run regressions of the form:

P ln(Pij ) = αP + βX Xij + δjP + Pij

(15)

where Pij is the rent or imputed rent for unit i in area j, XijP is a vector of unit characteristics, and δjP is a set of area fixed effects. We take δˆjP as our inter-area housing price indices, subtracting the national average so that the indices may be interpreted as the percentage by which housing prices in a given area differ from the national average. The appendix describes the covariates XijP in detail. Appendix table A1 displays the housing price indices indices at the metropolitan level using

14

the 2000 Census 5 percent data sample. The table displays indices calculated using only rental units and indices calculated using rental and owner-occupied units. In both cases the indices have been normalized so that the average index value weighted by (the square root of) MSA population is zero. Generally, large and coastal cities have more expensive housing by these measures. For instance, the New York, NY-NJ MSA has a rental index 44 log points higher than the national average and San Francisco-Oakland, CA has a rental index 66 log points higher than national average. Smaller cities and those located in the interior of the country, as well as non-metropolitan areas, tend to have below average rental indices. For instance, the non-metropolitan areas of Alabama have rents 64 log points below the national average. The standard deviation of the rental index across areas is 21.8 percent. The housing price index that includes owner-occupied units follows the same general pattern, although it is slightly more dispersed with a standard deviation of 24.5 percent.

4.2

Expenditure Shares

Appendix table A1 also displays the median expenditure shares on housing across metropolitan areas, calculated from the 2000 Census data. As noted above, these shares are calculated as shares of wage and salary income, and include expenditures on utilities as well as on contract rent. The table shows median expenditure shares for renters only and for the combination of renters and owners. For renters, the median housing expenditure share averages 22.2 percent across metro areas, with a standard deviation of 1.9 percent. Including owners in the expenditure share calculations lowers the average share to 17.6 percent but increases the standard deviation across metro areas to 2.5 percent. Additional specifications in the analysis make use of the aggregate expenditure share, calculated as the sum of all rental expenditures in the area divided by the sum of all incomes in the area. The aggregate expenditure shares are lower on average than the median expenditure shares. For renters only the average share is 19.3 percent with a standard deviation of 1.5 percent, while for renters and owners the average share is 15.9 percent, with a standard deviation of 2.2 percent. 15

4.3

Predicted Wage Indices

The compensated regressions use an index of predicted wage differentials in order to measure the variation in earnings potential, and thus utility levels, among residents of different metro areas. We construct the inter-metropolitan predicted wage indices using hedonic regression methods, running regressions of the form: W ln(Wij ) = αW + βX XijW + δjW + W ij

(16)

where Wij is the hourly wage for person i in area j (calculated as annual wage income divided by the product of number of weeks worked and usual hours per week), XijW is a vector of personal characteristics, and δjW is a set of area fixed effects. We take the median or mean of the βˆW XijW as our predicted wage indices, again subtracting the national average so that the indices may be interpreted as the percentage deviation from the national average.15 The appendix describes the covariates XijW in detail. These wage indices capture the idea that observed wage differentials across metro areas reflect both compensating differentials for workers and firms that are unrelated to the skill compositions in those areas, and differentials that reflect the different compositions of skills and worker types across areas. Put simply, San Jose, CA offers high wages both to compensate workers for the high costs of living there, and because the workers there have characteristics that would predict high wages no matter where those workers lived. The predicted wage indices are meant to capture the latter element while omitting the former element, which is captured by the area fixed effects δjW . Using the predicted wage indices also helps us to avoid a problem of measurement error associated with using actual wages: the relevant income measure in the context of our static theoretical model is permanent income, which may not be well-measured by current income. Using wages predicted from observed worker characteristics rather than actual wages or incomes should help to alleviate this problem. Appendix table A1 displays the predicted wage indices at the metropolitan level using the 2000 Census 5 percent data sample. 15

Additional specifications use the average predicted values from the wage regressions.

16

5

Cross-sectional Results

Figures 3A and 3B illustrate the inter-metropolitan relationship between median expenditure shares and relative housing costs using the Census data: 3A is for renters only; 3B, for renters and owners. The former uses a price-index for rental units, while the latter uses a price index for all housing units. The regression line has a slope equal to β1 = −β2 in the empirical regression (10) where it is imposed that β3 = β4 = 0. In both cases, the relationship is positive and statistically significant, indicating that housing demand is price-inelastic. The slope is much steeper in figure 3B, which includes home-owners, and the fit is tighter as well. Table 1 presents estimates using the full compensated model from (10), starting with the median expenditure share for renters in columns 1-4. Column 1 displays the results of a simple regression of the median housing expenditure share on the log rental price index. The geometric mean of the median expenditure share across areas is 22.1 percent, while the implied price elasticity of housing demand is -0.82, statistically different from unity. As predicted, the regression coefficient on the housing price index increases as we add the non-housng price index and the predicted wage index in column 2. The implied price elasticity of demand is -0.68, while the implied income elasticity of demand is 0.66. The coefficients on the two price indices have opposite and roughly equal magnitudes, so that the regression does not reject the rationality restriction of demand theory; the p-value on the test that the coefficients on the housing and non-housing price indices sum to zero is 0.66. The failure to reject this restriction motivates restricting the coefficients to sum to zero in column 3, which is our preferred specification. The price and income elasticities do not change much from column 2, while the implied elasticity of substitution between housing and nonhousing goods is 0.70, significantly less than unity. These results are consistent with Schwabe’s Law that housing is a necessity, but are not consistent with Cobb-Douglas, quasilinear, or inelastic preferences for housing. These results appear largely unaffected by the inclusion of two natural amenities in column 4, distance to a coast line and the average slope of the land within a metro area. The housing expenditure share is not statistically related to distance to the coast, but is positively related to the 17

average slope of the land. This novel result appears justified: housing on hillier terrain has better views and is more easily seen. Households may choose to spend more on their houses to take advantage of the better views, or to impress their neighbors.16 One limitation of including these amenities, however is that it causes the homogeneity restriction to fail, and so we leave this issue for further research. Column 5 includes the expenditures of homeowners as well as renters. In this case, the homogeneity restriction continues to hold, the expenditure share is lower on average, and rises more strongly with prices, as already seen in figure 3B. The estimates suggest a similar but lower income elasticity, and a significantly lower price elasticity of -0.45. The elasticity of substitution is also substantially lower at 0.42. The preceding estimates apply to the median household. However, the market response depends on the overall amount of spending on rent, putting more weight on high-income renters. Therefore, column 6 displays a regression with the log aggregate expenditure share on housing as the dependent variable; this specification also uses the average rather than median of the predicted wage index across MSAs. This specification produces an estimated price elasticity of demand equal to -0.77, an income elasticity of 0.84, and an elasticity of substitution of 0.76. Therefore, the results using the aggregate expenditure share are closer to being consistent with the predictions of Cobb-Douglas preferences than the results using the median expenditure share, although Cobb-Douglas preferences can still be rejected statistically. The rationality restriction continues to hold. Finally, column 7 presents results using the aggregate expenditure share for homeowners and renters. These results suggest less price- and income-elastic demand and a lower elasticity of substitution between housing and non-housing goods than results in column 6. Therefore, the results are less consistent with Cobb-Douglas preferences. This specification rejects the rationality restriction that the unrestricted coefficients on the housing and non-housing price indices sum to zero. Table 2 assesses the robustness of the results in table 1 by examining data from the 1990 and 16

This finding is complementary to that in Albouy and Ehrlich (2012) that housing is more expensive to construct in hilly terrain.

18

1980 Censuses. It uses our preferred specification from column 3 of table 1. The results are largely consistent across years. In both cases, the the price elasticity of housing demand is less than one in absolute value and the income elasticity of demand is less than one, although the latter difference is not statistically significant in the 1980 Census. However, the elasticity of substitution is significantly less than one in both years. Table 3 presents specifications corresponding to equation 4, which feature a more traditional raw measure of household income. Column 1 shows results from an unrestricted regression of the log median expenditure share on the housing and non-housing price indices and the household income index. The three coefficients sum to -0.08, a number that is insignificantly different than zero, passing the homogeneity restriction with a p-value of 7 percent. Unsurprisingly, the results are similar in the restricted regression displayed in column 2. The uncompensated price elasticity is -0.40 and the income elasticity is 0.34. However, in contrast to the compensated regressions in tables 1 and 2, measurement error in the household income measure used in table 3 will likely lead to division bias, causing the coefficient to be too low and underestimating the income elasticity. Columns 4 through 6 of table 3 demonstrate that the results are relatively insensitive to alternative specifications. However, the implied elasticity of substitution between renters and owners is higher in the columns that include both renters and owners than in the columns that include only renters.

6

Household Demand for Housing over Time

Table 4 estimates an uncompensated housing demand using the time series data presented in table 1. In all of the cases we use purely nominal values of prices and income. However, we include separately the log CPI-U for shelter and the log CPI-U for all items less shelter, thereby remaining agnostic about the proper deflator, which should be revealed by the behavior we observe. In the bottom panel, we provide a decomposition to explain the growing share of income spent

19

on housing discussed in the introduction. Rearranging (7) and replacing Q and j with t, we have

pt − cˆt ) + (y,m − 1)[(m ˆ t − cˆt ) − s¯y (ˆ pt − cˆt )] + αt t + et sˆty = (1 − s¯y + H y,p )(ˆ

(17)

The first component represents the change in the income effect due to the pure compensated price effect. This effect is positive when the relative price of houisng increases if σ < 1, as 1−sy +H y,p ) = (1 − sy )(1 − σ). The second component is the income effect, making the proper adjustment for changes in relative prices, from a parallel rise in the budget set. The third component is due to the time trend, which could represent a change in household preferences. Indeed, number of people in a household has fallen from 3.2 to 2.5 since 1970. Another possibility could be increasing complementarity of housing with local amenities as households have shifted locations. The time trend may also reflect limitations in the data and its ability to identify low-frequency responses in housing consumption from shifting prices and income. The first column applies to owners and renters using the BEA numbers, using nominal GDP per household as the income measure. As required by demand theory, the three coefficients add up to a number not significantly different from zero. In the restricted model, the implied uncompensated own-price elasticity is -0.68, which is not significantly different from the preferred regression in Table 1. The income elasticity is slightly smaller at 0.55. Its estimate may be confounded by income’s collinear time trend, which is statistically significant. It suggests a secular trend towards greater housing consumption independent of price and income movements. In the BEA numbers, the overall increase in the housing share over the sample period was 8 percent (just under 2 percentage points). According to these estimates, the increase in the relative price of housing raised the income share devoted to it by 7 percent, while rising incomes reduced it by 12 percent. The positive time trend over-predicts the change at 15 percent, raising questions. Columns 2 and 3 use average expenditure shares from the CES for renters, and for renters and owners combined, using measures of average income. These estimates are less precise, although the point estimates suggest elasticities closer to one, especially when owners are included. Conse-

20

quently, the decomposition suggests that changes in the expenditure share on housing were raised somewhat by the rising price of housing, but even more by by the time trend, which ultimately explains little. In interpreting this trend, one should bear in mind reporting issues in the CEX with households reporting fewer and fewer of their non-housing expenditures (citation?). In column 4 and 5, we examine the median income shares from the AHS for renters and renters and owners. These estimates are even less precise, and suggest that demand is fairly elastic in prices and rather less elastic in income. The decomposition reveals that the rather large increase in housing expenditures for renters is due somewhat to increases in prices, and also to falling median incomes. When home-owners are included, the income effect is again negative as median incomes as a whole rose. Nevertheless, much of the change is explained by the secular time trend. Relative to the cross-sectional point estimates, the time series estimates are slightly more elastic with respect to prices and less with respect to incomes. If we were to apply the cross-sectional parameters, we would find (positive) relative-price effects further from zero and (negative) income effects closer to zero. This would explain more of the observed change, and reduce the change attributed to the time trend.

