- Email: [email protected]
A generalization of residential location theory
Regional Science and Urban Economics 7 (1977) 251-266. © North-Holland
A GENERALIZATION OF RESIDENTIAL LOCATION THEORY* Harry W. RICHARDSON University of Southern Cahfornia, los Angeles, CA 90007, U.S.A. Received December 1975, final version received September 1976 This paper offers a more ger'~ral theory of residential location as an alternative ~o the 'tradeoff' model. The theory takes acceant of m u h i # e employment centers, time as wel~ as income constraints, environmental quality, non-work trips, local jurisdictions, the restricted number of houses available for occupation at any time and the constraints on choice imposed by the search process. The result is a very irregular bid rent surface. Its implications for the aggregate rent surface are discussed.
1. I n t r o d u c t i o n
The 'trade-off' theory of residential location, namely the idea that households choose where to live by trading-off CBD accessibility against mere space, is the micro-behavioral analogue of the standard monocentric urban land use model. Although there have been some criticisms of the trade-off model, it has remained remarkably resilient. But it has served its purpose and is now an obstacle to progress in urban economics. Its underlying assumptions (a single employment center, no locationai externalities - or spatially invariant externalities, transport cost savings as the sole determinant of location rent, ~ the neglect of information and search costs in finding a house, and the fact that many urban residential sites are occupied) make the trade-off model a very special ease that is of little help ir~ l, lderstanding residential !ocation behavior in a modern city. The modest a~m of this paper is to begin the development of a more general, and hopefully ri, her, model. If peripheral locations are ~onsidered desirable for reasons external to the land market (i.e., regardles~ of rent leve~s and plot size,'O, the door is open to much more complex residendali location decisions than implied in the trade-~fl" model. Introducing public goods and e×ternalities into residential loc~tion ®Thanks are due to an anonymous refer~ for very helpful criticisms. 1In general, this paper u ~ s P e term 'rent', even though the model developed is a buyer mode|. This is appropriate since mo,~t of the discussion focuses on housing cxpendi~urc~ rather than h o u ~ prices.
252
1t".IV. Richardson, Reside:~tiallocation theory
theory is an important step in its generalization. In addition, the marginal utility of distance in the standard model refers to distance from the CBD of the metropolitan area. In a multicentric model a distant location may offer both closer accessibility to jobs (at one of the subcenters) and escape from central city congestion. From the perspective and needs of some households, a peripheral location may offer both more access and more space. A serious limitation of the standard model is that it implicitly assumes that the household is free to locate anywhere. All households are assumed to be locating simultaneously and inslantaneously. In fact, only a small minority of households (movers) choose a site at any one time and only a small proportion of sites (vacant-including new-houses and lots) are available at that time. Moreover, the search process is constrained by limit¢ 4 time, a restricted budget and imperfect information, hard facts which force the houseseeker to narrow down the spatial focus of his search area very quickly. These considerations may force the residential locater to cheese a second-bes~ site, a decision which is unlikely to be rever~_~-dquickly because of locational inertia and (especially for owner occupiers) high relocation costs. The fact that t],e standard model usually treats housing demand as a demand for land is not pa~-ficularly serious, since housing characteristics may easily be included within the c~,mposite commodity set [Papageorgiou (1976)], particularly if construction costs can be assumed invariant with distance, while most households have relatively fixed house type preferences in mind so that the location model, in effect, standardizes for hous',ag characteristics. On the other hand, ff certain house types exist only at a few locations, this may become a severe constraint on the residential location decision (for instance, if the demand for new houses is income-elastic and if new houses are built only on the urban periphery). 2. Precedents Degpite the near monopoly of the standard model, there have been some hints of how a less restrictive theory of residential location might be developed. Ellis (1967) emphasized the importance of environmental attd neighborhood characteristics in residential location decisions and viewed the housing market as a process in which the heterogeneous preferences of households are matched up with the characteristics of the available stock. Stegman (1969) showed that accessibility could no longer be considered in unidimensional terms as minimizing distance from the CBD. Senior and Wilson (1974) similarly demonstlated that the residence-workplace nexus need not depend on the relationship bctween the CBD and radial locations, but could assume a cris~;-cross pattera between any pair of zones. Also they pointed ol~t the obvi,)us but longneglected fact that for some households (i.e., retired househok,~s) access to a workplace does not matter al all. Beckmann (1973) made a few c.oneessions to the staadard mo~el by allowing
