Really Useful Tests of the Monocentric Model Author(s): N. Edward Coulson Source: Land Economics, Vol. 67, No. 3 (Aug., 1991), pp. 299-307 Published by: University of Wisconsin Press Stable URL: http://www.jstor.org/stable/3146425 Accessed: 15/02/2009 16:48 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=uwisc. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected].

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Really Useful

Tests

of the Monocentric

Model

N. Edward Coulson

I. INTRODUCTION

The models of residentiallocation in urban land markets originally developed by Alonso (1964) and Muth (1969) have long since establishedthemselves as the foundations of urbaneconomic theory. The fundamental results of the basic model are that the spatial distributionof land and housing prices, consumption of land (and other housingattributes),and the spatialarrangement of residents (by income or some other characteristic)are determinedby the transportationcosts to the central business district (CBD). In particular, the principal qualitative hypotheses which the standard model generates are: (1) The price of land declines with distance from the CBD (the negative rent or land price gradient); and (2) the consumptionof land per household increases with distance from the CBD (the negative density gradient).' Prior research, with the exception of Yinger (1979), has had great difficulty in verifying predictions of the model, mainly because the data which has been used to test these predictions has not at all conformed to the assumptions used to derive them. This has been most clear in tests of the rent gradient, where a positive and/or insignificantgradient is usually estimated. In Section II, the monocentric model is brieflyreviewed, along with the difficulties of verifyingits predictions. In any case, a useful test of the monocentric model must arise through the use of data which both contains the variables described by the model, and is gathered from a housing market which conforms as closely as possible to its assumptions. In Section III, I describe a data set which does in fact conformvery closely to these conditions on both counts. In Section IV, the results are presented. The key result is that the existence of a negative rent gradientis confirmed with an implied total cost of

transportation of around $.50 per mile, which seems fairly plausible. The negative density gradientis also confirmedalthough the decay rate seems to be less than that previouslyestimated. Section V concludes. II. TESTING THE MONOCENTRIC MODEL

In this section I wish to outline the difficulties which have been typically encountered when attemptsare made to verify the predictionsof the monocentric model. Attention is mainly directed at the rent gradient, since this has been the predictionmost difficultto verify. Richardson(1977) noted the empiricalexistence of positive rent gradients and speculatedthat neighborhoodeffects were the cause. However, it is often the case that positive rent gradientsare still found even when measures of neighborhood qualityare includedamongthe conditioning variables which suggests perhaps that observable measures do not adequately capture neighborhood quality. If these omitted neighborhoodattributes are negatively correlatedwith distance, then a positive rent gradientcan be estimated due to this specification error, even when the monocentricmodel is true. It is also probably true that congestion externalities, such as pollution, traffic, and noise are indeed more severe as distance to the CBD decreases. Departmentof Economics, Penn State University. This paper was preparedfor the Annual Meeting of the AmericanReal Estate and Urban Economics Association, Atlanta, December 1989. I would like to thank Austin Jaffee for providing access to the multilist;Alex Anas, John Yinger, and an anonymous refereefor helpfulcomments;and Ara Khatchaturian for excellent researchassistance. 'The monocentricmodel yields other predictions which cannot be tested with the transactionsdata at hand,includingthe positive incomegradient(Wheaton 1977)and the negative wage gradient(Madden1986). Land Economics * August 1991 * 67(3): 299-307