7 7.1

Putting the Parameter Estimates into Use Utility and Expenditure Functions

The estimates from the previous sections are sufficient to identify the utility and expenditure outlined in section 3.5. To illustrate what may be a realistic example, we take inspiration from column 3 of table 1, and use slightly rounded rational numbers for the parameters, setting σ = 2/3, δ = 7/8, γ = 3/2. Substituting in these parameter values into equations (13) and (14) yields nonhomothetic separable CES utility and expenditure functions ready for quantitative analysis:

" U (x, y; 1) =

25 − 6y 42x

− 12

− 12

− 47

# 43

3

 2 3

1 3

1 2

1 3

 7 c u +p  , e(p, c, u; 1) = 36    23  3 4 47u + 25 21

(18)

In the utility function, the units of x and y are as median income shares, with baseline values at x = 0.78 and y = 0.22. Thus, a value of y = 1 would correspond to housing consumption 1/0.22 = 4.54 times that of a median renter. These functions could be applied immediately in a number of models in urban, macro, or public economics involving the housing sector.

7.2

Cost-of-Living Indices over Space and Time

As we have seen, the relative price of housing can vary considerably over space and time. Since housing is a major consumption item across households whose income varies widely, it is sensible to construct a cost-of-living index that incorporates realistic substitution and non-homothetic consumption behaviors. Below we compare four different cost-of-living indices (COLIs) for housing and non-housing goods. When demand is homothetic and exhibit constant elasticity of substitution the ideal cost-ofliving index is defined by fixed weight Lespeyres index in the zero-substitution case (σ = 0):

COL1 =

δcj + (1 − δ)pj δ¯ c + (1 − δ)¯ p

(19)

where δ = 1 − s¯y . In the unit-elastic (σ = 1) case it is the Cobb-Douglas index: δ

COL2 =

1−δ

(cj ) (pj )

(20)

(¯ c)δ (¯ p)1−δ

While for arbitrary values of σ, the ideal index is "

δ σ c1−σ + (1 − δ)σ p1−σ j j COL3 = σ 1−σ σ 1−σ δ c¯ + (1 − δ) p¯ where the distribution parameter is set as δ =

1 1

s¯

1+( 1−ys¯ ) σ y

22

1 # 1−σ

(21)

. When we allow for non-homotheticity,

we obtain an ideal cost-of-living index using the separable CES function that is remarkably similar "

δ σ uγ(1−σ) cj1−σ + (1 − δ)σ pj1−σ COL4 = σ γ(1−σ) 1−σ δ u c¯ + (1 − δ)σ p¯1−σ

1 # 1−σ

(22)

where the base utiity level is given by uγ(1−σ) = [(1 − s¯y )(1 − δ)σ p¯1−σ ] / (s¯y δ σ c¯1−σ ) . While there is no explicit indirect utility function, the expenditure function may be solved for numerically to provide a value of u for any income m. Figures 4A and 4B plot these four cost-of-living indices against the relative price of housing (pj /cj ), with 4B adjusting for a base level of income that is one half the median represented in 4A. To underscore the relevance of these measures, the relative rents in 2009 were 1.4 times higher than in 1970; the same as the average difference between New York and St. Louis in 2000. In San Francisco rents were 1.7 times and Oklahoma City in 2007. In 4A, we see how the fixed housing demand measure overstates differences in cost-of-living by ignoring households’ ability to substitute between housing and other goods according to their relative prices, while the Cobb-Douglas preference measure understates these differences by assuming that substitution is easier than it is. Our intermediate value of substitution of σ = 2/3 more accurately reflects substitution possibilities. For example, when housing rents are double the national average (i.e. pj /cj = 2), the fixed demand measure overstates the true cost-of-living differential by 3.3 percentage points, while the Cobb-Douglas measure understates it by 1.4 percentage points. In figure 4B, we see how the non-homothetic CES cost-of-living index has a much greater slope than the one that fails to account for housing being a necessity. For poorer households, the other COL indices understate the burden of living in expensive areas, and overstate it in poorer areas. In other words, the correct index accounts for how high-rent cities are especially expensive for the poor. Of course, the regular CES function could be adapted to poorer households simply by changing its distribuiton parameter δ. The advantages of the non-homothetic CES function is that it offers a continuous mapping of cost-of-living for any level of well-being, based on income at some

23

reference city at a given point in time.

8

Conclusion

Analyzing the cross-sectional relationship between housing rental prices and housing expenditure shares suggests that housing demand is both price and income inelastic, albeit with elasticities closer to one than to zero. The so-called “affordability” crisis in rental units in recent data appears to stem somewhat from the rising relative and stagnant or declining incomes among those who tend to rent. Nevertheless, there appears to be a secular rise in housing consumption that may be due to shifts in household preferences, as households themselves have been changing. For researchers interested in the role of housing, we offer a plausible ideal cost-of-living index that improves on traditional CPI-style indices which assume fixed housing demand, may overstate inflation or differences in housing-cost across cities, or misrepresent them for the poor. Indeed, we find that expensive cities are even more expensive for the poor, thereby exacerbating affordability problems. We also believe that the non-homothethic CES function may help provide economists across fields, particularly urban and macro, with a better baseline to perform analyses and simulations.17

References [1] Albouy, David. 2008. “Are Big Cities Really Bad Places to Live? Improving Quality-of-Life Estimates across Cities.” NBER Working Paper No. 14981. [2] Albouy, David, and Gabriel Ehrlich. 2012. “Land Values and Housing Productivity across Cities.” NBER Working Paper No. 18110. [3] Albouy, David, and Hanson Andrew. 2014. “Are Houses Too Big or in the Wrong Place? Taxes Benefits to Housing and Inefficiencies in Location and Consumption.” NBER Tax Policy and the Economy. 17

Our results are consistent with the assumptions made by of Albouy and Stuart (2014) and Rappaport (2008a), should help researchers to predict population densities both within and across metropolitan areas

24

[4] Albouy, David, and Lue Bert. 2014. “Driving to Opportunity: Local Rents, Wages, Commuting Costs and Sub-Metropolitan Quality of Life.” NBER Working Paper No. 19922. [5] Albouy, David, and Bryan Stuart. 2014. “Urban Population and Amenities.” NBER Working Paper No. 19919. [6] Baum-Snow, Nathaniel and Ronni Pavan. 2013. ”Inequality and City Size.” Review of Economics and Statistics 95(5): 1535-1548. [7] Black, Dan, Gates Gary, Sanders Seth, and Taylor Lowell. 2002. “Why Do Gay Men Live in San Francisco ?” Journal of Urban Economics 51(1): 54-76 [8] Black, Daniel, Natalia Kolesnikova, and Lowell Taylor (2009)“Earnings Functions When Wages and Prices Vary by Location.” Journal of Labor Economics 27(21). [9] Blundell, Richard, Panos Pashardes and Guglielmo Weber “What do we Learn About Consumer Demand Patterns from Micro Data?” The American Economic Review 83(3): 570-597. [10] Carrillo, Paul, Dirk Early, and Edgar Olsen. 2013. “A Panel of Interarea Price Indices for All Areas in the United States 1982-2012” Available at SSRN: http://ssrn.com/abstract=2378028 or http://dx.doi.org/10.2139/ssrn.2378028. [11] Cheshire, Paul, and Sheppard Stephen. 1998. “Estimating the Demand for Housing, Land, and Neighbourhood Characteristics.” Oxford Bulletin of Economics and Statistics. Department of Economics, University of Oxford, vol. 60(3): 357-82, August. [12] Davis, Morris A., and Jonathan Heathcote. ”The price and quantity of residential land in the United States.” Journal of Monetary Economics 54.8 (2007): 2595-2620. [13] Davis, Morris, and Francois Ortalo-Magne. 2011. “Household expenditures, wages, rents.” Review of Economic Dynamics(14): 248-261. [14] Davis, Morris and Michael Palumbo (2007) “The Price of Residential Land in Large U.S. Cities.” Journal of Urban Economics, 63: 352-384. [15] Deaton, Angus. 1986. “Demand Analysis” in Z. Grilliches and M.D. Intriligator Handbook of Econometrics, 3: 1767-1839. [16] De Leeuw, Frank. 1971. “The Demand for Housing: A Review of Cross-section Evidence.” The Review of Economics and Statistics 53 (1): 1-10. [17] Desmet, Klaus and Esteban Rossi-Hansberg. 2013. “Urban Accounting and Welfare.” American Economic Review 103(6): 2296-2327. [18] Eckhout, Jan. 2004. “Gibrats law for all cities.” American Economic Review 94 (5): 14291451. [19] Ermisch, J., J. Findlay, and K. Gibb, 1996, “The Price Elasticity of Housing Demand in Britain: Issues of Sample Selection. Journal of Housing Economics 5(1): 64-86. 25

[20] Engel, E. 1857. “Die Produktions- und Consumtionsverhaltnisse des Konigreichs Sachsen.” reprinted with Engel (1895), Anlage I, 1-54. [21] Guerreri, Veronica, Daniel Hartley, and Erik Hurst. 2013. “Endogenous Gentrification and Housing Price Dynamics.” Journal of Public Economics 100: 45-60. [22] Goodman, Allen C. 1988. “An Econometric Model of Housing Price, Permanent Income, Tenure Choice, and Housing Demand.” Journal of Urban Economics 23 (May): 327353. [23] Goodman, Allen C. 2002. “Estimating Equilibrium Housing Demand for “Stayers” of Housing Price, Permanent Income, Tenure Choice, and Housing Demand.” Journal of Urban Economics51: 1-24. [24] Goodman, Allen C, and Kawai Masahiro. 1986. “Functional Form, Sample Selection, and Housing Demand.” Journal of Urban Economics 20: 155-167. [25] Gyourko, Joseph, Christopher Mayer, and Todd Sinai, 2013. ”Superstar Cities,” American Economic Journal: Economic Policy 5(4): 167-99. [26] Handbury, Jessie (2013) “Are Poor Cities Cheap for Everyone? Non-homotheticity and the Cost-of-Living across U.S. Cities.” mimeo. [27] Hansen, Julia L. and Formby John P. 1998. “Estimating the Income Elasticity of emand for Housing:A Comparison of Traditional and Lorenz-Concentration Curve Methodologies.” Journal of Housing Economics 7: 328-342. [28] Hanushek, Eric, and Quigley John. 1980. “The Dynamics of housing market: a stock adjustment model of housing consumption.” Journal of Urban Economics 6(1), pp. 90-111. [29] Harmon, Oskar R. 1988. “The Income Elasticity of Demand for Single-Family OwnerOccupied Housing: An Empirical Reconciliation.” Journal of Urban Economics 24: 173185. [30] HUD PD & R Edge. “Secretary Donovan Highlights Convening on State of Americas Rental Housing.” January 2014. http://www.huduser.org/portal/pdredge/ pdr_edge_featd_article_012714.html. [31] Ioannides, Yannis M., and Jeffrey E. Zabel. 2003. “Neighborhood Effects and Housing Demand.” Journal of Applied Econometrics 18 (Sep): 563-584. [32] Joint Center for Housing Studies of Harvard University. 2013. “America’s Rental Housing: Evolving Markets and Needs.” Mimeograph. Harvard University. [33] Kau, James B., and Sirman C. F. 1979. “Urban Land Value Functions and the Price Elasticity of Demand for Housing.” Journal of Urban Economics 6(4): 112-121. [34] Larsen, Erling Red. 2002. “Searching for Basic Consumption Patterns:is the Engel Elasticity of Housing Unity?” Discussion Papers No. 323, August 2002 Statistics Norway, Research Department. 26