for d
&&ed emp~o~ent (though distributed in pr oportion to p~J~u~ati0~ density), non-work trips and differences in family size core $evastatiq to the chances of survival of this model wab the introduction of public gA)cds aad choice of fiscal jurisdiction into residattial location decisions [Ellicks%~!(197]), O&S (196% BAR (1973)f. When public goods and other extern,~!ities are introduced, the familiar predictions ef negative rental (land value) and wage gradients @ve way to much more irre lar spatial patterns [an implication of the models of Polinsky and ShatjeU (1 ), and Polinsky and Rubinfgld (1974)]. Yamada (1972) also gave explicit atten to the role of environmental externalities in the residential site choice, but unnecessarily restricted the scope of the amxlysis by ussunling negative rent and density gradients and by makmg environmental quality a function of density and distance. Once irregular spatA variations in rents and densitics are permited, the problem of reconciling household equilibrium and overall macrospatial equilibrium may become seriorls. A possible solution, suggested by Ingram et al. (1972), is to argue that the housing market is always in a disequilibrium state, moving towards but never reaching an equilibrium. This kind of modification relieves the pressure on models of individual household location decisions; in particular, they no longer have to be so simple as to make it feasible to derive aggregate equilibrium conditiors. As a result, the compromise between simplirity ana realism may be allowed to shift a little in favor of the latter. There have been several recent papers that have stressed the need for ;I wider concept of urban rent than merely treating it as 1ocationaP rent (or even more narrowly, as CBD-accessibility rent). I,ittle (1374) suggested four dimensions to residential location (structure characteristics, accessibility, neighborhood, and local government jurisdiction) and argues th,,t these should be handled simulraneously. Particular attention was given to the desire of most famil’es to live in homogeneous neighborhoods [a point echoed by Kirwan and Bal’ (1973)J. Richardson (1977) has shown how introducing environmental externa.ities into t:le sundard model may generate a positive rent gradient’ while Ams(+n (1974), with a subsequent extension by Richardson (1975), demonstrated that minor cdganges in assumptions (specifically, a preference for low densiti:s and a minimum space requirement constraints) can lead to discontinuous density nctions (and by implication disconti Papageorgiou (19’76)has moved fu om the assumptic ns of the
254
H.W. Richardson, Residential location ttzee "y
standard model. The most drastic of his revisions ate the introduction of multicentric spatial structures (in particular, an intraurb~m hierarchy of ~nters on Christaller-L~Ssch lines) and a residential attractiveness variable, embracing housing quality, social prestige of neighborhvods and other factors, l'he multicentric environment generates irregular rent gradients while residential attractiveness leads to further irregularities, especially as it is adversely affe.-ted by proximity to city centers. In many sets of circumstances the location deci fion may be indeterminate; for example, the bid rent function of an indivitual household may coincide with the aggregate rent surface at more than on~ p ~int. In such a case, the residential location decision will be discontinuous even under the assumption of continuous space. The choice between two or more spatially separated residential sites may be determined by random factors- such a.~ the subjective price flexibility of the seller.
3. A generalized residential location model A more broadly defined theory of residential location needs to take account of the following considerations: (1) Housing tastes and preferences differ among household types, stratified by income, family size, stage of family life cycle, social class, race, etc. Here it is assumed that there are many household groups (i = 1). (2) Location r is an argument in the utility func',tion. Location is two dimensional so that the household may choose one cf several workplac,;s (2 = 1) in addition to the CBD. Also, the household interacts with ~ .weral other locations (j = 1) for shopping, school trips, leisvre, etc. However, locations are desirable not for their own sake but for their attributes, including accessibility and environmental qualities. 4 (3) Time is a scarce resource to be allocated, much the same way as income. Leisure is assumed to enter the utility function. Working time is also assumed to be variable, though the special (thovgh common) case of fixed working hours can be easily analyzed. (4) A distinction is drawn between the price of land and the price ofthe structure. If a house is a bundle of utility-generating characteristics, it can be broken down via use of a hedonic price index into a set of dwelling ch~:racteristics each with its own price. These prices are assr,med to be invar,:~nt over space [it seems reasonable to assume little variation in construction costs with a metropolitan region; but see Muth (1969 pp. 46-69)]. *if these attributes are correctly and fully specified, location per se makes no separate contribution to utility. Otherwise, it acts as a residual picking up unspecified locational characteristics.
