Revealed Preference and Location Choice

JOURNAL OF URBAN ECONOMICS ARTICLE NO.

41, 358]376 Ž1997.

UE962005

Revealed Preference and Location Choice1 Geoffrey K. Turnbull Department of Economics, Louisiana State Uni¨ ersity, Baton Rouge, Louisiana 70803 Received March 9, 1995; revised January 29, 1996 The assumption that population and structural densities are defined over parcels of land with zero area is problematic when interpreting the standard residential land use model as approximating a large finite economy. This paper extends revealed preference theory to depict a spatial economy with land parcels of finite area. The model generalizes the standard utility function approach, allowing for many consumer types with different tastes and incomes and labor-leisure choice, to show that the most important predictions of the standard theory hold in the general finite framework studied here. Q 1997 Academic Press

I. INTRODUCTION The calculus-based utility function model of urban residential land use provides key results underlying most urban economic analyses of spatial housing and land markets. The model has been criticized, however, for its treatment of land consumption. One such complaint is that the typical characterization of price and consumption gradients as smooth functions over space is based upon the assumption that each household’s land consumption is a length of arc with infinitesimal width. As Berliant w5x points out, the standard interpretation of the model is logically inconsistent in that structural and population densities are defined over residential land parcels with area of measure zero, with the further implication that, contrary to usual practice, the standard model cannot be construed as an approximation to a large finite spatial economy. Although a variety of justifications for interpreting the standard model as such an approximation have been offered in the literature w4, 18, 22x, an alternative approach is to refocus formal analysis to directly address land use theory within the strictures imposed by the finite spatial economy. The contribution of this paper is to present one alternative formulation of the standard model depicting a large finite spatial urban residential economy 1 I thank the participants in the Urban Land Economics Seminar at the University of British Columbia, the Editor and reviewers for helpful comments.

358 0094-1190r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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with land parcels of positive area. This paper extends revealed preference theory to deal with intra-urban residential location choice, exploiting the ease with which the revealed preference method allows configurations wherein consumers occupy residential lots with positive area, to depict the spatial market outcome as a large finite economy in equilibrium. A disadvantage of taking the revealed preference approach is that it explicitly sidesteps traditional equilibrium existence issues. The restriction that consumers reside at only one location introduces a nonconvexity that, with the introduction of spatially distinct parcels of land, creates compelling difficulties when characterizing or demonstrating the existence, uniqueness, or efficiency of market equilibrium w6, 9, 10x. On the other hand, Berliant and Fujita w7x provide existence and efficiency results for a simple utility function model that suggest that similar conclusions may be eventually obtained for generalized finite spatial economy models. Nontheless, the revealed preference approach taken here has several advantages for our purposes. It conveniently allows us to address the finite spatial economy directly, rather than with models justified as approximate outcomes. The revealed preference approach allows for a grainy spatial structure implied by residential lots of finite area, while retaining the interpretations of the standard model. In addition, the model structure is parsimonious, yet more general than the standard versions of the utility function model. The simplest revealed preference model considered in this paper allows for many types of consumers with different tastes and incomes, general transportation technology, and labor-leisure choice}a set of complications that leaves the standard model intractable. Turnbull w26x uses revealed preference to study the partial equilibrium theory of urban consumer demand. The focus of the current paper differs from the earlier study. Whereas the earlier paper is concerned solely with demand properties for goods like housing, whose prices vary over space, this paper is concerned with the spatial characteristics of the observable price and consumption outcomes implied by market equilibrium. Given the different focus, it is perhaps not surprising that the revealed preference axiom used in the earlier study, which is defined over both quantities of goods as well as location per se, must be modified to a more conventional revealed preference axiom over quantities of goods alone in order to derive meaningful results for the spatial market equilibrium. Remarkably, we find that the most important predictions of the standard utility function model hold in the generalized framework studied here. From a broader perspective, this suggests that the assumptions of smooth densities and infinitesimal width plots of land underlying the standard utility function model can be interpreted as convenient assumptions which simplify exposition but do not drive the key implications of land use theory.

