Wage gradients, rent gradients, and the price elasticity of demand for housing: An empirical investigation PDF

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JOURNAL OF URBAN ECONOMICS 12, 168- 176 ( 1982) Wage Gradients, Rent Gradients, and the Price Elasticity of Demand for Housing: An Empirical Investigation’ RANDALL W. EBERTS Department of Economics, University of Oregon, Eugene, Oregon 97403 AND TIMOTHY J. GRONBERG Department of Economics, Texas A &...

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JOURNAL

OF URBAN

ECONOMICS

12, 168- 176 ( 1982)

Wage Gradients, Rent Gradients, and the Price Elasticity of Demand for Housing: An Empirical Investigation’ RANDALL W. EBERTS Department

of Economics, University of Oregon, Eugene, Oregon 97403 AND TIMOTHY J. GRONBERG

Department of Economics, Texas A & M University, College Station, Texas 78363, and Northwestern University, Evanston, Illinois 60201 Received September 24, 1980; revised June 1, 1981 As developed in Muth’s “Cities and Housing,” attainment of locational equilibrium within an urban area implies a necessary functional correspondence between wage and price gradients and the compensated price elasticity of demand for housing. In this paper estimates of the rent and wage gradients are utilized to generate price-elasticity estimates via this equilibrium correspondence. The Box-Cox transformation technique is used with data from the metropolitan Chicago area to test for the functional forms of the wage and rent gradients. The optimal maximumlikelihood functional forms for both gradients yield a price-elasticity estimate of - 0.40.

INTRODUCTION Richard Muth’s pioneering volume, “Cities and Housing” [7], has provided the basis for much of the theoretical and empirical work in urban economics during the past decade. Almost any investigation into the effects of space upon the decision-making processes of consumer and producer agents in the market for housing has benefited and can continue to benefit from Muth’s groundbreaking analysis. As with many fundamental works, reexamination of the text continues to yield new and useful insights and thereby suggests additional lines of study. This paper draws upon a portion of the Muth monograph that has remained unexplored, namely the relationship between the price-distance function for housing and the wage-distance function for labor that obtains in locational equilibrium. We suggest a methodology for utilizing Muth’s equilibrium conditions to draw inferences ‘The authors thank G. S. Goldstein and an anonymous referee for helpful comments on an earlier draft of this paper. 168 0094 1190/82/050168-09$02.00/O Copyright All rights

0 19X2 by Academic Press. Inc. of reproduction in any form reserved.

PRICE ELASTICITY

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concerning the compensated price elasticity of demand for housing services. This new application of the Muth conditions is accomplished by employing the Box-Cox transformation technique to test for the functional forms of the two price-distance relationships (housing and labor). Specification of the estimating equations follows the established hedonic-price literature in the caseof housing prices and the work of Eberts [2] in the caseof wages. The empirical results generated employing this general spatial-equilibrium methodology yield a maximum-likelihood mean-price elasticity estimate of -0.40. This value accords well with findings in recent studies by Polinsky and Ellwood [9] and by Gerking and Boyes [4]. In addition to the optimal form results, certain restricted functional-form hypotheses are investigated. The importance of assumptions concerning functional form to the resulting elasticity estimates is established. Of particular interest is the finding that, for our Chicago sample, the functional-form combinations employed in the Muth monograph are not rejected at the 95% confidence level. THEORETICAL FOUNDATIONS The theoretical underpinnings for the empirical analysis in this paper are found in Muth’s [7] model of residential location within a core-dominated city with uniformly distributed local employment opportunities. Locational equilibrium within this model implies the existence of a negative relationship between distance from the CBD and both housing price and nominal wages. Both of these relationships are derived from the assumptions of perfect intraurban mobility, which requires that identical consumers achieve identical utility in spatial equilibrium. Thus, core workers with identical skills living at different distances from the CBD must pay compensating differential prices for housing to offset differences in commuting costs. This implies that the price of housing must decline as distance from the employment center increases.Given a negative price-distance function for housing, locally employed workers within a given labor class must also face a negative wage-distance function for their labor servicesin order to equalize real incomes over space. The functional form of the wage gradient for the local employment sector is thus related to the functional form of the housing-price gradient. The precise nature of that relationship is derived from the first- and second-order conditions of the constrained utility-maximization problem facing a locally employed worker. Local workers are assumed to maximize a two-good utility function, V-F Ma ,, . . . ,a,)), where X is a composite nonhousing private commodity and h represents housing services. Housing services are assumed to be a function of a number of attributes, a ,, . . . ,a,, such as structural quality and neighborhood amenities. The budget constraint takes the form

g = w(k) - x- P(u,,..., (I,, k)h(u,,..., a,).