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An alternative and important explanation of the positive gradient is the suburbanizationof jobs. Most metropolitanareas are not monocentric, but have subcenters at which a substantial amount of employment is located. Each of these subcenters may have rent gradients of their own, which will collide with the CBD's gradients, and each other's. This will have a confounding effect on any estimate of "the" rent gradientfor that city. To aggregateobservations on house prices from locationally distinct submarketsin the estimation of a single rent gradient is to force linearity (or at least monotonicity) on a functionwhich is really piecewise linear(or perhaps nonmonotonic). Thus, while the monocentricmodel can be true within each submarket, it appears to be false for the metropolitanarea as a whole. A third problem is the measurementof distance. Many data sets do not have precise measures of the location of the house. They are restrictedto gross measures such as the census tract, or to a centralcity/suburb binary measure. Taking the center of the tract (say) as the location introduces a potentiallysevere error in measurement.2 Fourth, there is the question of the context in which the test is performed. The most appropriatewould seem to be to treat the housing marketas a kind of putty-clay environment, where input quantities can perhaps be freely chosen when construction first takes place, but are very difficult to change ex post. This implies that the assumption of unit prices of the inputs and outputs, while appropriatefor the analysis of housing marketsin long-runequilibrium or in any scenario where inputs are separable, is not so for cross-section regression analysis where attributeprices, while fully flexible, may not be constant. Thatis, given the nonseparabilityof housing attributesin the existing housing stock, arbitrage.will not occur and attributeprices may not be linear (Rosen 1974; Coulson 1989). The monocentric model is then best tested in the frameworkof the hedonic model, where the sales price of a house is regardedas a possibly nonlinearfunctionof each housing attribute, including distance. This has the

August 1991

additional advantage of not requiring aggregation of different kinds of "housing capital";each attributeis simply treatedas a different argumentin the hedonic function. Nor does it require analysis of supplier behavior, since supply is treated as fixed in place. Let P(K, L, t) be the hedonic price function which maps housing attributescapital (a vector), land, and distance from the CBD, respectively into a sales price. With Y as household income, Z as consumption of the composite commodity, and k as the unit distance cost of transportationper unit time, the per period budget constraint is given as Z= Y- P(K,L,t)

- kt

[1]

and the utility function is U = U(K, L, Z).

[2]

Substituting[1] into [2], totally differentiating with respect to t, invokingthe envelope theorem, and noting that with only one income group, dU = dY = 0 in equilibrium,

this equilibriumyields the condition P/9at= -k.

[3]

Given the standardinterpretationof the derivatives of the hedonic price function, we can think of [3] as a statement about the hedonic (marginal)price of location. Note that this is not the standardsetup of the monocentric model as in say Muth or more recently Henderson (1985)and Fujita (1989). Those models typically assume a linear price schedule for the commodity "housing" (H), a unit of which is an aggregate of all housingattributes.From this and the assumptionof builderprofit maximization, one can derive relations between distance and both the price of housing (P) P/lat = -k/H

[4a]

2Anothermethod of verifyingthe rent gradientis to examinechanges in the relativeprice of housingas the cost of transportationchanges. The naturalperiod to examine is the oil shock years, and Small (1986) and Coulsonand Engle (1987)find some evidence of a shiftingslope.

Coulson: Monocentric Model

67(3)

and the price of land (PL) aPL/at = -

kIL

[4b]

ratherthan the hedonic price of location in [3]. Either [3] or [4] are possible tests of the monocentricmodel because both are statements about equilibriumprices of housing attributes when the assumptions of the model hold. The argumentmade below is that [3] is a more useful test. There are several reasons for this. First, the ceterus paribus conditions are different. In [3] all other housing attributes are held constant as distance is perturbed, thus its interpretationas a price of location. Equations [4] on the other hand, are not holding quantity of housing or land constant;on the contrary, they invoke the theoretical predictions of the Alonso-Muth model regardingchanges in land and capital consumptionas distance changes. Furthermore, in a hedonic regression where distance and other housing attributes are included on the right-hand side, the coefficient on location must be interpreted in precisely the ceterus paribus context noted for [3]. A second related reason is that [3] remains a test of the importanceof transport costs even in cases where the density gradient deviates from the predictionof the standard model. This is important because when developers make mistakes-for example, buildingtoo dense or too sparse for the location-or when the economic environment stochastically changes, the housing in place is ex post suboptimal.Such deviations from optimalbehaviorwill be very persistent, and this may very well destroy the observed land consumption-distance relationship. However, if the assumptions of the monocentric model hold, [3] should still be observed. Third,an important,thoughimplausible, assumption of the standard model is that all residents have the same income. This assumption can be eliminated if assumptions are made about the form of the utility function,following the method describedin Epple (1987)3 or alternatively Montesano