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The paper is organized as follows. Section II presents the model and the revealed preference axioms relevant to the spatial demand context. Section III derives implications of the model, showing how revealed preference extends the simpler standard model implications to economies with varieties of consumer types, a generalized transportation technology, and labor-leisure choice. Section IV addresses the complications introduced by revealed preference for location per se. Concluding remarks are contained in Section V. II. THE REVEALED PREFERENCE MODEL Following the conventional urban spatial model, all variables are real valued. The urban area is monocentric, with a concentration of employment in the single CBD. The urban area comprises a compact land area in the Euclidean plane, completely partitioned in equilibrium into a set of parcels, A s  a1 , a2 , . . . , a N 4 , where each a i denotes a subset comprising a contiguous area in the urban plane. Parcels do not overlap in equilibrium. As in the standard model, consumer decisions are concerned with the size Žarea. of a i and its proximity to the CBD, but not its geometric shape.2 The CBD occupies one parcel in A. The revealed preference assumption maintains that the observed urban area configuration is an equilibrium, an outcome reflecting consistent consumer behavior. All of the properties of the spatial equilibrium model are deduced by exploiting the consistency condition on individual consumers. Denote the consumption vector x s w x 1 ??? x n x9, in which x 1 is housing and x n is leisure time. Housing services, x 1 , are derived from the combination of land and capital; the housing production sector is relegated to the background in this model.3 Consumers are price takers, with price vector 2 See Berliant and Ten Raa w8x for an extension of utility theory to the problem of lot shapes and configurations. Consumers in this model make their choices over the level of housing services being produced with the land parcel and its attached structure Žcapital.. Thus, consumers in this model reveal preference over lot shape only to the extent that the lot configuration affects housing services derived from a lot of specified size. 3 Housing services are produced by competitive profit maximizing firms and land is allocated among users by maximum bid rents. An outline of this sector is the following. The land area of parcel a i is a i . Housing services supplied on parcel a i with capital k i are si s f Ž a i , k i ., where f need not be differentiable nor even continuous, but must exhibit D frD a ) 0 and D frD k ) 0 over some range of a and k values. Denote the input or bid price of the land in parcel a i by r Ž a i .. ŽBerliant and Fujita w7x provide an example of a finite model with a similar additive land price construction.. The per unit price of capital is r , so that bid price equilibrium in the land market requires zero housing producer profits, so that rŽa ˜i . s p1Ž d i . f Ž a˜i , ˜k i . y r ˜k i , where p1Ž d i . is the price of housing services at a parcel with location index d i , as explained below, and tilde denotes input choices satisfying the Weak Axiom of Profit Maximization w30x. Housing supply, s, satisfies demand, x 1 , on each parcel in equilibrium, so that x 1 s s ; a i g A.

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p s w p1 ??? pny1 w x9, where p1 is housing price or rent, pi is the price of good i, and w is the wage rate. In some applications it is convenient to partition the price vector as p s wŽq w x9. The conformable partition of x s wv x n x. Nonwage income, including the consumer’s share of land rent income in equilibrium, is m. Note that m and w may vary across consumer types, as will become clear below. All employment is located in the CBD. Each consumer chooses both x and a residence site; the choice of house location determines the length of the consumer’s commute to the CBD. All residential parcels are indexed by their effective Ži.e., minimum travel cost. distance from the CBD, denoted d. The set of all distances in the urban area is denoted D.4 Commuting distance d takes time t Ž d . and out-of-pocket money cost Ž c d ., where D tD d ) 0 and D cD d ) 0 indicate greater time and money cost of commuting longer distances. Neither t Ž d . nor cŽ d . need be differentiable, continuous, or exhibit other restrictions typically assumed in parametric theory unless otherwise stated. As a final spatial component of the theory, housing price may vary across the urban area, according to p1Ž d ., where the precise manner of variation is an implication of the theory. As in the travel cost case, the housing price function need not be differentiable or continuous. The spatially varying price vector is pŽ d . s w p1Ž d . p 2 ??? pny1 w x. The consumer can allocate total time available to three activities: work Ž L., commuting Ž t ., and leisure Ž x n .. Defining time allocations as proportions, the time constraint is L q t Ž d . q x n s 1. In order to impose the time constraint directly into the budget constraint, note that total expenditures on all goods except for leisure time are qŽ d .v. The money cost of cummuting from d is cŽ d . so that the constraint that total spending not exceed total income m q wL requires qŽ d .v q cŽ d . F m q wL. Solving the time constraint for L s 1 y t Ž d . y x n , the budget constraint at d becomes qŽ d .v q wx n F m q ww1 y t Ž d .x y cŽ d ., i.e., pŽ d .x F m q ww1 y t Ž d .x y cŽ d .. The budget set facing the consumer at residence location d is therefore defined as B Ž d . ' x N p Ž d . x F m q w 1 y t Ž d . y c Ž d . 4 . The consumer is free to choose from among locations in D, however, so the consumption opportunity set is B ' D d g D B Ž d .. Each consumer selects some consumption-location vector y s wx d x from B. The set of consumption-location bundles chosen by the consumer 4 Note that d is the minimum travel cost distance from the parcel, that is, the distance from one reference point in the finite parcel, say a point on the frontage facing the CBD. The transportation technology defines a mapping from A into D. Notice, however, that it is not necessary to assume that each parcel the same distance from the CBD must be the same size.