H. 14". R~chardson, Residential location theory
2.55
(5) The location decision involves not only choosing a house but also an area in which to live. Although land prices and travel costs are elements in this choice, neighborhood and environmental characteristics are also important. These include public services (and taxes), air quality, the quality of schools, the average density of the neighborhood, and social or ethnic homogeneity (or, since preferences vary, heterogeneity). (6) The weights in the utility function differ" between buyers and renters. Buyers :ire also subject to an additional constraint, the availability of capital. The model explicitly discussed is a buyer model, though with ::ertain obvious differences it is applicable to renters. (7) The residential location decision is severely constrained by the relatively small proportion of houses vacant, ih: time available for the ~earch process, the relatively narrow spatial limi~ s to the search field and the costs of searching. The result is that the actual residential choice is much more restricted than if the total housing stock were axailable or ir transaction costs were zero. (8) Income-earning opportunities vary over space. Wage rates may vary among employment centers (even for the same occupational group), while wage income may also change becau.~e of variable working hours. A model reflecting these attributes is easy to develop, though it is not as readily manipulable mathematically as the standard trade-off model. The residential locator maximizes his utility function subject to constraints in the usual way. However, the utility function is more complicated than usual, and the constraints are of two different types, tone set refers to resource constraints (income, time and capital), while the other set refers to the searcil process. The resource constraints determine ,~here and 2: ";,hat type of housing the household would like to live given its utility function, income and access to capital. The search process constraints determine the housing opportunities available to the household, how it responds to these constraints, and hence what the household will settle for within a limited time period. Even if the chosei: location is less than ide~ I, it will tend to bc an equilibrium location because of the existence of relocatioa costs (buying and selling costs for owners, leasing restrictions and costs for renters). 'rhe r,:~idential site choice involves maximizing a utility function of the followinj type: .. L, {.~,, M a x U = U[c, r, s, .., ~ ~,,,.., 2 .~,,),(qt , q 2 , ..., q=}],
{l}
y ~ - ( l + g,}pJ, - C~, - T~Pdr >_ O,
{2)
subj{~*c t 0
6 i
(3}
H.W. Richardson, Residential location theory
D
D,,
(4)
Di + F i ~_ Ai-A N-Wi--T~-L
*i, ~ ~_ O,
t ~ ~_ (A i - [ A ' i + Dt])~ l,
(5) (6)
(7)
where c r s z
ql
,
composite consumption, distance from CBD, housing space, workplace (measured as distance from the CBD; if the workplace is the CBD z drops out of the utility function), y~ = income received by household i living at r and working at z, L l = leisure of household i, set of neighborhood and environmental characteristics as:~ociated with site r, set of dwelling characteristics, property tax rate (i.e., local property taxes as a fraction of house price), gr = c',= consumption expenditure of household i located at r, T~M - total money costs of travel of household i at r, e i = : price of house purchased by i, D i = housing deposit (downpayment) made available by i, Op downpayment required by lender at location r, according to the risks of the neighborhood, K i = capitalization factor, a variable determined by the rate of interest, the period of the mortgage loan and the credit rating of i, l = search costs per hour for household i, E F i = search costs made available by i, expenditure on housing of i at r, Ai= assets of i, A *i -_ minimum reserve assets of i, N = total time available (net of sleeping time), W i = working time of i, time spent on travel by household i.
....