301

(1972), though this is by no means an easy problem. In any case, the empirical existence of multipleincome groupswill clearly have implicationsfor the estimationof land or housing price gradients. From [4] it is clear that if housing and land are normal goods, then the observed land price gradient (say) will be the upperenvelope of each group's bids at various distances. However, even in the presence of several groups, [3] should still hold. What [3] says in this context is that bids for otherwise identicalhouses at differentdistances must differ by the difference in transportation costs. If more than one income groupexists in the market for the particular housing type, their bid-gradientsmust lie atop one another.4 As noted, the monocentric model also makes a predictionon the relationbetween land consumptionand location. In this context, builder behavior is explicit, and the profit-maximizingbuilder will increase lotsize with distance to the CBD (Henderson 1985).The easiest way to see this is by differentiating the builder's profit function with respect to land, setting this equal to zero and obtaining HL = PL/P where HL

is the marginalproduct of land in housing production.Then take the total differential with respect to t and substitutefrom [4] to get, with some manipulation, aHL/at = k(PLL - PH)IHLP2

which is negative, since the parentheticexpressionis the negative of paymentsto capital. If land has decreasing marginalproduct, then land per household must be 3Epple's analysis was actually carried out using differencesin the utilityfunctionto generateheterogeneity, but income differences seem amenable to his method. 4A useful thought experiment is to imagine that differenttypes of housing are randomlyarrangedby location. Then the prices of these otherwise identical houses can only be the difference in transportation costs, regardlessof what type of household occupies them. This is destroyed if differentincome groups have differenttransportationcosts. Deacon and Sonstelie (1985) do not seem to find a correlation between waiting-timecosts and income.

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increasingwith distance. Under additional assumptions,it is possible to show that the land consumption gradient has a negative exponential form.5 The negative density gradient has been estimated with rather more success, going back to Muth (1969), through,for example, McDonaldand Bowman (1976). The difficultyin tests by these authors,at least, is that the unit of observation is the census tract, rather than the housingunit. Hence the regressionis of the tract population density on tract distance, whereas the monocentric model describes a relationshipbetween household land consumption and distance. Hence if households at all distances consume identical amountsof land, but the proportionof land devoted to nonresidential uses (including vacant land) increases with distance, then the regressionwill confirmthe monocentric model when it is not true. Even when data on land consumptionis available, problems of location measurementand suburbanization remain. III. THE DATA

The data used here is ideal for the purpose of testing the theorems of the monocentric model. The units of observationare 406 home sales gatheredfrom the multiple listing service for the calendar year 1987 within the State College metropolitanarea. State College is a quintessentialmonocentric city since a substantialmajorityof the commutingis to and from a relatively compact and well-defineddowntown area. The foremost reason for this is that Penn State University is located in this core area and the university is far and away the largest employer in the region. Data indicate that the universityemploys over 14,000people, while the next largest employer (located aboutone mile to the west of the downtown area) employs about 1,000. Furthermore, the CentralBusiness District is located directly adjacent to the university and contains the majority of professional offices, consumerand business services, and retail establishments.Thus, this core area of the universityand downtown is the destination for both commuting and other intra-area

August 1991

trips, as postulated by the monocentric model. Furthermore, the spatial range of the sample is large enough to encompass what is putatively an entire housing market, which is about a ten- to fifteen-mileradius aroundthe core. It leaves out that portion of the MetropolitanStatisticalArea (Centre County, PA) which is not occupied by commutersto the core. On the other hand, this MSA is one of the smallest so designated by the Census Bureau. This of course contributes to its monocentricity, but equally importantis the fact that the compactness of the housing market makes it relatively straightforwardto measure distance from place of residence to the center of the city using ordinarystreet maps. In fact this is not quite what was done. Instead, the 406 home sales were grouped into 80-90 developments and subdivisions (this datumis included in the multilist)and distance to the CBD was measured from the center of the development to the edge of the university campus. Since nearly all of the subdivisions are quite small relative to the town as a whole, this should give quite accurate measurement.Distance was measured by ruler along the appropriate streets from trip origin to destination and the ruler measurementhas been converted to miles in the calculationsreportedbelow. In addition, time costs are thought to play a majorrole in commutingcosts. This would be unimportantif time were proportional to distance, but some routes to the downtown area may be more congested than others, and time cost per mile may be higher. Here, this is dealt with by noting that the line hauls to the center mostly proceed from each of the four compass directions and these four routes have substantially different amounts of congestion duringpeak commuting hours. As part of the record for each observation, I note the directionof the developmentrelative to the core and in the next section test whether

