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Externalities, preferences, and urban residential location: Some empirical evidence
JOURNAL
OF URBAN
ECONOMICS
14, 338-354 (1983)
Externalities, Preferences, and Urban Residential Location: Some Empirical Evidence* MINH CHAU To, ALAIN LAPOINTE, AND LAWRENCE KRYZANOWSKI’ Centre de recherche en gestron, Unioersitb du Quhbec 6 Montreal, Cme Postale 8888, succ. A, Montreal, Quebec H3C 3P8, Canada Received December 11, 1981; revised May I982
INTRODUCTION One of the most important phenomena to affect North American cities has been the movement of high-income households to the suburbs from the city core. It has been alleged that this movement of households has both induced and aggravated many of the social and economic problems that are found in the cities, such as the geographic concentration of households with similar ethnic and/or wealth characteristics in distinct sections of the cities, and the relative economic decline and increase in the poverty of the urban core. Two distinct explanations have been proposed for the current spatial location of the residences of households with different income levels within North American urban areas. Although these explanations are either implicitly or explicitly deemed to be competing viewpoints in the literature, there is reason to believe that they may be joint (and somewhat complementary) explanations for the same phenomena. The first explanation is based on the existence of important racial and social externalities, and on the supposedly fragmented organization of local government in North America. Thus, the movement of higher-income households to the suburbs is both a cause and a direct consequence of these externalities, especially for the cores of most urban centers. In addition, it is believed that such movement is *The authors thank the members of Le Centre de recherche en Economique et Statistiques, and of Le Centre de recherche sur la Micmeconomie et I’Optimisation, Faculte des Sciences kconomiques, Universite de Toulouse, France, for their comments on an earlier draft of this paper. We also thank an anonymous referee of this journal for his helpful comments on a previous draft of this paper. Any errors which remain are, of course, the responsibifity of the authors. ‘The authors are (Assistant) Professor of Finance, Universite du Quebec i Montreal, Associate Professor of Economics, Ecole des Hautes Etudes Commerciales and Associate Professor of Finance, Concordia University, respectively. This paper was written while the first two authors were fellows of the Laboratoire de Recherche en Sciences immobilieres, Universitc du Quebec a Montreal. 338 0094- 11W/83 $3.00 Copyright All tights
0 1983 by Academx Press, Inc of reproduction in any form reserved.
URBAN RESIDENTIAL
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339
further aggravated by the relative real estate tax savings which accrue to suburban residences, due primarily to the need to provide fewer, and less necessary, services in the suburbs.’ The second explanation, which was developed by Alonso [l] and by Muth [7], is based on spatial equilibrium theory. More specifically, a households optimal residential location decision is based on arbitrage between the cost of travel and land, where the cost of the former rises and the cost of the latter declines for households who work in the city core as the travelling distance (or time) from their place of residence to place of work increases. In Muth’s [7] model, a household chooses a residential location by comparing the income elasticity of its demand for land with the income elasticity of its travel cost. Thus, if higher incomes are associated with higher income elasticities for land than for travel cost, then higher-income households will desire residential locations outside of the city core; and if lower incomes are associated with lower income elasticities for land than for travel cost, then lower-income households will desire residential locations within the city core. Although Alonso’s [l] model is essentially the same as Muth’s model, it is based on a bid-price approach.3 More specifically, since the owner of each parcel of urban land is viewed as a monopolistic seller facing atomistic buyers, land is taken as a good which is purchased by the household which bids the highest price for it. Thus, if a high income results in a greater demand for land relative to the cost of increased travel, then higher-income households will bid relatively more than lower-income households for land as the distance of the land from the city core increases. Since this implies that a household will locate farther from the city core the higher its income is, the second explanation attributes the movement of higher-income households to the suburbs to preferences, and not to externalities. Thus, the second explanation is not only more general than the first but it may be more attractive because it can be used to account for the opposite urban residential location pattern in European cities. More specifically, the tendency for higher-income households to locate in the core of European cities and for lower-income households to locate in the suburbs can be attributed to a different preference ordering of land and travel costs by Europeans as compared to North Americans. Although the merits of each explanation can be debated at length, the validity of each position can be determined only by empirical tests. Thus, the purpose of this paper is twofold: first, to formulate a more operational preference-based model of residential location that alleviates some of the 2For a more extensive discussion of this position, see Haugen and Heins [2]. 3Although Alonso [I] and Muth [7] differed in their model formulations, Wheaton [9] has shown that the two models are essentially equivalent since both are based on a spatial equilibrium. In turn, such an equilibrium depends upon arbitrage between the demand for land and travel cost.
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AND KRYZANOWSKI
criticisms of a more elaborate Alonso-Muth model, and second, to empirically test the validity of the preference explanation for the pattern of residential urban location using both an Alonso-Muth model and the model derived herein. Unlike Wheaton [ 1l] who found that the preference explanation was not empirically supported using an Alonso-Muth model for a large U.S. urban center, we find that the preference explanation is empirically supported using both models for a large Canadian urban center. Furthermore, if the preference and externalities explanations are competing alternatives, then our findings suggest that the latter is not supported empirically; if they are not mutually exclusive alternatives, then our findings shed no light on the validity of the latter explanation. The remainder of this paper is organized as follows. In the first section, the basic Alonso-Muth model is first presented and then extended to account for the relationships which are believed to exist between income, housing, and travel costs. Wheaton’s [ 1l] empirical test of the Alonso-Muth model is also summarized and reviewed in this section. In the second section, the data base, the empirical procedures, and the empirical results of an alternative test of the Alonso-Muth model (i.e., one based on data for Montreal and a Cobb-Douglas utility function) are presented and analyzed. In the third and fourth sections, an alternate model of urban residential location is developed and empirically tested, respectively. In the fifth and final section, some concluding remarks are presented. THE ALONSO-MUTH
MODEL
In the Alonso-Muth model, a household is assumed to have a monotonically increasing and strictly concave utility function which is given by u = u(z, x, k),
(1)
where u is the household’s utility; z is a composite good (that excludes shelter and travel) with an arbitrarily defined price of unity; x is the quantity of housing; 4 and k is a measure of travel (in particular, the distance of the households residence from the city core). “In their original models, Alonso and Muth measured x as being land surface area. Although this simplifies the theoretical development of a single arbitrage process between demand for land and travel cost, it ignores the heterogeneous nature of a unit of housing (see Lapointe [5] and Wheaton [9-l I]. In this study, the housing quantity x is defined as a scalar such that x=WA,
(N’)
URBAN
RESIDENTIAL
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341
Each household is also subject to a budget constraint given by y = z + xp(x)
+ WC, Y),
(2)
where y is the household’s level of income; p(x) is the price of a unit of housing; T(k, y) is the monetary cost of k units of travel for a household with a level of income of y; and all the other variables are as defined earlier. A household’s bid-price function ?r is then defined as the maximum price the household is willing to pay for a house, while simultaneously maintaining a constant utility level which is commensurate with that of households with similar income levels. Thus, the bid-price function is given by -n = dx,
k; Y,, u,,),
(3)
where y, is the (known) level of the household’s income; u. is the level of utility corresponding to an income level of yo; and all the other variables are as defined earlier. Since in the determination of its bid price function the household has to satisfy its budget constraint (2) (3) can be rewritten as T = ;[yo
- z - T(k
YO)].
(4)
Thus, each household will behave as if it attempts to maximize (4) subject to (5), where (5) is given by U(Z, x, k) = uo.