= := = =
?It
In addition, tbr completeness there are the foUowing definitional equations: = PL.S
, Pil.,
H . W . Pdcluu,dson, R e s i d e n t i a l location t h e o r y
2.¢7
g, = a,b,,
~1~9
c; = t, oc;,
(1 ])
r~,, = X n,J,~d~,s ' ' + 2f,~d!~,
(12)
J yt/z =
i l + y~.w, W~w,
(] 3)
where rent of land at location r, P~n, = expenditure of household i on s~ructure, price of dwelling characteristic m converted to a rent payment per period, a , = tax rate per dollar of assessed value at r, b r --- assessment-value ratio at r, distance between i's location and destination j, d',,- distance between i's location and workplace z, i lit1 --'- number of non-work trips to destination j, /rS = transport cost per unit of distance between r and j, f , : = transport rate between home and workplace, ),,~= wage rate of i at workplace z, .)'Nw = non-wage income of h,,a.iehold i. p/.,. =
The time and space search process constraints migl~t take the following form:
where t~= dt = ht = H = Vt, = AH,, = H~'=
H,, < E , H + A H , , ,
(14)
t i ~_ dih i,
(15)
H~' ~ R'H,,,
(16)
HI, ~ It,~,' c H,, c tl,
(17~
time available for household i's search, days available, average search time (numb~~ ~,~ hour:.) per day, housing stock of metropoi~ap, area, vacancy rate over ~eriod #, net addition to housing stock in period t ~, number of houses av~lable in spatial search field of household i during period t~,
258
H. W. Ri~ardson, Residemial location theory
R ~ = parameter of the search field, determining the proportion of available houses (H,,) considered by household, H~ = selected house and site of household i. These equations summarize the model. Eq. (2) is the budget constraint, similar to that employed in the standard trade-off model. The definitional equations (8)-(13) show how it differs from the standard case. As stated in (8), housing expenditures are divided into expenditures on land and on the dwelling, while dwelling price is merely the aggregation of price x quantity of separable housing characteristics (9). The property tax rate (g,), as given by (10), is the result of local government decision. It is a necessary element in the budget constraint when the residential location decision includes the possibility of a choice of local tax jurisdictions. Eq. (11) shows that household consumption varies over space (in addition to the fact that consumption is a function of income), though the f.o.b, price of composite consumption varies with location, partly because of substitution between goods and housing as their relative prices change, partly because changes in location alter the costs of acquiring goods (the money and time costs involved in shopping trips). In general, the more decentralized the city and the larger the number of subcenters, the smaller the spatial variation in these shopping costs. Eq. (12) is the aggregate travel cost function, allowing fcr a flexible number of nonwork trips to a variety of locations and assuming that the household contains one worker making a trip to and from work each day. Since the workplace need not be constrained to the ' C B D while the non-work trips may radiate out in all directions from the home size, the travel cost equation (12) is one of the major consequences of relaxing tlae monocentric city assumption. Fq. (13) shows how income-earning opport~:nities vary with location. For a given workplace working hours (I4;) will tend to vary with the length of the worktrip. The wage rate (w.) changes from one workplace to another. For example, if the employment centers are arranged in a size hierarchy, the wage rate may be positively related to the order of the hierarchy due to scale economy effects. Generalizing the wage gradient concept [Moses (1962)] to two-dimensional space, there will be a wage surface due to the fact that commuting costs give employers in low-order centers a degree of protection against wage equalization. The house price constraint (3) is important for housebuyers, since a housing deposit is the entry price to the buyer segment of the housing market while access to mortgage capital determines how much a household can spend on housing. However, since the deposit is frequently the result of past savings, ~hile the size of the mortgage granted tends to be linearly related to income, the house price constraint is predominantly a wealth and income constraint in disguise. Constraint (4) reflects the restriction placed by leaders on the mortgage loan/value ratio due to perceived variation in the riskiness of neighborhoods
IT. IV. Richardson, Residential location theory
259