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from B is denoted H; the consumer shall be identified by his choice set, H. Any value of y observed as a selection by the consumer must be in H. Although most revealed preference axioms in nonspatial theory assume that the choice set H has a unique element, multiple element choice sets will be typically required in spatial equilibrium models where identical consumers may reside at different distances from their job sites in the urban land market. The set hŽ d . is defined as the location-specific choice set from B Ž d .. In the parametric approach to location theory, the consumer’s choice can be thought of as comprising two dimensions: the choice of location d and the choice of the consumption bundle at that location. Muth’s w20x approach assumes utility is a function of x alone; the consumer has no preference for location per se. In the standard framework using Muth’s utility function, the consumer’s problem can be broken down into two stages. First, the consumer determines the utility maximizing bundle x at each feasible d. Second, the consumer selects from those locations the one that will yield the greatest utility when consuming the utility maximizing x for that location. Location choice is the result of the interplay between the benefits and costs implied by the spatially-varying budget constraint B Ž d . over D. The approach taken here is similar to Muth’s approach in that the consumer does not reveal preference for location per se, but rather for the consumption opportunities that may be available at the chosen location compared with the opportunities elsewhere. ŽRevealed preference for location per se will be addressed later.. Thus, the revealed preference axiom is defined for the choice of x, with the selection of d by the consumer arising from the interplay of revealed preference for x and the spatial configuration of B over D. Intuitively, a bundle x i is revealed preferred to some other bundle x if x i is chosen by the consumer when x was available as well. Using the above notation, define relations R D , R, and P D as follows. For an observed x i chosen by the consumer from B Ž d i . and some other x, x i R D x indicates that x i is directly re¨ ealed preferred Ž or re¨ ealed at least as good as . to x if pŽ d i .x F m q ww1 y t Ž d i .x y cŽ d i .. x i P D x indicates that x i is directly re¨ ealed strictly preferred to x if pŽ d i .x - pŽ d i .x i. Finally, R is the transitive closure of R D . x i Rx indicates that x i is re¨ ealed preferred Ž or re¨ ealed at least as good as . to x if for some sequence of observations x i,x j, . . . , x k ,x, we have x i R D x j,x j R D x t , . . . , x r R D x k , x k R D x. The consumer’s choice set at d is the set of bundles which are directly revealed preferred to all others in the budget set at d, or hŽ d . '  x9 N x9R D x ; x g B Ž d .4 . The consumer’s choice set over the urban area is the set of all consumption-location bundles such that the consumption bundles are revealed preferred to all other bundles available regardless of location,

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or H '  y9 s wx9 d9x N x9Rx ; y s wx d x g B4 . Clearly, y i s wx i d i x g H « x i g hŽ d i .. Further assuming re¨ ealed nonsatiation, y i g H « pŽ d i .x i s m q ww1 y t Ž d i .x y cŽ d i . and x j g hŽ d j . « pŽ d j .x j s m q ww1 y t Ž d j .x y cŽ d j .. Using these definitions, rational consumer behavior is that which is consistent with GARP. Generalized Axiom of Re¨ ealed Preference (GARP): x i Rx k « not x k P D x i The axiom is a generalization of the weak and strong axioms of revealed preference in the sense that it allows for multiple element choice sets. This property is analogous to flat regions on indifference surfaces in utility theory, which lead to multiple-valued demand functions in standard theory w23, 29, 31x. The nonuniqueness property is essential in the spatial environment when identical individuals reside at different locations in the urban area, a spatial equilbrium condition in the utility function model. III. IMPLICATIONS OF SPATIAL EQUILIBRIUM The predictions of the utility function monocentric city model include a convex declining housing price and increasing housing consumption with distance as well as spatial separation of consumer types in the urban area under certain assumptions w17, 19, 21x. This section examines the comparable results implied by revealed preference. It is convenient to begin with a basic result useful in subsequent derivations. LEMMA 1. pŽ d j .x i G m q ww1 y t Ž d j .x y cŽ d j . ; y i s wx id i x g H and d i , d j g D. Proof. Consider an arbitrary consumer y i s wx i d i x g H. To prove the lemma 1 by contradiction, suppose the contrary: pŽ d j .x i - m q ww1 y t Ž d j .x y cŽ d j . for d j / d i . Thus x i f hŽ d j . and ; x j g hŽ d j ., x j / x i and x jP D x i.

Ž 1.

x i Rx j

Ž 2.

However, wx i d i x g H, so that

regardless of whether or not hŽ d j . is a subset of H. Observe that Ž1. and Ž2. violate GARP, from which we conclude that our supposition cannot be true. Thus pŽ dj .x i G m q w 1 y t Ž dj . y c Ž dj . which is lemma 1

Ž 3. Q. E. D.

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The best-known result derived from the utility function location and land use theory is the declining housing price with distance from the CBD w7, 21x. A similar result holds under the generalized revealed preference axiom, as stated below. THEOREM 1 ŽMonotonic Declining Housing Price.. D p1 D d s w p1Ž d i . y p1Ž d j .xw d i y d j x - 0 ; d i / d j g D for the monocentric city under GARP. Proof. Consider a consumer y i s wx i d i x g H, x i g hŽ d i . and by nonsatiation pŽ di . x i s m q w 1 y t Ž di . y c Ž di . .

Ž 4.

By lemma 1, Ž3. holds at d j / d i . Subtracting Ž4. from Ž3. yields p Ž d j . y p Ž d i . x i G wt Ž d i . q c Ž d i . y wt Ž d j . y c Ž d j . .

Ž 5.