(5)
where A is a column vector of housing attributes, which include local amenities considered by the household in its locational decision; and
W is a line vector of the weights that the household places on each housing attribute in A. Thus, such a definition of x accounts for the heterogeneous nature of the good housing. More specifically, households must take the offered “packages” of housing attributes as givens, since households do not realistically have the opportunity of either (i) first purchasing each desired attribute separately and then packaging these desired attributes, or (ii) first “depackaging” existing packages and then “repackaging” the desired resultant attributes. This definition implies that the marginal rate of substitution between each pair of attributes is equal to the ratio of their shadow prices.
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Since (5) must be satisfied for each location, Lapointe [5, pp. S-91 has shown that the spatial equilibrium of each household implies that the bid price function q is identical to the Lagrangian (6) for each location, where (6) is given by
L = ;[a - z - T(k,y,)] + qt.+, x7k) - %I. Thus, the effect of a change in the distance of the household’s residence from the city core on the household’s bid-price function is given by’ 6nr/6k = 6L/6k
= - $ST,Sk)
+ h@u/ilk).
(7)
Based on the first-order conditions for the maximum L in (6), h can be obtained from h = l/[x(Su/iJz)J. Using this value for A, (7) can be rewritten as
z ) can be interpreted as the marginal value of distance (i.e., the marginal rate of substitution between distance and the composite good), and (ST/Gk) can be interpreted as the marginal monetary cost of travel. Thus, the term in parentheses in (8) is what Wheaton [ 1l] has referred to as the total net marginal cost of distance (TMCD). The effect of income on the bid-price function can then be found by differentiating (8) with respect to y to get
In Eq. (8), (Wak)/(W6
w.!&&(~)=
-TMCDy+;GTp.
(9)
If rewritten in terms of income elasticities, (9) becomes
(10) where ETMCD,y is the income elasticity of the total net marginal cost of distance; and Ex, y is the income elasticity of the demand for housing. ‘Wheaton [ 1 I, pp. 621-6221 estimated (7) by finding the envelope function the family of functions n’ (x, k; ub), i = I, 2, , where ub is a parameter.
n( x, k, uo) of
URBAN
RESIDENTIAL
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From IQ. (lo), it is obvious that S2a/6k Sy decreasesin terms of positive values as y increases if tTMCD,y > ,$?,Y.In this case, since the bid price decreasesas the distance from the city core decreasesfor higher levels of income, higher-income households will locate their residenceswithin the city core. On the other hand, 6*n/Sk 6y increases in terms of negative values as y increases if [rMCD, y < [,, Y. In this case, higher-income households locate their residences outside of the city core. Before proceeding, it should be noted that the comparison between the two income elasticities is valid only if the household is in spatial equilibrium.6 Using a constant elasticity function, Wheaton [ 10, 1I] estimated &,rCD, y and 5x,, for 1965 interview data for a screened sample of over 2000 households. To be included in the screenedsample, the household had to (1) be a resident of the San Francisco Bay area, (2) have a head who worked in the core of San Francisco, and (3) neither be nonwhite or a blue-collar worker. Although the remaining households were stratified into 128 subgroups based on size of the household and the age, race, and occupation of the household head, Wheaton retained only seven of the subgroups for estimation purposes because of the small number of observations in the excluded subgroups. For the seven subgroups retained, the sample size varied from 45 to 144 households. Wheaton used three housing attributes-number of rooms, age of building, and the general condition of housing (a scale variable). He found not only that the two income elasticities were approximately equal but that this result was confirmed by a simulation of the residential location pattern of the seven subsamples of households. Thus, Wheaton concluded that the Alonso-Muth model did not adequately explain the suburban location of higher-income households and that the underlying rationale was probably based on externalities. Although Wheaton’s study provided a number of important findings, it has four deficiencies. First, Wheaton used an exponential function of current income as a proxy for ua. As noted by Vaughn [8], the use of current income may induce both a downward bias in the estimates of the income elasticity of the demand for housing and an “errors-in-the-variable” problem in the demand equation. Second, Wheaton computed the housing costs of owner occupants by using a constant mortgage interest rate. Since mortgage interest is tax deductible in the United States, it is a benefit which increases in value as the income (and thus the tax bracket) of the household increases. Third, by excluding blue-collar workers and nonwhites who worked in the city core from his sample, Wheaton may have been using a sample which is nonrepresentative of the population of households being studied. Fourth, since Wheaton dealt with only one urban center (i.e., San Francisco), his findings might be city-specific. Although this limitation is 6Thus, the elasticity coefficients
should be estimated for constant uO.
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AND KRYZANOWSKI
understandable on practical grounds, the replicability of the findings for San Francisco with those of other urban cities needs to be empirically verified. AN ALTERNATIVE EMPIRICAL TEST OF THE ALONSO-MUTH MODEL Given the above limitations of Wheaton’s study, it seemedappropriate to attempt a validation of the Alonso-Muth model using household data for a different North American urban center. As a result, the city of Montreal was chosen for two reasons: first, the required data for such a test were readily available; and second, while Montreal exhibits the typical residential location pattern with respect to household income, it provides a somewhat different social and urban environment than does San Francisco. Due to the lack of a reliable statistical routine for estimating the parameters of a nonlinear utility function, each household is assumed to have a Cobb-Douglas utility function, which is given by’
where (Y,y, and /I are the parameters of the household’s utility function. Since each household is also assumed to be in equilibrium, the household’s bid-price function is: li = xrr = xP( k). Using (4) and (1 l), the equilibrium relationship for the bid-price function can also be determined as 7i = y, - T( k, y,,) - ( u;‘a)( k-Y’a)(x-p/Y).
(12)
As noted earlier in the paper, this function is identical to the Lagrangian (13), where (13) is given by ln[y,
- T( k, y,) - +] = d In u,, - : In k - t In x.
(13)
Since x is the result of a vector product,* Eq. (13) can be expanded into (13’): ln[y,, - T(k, y,) - +] = iln
u0 - :ln k - ~III
1 1 (13’) Cwiai
,
i
‘The Cobb-Douglas used by Wheaton. *See footnote 6.