[35] Mayo, Stephen K. 1981. “Theory and Estimation in the Economics of Housing Demand.” Journal of Urban Economics 10: 95-116. [36] Michaels, Guy, Ferdinand Raurch, and Stephen J. Redding. 2013. ”Urbanization and Structural Transformation.” The Quarterly Journal of Economics 127(2): 535-586. [37] Mills E.S. 1967. “An aggregative model of resource allocation in a metropolitan area.” American Economic Review 57: 197-210. [38] Moretti, Enrico 2013 ”Real Wage Inequality” American Economic Journal: Applied Economics 5(1): 65-103. [39] Muth, Richard F. 1960. “The Demand for Non-Farm Housing.” in The Demand for Durable Goods, edited by Arnold C. Harberger, 29-96. Chicago, IL: University of Chicago Press, 1960. (Ph.D. dissertation) [40] Piketty, Thomas. 2014. Capital in the Twenty-First Century. Cambridge, MA: Harvard University Press. [41] Pollinsky, A. Mitchell, and Ellwood David T. 1979. “An Empirical Reconciliation of Micro and Grouped Estimates of the Demand for Housing.” The Review of Economics and Statistics 61(2): 199-205 [42] Rappaport, Jordan. 2008a. “A Productivity Model of City Crowdedness.” Journal of Urban Economics 65: 715-722. [43] Rappaport, Jordan. 2008b. “Consumption Amenities and City Population Density.” Regional Science and Urban Economics 38: 533-552. [44] Reid, Margaret. 1962. Housing and Income. Chicago: University of Chicago Pres. [45] Roback, Jennifer. 1982. “Wages, Rents, and the Quality of Life.” Journal of Political Economy 90: 1257-1278. [46] Rosen, Harvey. 1985. “Housing Subsidies: Effects on Housing Decisions, Efciency, and Equity. In Handbook of Public Economics, vol. 1, edited by M. Feldstein and A. Auerbach. Amsterdam: North-Holland. [47] Rosen, Sherwin. 1979. “Wages-based Indexes of Urban Quality of Life.” In Current Issues in Urban Economics, edited by P. Mieszkowski and M. Straszheim. Baltimore: John Hopkins Univ. Press. [48] Ruggles, Steven, Matthew Sobek, Trent Alexander, Catherine A. Fitch, Ronald Goeken, Patricia Kelly Hall, Miriam King, and Chad Ronnander. 2004. Integrated Public Use Microdata Series: Version 3.0. Minneapolis: Minnesota Population Center. [49] Saiz, Albert. 2010. ”The geographic determinants of housing supply.” The Quarterly Journal of Economics 125.3: 1253-1296.

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28

TABLE 1: COMPENSATED DEMAND FUNCTIONS - 2000 CENSUS DATA

Dependent Variable: Regression Results: Housing Price Index

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Log Median Expenditure Share

Log Median Expenditure Share

Log Median Expenditure Share

Log Median Expenditure Share

Log Median Expenditure Share

Log Aggregate Expenditure Share

Log Aggregate Expenditure Share

0.181 (0.013)

0.242 (0.020) -0.274 (0.098) -0.340 (0.081)

0.236 (0.015) -0.236 (0.015) -0.345 (0.080)

0.481 (0.011) -0.481 (0.011) -0.393 (0.062)

0.195 (0.015) -0.195 (0.015) -0.157 (0.081)

0.445 (0.012) -0.445 (0.012) -0.276 (0.065)

-1.509 (0.000)

-1.509 (0.000)

-1.509 (0.000)

0.232 (0.017) -0.232 (0.017) -0.413 (0.077) -0.006 (0.088) 0.013 (0.002) -1.509 (0.000)

-1.749 (0.000)

-1.649 (0.000)

-1.851 (0.000)

332 0.378 No

332 0.425 No

332 0.426 Yes

328 0.487 Yes

332 0.850 Yes

332 0.350 Yes

332 0.814 Yes

-0.032 (0.083)

-0.032 (0.083)

-0.204 (0.087)

-0.041 (0.103)

-0.048 (0.082)

0.388 (0.086)

0.698

0.698

0.020

0.691

0.560

0.000

Renters Only

Renters Only

Renters Only

Renters Only

Renters and Owners

Renters Only

Renters and Owners

0.221 (0.001) -0.819 (0.013) 1.000 Restricted

0.221 (0.001) -0.683 (0.028) 0.660 (0.081)

0.221 (0.001) -0.687 (0.026) 0.655 (0.080) 0.697 (0.019) 0.885 (0.007) 1.459 (0.328)

0.221 (0.001) -0.676 (0.026) 0.587 (0.077) 0.702 (0.022) 0.884 (0.008) 1.779 (0.332)

0.174 (0.001) -0.451 (0.017) 0.607 (0.062) 0.418 (0.014) 0.985 (0.002) 0.816 (0.124)

0.192 (0.001) -0.774 (0.023) 0.843 (0.081) 0.758 (0.018) 0.886 (0.006) 0.805 (0.409)

0.157 (0.000) -0.511 (0.017) 0.724 (0.065) 0.472 (0.014) 0.980 (0.002) 0.619 (0.143)

Non-Housing Price Index Predicted Wage Index Inverse Distance to Coast Average Slope of Land Constant

N Adjusted R-squared Constrained Regression? Unconstrained Sum of Housing and NonHousing Price Index Coefficients P-value of Test of Homogeneity of Demand Sample Implied Demand Parameters: Geometric Mean Expenditure Share Uncompensated Own Price Elasticity of Housing Demand Income Elasticity of Housing Demand Elasticity of Substitution Between Housing and Consumption Goods Distribution Parameter Non-homotheticity Parameter

TABLE 2: COMPENSATED DEMAND FUNCTIONS - ADDITIONAL DATASETS

Dependent Variable: Regression Results: Housing Price Index Non-Housing Price Index Predicted Wage Index Constant

N Adjusted R-squared Constrained Regression? Unconstrained Sum of Housing and NonHousing Price Index Coefficients P-value of Test of Homogeneity of Demand Sample Data Source

(1)

(2)

(3)

Log Median Expenditure Share

Log Median Expenditure Share

Log Median Expenditure Share

0.199 (0.025) -0.199 (0.025) -0.056 (0.036) -1.285 (0.005)

0.204 (0.014) -0.204 (0.014) -0.258 (0.096) -1.510 (0.004)

0.172 (0.025) -0.172 (0.025) -0.150 (0.115) -1.539 (0.005)

274 0.185 Yes -0.181 (0.068)

287 0.437 Yes -0.238 (0.124)

288 0.144 Yes -0.238 (0.191)

0.008 Renters Only 2010 ACS

0.056 Renters Only 1990 Census

0.214 Renters Only 1980 Census

0.277 (0.001) -0.785 (0.029) 0.944 (0.036) 0.725 (0.034) 0.830 (0.014) 0.283 (0.175)

0.221 (0.001) -0.739 (0.025) 0.742 (0.096) 0.738 (0.018) 0.871 (0.007) 1.267 (0.483)

0.215 (0.001) -0.795 (0.034) 0.850 (0.115) 0.781 (0.032) 0.861 (0.012) 0.869 (0.690)

Implied Demand Parameters: Geometric Mean Expenditure Share Uncompensated Own Price Elasticity of Housing Demand Income Elasticity of Housing Demand Elasticity of Substitution Between Housing and Consumption Goods Distribution Parameter Non-homotheticity Parameter

TABLE 3: UNCOMPENSATED DEMAND FUNCTIONS - 2000 CENSUS DATA

Dependent Variable: Regression Results: Housing Price Index Non-Housing Price Index Household Income Index

(1)

(2)

(3)

(4)

(5)

(6)

Log Median Expenditure Share

Log Median Expenditure Share

Log Median Expenditure Share

Log Median Expenditure Share

Log Aggregate Expenditure Share

Log Aggregate Expenditure Share

0.617 (0.017) -0.033 (0.054) -0.665 (0.023)

0.604 (0.016) 0.061 (0.013) -0.665 (0.023)

0.642 (0.011) -0.181 (0.014) -0.462 (0.021)

0.557 (0.016) 0.076 (0.013) -0.634 (0.024)

0.595 (0.013) -0.188 (0.018) -0.407 (0.025)

-1.509 (0.002)

-1.509 (0.002)

0.583 (0.017) 0.068 (0.014) -0.652 (0.024) 0.098 (0.051) 0.004 (0.001) -1.509 (0.002)

-1.749 (0.002)

-1.649 (0.002)

-1.851 (0.002)

332 0.829

332 0.828

328 0.832

332 0.933

332 0.795

332 0.891

No

Yes

Yes

Yes

Yes

Yes

-0.080 (0.045)

-0.080 (0.045)

-0.183 (0.049)

0.159 (0.053)

-0.062 (0.046)

0.283 (0.064)

0.073

0.073

0.000

0.003

0.175

0.000

Renters Only

Renters Only

Renters Only

Renters and Owners

Renters Only

Renters and Owners

0.221 (0.000) -0.383 (0.017) 0.335 (0.023)

0.221 (0.000) -0.396 (0.016) 0.335 (0.023) 0.414 (0.015)

0.221 (0.000) -0.417 (0.017) 0.348 (0.024) 0.436 (0.016)

0.174 (0.000) -0.358 (0.011) 0.538 (0.021) 0.320 (0.010)

0.192 (0.000) -0.443 (0.016) 0.366 (0.024) 0.461 (0.015)

0.157 (0.000) -0.405 (0.013) 0.593 (0.025) 0.370 (0.013)

Inverse Distance to Coast Average Slope of Land Constant

N Adjusted R-squared Constrained Regression? Unconstrained Sum of Housing Price, Non-Housing Price, and Household Income Coefficients P-value of Test of Homogeneity of Demand Sample Implied Demand Parameters: Geometric Mean Expenditure Share Uncompensated Own Price Elasticity of Housing Demand Income Elasticity of Housing Demand Elasticity of Substitution Between Housing and Consumption Goods

TABLE 4: PRICES, INCOMES, AND HOUSING EXPENDITURE SHARES - TIME SERIES DATA

Dependent Variable:

Data Source: Unrestricted Regression Results: Log CPI-U: Shelter Log CPI-U: All Items Less Shelter Log Household Income Measure Year (Linear Time Trend) Constant

P-value of Test of Homogeneity of Demand Restricted Regression Results: Log CPI-U: Shelter minus Log CPI-U: All Items Less Shelter Log Average Household Income minus Log CPI-U: All Items Less Shelter Year (Linear Time Trend) Constant