170

EBERTS AND

GRONBERG

The relationship that is of concern to the present study is derived from the partial differentiation of the LaGrangian function L = U - Xg with respect to distance i3L ak=

-X(P,h

- W,) = 0

or W, = P,h.

(1)

This condition requires that in equilibrium the consumer cannot increase his real income by a marginal change in location. Under the assumption of homogeneous preferences, the corresponding second-order condition for locational stability requires that ah W,, - hPkk - Pkz = 0. Muth refers to this situation as one of neutral equilibrium where locally employed households are indifferent to all locations within the urban area. With utility constant over space,changes in housing consumption represent compensated demand changes,which implies that

Using this relationship (2) can be written as wkk/wk

=

(Pkk/Pk)

+

E(Pk/P)

(3)

where c is the compensated price elasticity of demand for housing services. Equation (3), which is identical to Muth’s [7] equation 4.4, forms the basis for the empirical estimation in the following sections. EMPIRICAL

METHODOLOGY

If locational equilibrium obtains, the wage and housing price-distance functions are linked to one another via (3), with the precise form of the relationship dependent upon consumer preferences as reflected in the price elasticity of demand for housing. Muth utilizes (3) to draw implications as to the form of the wage gradient. Employing existing demand estimates (Muth [8], Reid [lo]) of E = - 1, and assuming a negative exponential price-distance function, he concludes that money incomes of locally employed workers must decline by a constant absolute amount per unit

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OF DEMAND

distance, that is, IV,, = 0. In his analysis, Muth makes inferences conceming the wage-distance function conditional upon estimates of P(k) and E. There exists an interesting alternative approach to employing (3) for inferential purposes, however. If independent estimates of W(k) and P(k) are derived for a single urban area, point estimates for e can be obtained conditional upon the empirical results of the wage and price function estimations. Thus, estimation of the equilibrium structure of spatial prices for labor and housing conveys information concerning properties of the demand function for housing. Both W(k) and P(k) are reduced-form relationships that reflect the interaction of supply and demand forces in the labor and housing markets, respectively. To estimate these functional relationships, a particular functional form must be specified. The theory does not provide an indication of the most appropriate specification (for a discussion of this point for the case of the housing market, see Rosen [ 111). Fortunately, the specification of functional forms can be approached in a flexible fashion, which allows the data to indicate the optimal form, by utilizing the Box-Cox transformation technique (for applications of this technique to estimating hedonic housing-price regressions,seeGoodman [ 51,Bender et al. [ 11).In this paper, a variant of the quadratic Box-Cox transforms is employed that allows the wage and price equations to be written as 2

wt(x)= wO + aI4t(P) + 2 ak[ZktX~tl(P) + E k=2

j=r+l

"jzjt(P)

+ ‘t t4)

k=2

In the first equation, the wage observed in the t th community is denoted by w,, the distance of the tth community to the CBD by X,t, and the selected socioeconomic and organizational variables that affect both the supply and demand for labor are denoted by the Z’S.~ The housing-value *At the referee’s suggestion, we attempted a three-parameter grid search, allowing the transformation parameter on the distance variable to adjust independently from that on the remaining explanatory variables. The maximum-likelihood value for the distance-transformation parameter was not statistically different from that for the other regressors.We therefore include only the results from the two-parameter search procedure. ‘The explanatory variables are entered into the wage equation either in a multiplicative form with distance or additively. The noninteractive terms include the distance variable and three measures of organizational structure: population, per capita local government expenditures, and the length of the workweek. The interactive measures include per unit housing costs, employment concentration, and population change. The interested reader may refer to Eberts [2] for a more detailed explanation of the variables. The housing-price equation follows the standard model and includes the distance to the CBD, percentage of homes owner occupied, median number of rooms, median age of house, and local school expenditures per pupil.