function is a limit form of the more general constant elasticity function
URBAN
RESIDENTIAL
LOCATION
345
where ai is the ith housing attribute that is considered by the household in its locational decision; wi is the weight that the household places on a,; and all the other variables are as defined earlier. Since (13’) is linear, its parameters were estimated using ordinary least squares. The data were drawn from a 1972 survey of 1499 households in the Montreal area. The data base, commonly referred to as “SIMulation du LOgement a Montreal” (or SIMLOM), was assembled by the Institut National de la Recherche Scientifique (INRS). Households with heads who did not work in the core of Montreal were excluded from the estimation sample.’ A number of criteria, either individually or jointly, were used to stratify the resultant sample into 125 subsamples.These criteria included the age of the head of the household, the size of the household, and ethnic origin. Unfortunately, only three of the subsamples, which were in a consecutive sequenceof household income, had sufficient numbers of observations and thus could be retained. The three stratified subsamples that were retained consisted of three- and four-member households whose primary language of communication at home was French. The variable values required to estimate (13’) were computed as follows. The housing costs for a homeowner, ?i, were calculated as the sum of the homeowner’s annual mortgage payments and a return of 10% on the homeowner’s initial equity investment. lo The housing costs for a renter, 7j, were taken as being the renter’s total annual rental payment. To avoid the bias caused by using current income as a proxy for permanent income, the households were first grouped into four income classes. The appropriate midrange (i.e., $2500, $7500, $12,500, or $17,500) of each class was then used as a proxy for the household’s permanent income. Although Wheaton used a proxy (ai) for uo, none was used here for two reasons. First, separate estimates of (Y,/3, and y were not required; and second, the construction of such a proxy may result (as in the Wheaton study) in both the dependent variable and one of the independent variables being functions of yo. Since no proxy was constructed for uo, the estimated intercept of the OLS form of (13’) is an estimate of (1/a)log uo. T( k, yo) in (13’) was measured as being the annualized cash cost of transportation between a household’s residence 9For the purposes of the study, the core of Montreal was defined as the area bordered by the streets of Iberville, St. Joseph, Park, Pine, Sherbrooke, Atwater and Wellington, and by the St. Lawrence River and the track of the Canadian National Railway. ‘oAlthough it is possible, there is no a priori reason for believing that this introduced any systematic bias into the findings reported herein,
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and place of work. Two measures of travel were used: k, which was a measure of the distance in miles between the household’s residence and place of work, and k’, which was a measure of the travelling time in minutes between the household’s residence and place of work. Since there were 14 available housing attributes (such as number of rooms, number of toilets, neighborhood characteristics, and heating control) and since there was no theoretical basis for choosing one weighting scheme of these attributes over another, two procedures for generating estimates of the housing quantity x = Ciwiaj were used. The first estimation procedure used principal-components analysis to generate a linear combination of the housing attributes. The second estimation procedure used various ad hoc linear combinations of the housing attributes (such as y = l/14 Vi::;and wj = 1, w, = ov. ,, i * j). Of all the linear combinations that were generated, only one led to statistically significant empirical results. This was the linear combination where the weight on the number of rooms ( wj) and the weights on the remaining 13 attributes (w,, i = 1,. . . , 14, i * j) were set to one and a vector of zeroes,respectively. Fortunately, this was not totally unexpected, because of the primary (and possibly overriding) importance of the number of rooms in a household’s decision to purchase the housing good. Thus, in the remainder of the paper, the quantity of housing was proxied (and thus measured) by the number of rooms.” Regression results for each of the three stratified subsamples for each of the two measuresof travel (i.e., k and k’) are presented in Table 1. Based on Table 1, it appears that most of the estimated regression coefficients are significant. Values of (6u/6k)/(Su/6z) and (&/6k’)/(cYu/6z), which were computed from the estimated regression coefficients, the means of z = y,, - T( k, yO) - r and k (or k’), are also presented in Table 1.12 To estimate the marginal monetary cost of travel, equations for T(k or k’, y) and In T(k or k’, y) were estimated using ordinary least squares. The two estimated equations, which had both the highest R2 values and signifi“The appropriateness of using the number of rooms as a measure of the quantity of housing consumed was empirically supported. More specifically, the following cross-sectional regression was estimated for households whose head worked in the urban core x = 3.3136 + 0.0001y + O.O728k, (14.6747)(5.4214)(4.4055) where the sample size is 357, the Fvahre is 29.373, the adjusted R* is 0.137, and the r values are in parentheses. Thus, as expected, the quantity of housing consumed increased as income and distance increased. “The values of (SU/~~)/(C?U/~Z) and (tSu/t?k’)/(6u/tSz) were given by ya-‘(z/k) and ~‘a-‘(z/k’), respectively.
au/Sk 6 u/S2 6u/6k’ 6u/& hi Su/Gk Sk=--- Gu/Sr 61i 6u/6k’ _=--Sk’ 6u/6t
-p-‘(k) t value -?‘a-‘(k’) f value -pa-‘(X) f value cl - ’ log ug f value
ST 6k ST Sk’
Stratified sample: Income, midrange: N R2 F value
Travel measure:
TABLE
-0.041 (- 2.232) 8.798 (345.801)
- 0.228 (- 1.046) 1.529 (24.284)
I .22
4.33
4.51
- 0.009 (- 2.505)
-0.044 (- 1.407) -
k 2 $7500 31 0.303 6.111
1.46
1
37.84
-
37.86
-0.012 (- 2.695) - 0.043 (- 2.429) 8.785 (366.615)
2 $7500 31 0.323 6.694
3.08
3.09
-0.019 (-2.715) - 0.230 ( - 1.502) 7.414 (32.376)
-
1 $2500 7 0.756 6.213
k’
82.35
-
82.39
0.066 3.209) 0.003 0.709) 9.378 (316.251)
((-
3 $12,500 22 0.351 5.157
Model using Household Data for Montreal
29.65
29.89
0.007 (0.674) 9.344 (213.579)
- 0.059 (- 2.930) -
3 $12,500 22 0.350 5.120
Test of the Alonso-Muth
1 $2500 I .531 2.319
Summary of the Findings for the Alternative
348
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AND KRYZANOWSKI
cant regression coefficients at the 0.001 level, areI T = 0.822 + 0.236k + O.OOOly,
(14)
where the sample size is 340, R2 is 0.246, the F value is 54.908; and In T = - 1.798 + 0.043k’ + O.OOOly,
(15)
where the sample size is 340, R2 is 0.227, and the F value is 49.538. The marginal monetary cost of travel, 6T/6k, based on Eq. (14), is 23.6 cents per mile travelled. The marginal monetary cost of travel, GT/Sk’, based on Eq. (15), is obtained by solving
Using the mean values of k’ and y for each of the three classesof household income, the corresponding values of 6T/6k’ are 1,2, and 4 cents per minute of travel time, respectively. The slopes of the bid-price functions, G/Sk = x(&a/6k) = [(Su/Gk)/ (6u/6z)] (- 6T/6k) and &?/tSk’ = [(su/6k’)/(Su/tSz)] (- sT/Gk’), were computed using (8) and the values of 6T/6k and 6T/6k’, respectively. These values are presented in the last two lines of Table 1. From these values, it is evident that the slope of the bid-price function increases markedly with income for both the k and k’ measures of distance. Thus, unlike the findings of Wheaton for the San Francisco Bay area, these results are supportive of the Alonso-Muth model. AN ALTERNATIVE
MODEL OF URBAN RESIDENTIAL LOCATION
To further examine the validity of preferences as an explanation for the pattern of residential location of various income groups within an urban area, a simple model of arbitrage between distance and housing will be developed in this section of the paper and tested in the next section. One of the strengths of the model is that it yields easily testable relationships between variables using data which are readily available. As in the Alonso-Muth model, suppose that the household tries to maximize its utility function (1) subject to its budget constraint (2). The household’s Lagrangian is then L = u(z, x, k) + d[z + xp(k)
+ T(k, y) -y],
(17)
where all the terms are as defined earlier. 13The remaining 12 (unreported) equations do not differ materially in terms of the values of 6T/6k and ST/Gk’ that they imply.
from these two equations
URBAN
RESIDENTIAL
349
LOCATION
The following are derived from the first-order conditions: 6L/6x
= Gu/Sx + ep = 0;
8 =( SL/6k
= 6u/6k
-6u/Sx)/p;
+ (8x)(6p/6k)
(18)
and + 8(67’/6k)
(19) = 0.
(20)
Substituting (19) into (20), and rearranging (20), gives
where MRS, k is the marginal rate of substitution between distance and housing. Equation (21) can be interpreted as follows: the demand for housing is proportional to the product of its price and the marginal rate of substitution between housing and distance, minus the marginal monetary cost of travel, all divided by the residence’s relative price per unit of travel from the residence to the city core. Equation (21) can be estimated if it is assumed that MRS,,, = (Su/Gk)/(Su/Gx), which is less than zero, is constant. This assumption requires that the household’s relative preferences between housing and distance are invariant to a marginal increase in y. In other words, this implies that the optimal path of the household in the plane spanned by x and k is linear. Another equilibrium relationship can be obtained from the observation is that the elasticity of utility with respect to distance, .& = (su/Sk)(k/u), less than zero, and that the elasticity of utility with respect to housing, [, = (Su/Gx)(x/u), is greater than zero. Using these observations, (21) can be rearranged to obtain P = (W,)(k)[(Sp/~k)
+ (W6k)/xl.