over space. Constraint (5) shows that both the downpaymer, t and money search costs will be limited by the household's wealth and reserve needs. The time constraint (6) is introduced because leisure confers utility and because working time (and hence income) is variable. The allocation of" time is an important ingredient in the housing decision because substitution of work for leisure increases the ability to spen~l more on housing while the substitution of travel time for leisure widens the spatial range of the site choice. Constraint (7) shows that search time is limited by the allocation from savings for the search and by search costs per unit of time. Constraints (14)--(16) refer to the limitations of time and space on the search process. A potential buyer has, from (7), onl) a limited time available (t ~) to find a house, if the search process fails, he will shift into the rental market. Potential renters usually have a much tighter time constraiL,, and will take second- or third.-best choices if necessary. The stock of hou, ing available for occupation in period t ~is limited to the number of vacant houses on the market during that period plus net additions (new houses minus demolitions) to the stock in t ~. Con,traint (16) reflects the fact tbat because of limited time and/or area preferences, houseseekers will limit the areal extent of ,heir search to a very few neighborhoods within the metropolitan ar~.'a that match up with their x" preferences (some of which will be functionally related to income). If the workplace z has already been chosen, the houses Hf; will ali fall within commuting limits from z; if the workplace is to be selected after the residet~tiai site r, the spatial search area may be much wider. It is important to not~. that the areas within the search field need not be continuous. As shown by (17), the effect of the search process constraints is that the choice available to a residential locator is much narrower than the urban housing stock. Given the narrowness of choice and the spatial discontinuities in available supply, the 'optimal' choice is likely to be a second-best choice. However, the search process constraints are different in kind from the resovrce constraints. The latter determine housing expenditure, house and neighborhood type, locationai choices, i.e., the main parameters of the residential location decision, whereas the search constraints suggest that the actual housing choice will deviate, though within a relatively small range, from the ideal ~ c a u s e of supply limitations. The closeness of the actual to the ideal housing choice will be : .~unction of the time available and the boundaries of the search field, though also subject to a strong stochastic influence. In addition, the theory of search models may easily oe integrated into the analysis allowing more empt~asis on search cost~ and (,n the imperfect knowledge of participants in the housing market. Although the house search was listed as one of the more ot ~ous applications for optim~d stopping rules in an early paper by McCall (1965;, the idea has never received serious attention from residential location theorists. One reason raay be that the a~sumptions of the standard stopping rule model (an infinite time horizor, no discounting, random
260
H. IV. Richardson, Residential location theory
sampling from a known distribution but with one sample drawn per tlme period, no recall, indifference towards risk, no adaptation to new information, and a static framework) are too restrictive for analysis of the house-search process. However, it is not difficult to relax these assumptions as shown in studien of labor market behavior [see Lippman and McCall (1976a, b)]. In an elementary search model, the housebuyer would compare the expected utility from the last house seen, U] with the expected utility gained from continued searching according to the optimal stopping rule, U~. The optimal search policy is that the searcher should: Buy the house if
U,l > Ud,
Continue searching if
U,i < U~.
The ,,alue of U~ is, therefore, critical. In price-searching models its equivalent would be called the reservation price [Teiser (1973)], but in the house search context U~ represents the expected utility from the minimum acceptable house. This will be a function of price (p), location (r), neighborhood characteristics ( ~ , x ) and dwelling characteristics (~mq); thus U~ is directly related to characteristics of the individual's utility function. There is also an obvious relationship between search costs, F ~, and U~o; the higher these costs, the lower the value of U~. This permits a link between the search process and constraint (5), while yet another connection could be made by introducing F ~(negatively) as an argument of the utility function. The value of U~ is also affected by the nature and conditions of the search process. In this standard model Uo~ is the critical value associated with the o?,!mal stopping rule and it is chosen so as to equate the additional search costs incurred with the expected marginal return from looking at one more house; this allows the housebuyer to behave myopically and optimally by a sequer.tizl search. Thus, the virtue of this simple model, and the critical value concept, is that the optimal solution is obtained not by comparing the net bene~.s of stopping with those of continuing the search for an indefinite time but merely with those derived by looking at the next house on the viewing list. Realism may be introduced into the model in several ways. These include: (1) Discounting: The expected utility gained from the selected house is discounted over a much longer time horizon than the search costs. Thus, discounting tends to curtail the search period. (2) The time horizon is finite. This reduces U~. (3) Sampling with recall (i.e., houses seen in the past remain among the options to be chosen) is more appropriate than the sampling without recall (i.e., take it or leave it now) assumption of the basic model. Recall tends to increase U~. However, recall in the housing market is less than perfect, as ~ome past opportunities are sold to other buyers.