Now consider d j - d i . Using D tD d ) 0 and D cD d ) 0 the right hand side of Ž5. is strictly positive so that Ž5. implies wpŽ d j . y pŽ d i .xx i ) 0. Taking the inner product on the left hand side of this inequality and using the spatial invariance of nonhousing prices, this implies p1 Ž d j . y p1 Ž d i . x 1i ) 0

Ž 6.

from which p1Ž d j . ) p1Ž d i . immediately follows. Repeating the procedure for all d i g D yields monotonicity, D p1 D d - 0. Q. E. D. This result opens the way for Theorem 2, establishing the positive housing consumption gradient which underlies the declining structural and population densities in the utility function theory. THEOREM 2 ŽHousing Consumption Gradient.. For two identical households, i and j, for whom y i, y j g H satisfy d j - d i , housing consumption satisfies x 1i G x 1j under GARP. Proof. Define x iy as the vector in hŽ d i . with the minimum x 1 and x iq as the element in hŽ d i . with the maximum x 1. The stipulation y i, y j g H ensures that hŽ d i . and hŽ d j . are nonempty so that there exist x iy and x jq such that wx iy d i x, wx jq d j x g H. By lemma 1, wx iy d i x g H implies p Ž d j . x iyG m q w 1 y t Ž d j . y c Ž d j . .

Ž 7.

By lemma 1, wx jq d j x g H implies p Ž d i . x jqG m q w 1 y t Ž d i . y c Ž d i . .

Ž 8.

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Because x jqg hŽ d j ., by nonsatiation pŽ d j .x jqs m q ww1 y t Ž d j .x y cŽ d j ., which, with Ž7., reveals p Ž d j . w x iyy x jq x G 0.

Ž 9.

Similarly, x iyg hŽ d i . implies pŽ d i .x iys m q ww1 y t Ž d i .x y cŽ d i ., so that this with Ž8. reveals yp Ž d i . w x iyy x jq x G 0.

Ž 10 .

Adding the left hand sides of Ž9. and Ž10. and using the spatial invariance of nonhousing to simplify the vector inner product yields p Ž d j . y p Ž d i . w x iyy x jq x G 0 p1 Ž d j . y p1 Ž d i .

x 1iyy x 1jq G 0.

Ž 11 .

Theorem 1 requires p1Ž d j . ) p1Ž d i . for d j - d i , so that Ž11. implies x 1iyG x 1jq.

Ž 12 .

By definition x 1i G x 1iy and x 1jqG x 1j , which with Ž12. implies x 1i G x 1j . Q. E. D. In the standard utility function model, the consumer is in location equilibrium when situated such that the marginal benefit of moving farther out, in the form of savings on housing expenditures at the lower price farther away form the CBD, just equals the marginal cost of moving farther out, in the form of greater commuting costs. The revealed preference theory yields a related set of conditions, the finite analogs to Muth’s equaiton in the calculus models. For any given household, ; y i s wx i d i x g H condition Ž5. holds for all d j / d i . Taking the inner product of the left hand side of Ž5. and using the spatial invariance of nonhousing prices, yields w p1Ž d j . y p1Ž d i .x x 1i G wt Ž d i . q cŽ d i . y wt Ž d j . y cŽ d j . ; d j g D. Put somewhat differently, this result requires ; y i s wx i d i x g H, p1 Ž d j . y p1 Ž d i . x 1i G wt Ž d i . q c Ž d i . y wt Ž d j . y c Ž d j . ; d j - d i Ž 13 . p1 Ž d i . y p1 Ž d j . x 1i F wt Ž d j . q c Ž d j . y wt Ž d i . y c Ž d i . ; d j ) d i . Ž 14 . When interpreting these conditions, note that this version of Muth’s equation is a global, not a local, equilibrium condition. Condition Ž13. states that for any arbitrary consumer, all residential sites closer to the CBD than the choice site must entail greater housing costs that outweigh

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the commuting cost savings from making such a move. Likewise, condition Ž14. states that all residential sites farther from the CBD must entail an increase in commuting costs that outweigh the housing costs savings from making such a move. In contrast, the derivative version of Muth’s equation in the standard calculus model only requires that the marginal benefit-cost comparisons hold in the infinitesimal neighborhood of the consumer’s optimal location. Utility function models of location choice and land use predict convex housing Žor land. price with respect to distance in the monocentric city. Conditions Ž13. and Ž14. are useful for deriving a comparable result for the revealed preference theory. Notice that the transportation cost function in the following result includes the linear transportation cost function popular in utility function models as a special case w32x. THEOREM 3 ŽPrice Function Convexity.. Housing price is con¨ ex under GARP and nonincreasing incremental commuting costs wt Ž d . q cŽ d .: ; d i d j - d k g D,



wt Ž d j . q c Ž d j . y wt Ž d i . q c Ž d i . 4 r Ž d j y d i . G  wt Ž d k . q c Ž d k . y wt Ž d j . q c Ž d j . 4 r Ž d k y d j . « p1 Ž d i . y p1 Ž d j . r Ž d j y d i . G p1 Ž d k . y p1 Ž d j . r Ž d k y d j . .