5Residentialdensity is just the inverse of residentiallandconsumption;hence, the one gradientis easily derivedfrom the other.

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Coulson: MonocentricModel

the hedonic price of location varies with the compass. A few other aspects of the data need to be discussed. The spatial range of the data is encompassed by a single school district, so that differencesin the provisionof public education should not be important.On the other hand, within the range lie several local township governments and one larger borough government (State College), between which there are some small differences in taxes and public service levels. However, in the hedonic regressions reported below, the inclusion of 1986 property taxes in the list of housing attributes had a t-ratio of about 0.25, so that it may be the case that taxes and public services approximately wash out in the Tiebout equilibriumdescribed by Hamilton (1983). The other attributes to be included in the hedonic regression include: FLOORSPACE, squarefeet of living area including improvedbasement;LOT, also measuredin squarefeet, computed, when the size is not specifically given, as vertical dimension times horizontal;6NBATHS, the numberof baths, with half-bathscounting as one-half; YRBUILT,the year the house was constructed; GAR, the number of garage spaces, with carportscountingas zero; and dummy variables which equal one if the house has a fireplace (FP), woodburning stove (WB), electric heat (ELEC), hardwood floors (HW) and a public sewer connection (SEWER). Three sets of dummy variableswere found to have negligibleexplanatorypower and were omittedfrom the results reportedhere; they had no effect on these results: dummies for construction style (Cape Cod, Ranch, etc.), exterior material(wood, stone, etc.), and roofingmaterial (asphalt, synthetic, etc.). IV. THE RESULTS Attentionis now turnedto the estimation of the hedonic price index. One issue is the properfunctionalform, the choice of which shouldbe to some extent guided by the theoretical relationships described above. In equation [3], it was noted that linear transportation costs imply a constant hedonic

303

price of location, suggestingin turna linear hedonic price index. Transportationcosts may not be linear, of course, in which case the functional form may be misspecified, but the linear case will serve as a useful benchmark.Anotherpotentiallyuseful functional form is the semilog. The relationship between rent and distance is shown by Muth (1969) and Papageorgiouand Pines (1989) (in a model without capital) to be of this form when certain assumptions on the utility function are made. In addition, since there is no reason to expect prices to be linear in a marketwith nonseparability, the semilog has proven to be a popular choice among researchers. In his analysis of the rent gradient Yinger (1979) used a double log functional form which was derived from the assumptions of CobbDouglasutilityand productionfunctions. In fact, it turns out that all of these simple functionalforms are strongly rejected (via a LagrangeMultipliertest) in favor of the more flexible Box-Cox functional form.7 The Box-Cox transformationwas therefore carried out on the dependent variable and the four continuously measured attributes: YRB, FLRSPA, LOT, and DIST. The likeli-

hood function was maximized with transformation parameter 0.1 on the left-hand side and 0.7 on the right. The estimates under the linear and BoxCox functionalforms are given in Table 1.8 Each of the models fits the data well, thoughas noted the Box-Cox is more "justified" than the others. The coefficients of the linear model are sensible in size. Land prices are estimated to be about $.17 per 6Some lots are irregularlyshaped. In most cases the average of the two smallest dimensions was multipliedby the average of the two largest. When morethanfour dimensionswere reportedthe smallest one or two were thrownout, andthen the above procedure was applied. 7The test is a modifiedversion of the functional form test given by Godfrey and Wickens (1981). The modificationallows differenttransformationparameters for the left and right sides. 8The results of semilog and loglineargive essentiallythe same fit, thoughthe calculationof time costs are somewhat lower than the Box-Cox and linear forms.