N (years with non-missing data) Implied Demand Parameters from Restricted Regressions: Geometric Mean Expenditure Share

Uncompensated Own-Price Elasticity of Housing Demand Income Elasticity of Housing Demand

(1)

(2)

(3)

(4)

(5)

Log Aggregate Expenditure Share

Log Average Expenditure Renters Only

Log Average Expenditure Renters & Owners

Log Median Expenditure Share Renters

Log Median Expenditure Share Renters and Owners

BEA

CEX

CEX

AHS

AHS

0.313 (0.082) 0.138 (0.099) -0.449 (0.085) 0.004 (0.002) -1.718 (0.003)

0.253 (0.294) -0.055 (0.253) -0.238 (0.129) 0.009 (0.004) -1.232 (0.004)

-0.036 (0.237) 0.142 (0.210) -0.044 (0.091) 0.005 (0.002) -1.381 (0.003)

0.398 (0.484) 0.105 (0.608) -0.562 (0.166) 0.007 (0.005) -1.283 (0.010)

-0.163 (0.316) 1.022 (0.370) -0.758 (0.224) 0.008 (0.004) -1.605 (0.009)

0.95

0.78

0.34

0.82

0.53

0.318 (0.071) -0.453 (0.054) 0.004 (0.001) -1.718 (0.003)

0.249 (0.281) -0.234 (0.122) 0.008 (0.001) -1.232 (0.004)

0.152 (0.080) -0.132 (0.031) 0.007 (0.001) -1.381 (0.003)

0.262 (0.197) -0.522 (0.127) 0.006 (0.002) -1.283 (0.010)

0.103 (0.182) -0.889 (0.203) 0.010 (0.001) -1.605 (0.009)

42

28

29

21

23

0.179 (0.000) -0.682 (0.071) 0.547 (0.054)

0.292 (0.001) -0.751 (0.281) 0.766 (0.122)

0.251 (0.001) -0.848 (0.080) 0.868 (0.031)

0.277 (0.003) -0.738 (0.197) 0.478 (0.127)

0.201 (0.002) -0.897 (0.182) 0.111 (0.203)

Decomposition of Long-run Change in Expenditure Share on Housing: Total Change in Log Share 0.080 0.263 0.271 0.341 0.356 Change Attributed to Time Trend 0.150 0.286 0.281 0.231 0.396 Change from Compensated Relative Price Effect 0.065 0.052 0.031 0.034 -0.021 Change from Real Income Effect -0.140 -0.018 -0.028 0.032 -0.019 Residual 0.006 -0.057 -0.013 0.043 -0.001 Homogeneity of demand requires that the coefficients on log CPI-U for shelter, log CPI-U for all items less shelter, and log real household income sum to zero. The restricted regressions impose this constraint. Newey-West standard errors with a lag of three reported in parentheses. For non-BEA series, a moving average with weight of 0.5 for the year after and the year before is used to deal with noise in the data.

0.8

1.0 Relative Price of Shelter

Expenditure Share on Housing 0.2 0.3

1.2

0.4

Figure 1: Expenditure Share on Housing 1970-2013

1970

1975

1980

1985

1990 1995 Year

2000

2005

2010

Aggregate Housing Share-BEA Avg Rent Share-CEX

Avg Housing Share-CEX Median Housing Share-AHS

Median Rent Share-AHS

Median Rent Share-Census

Relative Price of Shelter For non BEA series, a moving average with weight of .5 for the year after and the year before is shown in the curve.

Figure 2: Housing Consumption with Production Possibility Expansions Non-housing

E

Income expansion path

Constant expenditure share at changing prices

Constant expenditure share at original prices

C B D

Original budget Constraint

A

Original production possibility frontier

New production possibility frontier

New budget Constraint

LEGEND A = original bundle C = new bundle A to B = income effect (necessity) B to C = substitution effect (inelastic) A to D = income effect (neutral) D to E = substitution effect (unit)

New Indifference Curve Equivalent new budget constraint at original relative prices

Original Indifference Curve

Housing

Los Angeles-Long Orlando, FL Beach, CA Riverside-San Bernardino,CA Denver-Boulder, CO Phoenix, AZ Portland, Norfolk-VA Beach--Newport News, VA OR-WA Seattle-Everett, WA Sacramento, CA Tampa-St. Petersburg-Clearwater, FL San Jose, CA Atlanta, GA San Francisco-Oakland-Vallejo, CA Minneapolis-St. Paul, MN San Antonio, TX Philadelphia, PA/NJ Boston, MA-NH Dallas-Fort Worth, TXDC/MD/VA NewWashington, York-Northeastern NJ Houston-Brazoria, TX IL Baltimore, MD Chicago, Cleveland, OH Kansas Indianapolis, City, MO-KS IN Pittsburgh, PA St. Louis, MO-IL Detroit, MI

0.25

San Francisco-Oakland-Vallejo, CA San Diego, CA Los Angeles-Long Beach, CA Seattle-Everett, WA Bloomington, IN Bryan-College Station, TX Boston, MA-NH Portland, OR-WA New York-Northeastern NJ Miami-Hialeah, FL Denver-Boulder, CO

0.20

Fort Lauderdale-Hollywood-Pompano FL San Beach, Diego, CA

San Jose, CA

Sacramento, CA Gainesville, FL Riverside-San Bernardino,CA

Norfolk-VA Beach--Newport News, VA Phoenix, AZ Chicago, IL Fort Lauderdale-Hollywood-Pompano Beach, FL Orlando, Atlanta,FLGADC/MD/VA Washington, Cleveland, OH Baltimore, MD Philadelphia, PA/NJ Tampa-St. Petersburg-Clearwater, FL Minneapolis-St. Paul, MN Dallas-Fort Worth, TX MI Detroit, Indianapolis, Kansas City, MO-KS IN San Antonio, TX Pittsburgh, PA St. Louis, MO-ILTX Houston-Brazoria,

0.15

Median Expenditures on Housing as a Share of Income

0.30

Gainesville, FL

Miami-Hialeah, FL

0.25

Figure 3B: Median Share of Income Spent on Housing and the Relative Price of Housing, All Households 2000

Bryan-College Station, TX

Bloomington, IN

0.20

Sheboygan, WI

Housing (Rental) vs. Non-Housing Price Index METRO POP

>5 Mil

1.5-5 Mil

Non Metro

Linear fit

Slope = 0.22

0.5-1.5 Mil

<0.5 Mil

2.0 2.2 2.4 2.6

1.8

1.6

1.4

1.2

1.0

0.8

0.6

2.0

1.8

1.6

1.4

1.2

1.0

0.8

Sheboygan, WI

0.6

Median Expenditure on Gross Rent as a Share of Income

0.35

Figure 3A: Median Share of Income Spent on Rent and the Relative Price of Housing, Renters Only 2000

Housing vs. Non-Housing Price Index METRO POP

>5 Mil

1.5-5 Mil

Non Metro

Linear fit

Slope = 0.46

0.5-1.5 Mil

<0.5 Mil

Figure 4B: Comparison of Cost-of-living Indices at one half of median household income

San Jose

Los Angeles

Change from 1970 to 2009;

National Average, Kansas City

.8

.8

1

1

Cost-of-living Indices

1.2

Johanstown, PA

San Jose

Changes from 1970 to 2009; Los Angeles

Cost-of-living Indices

1.2

Johnstown, PA

National Average, Kansas City

1.4

1.4

Figure 4A: Comparison of Cost-of-living Indices at median household income

.5

1

1.5

2

2.5

.5

1

Relative Price

1.5

2

2.5

Relative Price

COL1: Fixed Demand

COL2: Cobb-Douglas

COL1: Fixed Demand

COL2: Cobb-Douglas

COL3: Homothetic CES

COL4: Separable Family of CES

COL3: Homothetic CES

COL4: Separable Family of CES

Appendix A

Separable Family of CES

A.1

Formulation and Parameters

We use the simple “separable family” of CES utility function from Sato (1977), who writes it as  U=

δ1 xρ + θ1 δ2 y ρ + θ2

 αρ

where θi = − (α − δi ) ρ − δi is composed of more elementary parameters. These are δ = δ1 = 1 + δ2 = δ =the distribution parameter, σ = 1/(1 − ρ) =the substitution parameter, and γ = 1/α=the non-homotheticity parameter. Then using these parameters, we may rewrite the utility function as  U=

A.2

ρ

δx + θ1 (δ − 1)y ρ + θ2

 γρ1

" =

γσδx

σ−1 σ

+ 1 − γδ − σ

γσ(δ − 1)y

σ−1 σ

+ 1 − γ(δ − 1) − σ

σ # γ(σ−1)

Marginal rate of substitution

Taking the ratio of partial derivates, the marginal rate of subtitution is then M RSx,y

δ = 1−δ

 ρ−1  −1  − σ1 γ(1−σ) x δxρ + θ1 x δ u σ = ρ y (δ − 1)y + θ2 1−δ y

At the household optimum, we have c/p = M RSx,y , c δ = p 1−δ

A.3

 − σ1  −σ  −σ γ(1−σ) x x h c 1 − δ − γ(1−σ) i−σ c 1−δ · u σ = u σ ⇒ = uγ(1−σ) y y p δ p δ

Expenditure Share on Housing

To solve for the expenditure share, note that (d ln y)/d ln x = dy/dx(x/y) = −cx/py = sx /sy , the share spent on x relative to y is equal to sx d ln y δ = = sy d ln x 1−δ

 1− σ1  σ  1−σ γ(1−σ) x δ c u σ = uγ(1−σ) y 1−δ p

i

Then to solve the expenditure share for y only add one and invert 1 cx c1−σ δ σ uγ(1−σ) + p1−σ (1 − δ)σ = +1= sy py (1 − δ)σ p1−σ (1 − δ)σ p1−σ ⇒ sy = σ 1−σ γ(1−σ) δ c u + (1 − δ)σ p1−σ In logarithms, we obtain an only partly linear equation ln sy = σ ln(1 − δ) + (1 − σ) ln(p) − ln[δ σ c1−σ uγ(1−σ) + (1 − δ)σ p1−σ ]

(A.1)

To complete the log-linearization, we take the total derivative we get the approximation (1 − σ)δ σ c1−σ uγ(1−σ)b c + γ(1 − σ)δ σ c1−σ uγ(1−σ) u b + (1 − σ)(1 − δ)σ p1−σ pb δ σ c1−σ uγ(1−σ) + (1 − δ)σ p1−σ = (1 − sy )(1 − σ)b p − (1 − sy )(1 − σ)b c − γ(1 − sy )(1 − σ)b u

sby = (1 − σ)ˆ p−

This gives us that β0 = σ ln(1−δ) = ln s¯y , β1 = (1−¯ sy )(1−σ), β3 = −γ(1−¯ sy )(1−σ). Inverting, the parameters can be expressed recursively as σ = 1 − β1 /(1 − eβ0 ), δ = 1 − eβ0 /σ , γ = −β3 /β1 .