172

FiBERl-S AND

GRONBERG

equation regressesthe median house value in community t, denoted by rt, against distance from the CBD, X,,, and selected explanatory variables not necessarily included in the wage equation. Following the Box-Cox transformation technique, the dependent and independent variables are written in the general form

YW=(YX -m =logy,

A#0 A=0

(6)

where X takes on values between -2 and 2. The independent variables are written in the sameform but the parameter p is allowed to vary between - 2 and 2 independently of X. The Box-Cox estimation technique involves a search over alternative values of X and p so as to maximize the log-likelihood function. Conditional upon the maximum-likelihood values of X and ~1,standard errors of the regression coefficients are obtained in the usual fashion. Note further that several common regression specifications, that is, linear, log-linear, and semi-log-linear, can be obtained by placing specific restrictions upon the values of the Box-Cox parameters.4 An additional advantage afforded by the Box-Cox transformation technique is the ability to subject these restricted forms to hypothesis testing. Suppose the particular hypothesis being tested is the linear form restrictions of X = 1, p = 1. First calculate the maximum value of the likelihood function under the restricted hypothesis. Then calculate the ratio of the restricted maximum-likelihood value to the unrestricted “true” maximumlikelihood value obtained from the initial parameter-search procedure. The difference between the unrestricted and restricted likelihood values will depend upon the divergence of the restricted values of X and p from their optimal values, and upon the behavior of the likelihood function over the relevant range. It has been shown that minus two times the log of this ratio of likelihood values is distributed as chi squared with two degrees of freedom (see Zarembka [12]). Functional forms that correspond to likelihood statistics lying within a lOO(1 - (Y)%confidence region are considered not to be different from the null hypothesis at the (Ylevel of significance (0 -c a -=l 1). RESULTS The wage- and housing-price equations are estimated from a sample of communities in the Chicago SMSA. The house values and housing characteristics are obtained from 1970 census data. The wage levels for each community are acquired from the semiannual survey of local government 4 The restrictions are as follows: linear (I, 1); log-linear (0,O); semilog (0, 1)

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173

salaries and fringe benefits conducted by the Cook County Bureau of Administration. The wage equation is estimated utilizing the monthly salaries of municipal administrative personnel. This group appears to be in a reasonable financial position to purchase the median-valued house in the sample communities if one accepts the four to one ratio of house values to yearly income as a rule of thumb. Although public labor markets are used in the estimation, a nationwide study by Ehrenberg and Goldstein [3] and a Chicago study by Eberts [2] suggest that considerable mobility exists between public and private-sector labor markets that results in similar wage scales for comparable jobs in the two sectors. Many features make the Chicago area well suited for a price-distance gradient study. It has been a laboratory for a number of previous studies including the urban-density study by Muth [7] and the land value study by Mills [6]. Both Muth and Mills found that the economic and geographic features of Chicago comply very closely with the assumptions of the monocentric city model. Three features make Chicago particularly attractive. First, Chicago is situated on a flat, almost featurelessplain. Second, the City of Chicago contains a concentration of most major types of employment and dominates the area’s labor market. Third, the major transportation routes emanate radially from the city center. For our sample, the unrestricted maximum-likelihood estimates of X and p are (LO) for the wage equation and (0,0.9) for the rent equation.5 The individual coefficient estimates and r-statistics derived conditional upon these transformation parameter values are presented in Table 1.6 Utilizing these estimation results in conjunction with (3) yields the following expression for evaluating the compensatedprice elasticity of demand for housing:

Evaluated at the sample mean distance k = 23, the maximum-likelihood estimate of E is therefore equal to -0.40.’ The 95% confidence interval is ‘Bender et al. [I] find the (0,l) form to be optimal (restricted) for their estimate of the hedonic-price equation of housesin the Chicago area. %Sincewe are primarily interested in the functional forms and have only a secondary interest in the coefficient estimates, we list the results in Table 1 only to show that the distance coefficient is negative and statistically significant in all cases and that the other explanatory variables have the appropriate values and explain a large portion of the variation in the wage and house values across communities. ‘Note that the estimated elasticity values vary with distance. Since the price of housing varies with distance, we are observing households at different points on their housing-demand surface. Unless the underlying functional form for the housing-demand function is of the constant elasticity variety, then theory suggeststhat elasticity will vary with distance as well. (A similar