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Land Economics

304

TABLE 1 Linear CONSTANT DIST LOT YRB FLRSPA BATHS GAR HW WOODB SEWER FIREP ELEC

Box-Cox

2 Directions

4 Directions

8970.87 (1.11) -3472.68 (-1.85) .171 (5.54) 217.91 (3.06) 32.43 (11.77) 4365.75 (1.95) 8571.94 (4.95) 3658.53 (1.36) -7076.66 (-3.01)

.705

.755

.702

6843.24 (.866) -3410.16 (-4.66) .165 (5.34) 218.62 (3.06) 32.26 (11.67) 5021.19 (2.25) 8777.91 (5.06) 4419.78 (1.66) -7505.39 (-3.19) -11719.83 (-2.96) 2954.95 (1.13) 11524.18 (4.08) 765.60 (1.30) 1721.54 (2.15) -2309.12 (-1.26) 367.49 (.213) .700

1.93 x 101

133.84

1.88 x 10"

1.91 x 10I

4113.5 (.528) -2633.41 (4.18) .173 (5.64) 212.50 (2.96) 32.77 (11.85) 4971.38 (2.22) 8779.53 (5.04) 4571.38 (1.71) - 7995.63 (-3.40) -9595.23 (- 2.56) 2779.52 (1.07) 12127.31 (4.30)

16.63 (60.01) -.118

(-4.67) .00021 (7.85) .0486 (5.33) .0080 (11.01) .245 (4.15) .263 (5.68) .157 (2.18) - .214

(-3.41) .031 (.315) .188 (2.66) .257 (3.29)

DIST*NORTH DIST*SOUTH

- 11042.87

(-2.78) 2165.84 (1.00) 12224.61 (4.32) 547.98 (.330) 1410.20 (.811)

DIST*EAST DIST*WEST Adj R2 ErrorSum of Squares

square foot, and costs of floor space is about $33 per square foot. Woodburners and public sewer connections have negative weight in the linear model, which is counterintuitive,though the latter's coefficient is positive (thoughexceedingly small) in the Box-Cox. The depreciation rate as estimated from the YRB coefficient at the mean values is .27 percent per year which is in line with other hedonic studies. The coefficient of interest is that associated with distance. The figure of $2,633.41 in the linearmodel gives the discount associatedwith movinga house one mile further from the city center, other attributesbeing held equal. Since this attributeprice is derived from the sale of the asset, ratherthan a flow price for some period of time, this

estimateof k is theoreticallythe capitalized savings of all future transportcosts from a given location as opposed to one a mile further from the core. In order to make a back-of-the-envelope calculation of the plausibilityof this coefficient, assume that 500 trips per year (i.e., one round-tripper day, 250 days per year) are taken, and that the life of the asset, which is the property rightsto the location ratherthan the structure, is infinite,capitalizedat an annualrate of 10 percent. Then the implied cost of transportationimputedfrom this coefficient is 0.10*(2633.41/500) = $.527/mile. The Motor Vehicle Association of the United States (as quoted in the 1989StatisticalAbstract of the United States) estimates the typical out-of-pocket cost of owning and

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Coulson: Monocentric Model

operatingan automobile at about $.36 per mile in 1987. Thus, about one-third of the cost of transportationcan be attributedto time cost accordingto this test of the monocentric model. The calculationfrom the linear model is basicallymatchedby a benchmarkdiscount calculated from the Box-Cox model. The per-mile discount at the average distance (3.4 miles) for the mean price house is $2,524.05, which yields transport costs of $.51.