(24
Equation (22) can be interpreted as follows: the price of one unit of housing is proportional to its distance from the city core, multiplied by the ratio of the income (or utility) elasticities of housing and travel cost, multiplied by the sum of the marginal price of housing per unit of travel and the ratio of the marginal monetary cost of travel and the quantity of housing. Equation (22) can be estimated if it is assumed that [.J& is constant. This assumption is more general than the assumption that MRS,,, is constant, for it requires that the household’s relative preferences with respect to x and k are either accentuated or unchanged (i.e., do not reverse themselves) with a marginal increase in y. In other words, this implies that
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the optimal marginal path of the household in the plane spanned by x and k is an exponential curve. This condition can be further interpreted by first noting that
(23) where b is a constant. Then, rearranging (23) gives (Gk/Sx)/k
= b/x = b(l/x),
(24)
or log k = log Xb + a’,
(25)
k = axb,
(26)
or
where a’ and a = e”’ are constants. Furthermore, since a is presumably negative, Eq. (26) suggeststhat the undesirability of travel is a monotically increasing function of a household’s income. AN EMPIRICAL TEST OF THE ALTERNATIVE MODEL OF URBAN RESIDENTIAL LOCATION To empirically estimate Eqs. (21) and (22), it was assumed that MRS,, k and 6,/S, were constant. This differs from the implicit assumption involved in the estimation of the Alonso-Muth model, that is, that the utility level across households is constant. Although both approaches imply that the estimations should be conducted using stratified samples of households, the sample used to estimate Eqs. (21) and (22) was only stratified by income. This was done to increase the sample size without causing any loss of generality. In the first step of the empirical procedure, &p/&k was approximated by conducting a regression p(k, y). l4 For the screened subsample (i.e., the 14Regressionp(k’, JJ) was also run. However, since it was not statistically significant, only k (i.e., the distance in miles from the city core) was used to test the alternative residential location model.
URBAN
RESIDENTIAL
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351
sample that only includes households for which the head works in the urban core), the estimated equation is given by In p = 5.6315 - O.OOOly- O.O971k,
(27)
where the sample size is 340, R* is 0.073, the F value is 13.284, and all the regression coefficients are significant at the 0.001 level. In the second step of the empirical procedure, Eqs. (21) and (22) were estimated for the stratified samples four to seven (i.e., those samples with total estimated permanent incomes of $2500, $7500, $12,500, and $17,500, respectively). The results of these estimations are presented in Table 2 and can be summarized as follows. First, the R* and F values for all eight cross-sectional estimations are very high. Second, the estimates of MRS,, k for strata four and six are neither of the expected sign nor are they statistically significant. Third, although the estimates of c (that is, the coefficient of (6T/6k)(Sp/iSk)-‘) have the expected sign, they are much smaller than their expected value of - 1. Fourth, for strata five and seven, the estimates of MRS,,, are statistically significant and have the expected sign. Moreover, these estimates decrease significantly in value as income increases from $7500 to $17,500. Fifth, the estimates of [,./& are statistically significant and have the expected sign for all four strata. Moreover, all of these estimates decrease markedly and monotonically in value with increasing income. And finally, Eq. (22), and not Eq. (21), is the more appropriate representation of a household’s equilibrium housing relationship. This is consistent with the notion underlying the Alonso-Muth model that households use relative values when making arbitrage-type decisions between housing and distance. Thus, it appears that our alternative model of arbitrage between distance and housing is statistically acceptable, and that both MRS,, k and (,/Sk decreasewith increasing household income. Since these results corroborate the earlier findings for the Alonso-Muth model, they provide additional empirical support for the validity of the preferences explanation for the pattern of residential location in the Montreal urban area. CONCLUDING
REMARKS
For household data for the San Francisco Bay area, Wheaton [ 1l] found that the empirical evidence did not support the predictions of the Alonso-Muth model. Thus, since this was unsupportive of the preferences explanation for the pattern of urban residential location, Wheaton concluded that the externalities explanation was valid. In this paper, empirical evidence supporting the predictions of the Alonso-Muth model was presented for households in the Montreal urban
subsample
W5k T value R= F value Estimate
MR’% k T value
of c
Income Sample size
Stratified
0.909 85.428 - 3 18.284
-
$2500 18 0.080 1.370
4
0.616 102.346 - 59.069
-
$7500 128 - 0.017 - 1.508
5
x = MR.Sr,,p(6p/6k)-’
0.628 63.774 -46.615
$12,500 76 0.001 0.200
6 7
0.66 I 29.710 - 19.495
$17.500 31 -0.876 - 1.409
+ c(&T/h’k)(6p/6k)-’
- 4.132 - 7.595 0.772 57.688
$2500 18 -
4
- 8.420 - 16.326 0.677 266.548
$7500 128
5
P = (5,/t,)kK~p/6k)
- 24.833 - 5.489 0.501 30.132
-
$17,500 31
$12,500 76
- 16.779 - 15.939 0.772 254.045
I 6
+ (STPk)/xl
TABLE 2 Summary of the Findings for the Empirical Test of the Alternative Model of Urban Residential Location ___~.
URBAN RESIDENTIAL
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area. Furthermore, corroborating empirical evidence was obtained using an alternative model of urban residential location for the same urban center.15 Thus, unlike the Wheaton study, the major finding of this study is that consumers’ preferences can significantly determine residential location patterns in at least one urban center. The differences between Wheaton’s results and those reported herein were, however, not unexpected. Since locational patterns are strictly dependent upon the relative patterns of the utility preferences of the households in the urban centers studied, it is not surprising that such preferences were found to differ for households in different cities, in different countries, speaking different languages, and having different cultures.‘6 While the nature of the empirical results, and the congruence of the empirical results for alternate models of urban residential location for the same sample are reassuring, the lack of replicability of the Montreal findings with those for San Francisco suggests that the two explanations may not be mutually exclusive. Thus, in future studies, it would seem appropriate to develop and empirically test a model which allows for the joint and/or interactive effects of both preferences and externalities on the pattern of urban residential location.
REFERENCES 1. W. Alonso, “Location and Land Use: Toward a General Theory of Land Rent” Harvard University Press, Cambridge, Massachusetts (1964). 2. R. Haugen and A. J. Heins, A market separation theory of rent differentials in metropolitan areas, Q. J. &on. 83, pp. 660-672 (November 1969). 3. A. Lapointe, Le choix du mode d’occupation sur le marche du logement a Montreal: Influence de l’origine ethnique, Acruul. &on. pp. 46-58 (1978). 4. A. Lapointe, “La Segregation Residentielle Ethnique a Montreal,” These de doctorat, Universite des SciencesSociales de Toulouse (1978). 5. A. Lapointe, “Repartition intra-Urbaine des Revenus: Quelques Considerations Methodologiques,” working paper, Universiti du Quebec a Montreal (I 979). 6. D. McFadden, The measurement of urban travel demand, J. Public Econ., 3, pp. 303-328 (November 1974). 7. R. Muth, “Cities and Housing,” Chicago Univ. Press, Chicago (1969). 8. G. A. Vaughn, Sources of downward bias in estimating the demand income elasticity for urban housing, J. Urban Econ. 3, pp. 45-56 (1976).
“It should be noted that a less restrictive assumption about the form of each households utility function is invoked to empirically test this model than is required to empirically test the Alonso-Muth model. “jWe are grateful to an anonymous referee of this journal for providing us with the essence of this comment.
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9. W. C. Wheaton, A comparative static analysis of urban spatial structure, J. &on. Theory 9, pp. 223-237 (October 1974). 10. W. C. Wheaton, A bid rent approach to urban housing demand, J. Urban Econ., pp. 15-32 (April 1977). I I. W. C. Wheaton, Income and urban residence: An analysis of consumer demand for location, Amer. Econ. Rev. 67, pp. 620-631 (September 1977).