Proof. Consider a consumer with y j s wx j d j x g H. Substituting d i , d j , and d k into conditions Ž13. and Ž14. and rearranging yields p1 Ž d i . y p1 Ž d j . x 1j G wt Ž d j . q c Ž d j . y wt Ž d i . y c Ž d i .

Ž 15 .

wt Ž d k . q c Ž d k . y wt Ž d j . y c Ž d j . G p1 Ž d j . y p1 Ž d k . x 1j .

Ž 16 .

Divide both sides of Ž15. by Ž d j y d i . and both sides of Ž16. by Ž d k y d j . to find, respectively, p1 Ž d i . y p1 Ž d j . x 1j r Ž d j y d i . G wt Ž d j . q c Ž d j . y wt Ž d i . y c Ž d i . r Ž d j y d i .

Ž 17 .

wt Ž d k . q c Ž d k . y wt Ž d j . y c Ž d j . r Ž d k y d j . G p1 Ž d j . y p1 Ž d k . x 1j r Ž d k y d j .

Ž 18 .

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The right-hand side of Ž17. is weakly greater than the left hand side of Ž18., leaving p1 Ž d i . y p1 Ž d j . x 1j r Ž d j y d i . G p1 Ž d j . y p1 Ž d k . x 1j r Ž d k y d j . p1 Ž d i . y p1 Ž d j . r Ž d j y d i . G p1 Ž d j . y p1 Ž d k . r Ž d k y d j . Ž 19 . Q. E. D..

which is the result to be shown.

THEOREM 4 ŽWeak Spatial Separation.. Consider any two consumers with identical marginal commuting costs but with possibly different housing tastes or nonwage incomes, whose choice sets are H9 and H0 under GARP. If for y i s wx id i x g H9 and y j s wx j d j x g H0 such that x 1i ) x 1j then d i G d j .5 Proof. For the household that has chosen to consume x 1i at d i , wpŽ d j . y pŽ d i .xx i G wt Ž d i . q cŽ d i . y wt Ž d j . y cŽ d j . from condition Ž5.. Similarly, for the household that has chosen to consume x 1j at d j , wpŽ d j . y pŽ d i .xx j F wt Ž d i . q cŽ d i . y wt Ž d j . y cŽ d j . from multiplying condition Ž5. by minus one. Given the equality of the right-hand sides of both of these conditions, together they imply w p1Ž d j . y p1Ž d i .x x 1i G w p1Ž d j . y p1Ž d i .x x 1j so that p1 Ž d j . y p1 Ž d i .

x 1i y x 1j G 0.

Ž 20 .

To prove the theorem by contradiction, suppose d i - d j . But d i - d j requires w p1Ž d j . y p1Ž d i .x - 0 by Theorem 1 so that Ž20. implies x 1i F x 1j , which contradicts the initial condition in the theorem. Therefore, our supposition d i - d j cannot be true and we conclude d i G d j . Q. E. D. The spatial separation of consumers by type in the urban area is readily seen from this result. If, for example, consumers vary only by nonwage income Žwhich ensures that marginal commuting costs remain the same across types. so that housing consumption is greater for one consumer type, the theorem states that the higher income consumers will separate from the lower income consumers. Further, since higher income consumers will consume more housing they will live farther out from the CBD than their lower income counterparts. This type of separation and spatial pattern is a standard result in the utility function models.6 If, on the other hand, the only difference between consumers lies in their revealed behavior, i.e., their revealed ‘‘tastes’’ for housing versus other goods, the theorem states that the consumer types will spatially 5 The outcome d i s d j is in the theorem because it is possible that x i and x j are both elements in hŽ d i .. 6 e.g., w17x. Because they do not allow for either labor-leisure substitution or the time cost of travel, greater ‘‘income’’ in their model is equivalent to greater ‘‘nonwage income’’ here.

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segregate, with the consumers whose tastes for housing are stronger living farther away from the CBD. Even though the concept of housing ‘‘tastes’’ in a revealed preference framework is not strictly analogous to differences in tastes as reflected by different utility functions, this interpretation yields results similar to the partial equilibrium utility model prediction that stronger tastes for housing reduces consumer demand for CBD proximity w14, 25x. The weak separation result can be seen by applying Theorem 4 in following fashion. For the consumers above, select dy, the minimum distance in bundles from H9 and dq, the maximum distance in bundles q y q from H0, and compare: dy i G d j . Since all other d i G d i and d j F d j , the only pooling that can exist will be at one distance, the boundary when q dy i s d j . In all other cases, the spatial separation between consumer types will be strong. This is similar to utility function models where the boundaries between two consumer types have infinitesimal width and so may be occupied by consumers of either type or both w17, 19x. There is no spatial pooling of types in the parametric utility function models other than at the boundary, though. The contrast between Theorems 2 and 4 deserves emphasis. Theorem 2 establishes that identical consumers living at different distances will exhibit different housing consumption. Theorem 4, on the other hand, pertains to consumers with different housing consumption in equilibrium, whether they are identical or differ in terms of income of revealed housing tastes. Given the observed difference in housing consumption, the consumer with the greater consumption will live farther away form the CBD, regardless of the source of difference in housing consumption. IV. LOCATION REVEALED PREFERENCE: FIXED WORK HOURS Alonso’s w3x utility function model with preference for distance can be derived as a special case of Muth’s w20x utility function model without preference for distance but with labor-leisure choice w26x. Denoting utility by uŽx., where x n is leisure as before, with work time exogenously constrained to L0 , the time constraint x n q L0 q t Ž d . s 1 can be rewritten for leisure as x n s 1 y L0 y t Ž d .. In this form the selection of commuting distance d determines the amount of leisure consumption directly; additional commuting time must necessarily reduce leisure time when the work hours margin of adjustment is precluded. Substituting for leisure in the utility function yields uŽ x 1 , . . . , x ny1 , 1 y L0 y t Ž d .. s f Ž x 1 , . . . , x ny1 , d ., where ­ ur­ x n ) 0 ensures ­fr­ d - 0. The function f is recognized as Alonso’s utility function in which the consumer prefers locations with less travel to locations with more travel.