174

EBERTS AND GRONBERG TABLE 1 Estimates of the Wage and Housing Price Equations for the Chicago SMSA _--.-. Functional forms Explanatory variables U?O) (1,1) Dependent variable: monthly salaries of municipal administrators -4741.72 Constant 1556.48 Distance from CBD measured in airline miles Per capita municipal government expenditures Length of workweek Population of community (Per unit housing costs) (distance to CBD) (Employment concentration) (distance to CBD) (Population change) (distance to CBD) R2 (adj) ---Ll,(~~ PL)

(2.25) -588.20 (2.64)

182.89 (1.52) -569.92 (2.18) III.86 (1.25) 124.52 (3.23) -78.98 (1.72) -7.00

(5.29) -24.44 (3.04) .51

(.W

.43 -250.59

-11.02 (1.64) ,005 (2.29) .006 (3.32) -.96 (1.05) -a007 (.259) .26 -252.16

(090.9)

(0,1)

WI

Dependent variable: median house value Constant Median number of rooms Distance to the CBD (linear miles) Percentageof houses owner occupied Age of median house School expenditures per pupil R2(adj) - L,(L PI

(8::) -.09 (3.88) -.005 (1.23) -.002 (1.23)

8.71 (42.26) .34 (8.14) -.09 (3.89) -.003 (1.25) -.002 (1.25)

(3.84) .82 -362.01

(3:y .80 - -362.20

4.07 (16.69)

.ooo8

point is made in Gerking and Boyes [4].) The optimal functional forms also imply positive elasticity values for distances less than 13 mi from the CBD. Although this result is somewhat discomfiting, the use of OLS estimation techniques in the interactive search for the optimal functional forms implies that the elasticity estimates are most valid at the means. It is a well known econometric property of OLS regressions that the confidence interval of the predicted dependent variable expands as the explanatory variable (in this case distance) deviates from the mean.

PRICE ELASTICITY

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( -0.44, -0.36). This estimate is significantly lower than the - 1 figure found in Muth [8] and Reid [lo]. It is somewhat more consistent, however, with the results of the study by Polinsky and Ellwood [9] who show how a variety of specification errors have plagued the extant housing-demand literature. Using “correctly specified” grouped and ungrouped housingdemand equations, they estimate the price elasticity of demand to be approximately - 0.73. Our estimate also accords well with the reported findings of Gerking and Boyes [4]. Applying the Box-Cox technique to the standard housingdemand model, their optimal-form mean-price elasticity estimate for the Chicago SMSA is -0.28. It is interesting that our approach, which utilizes general equilibrium-type spatial relationships to derive information about price elasticities, yields complementary estimates to those found in these partial-equilibrium demand studies. As noted in the methodology section, the Box-Cox technique also allows for testing of restrictions upon the transformation parameters. Although our primary interest lies in the results obtained from the optimal functional forms, it is useful to explore the acceptability of certain restricted forms as well. In particular, note that Muth’s [6] usageof (3) involved the assumption of a (0,l) restriction upon X and p in the rent equation, and resulted in an implied restriction of (1,l) for the wage equation. For our sample, application of the &i-squared testing procedure described earlier reveals that Muth’s functional-form restrictions cannot be rejected at the 95% confidence level. Conditional upon the restricted X and p values, the price elasticity of demand for housing is - 1, independent of the values of individual coefficients from wage and rent equations. Therefore, even though the optimal functional-form results imply a variable elasticity of demand specification for the underlying housing-demand function, Muth’s assumed constant unitary elasticity specification cannot be rejected for our sample. A variety of functional combinations in addition to the one employed by Muth arise from nonrejected restrictions upon the Box-Cox parameters. The 95% confidence intervals for X and p are (0.7,1.3; - 1,l) and ( - 0.6,0.6; 0.2), respectively. The associatedestimates of the price elasticity of demand range from -12.3 to 6.5, depending upon the wage-rent combinations e...