OF URBAN
ECONOMICS
14, 338-354 (1983)
Externalities, Preferences, and Urban Residential Location: Some Empirical Evidence* MINH CHAU To, ALAIN LAPOINTE, AND LAWRENCE KRYZANOWSKI’ Centre de recherche en gestron, Unioersitb du Quhbec 6 Montreal, Cme Postale 8888, succ. A, Montreal, Quebec H3C 3P8, Canada Received December 11, 1981; revised May I982
INTRODUCTION One of the most important phenomena to affect North American cities has been the movement of high-income households to the suburbs from the city core. It has been alleged that this movement of households has both induced and aggravated many of the social and economic problems that are found in the cities, such as the geographic concentration of households with similar ethnic and/or wealth characteristics in distinct sections of the cities, and the relative economic decline and increase in the poverty of the urban core. Two distinct explanations have been proposed for the current spatial location of the residences of households with different income levels within North American urban areas. Although these explanations are either implicitly or explicitly deemed to be competing viewpoints in the literature, there is reason to believe that they may be joint (and somewhat complementary) explanations for the same phenomena. The first explanation is based on the existence of important racial and social externalities, and on the supposedly fragmented organization of local government in North America. Thus, the movement of higher-income households to the suburbs is both a cause and a direct consequence of these externalities, especially for the cores of most urban centers. In addition, it is believed that such movement is *The authors thank the members of Le Centre de recherche en Economique et Statistiques, and of Le Centre de recherche sur la Micmeconomie et I’Optimisation, Faculte des Sciences kconomiques, Universite de Toulouse, France, for their comments on an earlier draft of this paper. We also thank an anonymous referee of this journal for his helpful comments on a previous draft of this paper. Any errors which remain are, of course, the responsibifity of the authors. ‘The authors are (Assistant) Professor of Finance, Universite du Quebec i Montreal, Associate Professor of Economics, Ecole des Hautes Etudes Commerciales and Associate Professor of Finance, Concordia University, respectively. This paper was written while the first two authors were fellows of the Laboratoire de Recherche en Sciences immobilieres, Universitc du Quebec a Montreal. 338 0094- 11W/83 $3.00 Copyright All tights
0 1983 by Academx Press, Inc of reproduction in any form reserved.
URBAN RESIDENTIAL
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further aggravated by the relative real estate tax savings which accrue to suburban residences, due primarily to the need to provide fewer, and less necessary, services in the suburbs.’ The second explanation, which was developed by Alonso [l] and by Muth [7], is based on spatial equilibrium theory. More specifically, a households optimal residential location decision is based on arbitrage between the cost of travel and land, where the cost of the former rises and the cost of the latter declines for households who work in the city core as the travelling distance (or time) from their place of residence to place of work increases. In Muth’s [7] model, a household chooses a residential location by comparing the income elasticity of its demand for land with the income elasticity of its travel cost. Thus, if higher incomes are associated with higher income elasticities for land than for travel cost, then higher-income households will desire residential locations outside of the city core; and if lower incomes are associated with lower income elasticities for land than for travel cost, then lower-income households will desire residential locations within the city core. Although Alonso’s [l] model is essentially the same as Muth’s model, it is based on a bid-price approach.3 More specifically, since the owner of each parcel of urban land is viewed as a monopolistic seller facing atomistic buyers, land is taken as a good which is purchased by the household which bids the highest price for it. Thus, if a high income results in a greater demand for land relative to the cost of increased travel, then higher-income households will bid relatively more than lower-income households for land as the distance of the land from the city core increases. Since this implies that a household will locate farther from the city core the higher its income is, the second explanation attributes the movement of higher-income households to the suburbs to preferences, and not to externalities. Thus, the second explanation is not only more general than the first but it may be more attractive because it can be used to account for the opposite urban residential location pattern in European cities. More specifically, the tendency for higher-income households to locate in the core of European cities and for lower-income households to locate in the suburbs can be attributed to a different preference ordering of land and travel costs by Europeans as compared to North Americans. Although the merits of each explanation can be debated at length, the validity of each position can be determined only by empirical tests. Thus, the purpose of this paper is twofold: first, to formulate a more operational preference-based model of residential location that alleviates some of the 2For a more extensive discussion of this position, see Haugen and Heins [2]. 3Although Alonso [I] and Muth [7] differed in their model formulations, Wheaton [9] has shown that the two models are essentially equivalent since both are based on a spatial equilibrium. In turn, such an equilibrium depends upon arbitrage between the demand for land and travel cost.
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criticisms of a more elaborate Alonso-Muth model, and second, to empirically test the validity of the preference explanation for the pattern of residential urban location using both an Alonso-Muth model and the model derived herein. Unlike Wheaton [ 1l] who found that the preference explanation was not empirically supported using an Alonso-Muth model for a large U.S. urban center, we find that the preference explanation is empirically supported using both models for a large Canadian urban center. Furthermore, if the preference and externalities explanations are competing alternatives, then our findings suggest that the latter is not supported empirically; if they are not mutually exclusive alternatives, then our findings shed no light on the validity of the latter explanation. The remainder of this paper is organized as follows. In the first section, the basic Alonso-Muth model is first presented and then extended to account for the relationships which are believed to exist between income, housing, and travel costs. Wheaton’s [ 1l] empirical test of the Alonso-Muth model is also summarized and reviewed in this section. In the second section, the data base, the empirical procedures, and the empirical results of an alternative test of the Alonso-Muth model (i.e., one based on data for Montreal and a Cobb-Douglas utility function) are presented and analyzed. In the third and fourth sections, an alternate model of urban residential location is developed and empirically tested, respectively. In the fifth and final section, some concluding remarks are presented. THE ALONSO-MUTH
MODEL
In the Alonso-Muth model, a household is assumed to have a monotonically increasing and strictly concave utility function which is given by u = u(z, x, k),
(1)
where u is the household’s utility; z is a composite good (that excludes shelter and travel) with an arbitrarily defined price of unity; x is the quantity of housing; 4 and k is a measure of travel (in particular, the distance of the households residence from the city core). “In their original models, Alonso and Muth measured x as being land surface area. Although this simplifies the theoretical development of a single arbitrage process between demand for land and travel cost, it ignores the heterogeneous nature of a unit of housing (see Lapointe [5] and Wheaton [9-l I]. In this study, the housing quantity x is defined as a scalar such that x=WA,
(N’)
URBAN
RESIDENTIAL
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Each household is also subject to a budget constraint given by y = z + xp(x)
+ WC, Y),
(2)
where y is the household’s level of income; p(x) is the price of a unit of housing; T(k, y) is the monetary cost of k units of travel for a household with a level of income of y; and all the other variables are as defined earlier. A household’s bid-price function ?r is then defined as the maximum price the household is willing to pay for a house, while simultaneously maintaining a constant utility level which is commensurate with that of households with similar income levels. Thus, the bid-price function is given by -n = dx,
k; Y,, u,,),
(3)
where y, is the (known) level of the household’s income; u. is the level of utility corresponding to an income level of yo; and all the other variables are as defined earlier. Since in the determination of its bid price function the household has to satisfy its budget constraint (2) (3) can be rewritten as T = ;[yo
- z - T(k
YO)].
(4)
Thus, each household will behave as if it attempts to maximize (4) subject to (5), where (5) is given by U(Z, x, k) = uo.
(5)
where A is a column vector of housing attributes, which include local amenities considered by the household in its locational decision; and
W is a line vector of the weights that the household places on each housing attribute in A. Thus, such a definition of x accounts for the heterogeneous nature of the good housing. More specifically, households must take the offered “packages” of housing attributes as givens, since households do not realistically have the opportunity of either (i) first purchasing each desired attribute separately and then packaging these desired attributes, or (ii) first “depackaging” existing packages and then “repackaging” the desired resultant attributes. This definition implies that the marginal rate of substitution between each pair of attributes is equal to the ratio of their shadow prices.