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To what extent can this relationship between utility function models be exploited to provide a framework of revealed preference over location per se? Notice that the model described in the previous section is the revealed preference analog to Muth’s utility function model extended to include labor-leisure choice w15x. Therefore, imposing a work time constraint in the revealed preference model defined over goods as in previous sections should yield a framework analogous to Alonso’s utility function model, where the consumer prefers less commuting to more. This is the approach taken in this section. Assume that work hours are constrained to be L0 , an exogenous quantity describing a standard workday for some consumers. Standard workdays need not pertain to all consumers nor must the hours constraint be identical for all constrained workers. Subjecting leisure time to the constraint, x n F 1 y L0 y t Ž d . for a given L0 . When binding, the constraint implies that x has an explicit location component ¨ ia the revealed preference for leisure. Notice that the application of a free disposal of time assumption in this treatment of the time constraint is a convenient device for deriving results, although the time constraint always holds with equality in equilibrium. Neither the statement of GARP nor its interpretation need be altered for the analysis to follow. The money income net of money commuting cost is m q wL0 y cŽ d .. Partitioning the price and consumption vectors using q s w p1 p 2 ??? pny1 x and v s w x 1 ??? x ny1 x so that p s wq w x and x s wv x n x, the budget set at d is now B Ž d . s  x s w v x n x 9 N q Ž d . v F m q wL0 y c Ž d . , x n F 1 y L0 y t Ž d . 4 .

Ž 21 . The consumption opportunity set is B s D d g D B Ž d .. As before, nonsatiation is assumed, which in this application means that x i g hŽ d i . « qŽ d i .v i s m q wL0 y cŽ d i . and x ni s 1 y L0 y t Ž d i .. A. Housing Price and Consumption Patterns Extending the theory to incorporate revealed preference for location, we find a robust housing price gradient result. ŽBecause lemma 1 does not hold for the revealed preference theory under consumption constraints it is not used in the proof.. THEOREM 5 ŽMonotonic Declining Housing Price under Work Time Constraint.. Under GARP for the fixed work time case, D p1 D d s w p1Ž d i . y p1Ž d j .xw d i y d j x - 0 ; d g D, d i / d j .

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Proof. For an arbitrary consumer y i s wx i d i x g H, x i g hŽ d i . so that s 1 y L0 y t Ž d i . and q Ž d i . v i s m q wL0 y c Ž d i . .

Ž 22 .

Now show qŽ d j .v i ) m q wL0 y cŽ d j . for d j - d i by contradiction. Suppose the contrary; suppose qŽ d j .v i F m q wL0 y cŽ d j .. For all x j g hŽ d j ., x nj ) x ni so that x i g B Ž d j . and x i f hŽ d j .. Since x nj ) x ni , we have x j / x i and x jP D x i.

Ž 23 .

x i Rx j .

Ž 24 .

However, wx i d i x g H, so that

Observe that Ž23. and Ž24. violate GARP, from which we conclude our supposition cannot be true. Thus q Ž d j . v i ) m q wL0 y c Ž d j . .

Ž 25 .

Subtracting Ž22. from Ž25. and then performing the vector inner product yields qŽ d j . y qŽ di . v i G c Ž di . y c Ž d j . ) 0

Ž 26 .

p1 Ž d j . y p1 Ž d i . x 1i ) 0

Ž 27 .

from which p1Ž d j . ) p1Ž d i . follows. Repeating for all d i g D yields the monotonicity property, D p1 D d - 0. Q. E. D. Having established the usual declining housing price, consider the housing consumption gradient. No definitive results have yet been derived in the standard preference structure approach when consumers have preference for location per se w25, 27x, so the outlook for additional results might appear bleak in this version of the revealed preference model. What is interesting, though, is that more can be said. In particular, the analysis reveals a consumption gradient result analogous to the earlier theorem for the unconstrained work time model. The theorem and proof differ somewhat from the unconstrained labor time versions examined earlier, and so are presented in full here. One of the basic results of demand theory is the substitution theorem, that compensated own price effects are negative. A similar result is known