A.4

Hicksian Demand and Expenditure Function

It is possible to solve for this utility function for both Hicksian demands, given by p−σ (1 − δ)σ y= σ [c1−σ δ σ uγ(1−σ) + p1−σ (1 − δ)σ ] σ−1



1 − γ(δ − 1) − σ 1 − γδ − σ γ (1−σ) − uσ γσ γσ

c−σ δ σ uγ(1−σ) x= σ [c1−σ δ σ uγ(1−σ) + p1−σ (1 − δ)σ ] σ−1



1 − γ(δ − 1) − σ 1 − γδ − σ γ (1−σ) − uσ γσ γσ

σ  σ−1

σ  σ−1

as well as the expenditure function

e(p, c, u; 1) =

B

 

1−σ σ γ(1−σ)

1−σ

σ

 1  1−σ

c δ u + p (1 − δ)  iσ  1 γ + (1 − σ − γδ)(1 − u γ(1−σ)  σ ) γσ h

.

(A.2)

Data

We define cities at the Metropolitan Statistical Area (MSA) level using 1999 Office of Management and Budget definitions of consolidated MSAs (e.g., San Francisco is combined with Oakland and San Jose), of which there are 276. We use United States Census data from the 2000 Integrated Public-Use Microdata Series (IPUMS), from Ruggles et al. (2004), to calculate wage and housing

ii

price differentials. The wage differentials are calculated for workers ages 25 to 55, who report working at least 30 hours a week, 26 weeks a year. The MSA assigned to a worker is determined by their place of residence, rather than their place of work. The wage differential of an MSA is found by regressing log hourly wages on individual covariates and indicators for which MSA a worker lives in, using the coefficients on these MSA indicators. The covariates consist of • 12 indicators of educational attainment; • a quartic in potential experience, and potential experience interacted with years of education; • 9 indicators of industry at the one-digit level (1950 classification); • 9 indicators of employment at the one-digit level (1950 classification); • 4 indicators of marital status (married, divorced, widowed, separated); • an indicator for veteran status, and veteran status interacted with age; • 5 indicators of minority status (Black, Hispanic, Asian, Native American, and other); • an indicator of immigrant status, years since immigration, and immigrant status interacted with black, Hispanic, Asian, and other; • 2 indicators for English proficiency (none or poor). All covariates are interacted with gender. This regression is first run using census-person weights. From the regressions a predicted wage is calculated using individual characteristics alone, controlling for MSA, to form a new weight equal to the predicted wage times the census-person weight. These new income-adjusted weights are needed since workers need to be weighted by their income share. The new weights are then used in a second regression, which is used to calculate the city-wage differentials from the MSA indicator variables. In practice, this weighting procedure has only a small effect on the estimated wage differentials. Housing price differentials are calculated using the logarithm reported gross rents and housing values. Only housing units moved into within the last 10 years are included in the sample to ensure that the price data are fairly accurate. The differential housing price of an MSA is calculated in a manner similar to wages, except using a regression of the actual or imputed rent on a set of covariates at the unit level. The covariates for the adjusted differential are • 9 indicators of building size; • 9 indicators for the number of rooms, 5 indicators for the number of bedrooms, number of rooms interacted with number of bedrooms, and the number of household members per room; • 2 indicators for lot size; • 7 indicators for when the building was built;

iii

• 2 indicators for complete plumbing and kitchen facilities; • an indicator for commercial use; • an indicator for condominium status (owned units only). A regression of housing values on housing characteristics and MSA indicator variables is first run using only owner-occupied units, weighting by census-housing weights. A new value-adjusted weight is calculated by multiplying the census-housing weights by the predicted value from this first regression using housing characteristics alone, controlling for MSA. A second regression is run using these new weights for all units, rented and owner-occupied, on the housing characteristics fully interacted with tenure, along with the MSA indicators, which are not interacted. The houseprice differentials are taken from the MSA indicator variables in this second regression. As with the wage differentials, this adjusted weighting method has only a small impact on the measured price differentials.

iv

TABLE A1: INTER-METROPOLITAN INDICES, YEAR 2000 MSA Name San Jose, CA Stamford, CT Santa Barbara-Santa Maria-Lompoc, CA San Francisco-Oakland, CA Ventura, CA Santa Cruz, CA Salinas-Seaside-Monterey, CA Los Angeles-Long Beach, CA Austin, TX Honolulu, HI Danbury, CT Santa Rosa, CA San Diego, CA Fort Lauderdale-Hollywood, FL Washington, DC-MD-VA New York, NY-NJ Seattle-Everett, WA Trenton, NJ Naples, FL Monmouth-Ocean, NJ Ann Arbor, MI Phoenix, AZ Yolo, CA Manchester, NH Denver-Boulder, CO Miami, FL Colorado Springs, CO San Luis Obispo-Atascadero-Paso Robles, CA Las Vegas, NV Newburgh-Middletown, NY Boston, MA West Palm Beach-Boca Raton, FL Nashua, NH Atlanta, GA Orlando, FL Dutchess County, NY Madison, WI Sarasota, FL Reno, NV Bridgeport-Milford, CT Chicago, IL Fort Collins, CO

MSA Population

Relative Price of Housing Index

1,688,089 354,363 400,661 4,645,830 754,070 258,576 281,166 12,400,000 1,167,216 876,066 184,523 459,235 2,807,873 1,624,272 4,733,359 17,200,000 2,332,682 350,093 249,728 1,128,173 479,754 3,070,331 170,044 107,037 2,198,801 2,221,632 515,629 246,312 1,375,174 343,591 3,951,557 1,133,519 116,182 3,987,990 1,652,742 277,140 429,839 587,565 339,936 343,379 8,804,453 235,532

1.90 1.59 1.58 1.54 1.45 1.45 1.41 1.39 1.39 1.38 1.38 1.36 1.34 1.33 1.33 1.32 1.27 1.26 1.25 1.25 1.24 1.23 1.23 1.23 1.22 1.22 1.22 1.21 1.21 1.21 1.20 1.20 1.20 1.19 1.18 1.18 1.17 1.17 1.17 1.17 1.16 1.16

Median Expenditure Share on Housing Renters Only Renters & Owners 0.23 0.23 0.27 0.23 0.25 0.27 0.24 0.25 0.25 0.25 0.22 0.24 0.25 0.26 0.22 0.22 0.24 0.21 0.22 0.24 0.24 0.24 0.27 0.22 0.24 0.27 0.24 0.28 0.24 0.22 0.22 0.25 0.22 0.23 0.25 0.23 0.24 0.24 0.25 0.24 0.22 0.26

0.25 0.22 0.25 0.23 0.22 0.27 0.24 0.22 0.18 0.23 0.19 0.24 0.23 0.18 0.18 0.20 0.22 0.17 0.19 0.18 0.18 0.18 0.21 0.18 0.20 0.20 0.19 0.24 0.19 0.17 0.21 0.18 0.16 0.18 0.18 0.17 0.19 0.18 0.20 0.20 0.18 0.20

Housing Price Index Rentals Only Rentals & Owned 0.74 0.57 0.45 0.57 0.44 0.48 0.40 0.40 0.25 0.43 0.39 0.38 0.35 0.30 0.36 0.34 0.30 0.27 0.23 0.30 0.24 0.15 0.21 0.17 0.23 0.20 0.13 0.23 0.18 0.23 0.31 0.23 0.25 0.17 0.12 0.21 0.16 0.15 0.20 0.20 0.19 0.13

0.98 0.85 0.63 0.75 0.54 0.81 0.62 0.53 0.12 0.65 0.46 0.58 0.46 0.08 0.27 0.47 0.40 0.23 0.29 0.25 0.21 0.08 0.27 0.12 0.22 0.17 0.03 0.42 0.07 0.13 0.46 0.09 0.15 0.03 -0.05 0.15 0.12 0.08 0.20 0.36 0.24 0.12

Non-Housing Price Index 0.11 0.11 -0.01 0.14 0.06 0.12 0.05 0.07 -0.08 0.11 0.07 0.07 0.06 0.01 0.07 0.07 0.06 0.04 0.00 0.07 0.02 -0.06 0.00 -0.03 0.03 0.00 -0.06 0.04 -0.01 0.04 0.12 0.05 0.07 -0.01 -0.04 0.05 0.00 0.00 0.04 0.05 0.04 -0.02

Predicted Wage Index Renters Only Renters & Owners 0.10 0.11 0.01 0.09 0.01 0.04 -0.06 -0.06 0.05 0.09 0.01 0.04 0.02 -0.03 0.07 0.01 0.07 0.02 -0.10 0.02 0.12 -0.01 0.04 0.02 0.03 -0.11 0.06 0.04 -0.08 -0.02 0.09 -0.04 0.07 -0.01 -0.01 0.04 0.11 -0.02 -0.03 -0.08 0.00 0.09

0.08 0.18 -0.02 0.05 0.02 0.03 -0.10 -0.07 0.06 0.01 0.14 0.03 0.01 -0.03 0.08 0.00 0.07 0.06 -0.06 0.08 0.13 0.00 0.03 -0.03 0.05 -0.12 0.07 0.04 -0.12 0.01 0.09 0.00 0.07 0.02 -0.03 0.06 0.11 -0.01 -0.03 0.03 0.02 0.12

TABLE A1: INTER-METROPOLITAN INDICES, YEAR 2000 MSA Name Dalls-Fort Worth, TX Non-metropolitan CT Portland, OR-WA Houston, TX New Haven-West Haven, CT Minneapolis-St. Paul, MN-WI Raleigh-Durham, NC Philadelphia, PA-NJ Anchorage, AK Bryan-College Station, TX Santa Fe, NM Fort Myers-Cape Coral, FL Charlottesville, VA Tampa-St. Petersburg, FL Milwaukee, WI Wilmington, DE-NJ-MD Rochester, NY Stockton, CA Non-metropolitanNH Hartford, CT Sacramento, CA Daytona Beach, FL Atlantic City, NJ Flagstaff, AZ-UT Tacoma, WA Olympia, WA Jacksonville, FL Iowa City, IA Tucson, AZ State College, PA Barnstable-Yarmouth, MA Salt Lake City-Ogden, UT Albany-Schenectady-Troy, NY Riverside-San Bernardino-Ontario, CA Indianapolis, IN Charleston-North Charleston, SC Albuquerque, NM Omaha, NE-IA Detroit, MI Nashville-Davidson, TN Bellingham, WA Melbourne-Titusville-Cocoa, FL

MSA Population

Relative Price of Housing Index

5,043,876 1,350,818 1,789,019 4,413,414 358,125 2,856,295 1,182,869 5,082,137 259,063 153,194 148,785 440,333 160,421 2,386,781 1,499,015 499,454 1,030,303 562,377 1,011,597 708,743 1,632,863 445,477 359,167 117,109 706,103 210,011 1,101,766 108,518 843,732 134,971 144,360 1,331,833 796,100 3,253,263 1,603,021 454,054 712,937 584,099 4,430,477 1,234,004 169,001 479,298

1.16 1.14 1.13 1.13 1.12 1.12 1.11 1.11 1.11 1.11 1.11 1.10 1.09 1.09 1.08 1.08 1.08 1.08 1.08 1.07 1.07 1.07 1.07 1.07 1.06 1.06 1.06 1.06 1.06 1.05 1.05 1.05 1.05 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.03

Median Expenditure Share on Housing Renters Only Renters & Owners 0.22 0.21 0.24 0.22 0.23 0.23 0.24 0.22 0.24 0.35 0.24 0.24 0.24 0.23 0.21 0.21 0.24 0.23 0.22 0.21 0.24 0.25 0.25 0.25 0.23 0.24 0.23 0.30 0.25 0.29 0.25 0.23 0.22 0.25 0.21 0.24 0.25 0.21 0.20 0.23 0.27 0.23