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Since (5) must be satisfied for each location, Lapointe [5, pp. S-91 has shown that the spatial equilibrium of each household implies that the bid price function q is identical to the Lagrangian (6) for each location, where (6) is given by
L = ;[a - z - T(k,y,)] + qt.+, x7k) - %I. Thus, the effect of a change in the distance of the household’s residence from the city core on the household’s bid-price function is given by’ 6nr/6k = 6L/6k
= - $ST,Sk)
+ h@u/ilk).
(7)
Based on the first-order conditions for the maximum L in (6), h can be obtained from h = l/[x(Su/iJz)J. Using this value for A, (7) can be rewritten as
z ) can be interpreted as the marginal value of distance (i.e., the marginal rate of substitution between distance and the composite good), and (ST/Gk) can be interpreted as the marginal monetary cost of travel. Thus, the term in parentheses in (8) is what Wheaton [ 1l] has referred to as the total net marginal cost of distance (TMCD). The effect of income on the bid-price function can then be found by differentiating (8) with respect to y to get
In Eq. (8), (Wak)/(W6
w.!&&(~)=
-TMCDy+;GTp.
(9)
If rewritten in terms of income elasticities, (9) becomes
(10) where ETMCD,y is the income elasticity of the total net marginal cost of distance; and Ex, y is the income elasticity of the demand for housing. ‘Wheaton [ 1 I, pp. 621-6221 estimated (7) by finding the envelope function the family of functions n’ (x, k; ub), i = I, 2, , where ub is a parameter.
n( x, k, uo) of
URBAN
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From IQ. (lo), it is obvious that S2a/6k Sy decreasesin terms of positive values as y increases if tTMCD,y > ,$?,Y.In this case, since the bid price decreasesas the distance from the city core decreasesfor higher levels of income, higher-income households will locate their residenceswithin the city core. On the other hand, 6*n/Sk 6y increases in terms of negative values as y increases if [rMCD, y < [,, Y. In this case, higher-income households locate their residences outside of the city core. Before proceeding, it should be noted that the comparison between the two income elasticities is valid only if the household is in spatial equilibrium.6 Using a constant elasticity function, Wheaton [ 10, 1I] estimated &,rCD, y and 5x,, for 1965 interview data for a screened sample of over 2000 households. To be included in the screenedsample, the household had to (1) be a resident of the San Francisco Bay area, (2) have a head who worked in the core of San Francisco, and (3) neither be nonwhite or a blue-collar worker. Although the remaining households were stratified into 128 subgroups based on size of the household and the age, race, and occupation of the household head, Wheaton retained only seven of the subgroups for estimation purposes because of the small number of observations in the excluded subgroups. For the seven subgroups retained, the sample size varied from 45 to 144 households. Wheaton used three housing attributes-number of rooms, age of building, and the general condition of housing (a scale variable). He found not only that the two income elasticities were approximately equal but that this result was confirmed by a simulation of the residential location pattern of the seven subsamples of households. Thus, Wheaton concluded that the Alonso-Muth model did not adequately explain the suburban location of higher-income households and that the underlying rationale was probably based on externalities. Although Wheaton’s study provided a number of important findings, it has four deficiencies. First, Wheaton used an exponential function of current income as a proxy for ua. As noted by Vaughn [8], the use of current income may induce both a downward bias in the estimates of the income elasticity of the demand for housing and an “errors-in-the-variable” problem in the demand equation. Second, Wheaton computed the housing costs of owner occupants by using a constant mortgage interest rate. Since mortgage interest is tax deductible in the United States, it is a benefit which increases in value as the income (and thus the tax bracket) of the household increases. Third, by excluding blue-collar workers and nonwhites who worked in the city core from his sample, Wheaton may have been using a sample which is nonrepresentative of the population of households being studied. Fourth, since Wheaton dealt with only one urban center (i.e., San Francisco), his findings might be city-specific. Although this limitation is 6Thus, the elasticity coefficients
should be estimated for constant uO.
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AND KRYZANOWSKI
understandable on practical grounds, the replicability of the findings for San Francisco with those of other urban cities needs to be empirically verified. AN ALTERNATIVE EMPIRICAL TEST OF THE ALONSO-MUTH MODEL Given the above limitations of Wheaton’s study, it seemedappropriate to attempt a validation of the Alonso-Muth model using household data for a different North American urban center. As a result, the city of Montreal was chosen for two reasons: first, the required data for such a test were readily available; and second, while Montreal exhibits the typical residential location pattern with respect to household income, it provides a somewhat different social and urban environment than does San Francisco. Due to the lack of a reliable statistical routine for estimating the parameters of a nonlinear utility function, each household is assumed to have a Cobb-Douglas utility function, which is given by’
where (Y,y, and /I are the parameters of the household’s utility function. Since each household is also assumed to be in equilibrium, the household’s bid-price function is: li = xrr = xP( k). Using (4) and (1 l), the equilibrium relationship for the bid-price function can also be determined as 7i = y, - T( k, y,,) - ( u;‘a)( k-Y’a)(x-p/Y).
(12)
As noted earlier in the paper, this function is identical to the Lagrangian (13), where (13) is given by ln[y,
- T( k, y,) - +] = d In u,, - : In k - t In x.
(13)
Since x is the result of a vector product,* Eq. (13) can be expanded into (13’): ln[y,, - T(k, y,) - +] = iln
u0 - :ln k - ~III
1 1 (13’) Cwiai
,
i
‘The Cobb-Douglas used by Wheaton. *See footnote 6.