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to hold in nonspatial revealed preference theory. The spatial version presented here is needed for the analysis of housing consumption gradients in the labor-time constrained model, and so is offered as an intermediate result.7 THEOREM 6 ŽHousing Substitution Theorem under Work Time Constraint .. Let x i g hŽ d i . for B Ž d i . '  x N qŽ d i .v F m q wL0 y cŽ d i ., x n F 1 y L0 y t Ž d i .4 and let x*q be the x 1 ]maximal element in h* for B* '  x N qŽ d j .v F m q wL0 y cŽ d i . q d , x n F 1 y L0 y t Ž d i .4 for d j - d i and d s w p1Ž d j . y p1Ž d i .x x 1i . d is the compensation which allows the consumer with net income m q wL0 y cŽ d i . at location d i to purchase the original ¨ ector x i at price ¨ ector pŽ d j . pre¨ ailing at d j . Then x 1*qF x 1i under GARP. Proof. x i g hŽ d i . so that by nonsatiation x ni s 1 y L0 y t Ž d i . and q Ž d i . v i s m q wL0 y c Ž d i . .

Ž 28 .

Because x*qg h* implies xUq s 1 y L0 y t Ž d i . by nonsatiation, to prove n the theorem we need to show q Ž d i . v*qG m q wL0 y c Ž d i .

Ž 29 .

by contradiction. Suppose the contrary: qŽ d i .v*q- m q wL0 y cŽ d i .. Then x*qf hŽ d i . and x i P D x Uq .

Ž 30 .

By construction x i g BU . Since x Uqg hU , for both x i g hU and x i f h* possibilities we have x*qR D x i so that x*qR x i

Ž 31 .

However, Ž30. and Ž31. violate GARP; therefore, our initial supposition cannot be true and Ž29. holds. Subtracting Ž28. from Ž29. yields q Ž d i . w v*qy v i x G 0.

Ž 32 .

7 In the utility function theory of nonspatial demand, compensation can be made to ensure the same level of utility Žthe ‘‘Hicksian’’ method. or to ensure that the original bundle of commodities are attainable at the new price vector Žthe ‘‘Slutsky’’ method.. The Slutsky compensation method is used in Theorem 6 because the revealed preference approach does not formally introduce the notion of ‘‘utility’’ per se. Also notice that although this theorem superficially resembles the substitution theorems provided by Turnbull w28x, there is a significant difference. The result presented here examines consumer behavior while holding location Žand labor supply. constant as the consumer confronts the price vector from one other available site in the urban area. In contrast, the theorems provided by Turnbull w28x pertain to partial equilibrium demand changes with endogenous relocation Žand flexible labor supply. as the consumer confronts a new Žexogenous. price vector across all available sites.

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By construction qŽ d j .v i s m q wL0 y cŽ d i . q d so that x*qg h* implies qŽ d j .v*qs qŽ d j .v i ; hence q Ž d j . w v*qy v i x s 0.

Ž 33 .

Subtract Ž32. from Ž33. and use the spatial invariance of nonhousing prices to get p1 Ž d j . y p1 Ž d i .

xUq y x 1i F 0. 1

Ž 34 .

Because d j - d i by stipulation, w p1Ž d j . y p1Ž d i .x ) 0 by Theorem 6 so that Ž34. implies that xU1 F x 1i . Q. E. D. Using the result, we can now address the question of interest, whether or not the consumption gradient result established earlier still holds. With some additional structure, the result can be extended to the location preference environment implied by the work time constraint. To do so, consider the following definitions, which define normality in the usual sense and state that housing and leisure are complements when the revealed preferred consumption bundles all entail greater housing consumption when leisure increases. DEFINITION. For hŽ d k . i defined over B Ž d k . i s  x N qŽ d k .v F m i q wL0 ycŽ d k .; x n F 1 y L0 y t Ž d k .4 and H Ž d k . j defined over B Ž d k . j s  x N qŽ d k .v F m j q wL0 y cŽ d k .; x n F 1 y L0 y t Ž d k .4 with m i ) m j , let x iy and x iq be the consumption vectors with, respectively, the minimum and the maximum amounts of x 1 in hŽ d k . i and similarly, let x jy and x jq be the consumption vectors with the minimum and maximum amounts of x 1 in hŽ d k . j. Housing is normal if x 1iy) x 1jy and x 1iq) x 1jq; that is, housing is normal if the maximum and minimum quantities of housing in all location-specific choice sets of the consumer increase when nonwage money income increases. DEFINITION. Let x jy and x jq be the consumption vectors with, respectively, the minimum and the maximum amounts of x 1 in hŽ d j . for B Ž d j . '  x N qŽ d j .v F m q wL0 y cŽ d j ., x n F 1 y L0 y t Ž d j .4 . Let x ky and x kq be similarly defined in h k for B k '  x N qŽ d j .v F m q wL0 y cŽ d j ., x n F 1 y L0 y t Ž d i .4 . Notice that by construction B Ž d j . and Bk differ only by the maximum leisure time available. Then x n and x 1 are defined to be complements Ž substitutes. when x 1jy) x 1ky and x 1jq) x 1kq Ž x 1jy- x 1ky and x 1jq- x 1kq . for d j - d i . THEOREM 7 ŽHousing Consumption Gradient under Work Time Constraint .. For two identical households, i and j, for whom y i, y j g H satisfy