0.16 0.17 0.21 0.15 0.19 0.17 0.18 0.17 0.17 0.21 0.20 0.17 0.19 0.17 0.17 0.17 0.15 0.19 0.17 0.17 0.20 0.17 0.19 0.19 0.19 0.19 0.16 0.18 0.19 0.19 0.21 0.19 0.16 0.19 0.16 0.18 0.19 0.16 0.16 0.18 0.22 0.16

Housing Price Index Rentals Only Rentals & Owned 0.14 0.17 0.15 0.08 0.16 0.18 0.11 0.17 0.30 -0.01 0.11 0.06 0.07 0.06 0.04 0.12 0.08 0.07 0.08 0.11 0.15 -0.01 0.09 0.01 0.08 0.08 0.01 0.03 0.00 0.04 0.11 0.08 0.05 0.11 -0.03 -0.01 0.00 -0.05 0.02 -0.03 0.05 0.00

-0.02 0.17 0.21 -0.10 0.20 0.07 0.05 0.07 0.20 -0.12 0.29 -0.03 0.00 -0.09 0.04 0.07 -0.09 0.11 0.04 0.16 0.19 -0.17 0.10 0.03 0.11 0.08 -0.11 -0.01 -0.03 -0.06 0.28 0.04 -0.04 0.09 -0.10 -0.01 0.00 -0.16 0.05 -0.05 0.12 -0.17

Non-Housing Price Index -0.01 0.03 0.03 -0.04 0.04 0.06 0.00 0.06 0.19 -0.12 0.01 -0.04 -0.02 -0.02 -0.04 0.04 0.01 0.00 0.01 0.04 0.08 -0.08 0.03 -0.06 0.02 0.02 -0.05 -0.02 -0.05 -0.01 0.06 0.03 0.00 0.07 -0.07 -0.05 -0.04 -0.08 -0.02 -0.07 0.01 -0.04

Predicted Wage Index Renters Only Renters & Owners 0.00 0.04 0.03 -0.04 0.01 0.05 0.02 0.02 0.07 0.03 0.11 -0.04 0.04 0.00 0.02 0.01 0.02 -0.11 0.07 -0.02 0.03 0.01 -0.13 0.06 0.02 0.05 0.02 0.09 -0.01 0.14 0.04 0.02 0.06 -0.06 0.00 0.00 0.03 0.03 -0.02 0.01 0.04 0.06

0.01 0.07 0.05 -0.02 0.04 0.08 0.07 0.03 0.05 0.04 0.10 -0.05 0.06 -0.01 0.04 0.05 0.06 -0.08 0.07 0.03 0.03 -0.03 -0.08 0.01 0.00 0.08 0.00 0.09 0.00 0.07 0.05 0.04 0.07 -0.07 0.02 -0.01 0.02 0.05 0.02 0.01 0.02 0.04

TABLE A1: INTER-METROPOLITAN INDICES, YEAR 2000 MSA Name Charlotte-Gastonia, NC Non-metropolitan RI Bremerton, WA Brockton, MA Baltimore, MD Lafayette-West Lafayette, IN Non-metropolitan HI Des Moines, IA Lancaster, PA Portland, ME Norfolk-Virginia Beach-Portsmouth, VA-NC San Antonio, TX Killeen-Temple, TX Memphis, TN-AR-MS Richmond, VA Rochester, MN Kenosha, WI Chico, CA Columbus, OH Kansas City, MO-KS Gainesville, FL Galveston-Texas City, TX Provo-Orem, UT Fort Pierce, FL Allentown-Bethlehem, PA-NJ Eugene-Springfield, OR Redding, CA Columbia, SC Medford, OR Non-metropolitan VT Lansing-East Lansing, MI Wilmington, NC Modesto, CA Non-metropolitan AK South Bend, IN Bloomington, IN Non-metropolitan CO Lexington-Fayette, KY Richland-Kennewick-Pasco, WA Non-metropolitan CA Worcester, MA Savannah, GA

MSA Population

Relative Price of Housing Index

1,499,677 258,023 234,652 258,188 2,513,661 181,493 335,651 375,685 464,550 241,693 1,553,838 1,551,396 313,151 998,698 995,112 122,319 148,260 202,375 1,443,293 1,682,053 219,795 249,853 367,035 323,090 641,637 324,317 162,160 544,165 179,811 608,387 445,925 233,637 450,865 367,124 266,264 122,388 924,086 258,129 191,186 1,249,739 282,673 232,087

1.03 1.03 1.03 1.03 1.03 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.01 1.01 1.01 1.01 1.01 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.99 0.99 0.99 0.99 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.97 0.97 0.97 0.97

Median Expenditure Share on Housing Renters Only Renters & Owners 0.21 0.22 0.23 0.22 0.22 0.27 0.23 0.21 0.21 0.22 0.24 0.22 0.22 0.23 0.23 0.21 0.21 0.28 0.22 0.21 0.30 0.22 0.24 0.25 0.21 0.27 0.26 0.22 0.26 0.23 0.23 0.25 0.22 0.21 0.21 0.32 0.23 0.24 0.21 0.24 0.21 0.24

0.17 0.19 0.19 0.19 0.17 0.18 0.22 0.16 0.17 0.18 0.18 0.15 0.17 0.17 0.17 0.15 0.16 0.21 0.17 0.16 0.19 0.15 0.21 0.16 0.16 0.21 0.20 0.16 0.21 0.18 0.16 0.20 0.18 0.18 0.15 0.21 0.20 0.17 0.16 0.20 0.18 0.18

Housing Price Index Rentals Only Rentals & Owned 0.01 0.08 0.04 0.09 0.05 -0.05 0.12 0.00 0.01 0.05 -0.03 -0.07 -0.10 -0.07 -0.02 0.02 0.03 -0.02 -0.02 -0.01 -0.04 -0.03 -0.01 -0.03 0.00 0.01 -0.04 -0.09 -0.02 -0.02 -0.03 -0.04 0.04 0.17 -0.13 -0.02 0.03 -0.06 -0.04 -0.03 0.02 -0.06

-0.04 0.21 0.11 0.18 0.03 -0.12 0.35 -0.07 -0.03 0.07 -0.07 -0.25 -0.25 -0.16 -0.10 -0.14 0.02 0.03 -0.06 -0.12 -0.16 -0.15 -0.02 -0.12 -0.04 0.08 0.00 -0.15 0.06 -0.06 -0.12 0.00 0.05 0.11 -0.24 -0.08 0.13 -0.11 -0.11 0.13 0.10 -0.08

Non-Housing Price Index -0.02 0.05 0.01 0.06 0.02 -0.08 0.10 -0.02 -0.01 0.03 -0.05 -0.08 -0.12 -0.09 -0.03 0.01 0.02 -0.03 -0.02 -0.02 -0.04 -0.03 0.00 -0.02 0.01 0.01 -0.04 -0.09 -0.01 0.00 -0.01 -0.02 0.06 0.19 -0.11 0.01 0.06 -0.03 -0.02 0.00 0.05 -0.03

Predicted Wage Index Renters Only Renters & Owners -0.02 0.08 0.09 -0.02 0.02 0.05 -0.01 0.04 -0.04 0.07 0.00 0.00 0.05 -0.04 0.01 0.02 0.01 0.02 0.02 0.02 0.07 0.02 0.04 -0.07 -0.01 0.04 0.01 0.01 0.03 0.06 0.03 0.00 -0.08 0.08 -0.02 0.05 0.05 0.03 -0.07 -0.04 0.02 -0.02

0.00 0.11 0.08 0.00 0.03 0.01 -0.09 0.05 -0.01 0.07 0.00 -0.04 -0.03 -0.03 0.03 0.07 0.03 0.00 0.03 0.05 0.07 0.02 0.08 -0.04 0.01 0.04 0.01 0.02 0.03 0.05 0.05 0.00 -0.09 0.02 0.01 0.07 0.06 0.05 0.00 -0.05 0.04 -0.02

TABLE A1: INTER-METROPOLITAN INDICES, YEAR 2000 MSA Name Reading, PA Racine, WI Green Bay, WI Myrtle Beach, SC Non-metropolitan NV Champaign-Urbana-Rantoul, IL Cincinnati, OH-KY-IN Lincoln, NE Boise City, ID St. Louis, MO-IL Springfield, IL Corpus Christi, TX Salem, OR Yuma, AZ Non-metropolitan MA Punta Gorda, FL Hamilton-Middletown, OH Syracuse, NY Tallahassee, FL Fort Walton Beach, FL Jackson, MS Athens, GA Tulsa, OK Sioux Falls, SD Cedar Rapids, IA Elkhart, IN Harrisburg, PA Wichita, KS Fitchburg-Leominster, MA Tyler, TX Greensboro--Winston-Salem--High Point, NC Amarillo, TX Akron, OH Springfield-Chicopee-Holyoke, MA-CT Appleton-Oshkosh, WI Buffalo, NY York, PA Fayetteville-Springdale, AR Panama City, FL New Orleans, LA Glens Falls, NY Greeley, CO

MSA Population

Relative Price of Housing Index

368,284 185,041 227,296 195,205 285,196 181,422 1,473,012 246,945 430,161 2,602,448 112,222 261,023 282,595 160,196 569,691 141,080 334,518 731,789 286,063 171,551 438,789 153,445 694,760 124,076 188,914 182,252 629,304 543,518 141,969 174,917 1,252,554 215,463 692,912 594,643 357,928 1,175,089 383,994 309,915 146,122 1,246,651 123,609 178,872

0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.93 0.93 0.93 0.93 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.91 0.91

Median Expenditure Share on Housing Renters Only Renters & Owners 0.21 0.21 0.19 0.22 0.22 0.27 0.21 0.23 0.24 0.21 0.20 0.22 0.23 0.23 0.21 0.23 0.21 0.23 0.30 0.23 0.23 0.27 0.22 0.21 0.21 0.20 0.21 0.20 0.20 0.22 0.21 0.22 0.22 0.22 0.19 0.23 0.20 0.22 0.23 0.23 0.23 0.24

0.16 0.16 0.16 0.17 0.17 0.17 0.16 0.17 0.17 0.15 0.15 0.15 0.19 0.17 0.18 0.16 0.16 0.15 0.18 0.17 0.16 0.19 0.15 0.16 0.15 0.15 0.16 0.15 0.17 0.15 0.16 0.15 0.17 0.18 0.15 0.16 0.16 0.17 0.17 0.18 0.17 0.20

Housing Price Index Rentals Only Rentals & Owned -0.05 -0.06 -0.07 -0.08 -0.04 -0.01 -0.09 -0.06 -0.04 -0.07 -0.14 -0.10 -0.03 -0.14 -0.03 -0.07 -0.07 -0.07 -0.02 -0.05 -0.18 -0.11 -0.12 -0.10 -0.10 -0.14 -0.08 -0.12 -0.05 -0.13 -0.14 -0.15 -0.09 -0.07 -0.12 -0.11 -0.09 -0.19 -0.12 -0.11 -0.05 -0.07

-0.11 -0.07 -0.07 -0.13 -0.04 -0.12 -0.06 -0.15 -0.11 -0.11 -0.21 -0.24 0.03 -0.18 0.11 -0.15 -0.08 -0.21 -0.10 -0.14 -0.26 -0.16 -0.22 -0.18 -0.13 -0.20 -0.08 -0.25 0.02 -0.25 -0.14 -0.27 -0.08 0.02 -0.14 -0.16 -0.12 -0.22 -0.17 -0.09 -0.17 -0.02