function is a limit form of the more general constant elasticity function
URBAN
RESIDENTIAL
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where ai is the ith housing attribute that is considered by the household in its locational decision; wi is the weight that the household places on a,; and all the other variables are as defined earlier. Since (13’) is linear, its parameters were estimated using ordinary least squares. The data were drawn from a 1972 survey of 1499 households in the Montreal area. The data base, commonly referred to as “SIMulation du LOgement a Montreal” (or SIMLOM), was assembled by the Institut National de la Recherche Scientifique (INRS). Households with heads who did not work in the core of Montreal were excluded from the estimation sample.’ A number of criteria, either individually or jointly, were used to stratify the resultant sample into 125 subsamples.These criteria included the age of the head of the household, the size of the household, and ethnic origin. Unfortunately, only three of the subsamples, which were in a consecutive sequenceof household income, had sufficient numbers of observations and thus could be retained. The three stratified subsamples that were retained consisted of three- and four-member households whose primary language of communication at home was French. The variable values required to estimate (13’) were computed as follows. The housing costs for a homeowner, ?i, were calculated as the sum of the homeowner’s annual mortgage payments and a return of 10% on the homeowner’s initial equity investment. lo The housing costs for a renter, 7j, were taken as being the renter’s total annual rental payment. To avoid the bias caused by using current income as a proxy for permanent income, the households were first grouped into four income classes. The appropriate midrange (i.e., $2500, $7500, $12,500, or $17,500) of each class was then used as a proxy for the household’s permanent income. Although Wheaton used a proxy (ai) for uo, none was used here for two reasons. First, separate estimates of (Y,/3, and y were not required; and second, the construction of such a proxy may result (as in the Wheaton study) in both the dependent variable and one of the independent variables being functions of yo. Since no proxy was constructed for uo, the estimated intercept of the OLS form of (13’) is an estimate of (1/a)log uo. T( k, yo) in (13’) was measured as being the annualized cash cost of transportation between a household’s residence 9For the purposes of the study, the core of Montreal was defined as the area bordered by the streets of Iberville, St. Joseph, Park, Pine, Sherbrooke, Atwater and Wellington, and by the St. Lawrence River and the track of the Canadian National Railway. ‘oAlthough it is possible, there is no a priori reason for believing that this introduced any systematic bias into the findings reported herein,
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AND KRYZANOWSKI
and place of work. Two measures of travel were used: k, which was a measure of the distance in miles between the household’s residence and place of work, and k’, which was a measure of the travelling time in minutes between the household’s residence and place of work. Since there were 14 available housing attributes (such as number of rooms, number of toilets, neighborhood characteristics, and heating control) and since there was no theoretical basis for choosing one weighting scheme of these attributes over another, two procedures for generating estimates of the housing quantity x = Ciwiaj were used. The first estimation procedure used principal-components analysis to generate a linear combination of the housing attributes. The second estimation procedure used various ad hoc linear combinations of the housing attributes (such as y = l/14 Vi::;and wj = 1, w, = ov. ,, i * j). Of all the linear combinations that were generated, only one led to statistically significant empirical results. This was the linear combination where the weight on the number of rooms ( wj) and the weights on the remaining 13 attributes (w,, i = 1,. . . , 14, i * j) were set to one and a vector of zeroes,respectively. Fortunately, this was not totally unexpected, because of the primary (and possibly overriding) importance of the number of rooms in a household’s decision to purchase the housing good. Thus, in the remainder of the paper, the quantity of housing was proxied (and thus measured) by the number of rooms.” Regression results for each of the three stratified subsamples for each of the two measuresof travel (i.e., k and k’) are presented in Table 1. Based on Table 1, it appears that most of the estimated regression coefficients are significant. Values of (6u/6k)/(Su/6z) and (&/6k’)/(cYu/6z), which were computed from the estimated regression coefficients, the means of z = y,, - T( k, yO) - r and k (or k’), are also presented in Table 1.12 To estimate the marginal monetary cost of travel, equations for T(k or k’, y) and In T(k or k’, y) were estimated using ordinary least squares. The two estimated equations, which had both the highest R2 values and signifi“The appropriateness of using the number of rooms as a measure of the quantity of housing consumed was empirically supported. More specifically, the following cross-sectional regression was estimated for households whose head worked in the urban core x = 3.3136 + 0.0001y + O.O728k, (14.6747)(5.4214)(4.4055) where the sample size is 357, the Fvahre is 29.373, the adjusted R* is 0.137, and the r values are in parentheses. Thus, as expected, the quantity of housing consumed increased as income and distance increased. “The values of (SU/~~)/(C?U/~Z) and (tSu/t?k’)/(6u/tSz) were given by ya-‘(z/k) and ~‘a-‘(z/k’), respectively.
au/Sk 6 u/S2 6u/6k’ 6u/& hi Su/Gk Sk=--- Gu/Sr 61i 6u/6k’ _=--Sk’ 6u/6t
-p-‘(k) t value -?‘a-‘(k’) f value -pa-‘(X) f value cl - ’ log ug f value
ST 6k ST Sk’
Stratified sample: Income, midrange: N R2 F value
Travel measure:
TABLE
-0.041 (- 2.232) 8.798 (345.801)
- 0.228 (- 1.046) 1.529 (24.284)
I .22
4.33
4.51
- 0.009 (- 2.505)
-0.044 (- 1.407) -
k 2 $7500 31 0.303 6.111
1.46
1
37.84
-
37.86
-0.012 (- 2.695) - 0.043 (- 2.429) 8.785 (366.615)
2 $7500 31 0.323 6.694
3.08
3.09
-0.019 (-2.715) - 0.230 ( - 1.502) 7.414 (32.376)
-
1 $2500 7 0.756 6.213
k’
82.35
-
82.39
0.066 3.209) 0.003 0.709) 9.378 (316.251)
((-
3 $12,500 22 0.351 5.157
Model using Household Data for Montreal
29.65
29.89
0.007 (0.674) 9.344 (213.579)
- 0.059 (- 2.930) -
3 $12,500 22 0.350 5.120
Test of the Alonso-Muth
1 $2500 I .531 2.319
Summary of the Findings for the Alternative
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AND KRYZANOWSKI
cant regression coefficients at the 0.001 level, areI T = 0.822 + 0.236k + O.OOOly,
(14)
where the sample size is 340, R2 is 0.246, the F value is 54.908; and In T = - 1.798 + 0.043k’ + O.OOOly,
(15)
where the sample size is 340, R2 is 0.227, and the F value is 49.538. The marginal monetary cost of travel, 6T/6k, based on Eq. (14), is 23.6 cents per mile travelled. The marginal monetary cost of travel, GT/Sk’, based on Eq. (15), is obtained by solving
Using the mean values of k’ and y for each of the three classesof household income, the corresponding values of 6T/6k’ are 1,2, and 4 cents per minute of travel time, respectively. The slopes of the bid-price functions, G/Sk = x(&a/6k) = [(Su/Gk)/ (6u/6z)] (- 6T/6k) and &?/tSk’ = [(su/6k’)/(Su/tSz)] (- sT/Gk’), were computed using (8) and the values of 6T/6k and 6T/6k’, respectively. These values are presented in the last two lines of Table 1. From these values, it is evident that the slope of the bid-price function increases markedly with income for both the k and k’ measures of distance. Thus, unlike the findings of Wheaton for the San Francisco Bay area, these results are supportive of the Alonso-Muth model. AN ALTERNATIVE
MODEL OF URBAN RESIDENTIAL LOCATION
To further examine the validity of preferences as an explanation for the pattern of residential location of various income groups within an urban area, a simple model of arbitrage between distance and housing will be developed in this section of the paper and tested in the next section. One of the strengths of the model is that it yields easily testable relationships between variables using data which are readily available. As in the Alonso-Muth model, suppose that the household tries to maximize its utility function (1) subject to its budget constraint (2). The household’s Lagrangian is then L = u(z, x, k) + d[z + xp(k)
+ T(k, y) -y],
(17)
where all the terms are as defined earlier. 13The remaining 12 (unreported) equations do not differ materially in terms of the values of 6T/6k and ST/Gk’ that they imply.
from these two equations
URBAN
RESIDENTIAL
349
LOCATION
The following are derived from the first-order conditions: 6L/6x
= Gu/Sx + ep = 0;
8 =( SL/6k
= 6u/6k
-6u/Sx)/p;
+ (8x)(6p/6k)
(18)
and + 8(67’/6k)
(19) = 0.
(20)
Substituting (19) into (20), and rearranging (20), gives
where MRS, k is the marginal rate of substitution between distance and housing. Equation (21) can be interpreted as follows: the demand for housing is proportional to the product of its price and the marginal rate of substitution between housing and distance, minus the marginal monetary cost of travel, all divided by the residence’s relative price per unit of travel from the residence to the city core. Equation (21) can be estimated if it is assumed that MRS,,, = (Su/Gk)/(Su/Gx), which is less than zero, is constant. This assumption requires that the household’s relative preferences between housing and distance are invariant to a marginal increase in y. In other words, this implies that the optimal path of the household in the plane spanned by x and k is linear. Another equilibrium relationship can be obtained from the observation is that the elasticity of utility with respect to distance, .& = (su/Sk)(k/u), less than zero, and that the elasticity of utility with respect to housing, [, = (Su/Gx)(x/u), is greater than zero. Using these observations, (21) can be rearranged to obtain P = (W,)(k)[(Sp/~k)
+ (W6k)/xl.