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d j - d i , housing consumption satisfies x 1i ) x 1j under GARP with housing normality and when housing is a re¨ ealed substitute or unrelated with leisure consumption. Proof. Define B Ž d i . '  x N q Ž d i . v F m q wL0 y c Ž d i . , x n F 1 y L0 y t Ž d i . 4 B* '  x N q Ž d j . v F m q wL0 y c Ž d i . q d , x n F 1 y L0 y t Ž d i . 4 B k '  x N q Ž d j . v F m q wL0 y c Ž d j . q d , x n F 1 y L0 y t Ž d j . 4 B Ž d j . '  x N q Ž d j . v F m q wL0 y c Ž d j . , x n F 1 y L0 y t Ž d j . 4 for which y i, y j g H, d j - d i , and x i g hŽ d i ., x j g hŽ d j ., x* g h*, and x k g B k . Notice that B* is the compensated set for x i, so that Theorem 6 implies x*1qF x 1i .

Ž 35 .

By construction, the only difference between B k and B* is the leisure time constraint; available leisure time is greater under B k than B*. The condition that housing and leisure are substitutes or unrelated goods yields x 1kq F xUq 1 .

Ž 36 .

Now show that the net income in B k exceeds that in B Ž d j .. The compensation is d s wqŽ d j . y qŽ d i .xv i. However, x k g h k « qŽ d j .v k s m q wL0 y cŽ d i . q d and x i g hŽ d i . « qŽ d i .v i s m q wL0 y cŽ d i . so that m q wL0 y c Ž d i . q d s q Ž d i . v i q d s q Ž d j . v i ,

Ž 37 .

where the second equality follows from the construction of d . Condition Ž29. in the proof of Theorem 6, however, leaves q Ž d j . v i G m q wL0 y c Ž d j .

Ž 38 .

which, with Ž37. requires m q wL0 y c Ž d j . F m q wL0 y c Ž d i . q d .

Ž 39 .

What Ž39. establishes is that the only difference between B Ž d j . and B k is the amount of net income available for goods consumption; the net income in B k will be at least as great as in B Ž d j .. Normality of housing therefore implies. x 1jqF x 1kq

Ž 40 .

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GEOFFREY K. TURNBULL

The definition x 1j F x 1jq, with Ž35., Ž36., and Ž40., yields x 1j F x 1i , which is the theorem. Q. E. D. B. Spatial Separation of Households by Type Recall that the analysis of spatial pooling and separation possibilities hinges critically on condition Ž5., which in turn follows from lemma 1. Unfortunately, because the lemma no longer holds when we introduce revealed preference for distance as we have in this section, no pooling or separation results can be obtained here unless we impose further restrictions on the model. In Alonso’s w3x utility theory with preference for travel distance, separation and pooling results can be derived only by further specifying the marginal disutility of commuting and other critical parameter values w11x. Thus, the inability to establish separation results in the general revealed preference framework used here is not too surprising. V. CONCLUDING REMARKS This paper presents a revealed preference theory of the urban spatial housing market. The model assumes a large finite economy with land parcels of positive area, thereby circumventing the problems associated with the calculus based continuum of consumers model wherein population and structural densities are defined over land parcels of measure zero. The revealed preference approach is simple yet robust; compared with the standard utility function approach, the model is very general, with multiple goods, multiple consumer types, and labor-leisure choice all incorporated into the version studied here. This represents a clear advantage over the utility function approach, since the utility function models often become intractable even when minor extensions are attempted. Perhaps more importantly, the revealed preference model formally demonstrates that many of the standard predictions of utility theory do pertain to large finite economies, thereby providing some level of comfort with the ad hoc interpretation of continuum models as approximating such economies. Finally, in a different vein, the results derived in the paper provide a foundation for nonparametric empirical tests of the residential location and land use theory. The existing literature testing urban land use theory focuses on the implications for housing prices, commuting patterns, spatial separation, etc. w16, 24, 33, 34x. None of the tests directly address the utility maximization assumption. Important results by Afriat w1, 2x establish that the utility maximization assumption can be tested directly by evaluating the conformance of the data with the nonspatial version of GARP.8 The Varian w29, 31x provides results making Afriat’s theorem empirically operational. See DeBoer w13x and Chalfant and Alston w12x for specific applications in the nonspatial environment. 8

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spatial version of GARP employed here is closely related to the nonspatial version. This implies that, at least in principle, Afriat’s results for the nonspatial economy can be extended to the spatial environment to test the utility maximization assumption in the theory of location choice. More work is needed to make the approach operational, but such nonparametric tests will add a new dimension to the growing literature concerned with testing the neoclassical urban theory.

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