Takingtime costs of commutingin State College to then be $.15/mile, and assuming an average driving speed of 40 miles/hour, the implicit value of time is estimated at $6.00/hour. This would seem to be on the low end of previously estimated time costs (Deacon and Sonstelie 1985). There are a couple of potentialexplanationsfor this low estimate. First, the automobile operating costs cited above are average, rather than marginal,costs. To the extent that this includes costs which have a fixed component (insurance,maintenance,etc.) the marginal costs of operation may be lower than the average. Since the coefficient is an estimate of the marginalcost (of one extra mile), the overestimate of out-of-pocket marginal costs is accompaniedby an underestimate of marginal time costs in the calculation above. Alternatively,suppose that the marginal cost of time is rising in commuting time. The linear approximation of time costs generated from a small housing market such as the one studied here will then be smaller than what might be expected in anotherenvironment. As noted above, the possibility exists that the time costs may vary with the route taken to the center. In orderto address this possibility, each developmentwas assigned to one of seven groups according to whether the development would use a northern, southern, western, eastern, or northeastern(only a few of the observations fit this category) line haul, or use surface streets to the north or south. In these latter two cases, these designations were limited to the two developments immediately to the north and south of the core. Dummy variables were created for the

305

north, south, east, and west houses. Dummies were not created for the two surface street developments, nor were they created for the northeastroute. In the formercase, this was because these houses are not particularly subject to congestion, and in the latter because few houses had this characteristic (so that congestion would not seem to be a problemhere either). The results of this estimation are in the thirdand fourth columns of Table 1. In the fourth,all four dummiesare interactedwith the distance variable and in the third only those in the north and south direction are differentiated,these being the most developed quadrants,and so possibly the most congested. The linear functional form was used in both cases, since it gave results similar to the Box-Cox, even though the BoxCox is theoretically superior. Examining column 4, the results are basically insignificantand of the wrong sign accordingto the priors expressed above. The north and south developmentsapparentlyhave a flatter gradient,indicative of a lower transport cost, and the eastern has a steeper gradient. None of these individualcoefficients has a t-ratio which would lead to rejection of a hypothesized zero value at conventional significancelevels. The restriction that all four are equal to zero is not rejected at the 1 percentlevel though it is at the 5 percent. The model of column 3 gives similarresults though it is not significantlydifferent than either the model with four dummies or the model with none. The overall conclusion is that in this case the direction does not change the steepness of the gradient.9 Attention is now turned to the density gradient.In this case, the traditionalmodel suggests a regressionof the log of land consumptionon distance only.'?This form per9A referee asks whetherthere is a correlationbetween directionand income. If so, the congestionand differentialtime costs could be cancelingout. I do not have dataon income, but the priceof housingis essentially the same in three of the directionsand is somewhat less in the eastern quadrant.Thus, this would not seem to be the explanationfor the nonresult. ?'Evenif transportationcosts are linear, the linear model is invalidhere since this is not conditionedon the existence of other attributes.

306

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Land Economics

formed well in comparison to other functions in McDonald and Bowman (1976). Estimatingthis model yields log(LOT) = 9.259 + .1198*DIST

(124.73) (6.18)

which seems to give strong support to the generalidea of a negative density gradient. The decay rate of density (- .1198)is rather smallerthan those estimated by McDonald and Bowman and this provides supportfor the idea of lower than average travel costs in this market. However, it is also the case thatthey follow the practiceof using census tract density ratherthan land consumption as the dependentvariable.As noted above, populationdensity is not the variablewhich is generally predicted by the monocentric model, since in its standardform, nonresidential land use is not incorporated, and this may account for the sharper decay rates that they estimate. On the other hand, the sample used here does not include multiunit structures. To the extent that these structuresare near the core, the decay rate will be underestimated. V. CONCLUSIONS An empiricaltest of a theory can be useful if the circumstances are such that the theory should be expected to pass if it is true. If it fails the test, then the theory would seem to have no empiricalcontent at all. If it passes, then it would seem to be a useful theory, at least under circumstances described by its assumptions. It is in that context that the above analysis is carried out. The State College area provides an ideal laboratoryfor testing the theorems of the monocentric model because more so than other, larger, areas, the assumptions of the monocentricmodel are basically fulfilled, and unlike smaller communities the turnoverand new constructionin the market yields a large enough sampleof transactions to make econometric analysis feasible. The results of these tests indicate that a rent gradient exists. Holding other attributes constant, the price of housing falls