(24
Equation (22) can be interpreted as follows: the price of one unit of housing is proportional to its distance from the city core, multiplied by the ratio of the income (or utility) elasticities of housing and travel cost, multiplied by the sum of the marginal price of housing per unit of travel and the ratio of the marginal monetary cost of travel and the quantity of housing. Equation (22) can be estimated if it is assumed that [.J& is constant. This assumption is more general than the assumption that MRS,,, is constant, for it requires that the household’s relative preferences with respect to x and k are either accentuated or unchanged (i.e., do not reverse themselves) with a marginal increase in y. In other words, this implies that
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the optimal marginal path of the household in the plane spanned by x and k is an exponential curve. This condition can be further interpreted by first noting that
(23) where b is a constant. Then, rearranging (23) gives (Gk/Sx)/k
= b/x = b(l/x),
(24)
or log k = log Xb + a’,
(25)
k = axb,
(26)
or
where a’ and a = e”’ are constants. Furthermore, since a is presumably negative, Eq. (26) suggeststhat the undesirability of travel is a monotically increasing function of a household’s income. AN EMPIRICAL TEST OF THE ALTERNATIVE MODEL OF URBAN RESIDENTIAL LOCATION To empirically estimate Eqs. (21) and (22), it was assumed that MRS,, k and 6,/S, were constant. This differs from the implicit assumption involved in the estimation of the Alonso-Muth model, that is, that the utility level across households is constant. Although both approaches imply that the estimations should be conducted using stratified samples of households, the sample used to estimate Eqs. (21) and (22) was only stratified by income. This was done to increase the sample size without causing any loss of generality. In the first step of the empirical procedure, &p/&k was approximated by conducting a regression p(k, y). l4 For the screened subsample (i.e., the 14Regressionp(k’, JJ) was also run. However, since it was not statistically significant, only k (i.e., the distance in miles from the city core) was used to test the alternative residential location model.
URBAN
RESIDENTIAL
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351
sample that only includes households for which the head works in the urban core), the estimated equation is given by In p = 5.6315 - O.OOOly- O.O971k,
(27)
where the sample size is 340, R* is 0.073, the F value is 13.284, and all the regression coefficients are significant at the 0.001 level. In the second step of the empirical procedure, Eqs. (21) and (22) were estimated for the stratified samples four to seven (i.e., those samples with total estimated permanent incomes of $2500, $7500, $12,500, and $17,500, respectively). The results of these estimations are presented in Table 2 and can be summarized as follows. First, the R* and F values for all eight cross-sectional estimations are very high. Second, the estimates of MRS,, k for strata four and six are neither of the expected sign nor are they statistically significant. Third, although the estimates of c (that is, the coefficient of (6T/6k)(Sp/iSk)-‘) have the expected sign, they are much smaller than their expected value of - 1. Fourth, for strata five and seven, the estimates of MRS,,, are statistically significant and have the expected sign. Moreover, these estimates decrease significantly in value as income increases from $7500 to $17,500. Fifth, the estimates of [,./& are statistically significant and have the expected sign for all four strata. Moreover, all of these estimates decrease markedly and monotonically in value with increasing income. And finally, Eq. (22), and not Eq. (21), is the more appropriate representation of a household’s equilibrium housing relationship. This is consistent with the notion underlying the Alonso-Muth model that households use relative values when making arbitrage-type decisions between housing and distance. Thus, it appears that our alternative model of arbitrage between distance and housing is statistically acceptable, and that both MRS,, k and (,/Sk decreasewith increasing household income. Since these results corroborate the earlier findings for the Alonso-Muth model, they provide additional empirical support for the validity of the preferences explanation for the pattern of residential location in the Montreal urban area. CONCLUDING
REMARKS
For household data for the San Francisco Bay area, Wheaton [ 1l] found that the empirical evidence did not support the predictions of the Alonso-Muth model. Thus, since this was unsupportive of the preferences explanation for the pattern of urban residential location, Wheaton concluded that the externalities explanation was valid. In this paper, empirical evidence supporting the predictions of the Alonso-Muth model was presented for households in the Montreal urban
subsample
W5k T value R= F value Estimate
MR’% k T value
of c
Income Sample size
Stratified
0.909 85.428 - 3 18.284
-
$2500 18 0.080 1.370
4
0.616 102.346 - 59.069
-
$7500 128 - 0.017 - 1.508
5
x = MR.Sr,,p(6p/6k)-’
0.628 63.774 -46.615
$12,500 76 0.001 0.200
6 7
0.66 I 29.710 - 19.495
$17.500 31 -0.876 - 1.409
+ c(&T/h’k)(6p/6k)-’
- 4.132 - 7.595 0.772 57.688
$2500 18 -
4
- 8.420 - 16.326 0.677 266.548
$7500 128
5
P = (5,/t,)kK~p/6k)
- 24.833 - 5.489 0.501 30.132
-
$17,500 31
$12,500 76
- 16.779 - 15.939 0.772 254.045
I 6
+ (STPk)/xl
TABLE 2 Summary of the Findings for the Empirical Test of the Alternative Model of Urban Residential Location ___~.
URBAN RESIDENTIAL
LOCATION
353
area. Furthermore, corroborating empirical evidence was obtained using an alternative model of urban residential location for the same urban center.15 Thus, unlike the Wheaton study, the major finding of this study is that consumers’ preferences can significantly determine residential location patterns in at least one urban center. The differences between Wheaton’s results and those reported herein were, however, not unexpected. Since locational patterns are strictly dependent upon the relative patterns of the utility preferences of the households in the urban centers studied, it is not surprising that such preferences were found to differ for households in different cities, in different countries, speaking different languages, and having different cultures.‘6 While the nature of the empirical results, and the congruence of the empirical results for alternate models of urban residential location for the same sample are reassuring, the lack of replicability of the Montreal findings with those for San Francisco suggests that the two explanations may not be mutually exclusive. Thus, in future studies, it would seem appropriate to develop and empirically test a model which allows for the joint and/or interactive effects of both preferences and externalities on the pattern of urban residential location.
REFERENCES 1. W. Alonso, “Location and Land Use: Toward a General Theory of Land Rent” Harvard University Press, Cambridge, Massachusetts (1964). 2. R. Haugen and A. J. Heins, A market separation theory of rent differentials in metropolitan areas, Q. J. &on. 83, pp. 660-672 (November 1969). 3. A. Lapointe, Le choix du mode d’occupation sur le marche du logement a Montreal: Influence de l’origine ethnique, Acruul. &on. pp. 46-58 (1978). 4. A. Lapointe, “La Segregation Residentielle Ethnique a Montreal,” These de doctorat, Universite des SciencesSociales de Toulouse (1978). 5. A. Lapointe, “Repartition intra-Urbaine des Revenus: Quelques Considerations Methodologiques,” working paper, Universiti du Quebec a Montreal (I 979). 6. D. McFadden, The measurement of urban travel demand, J. Public Econ., 3, pp. 303-328 (November 1974). 7. R. Muth, “Cities and Housing,” Chicago Univ. Press, Chicago (1969). 8. G. A. Vaughn, Sources of downward bias in estimating the demand income elasticity for urban housing, J. Urban Econ. 3, pp. 45-56 (1976).
“It should be noted that a less restrictive assumption about the form of each households utility function is invoked to empirically test this model than is required to empirically test the Alonso-Muth model. “jWe are grateful to an anonymous referee of this journal for providing us with the essence of this comment.
354
TO, LAPOINTE, AND KRYZANOWSKI
9. W. C. Wheaton, A comparative static analysis of urban spatial structure, J. &on. Theory 9, pp. 223-237 (October 1974). 10. W. C. Wheaton, A bid rent approach to urban housing demand, J. Urban Econ., pp. 15-32 (April 1977). I I. W. C. Wheaton, Income and urban residence: An analysis of consumer demand for location, Amer. Econ. Rev. 67, pp. 620-631 (September 1977).