Non-Housing Price Index -0.01 -0.02 -0.03 -0.04 0.00 0.03 -0.05 -0.02 0.01 -0.02 -0.09 -0.05 0.03 -0.09 0.02 -0.02 -0.01 -0.01 0.04 0.00 -0.12 -0.05 -0.06 -0.03 -0.04 -0.07 0.00 -0.05 0.03 -0.05 -0.06 -0.07 -0.01 0.02 -0.03 -0.03 -0.01 -0.11 -0.04 -0.03 0.04 0.02

Predicted Wage Index Renters Only Renters & Owners -0.02 0.02 -0.01 -0.04 0.02 0.08 0.00 0.04 0.02 0.02 0.04 0.02 -0.03 -0.05 0.04 0.00 0.03 0.01 0.07 0.09 -0.03 -0.03 0.00 -0.02 0.05 -0.04 0.03 0.02 -0.03 -0.02 -0.05 -0.02 0.01 0.01 0.04 0.02 0.00 0.01 0.00 -0.04 0.00 -0.03

-0.02 0.04 0.03 -0.05 -0.02 0.06 0.03 0.06 0.05 0.03 0.05 -0.01 -0.02 -0.12 0.06 -0.03 0.05 0.03 0.06 0.04 0.00 0.00 0.04 -0.01 0.06 -0.05 0.01 0.04 0.00 -0.02 -0.02 -0.02 0.03 0.02 0.05 0.04 0.00 -0.02 -0.02 -0.04 0.00 0.01

TABLE A1: INTER-METROPOLITAN INDICES, YEAR 2000 MSA Name Ocala, FL Little Rock-North Little Rock, AR Evansville, IN-KY Lakeland-Winter Haven, FL Bakersfield, CA Dover, DE Grand Junction, CO Cleveland, OH Grand Rapids, MI Lubbock, TX Vineland-Millville-Bridgeton, NJ Janesville-Beloit, WI Oklahoma City, OK Jacksonville, NC Kalamazoo-Portage, MI Dayton, OH Yuba City, CA Asheville, NC Clarksville-Hopkinsville, TN-KY Waterbury, CT Providence-Warwick-Pawtucket, RI-MA Topeka, KS Merced, CA Birmingham, AL Columbia, MO Roanoke, VA Pensacola, FL Sheboygan, WI Fresno, CA Greenville-Spartanburg, SC Binghamton, NY-PA Kankakee, IL Spokane, WA Bloomington-Normal, IL Davenport-Rock Island-Moline, IA-IL Fayetteville, NC Wichita Falls, TX Non-metropolitan OR Rockford, IL Pueblo, CO Montgomery, AL La Crosse, WI

MSA Population

Relative Price of Housing Index

259,712 584,977 252,410 482,562 650,891 125,613 111,922 2,255,480 984,107 243,899 146,275 151,640 892,347 149,091 451,406 954,465 137,870 225,195 134,209 108,117 1,025,944 168,994 209,707 803,700 136,063 236,363 411,270 111,021 924,612 796,528 254,116 104,042 418,375 152,616 268,781 299,932 131,595 1,194,699 319,846 135,990 333,479 105,700

0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.89 0.89 0.89 0.89 0.89 0.89 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.87 0.87 0.87

Median Expenditure Share on Housing Renters Only Renters & Owners 0.22 0.23 0.20 0.21 0.23 0.22 0.25 0.21 0.21 0.25 0.25 0.20 0.22 0.22 0.21 0.21 0.22 0.23 0.21 0.22 0.21 0.21 0.24 0.21 0.24 0.21 0.24 0.17 0.24 0.20 0.23 0.21 0.25 0.22 0.20 0.23 0.22 0.24 0.20 0.25 0.23 0.21

0.16 0.16 0.15 0.15 0.17 0.17 0.20 0.17 0.16 0.17 0.16 0.15 0.16 0.17 0.16 0.16 0.19 0.18 0.17 0.17 0.18 0.15 0.20 0.16 0.18 0.16 0.17 0.15 0.19 0.16 0.14 0.16 0.18 0.15 0.15 0.18 0.15 0.19 0.15 0.18 0.17 0.16

Housing Price Index Rentals Only Rentals & Owned -0.15 -0.15 -0.21 -0.15 -0.11 -0.09 -0.13 -0.06 -0.07 -0.15 -0.05 -0.12 -0.15 -0.16 -0.13 -0.14 -0.15 -0.11 -0.18 -0.04 -0.09 -0.17 -0.13 -0.16 -0.14 -0.21 -0.15 -0.18 -0.06 -0.21 -0.16 -0.13 -0.10 -0.08 -0.17 -0.12 -0.19 -0.11 -0.14 -0.22 -0.18 -0.18

-0.33 -0.20 -0.21 -0.27 -0.14 -0.14 -0.08 -0.03 -0.11 -0.30 -0.12 -0.16 -0.24 -0.25 -0.19 -0.13 -0.09 -0.06 -0.30 -0.02 0.04 -0.28 -0.06 -0.16 -0.21 -0.22 -0.24 -0.14 -0.04 -0.21 -0.28 -0.12 -0.13 -0.14 -0.20 -0.21 -0.36 -0.03 -0.20 -0.23 -0.22 -0.19

Non-Housing Price Index -0.06 -0.06 -0.11 -0.06 -0.01 0.01 -0.03 0.04 0.03 -0.05 0.05 -0.02 -0.05 -0.05 -0.03 -0.03 -0.04 0.00 -0.08 0.07 0.02 -0.06 -0.02 -0.05 -0.02 -0.09 -0.03 -0.06 0.06 -0.08 -0.04 0.00 0.02 0.05 -0.05 0.01 -0.06 0.03 -0.01 -0.09 -0.04 -0.04

Predicted Wage Index Renters Only Renters & Owners -0.04 0.00 0.02 -0.05 -0.08 0.01 -0.02 0.00 -0.05 -0.02 -0.19 -0.03 0.01 0.06 -0.02 0.00 -0.03 0.02 0.05 -0.08 -0.03 0.02 -0.15 0.02 0.10 0.00 0.03 0.03 -0.13 -0.02 0.01 -0.08 0.04 0.05 0.01 0.05 0.03 0.01 -0.02 -0.06 0.02 0.04

-0.08 0.01 0.02 -0.08 -0.09 -0.02 0.03 0.02 0.03 -0.03 -0.12 0.00 0.01 -0.02 0.02 0.02 -0.06 -0.01 -0.01 -0.12 -0.01 0.03 -0.16 0.03 0.07 0.00 0.00 0.02 -0.10 -0.01 0.04 -0.03 0.04 0.09 0.01 -0.03 0.00 0.00 0.00 -0.05 0.00 0.05

TABLE A1: INTER-METROPOLITAN INDICES, YEAR 2000 MSA Name Fort Wayne, IN Wausau, WI Yakima, WA Biloxi-Gulfport, MS Louisville, KY-IN Pittsburgh, PA Non-metropolitan UT Sioux City, IA-NE Eau Claire, WI El Paso, TX Waterloo-Cedar Falls, IA Waco, TX Columbus, GA-AL Non-metropolitan NY Knoxville, TN Non-metropolitan WA Non-metropolitan MD Odessa, TX Benton Harbor, MI Muncie, IN Toledo, OH-MI Utica-Rome, NY Greenville, NC Baton Rouge, LA Kokomo, IN Abilene, TX Tuscaloosa, AL Youngstown-Warren, OH Fargo-Moorhead, ND-MN Billings, MT Auburn-Opelika, AL Longview-Marshall, TX Shreveport, LA Flint, MI Mobile, AL Augusta, GA-SC Erie, PA Saginaw, MI Rocky Mount, NC Hickory, NC Jackson, MI Macon, GA

MSA Population

Relative Price of Housing Index

460,349 127,099 223,726 318,936 921,599 2,285,064 531,967 103,140 147,758 676,220 124,908 212,313 186,426 1,744,930 576,512 1,063,531 666,998 238,692 163,682 119,028 617,883 300,337 134,932 604,708 100,506 126,952 164,875 593,100 121,173 128,660 116,435 170,557 393,700 240,153 540,100 451,061 279,521 400,853 143,674 342,072 160,391 321,450

0.87 0.87 0.87 0.87 0.86 0.86 0.86 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.83 0.83 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.81 0.81 0.81 0.81 0.81

Median Expenditure Share on Housing Renters Only Renters & Owners 0.20 0.19 0.25 0.22 0.21 0.21 0.22 0.21 0.21 0.24 0.24 0.23 0.22 0.23 0.23 0.23 0.22 0.21 0.20 0.24 0.21 0.22 0.26 0.24 0.20 0.22 0.27 0.20 0.22 0.22 0.30 0.21 0.22 0.23 0.24 0.22 0.21 0.22 0.21 0.19 0.20 0.22

0.15 0.15 0.19 0.16 0.16 0.15 0.19 0.15 0.16 0.16 0.15 0.16 0.17 0.15 0.17 0.19 0.17 0.14 0.16 0.16 0.16 0.15 0.17 0.16 0.13 0.16 0.18 0.15 0.16 0.16 0.17 0.15 0.16 0.14 0.17 0.16 0.16 0.15 0.16 0.15 0.15 0.15

Housing Price Index Rentals Only Rentals & Owned -0.18 -0.16 -0.14 -0.17 -0.18 -0.16 -0.17 -0.20 -0.20 -0.25 -0.20 -0.22 -0.23 -0.13 -0.25 -0.13 -0.18 -0.22 -0.20 -0.24 -0.20 -0.20 -0.22 -0.17 -0.20 -0.21 -0.20 -0.29 -0.21 -0.22 -0.27 -0.26 -0.29 -0.21 -0.28 -0.25 -0.25 -0.21 -0.27 -0.25 -0.22 -0.25

-0.27 -0.25 -0.07 -0.24 -0.15 -0.19 -0.17 -0.27 -0.24 -0.38 -0.27 -0.33 -0.25 -0.26 -0.24 -0.03 -0.11 -0.39 -0.20 -0.31 -0.17 -0.30 -0.19 -0.18 -0.23 -0.36 -0.20 -0.28 -0.25 -0.25 -0.25 -0.36 -0.33 -0.32 -0.27 -0.28 -0.26 -0.24 -0.24 -0.23 -0.21 -0.31

Non-Housing Price Index -0.04 -0.02 0.00 -0.03 -0.03 0.00 -0.02 -0.04 -0.05 -0.09 -0.04 -0.06 -0.07 0.03 -0.08 0.04 -0.01 -0.05 -0.03 -0.07 -0.02 -0.03 -0.05 0.01 -0.02 -0.03 -0.02 -0.10 -0.01 -0.03 -0.07 -0.07 -0.09 -0.01 -0.08 -0.05 -0.04 0.00 -0.06 -0.04 -0.01 -0.04

Predicted Wage Index Renters Only Renters & Owners 0.00 0.03 -0.12 -0.02 -0.02 0.05 0.04 -0.04 0.04 -0.04 -0.01 -0.03 -0.03 0.02 0.02 -0.01 -0.03 0.01 -0.01 -0.03 -0.01 0.00 -0.03 -0.03 0.00 0.01 -0.01 0.00 0.04 0.06 0.04 0.01 -0.06 -0.09 -0.02 -0.03 0.02 -0.05 -0.11 -0.04 -0.04 -0.03