with distance to the CBD, and more importantly, falls at a rate approximatelyequal to the increase in transportationcosts, as predictedby the model. The prediction of land consumptionis also qualitativelyverified. The result on the rent gradient is of importance not just because it verifies a theory under ideal conditions, but also because urban researchers can have more confidenceabout predictionsin more complex housing markets. If operating costs (includingtime costs) can be estimated,one may be able to make reasonably accurate predictions concerning the marginaleffect of, say, subcenterdevelopmenton the surroundinghousing market. It shouldbe noted that the hedonic interpretation of the gradient leads to further questions about the actual demand curve for location. Hedonic functions are not demand (or supply) curves, and the demand for location has not been calculated. That would requirethe instrumentalvariableestimationlaid out in Epple (1987) and elsewhere. This in turn requiresdata on household characteristics(like any other demand function)which is not availablein this data set. This is an obvious area for future research. References Alonso, William.1964.Location and Land Use. Cambridge:HarvardUniversity Press. Coulson, N. Edward. 1989. "The Empirical Contentof the Linearityas RepackagingHypothesis." Journal of Urban Economics 25 (3):295-309. Coulson, N. Edward, and Robert F. Engle. 1987. "TransportationCosts and the Rent Gradient."Journal of Urban Economics 21 (3):287-97. Deacon, Robert, and Jon Sonstelie. 1985. "Rationing by Waiting and the Value of Time: Resultsfroma NaturalExperiment."Journal of Political Economy 93 (4):627-47.

Epple, Dennis. 1987. "Hedonic Prices in Implicit Markets:EstimatingDemandand Supply Functions for DifferentiatedProducts." Journal of Political Economy 95 (1):59-80.

Fujita, Masahiro. 1989. Urban Economic Theory. Cambridge:Cambridge University Press. Godfrey, L. G., and Michael Wickens. 1981.

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"Testing Linear and Log-linearRegressions for FunctionalForm." Review of Economic Studies 48 (3):487-96. Hamilton,Bruce. 1983. "Is the PropertyTax a Benefit Tax?" In Local Provision of Public Services, ed. G. Zodrow. Orlando,FL: Academic Press. Henderson,J. Vernon. 1985. Economic Theory and the Cities. Orlando, FL: Academic Press. Madden,Janice. 1985."UrbanWageGradients: EmpiricalEvidence." Journalof UrbanEconomics 18 (3):291-301. McDonald,John, and H. Woods Bowman. 1976. "Some Tests of Alternative Density Functions." Journal of Urban Economics 3 (3):242-52. Montesano, Aldo. 1972. "A Restatement of Beckmann's Model." Journal of Economic Theory4 (2):329-54. Muth, Richard. 1969. Cities and Housing. Chicago: University of Chicago Press. Papageorgiou,Yorgos, and David Pines. 1989.

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"The Exponential Density Function: First Principles,ComparativeStatics, and Empirical Evidence." Journalof UrbanEconomics 26 (2):264-68. Richardson,Harry. 1977. "On the Possibilityof Positive Rent Gradients."Journal of Urban Economics 4 (1):60-68. Rosen, Sherwin. 1974."Hedonic Prices and Implicit Markets: Product Differentiation in Pure Competition." Journal of Political Economy 82 (1):34-55. Small, Kenneth A. 1986. "Effects of the 1979 Gasoline Shortage on PhiladelphiaHousing Prices." Journal of Urban Economics 19 (3):371-81. Wheaton, William. 1977. "Income and Urban Residence: An Analysis of Consumer Demand for Location." American Economic Review 67 (4):620-31. Yinger, John. 1979. "Estimating the Relationship Between Location and the Price of Housing." Journal of Regional Science 19 (3):271-89.