0.01 0.02 -0.10 -0.02 0.00 0.05 0.05 -0.03 0.02 -0.10 0.02 -0.04 -0.03 0.01 0.03 0.00 0.00 -0.03 0.01 0.00 0.01 0.01 -0.01 0.00 0.00 0.00 -0.01 0.00 0.03 0.03 0.00 0.00 -0.04 -0.12 -0.01 -0.02 0.01 0.01 -0.08 -0.06 0.01 -0.02

TABLE A1: INTER-METROPOLITAN INDICES, YEAR 2000 MSA Name Canton, OH Chattanooga, TN-GA St. Joseph, MO Visalia-Tulare-Porterville, CA Albany, GA Laredo, TX Springfield, MO Hattiesburg, MS Las Cruces, NM Hagerstown, MD Non-metropolitan AZ Beaumont-Port Arthur-Orange, TX Jamestown-Dunkirk, NY Huntsville, AL New Bedford, MA Non-metropolitan ME Non-metropolitan ID Lynchburg, VA St. Cloud, MN Fort Smith, AR-OK Non-metropolitan MI Lake Charles, LA Johnson City-Kingsport-Bristol, TN-VA Scranton-Wilkes-Barre, PA Williamsport, PA Decatur, IL Peoria, IL Non-metropolitan FL Non-metropolitan WI Non-metropolitan IN Joplin, MO Lafayette, LA Mansfield, OH Non-metropolitan KS Sharon, PA Non-metropolitan WY Non-metropolitan TX Houma-Thibodaux, LA Non-metropolitan VA Duluth-Superior, MN-WI Lima, OH Non-metropolitan IA

MSA Population

Relative Price of Housing Index

408,072 434,752 101,442 367,566 120,551 190,074 327,829 111,694 173,843 128,316 942,343 381,559 140,116 344,491 174,864 1,033,664 863,855 213,723 168,856 169,401 2,178,963 183,144 314,402 624,276 121,501 114,926 346,102 1,222,532 1,866,585 1,791,003 155,401 247,230 130,084 1,366,517 120,147 493,849 4,030,376 103,563 1,640,567 199,548 156,274 1,863,270

0.81 0.81 0.81 0.81 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.78 0.78 0.78 0.78 0.78 0.78 0.78 0.77 0.77 0.77 0.77 0.76 0.76 0.76 0.76 0.75 0.75 0.75 0.75 0.75

Median Expenditure Share on Housing Renters Only Renters & Owners 0.21 0.21 0.22 0.24 0.22 0.23 0.23 0.26 0.25 0.19 0.22 0.22 0.22 0.20 0.19 0.21 0.22 0.20 0.20 0.19 0.21 0.20 0.21 0.20 0.20 0.22 0.20 0.22 0.19 0.19 0.21 0.22 0.19 0.21 0.19 0.20 0.20 0.20 0.20 0.21 0.18 0.19

0.16 0.16 0.16 0.19 0.15 0.18 0.17 0.17 0.17 0.17 0.17 0.13 0.15 0.15 0.18 0.17 0.17 0.16 0.15 0.15 0.15 0.14 0.16 0.17 0.16 0.14 0.15 0.16 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.16 0.14 0.15 0.16 0.14 0.14 0.14

Housing Price Index Rentals Only Rentals & Owned -0.25 -0.23 -0.29 -0.16 -0.30 -0.29 -0.25 -0.29 -0.31 -0.23 -0.24 -0.28 -0.25 -0.24 -0.20 -0.24 -0.26 -0.30 -0.24 -0.35 -0.28 -0.33 -0.38 -0.27 -0.28 -0.29 -0.25 -0.22 -0.27 -0.31 -0.36 -0.31 -0.32 -0.32 -0.32 -0.30 -0.39 -0.34 -0.35 -0.30 -0.35 -0.35

-0.20 -0.25 -0.34 -0.11 -0.33 -0.34 -0.28 -0.37 -0.27 -0.14 -0.20 -0.40 -0.39 -0.26 0.04 -0.22 -0.24 -0.30 -0.27 -0.36 -0.26 -0.32 -0.33 -0.21 -0.26 -0.35 -0.23 -0.26 -0.25 -0.30 -0.42 -0.27 -0.28 -0.43 -0.32 -0.26 -0.47 -0.33 -0.34 -0.31 -0.33 -0.37

Non-Housing Price Index -0.04 -0.02 -0.08 0.06 -0.08 -0.07 -0.04 -0.07 -0.09 -0.01 -0.02 -0.06 -0.03 -0.01 0.03 -0.01 -0.03 -0.07 0.00 -0.11 -0.05 -0.09 -0.13 -0.03 -0.03 -0.05 0.00 0.03 -0.02 -0.06 -0.10 -0.05 -0.06 -0.05 -0.05 -0.03 -0.11 -0.06 -0.06 -0.02 -0.06 -0.06

Predicted Wage Index Renters Only Renters & Owners -0.01 0.01 0.01 -0.20 -0.02 -0.09 0.01 0.03 -0.02 -0.02 0.00 -0.05 -0.01 0.01 -0.09 0.04 0.01 -0.04 0.02 -0.03 0.01 -0.04 0.03 0.02 0.01 0.02 0.03 -0.08 0.01 -0.01 -0.02 0.02 -0.01 0.04 0.05 0.04 -0.04 -0.07 -0.02 0.04 0.01 0.02

0.00 0.00 -0.01 -0.19 -0.05 -0.13 0.00 0.02 -0.08 -0.02 -0.07 -0.02 0.00 0.06 -0.07 0.02 0.01 -0.02 0.01 -0.05 0.00 -0.03 0.01 0.01 -0.01 0.00 0.04 -0.08 0.00 -0.02 -0.01 0.00 -0.01 0.01 0.01 0.02 -0.06 -0.07 -0.05 0.06 0.01 0.00

TABLE A1: INTER-METROPOLITAN INDICES, YEAR 2000 MSA Name

MSA Population

Relative Price of Housing Index

Median Expenditure Share on Housing Renters Only Renters & Owners

Housing Price Index Rentals Only Rentals & Owned

Non-Housing Price Index

Predicted Wage Index Renters Only Renters & Owners

Terre Haute, IN 149,397 0.75 0.22 0.15 -0.34 -0.37 -0.05 0.01 0.00 Non-metropolitan MN 1,565,030 0.74 0.20 0.15 -0.34 -0.35 -0.05 0.00 0.01 Monroe, LA 146,975 0.74 0.23 0.16 -0.35 -0.32 -0.05 -0.05 -0.04 Non-metropolitan PA 2,023,193 0.74 0.20 0.15 -0.34 -0.34 -0.04 0.01 -0.02 Goldsboro, NC 113,118 0.74 0.20 0.16 -0.38 -0.30 -0.08 0.01 -0.05 Non-metropolitan DE 158,149 0.73 0.20 0.17 -0.33 -0.09 -0.02 -0.06 -0.03 Non-metropolitan OH 2,548,986 0.73 0.19 0.15 -0.34 -0.31 -0.03 0.00 -0.02 Non-metropolitan IL 2,202,549 0.73 0.20 0.14 -0.36 -0.37 -0.05 0.02 0.00 Sumter, SC 104,047 0.72 0.21 0.15 -0.35 -0.39 -0.03 0.03 -0.06 Altoona, PA 131,023 0.72 0.19 0.15 -0.37 -0.34 -0.04 -0.02 -0.02 Alexandria, LA 128,075 0.72 0.22 0.16 -0.39 -0.38 -0.06 -0.10 -0.06 Non-metropolitan MT 774,080 0.72 0.23 0.18 -0.31 -0.25 0.01 0.04 0.03 Jackson, TN 107,550 0.72 0.22 0.16 -0.33 -0.36 0.01 -0.10 -0.03 Non-metropolitan NE 878,760 0.71 0.19 0.14 -0.38 -0.45 -0.05 0.02 0.00 Decatur, AL 145,469 0.71 0.20 0.15 -0.42 -0.34 -0.08 -0.04 0.00 Non-metropolitan NC 2,632,956 0.71 0.21 0.16 -0.37 -0.28 -0.03 -0.07 -0.07 Non-metropolitan GA 2,744,802 0.70 0.20 0.15 -0.40 -0.35 -0.05 -0.07 -0.07 McAllen-Pharr-Edinburg, TX 565,800 0.70 0.22 0.16 -0.45 -0.58 -0.09 -0.11 -0.19 Non-metropolitan WV 1,809,034 0.70 0.21 0.15 -0.43 -0.44 -0.07 0.02 0.00 Non-metropolitan ND 521,239 0.70 0.20 0.14 -0.44 -0.51 -0.08 0.07 0.02 Dothan, AL 138,133 0.69 0.19 0.15 -0.44 -0.42 -0.06 -0.03 -0.04 Danville, VA 109,618 0.68 0.20 0.15 -0.44 -0.41 -0.06 -0.12 -0.10 Anniston, AL 110,594 0.68 0.21 0.15 -0.45 -0.44 -0.07 0.00 -0.03 Florence, AL 142,703 0.68 0.22 0.16 -0.46 -0.36 -0.07 -0.01 0.00 Brownsville-Harlingen-San Benito, TX 336,631 0.68 0.22 0.16 -0.41 -0.50 -0.03 -0.09 -0.16 Non-metropolitan OK 1,862,951 0.68 0.20 0.14 -0.46 -0.52 -0.07 0.00 -0.03 Non-metropolitan NM 783,050 0.67 0.21 0.16 -0.44 -0.36 -0.04 0.01 -0.06 Non-metropolitan MO 1,798,819 0.67 0.20 0.15 -0.48 -0.47 -0.08 0.01 -0.04 Non-metropolitan AR 1,607,993 0.66 0.21 0.15 -0.47 -0.47 -0.06 -0.05 -0.06 Non-metropolitan SC 1,616,255 0.66 0.20 0.15 -0.39 -0.32 0.02 -0.05 -0.08 Non-metropolitan KY 2,828,647 0.65 0.19 0.15 -0.47 -0.47 -0.05 0.01 -0.03 Non-metropolitan TN 2,123,330 0.65 0.19 0.15 -0.50 -0.43 -0.07 -0.03 -0.06 Non-metropolitan SD 629,811 0.64 0.21 0.15 -0.45 -0.46 0.00 0.05 0.01 Gadsden, AL 102,183 0.63 0.18 0.15 -0.53 -0.45 -0.07 -0.01 -0.01 Non-metropolitan MS 1,869,256 0.63 0.21 0.15 -0.53 -0.53 -0.07 -0.08 -0.08 Non-metropolitan LA 1,415,540 0.62 0.21 0.15 -0.56 -0.46 -0.08 -0.05 -0.05 Johnstown, PA 233,942 0.62 0.19 0.15 -0.53 -0.44 -0.04 0.02 -0.01 Non-metropolitan AL 1,504,381 0.58 0.19 0.15 -0.64 -0.51 -0.10 -0.03 -0.06 Rental price indices constructed from 2000 Census 5% microdata samples. Aggregate expenditure shares constructed by dividing some of all rental expenditure by sum of all income in MSA. Relative price of housing index is ratio of expontiated rental housing price index to exponentiated non-housing price index.