It. W. Richardson, Residential location theory
261
(4) In a dynamic economy, Uo~ will vary over time under the influence of a myriad of factors, especially the relative rates of change in house prices and incomes. (5) Also, the number of houses viewed may be unevenly distributed ove- ~ime, partly because of fluctuations in the intensity of the search, partly because of variation in the rate of new listings. As a result, the costs per ~,using opportunity are variable. (6) The risk-averse buyer has a lower U~, and stops the search proce~, s,~oner. On the other hand, the sampling with recall model provides a tJ~-~.ree of insurance since the best house seen in the past provides some protection against an unsuccessful search. (7) As implied in constraint (5), savings or ~ealth influence the search proc.'ss. Assets decline as the search process continues, and this i~lduces the ~earcher to accept a lower U~. (8) The house-search process is far from the random search assumed in the standard model. It is an adaptive process where each house seen ! a new piece of information used to update knowledge of the available properties and to revise U~. More particularly, it is reasonable to assume that the search process is systematic, starting from the apparently best opportunities and working down. In consequence, U~ tends to decline as the search continues. The upshot of this analysis is clear. Knowledge ls imperfect, information is costly, the future is uncertain, time is limited and search funds are scarce. An optimal search policy under these conditions may lead a residential locator to buy a house that confers lower expected utility than would be derived from the house chosen in the traditional zero-transaction costs model.
4. Loc~tioml equilibrium This model is much more complicated than the standard model, both because of its more complex ,.ttility function and lhe additional constraints. It is doubtful whether a general e, tuilibrium version ,:f this model can be solved, but in any event there are som,: sound reasons why a general equilibrium approach should be broadened to include non-residential location behavior, while in this model s,~erai important, ariables such as prices and wage rates have been assumed exogenous. In oth,:r words, this model would need considerable extension to make it a truly ge~leral equilibrium model, while it requires simplific; ~ion and r~stric, ion to obt:~ia a mathematical solution. Accordingly, it is wvser t~ a~sume that the individual residential locator faces a given - but imperfectly perceived lent (and land valt~e) surface, subject to the qualification that this rent.~l surface ~eflects, in part, past locational behavior of households with simi1~r utility ';unctions to thos~ outlined in the model. This means that the aggreg:~te renta!i
H. IV. Richardson,Residentiallocation theory
262
surfac~ may have irregularities and discontinuities of a kind not encountered in the standard monocentric model (tLe latter either generates or assumes a nicely smooth, differentiable and cc.ntinuous negative rent gradient in onedimen~gional space). Despite the complications of the model, it is possible to derive certain marginal conditions, e.g., substitutions between housing, other goods and land or between work and leisure. However, the critical utility maximizatign condition refers to locational equilibrium, and this is the only one to be discussed here. This condition may be stated (with superscripts i omitted) as t
¢
I
t
¢
~
+
f
B
F
[W;w.+ W,w'.+ W;wA+p c,+ 2 p , . q " +[p,. . pL.S,+PL, ,I m
+ ~ v~,.d"+ T/a+ vrT~, + VLL', = O, ~ n
where v~, vr and vL are the values (shadow prices) of neighborhood characteristic n, travel timeand leisure, respectively. From (18), the slf,p~ of the individual's bid-rent function (d/3,./dr) is given by the equation
~PL
I
1
Or "
t'wO W Wa~, OWOw\ +T"
Os +PLT,+ E
Oxn f~TM TT,+ -fTr
Oc+Ep,- o¢' nl
OTN
OL1 T;r "
(19)
Assuming that s > 0, the slope of the bid-rent function will be negative only if the sum of the terms in the square brackets is positive. In fact, the sign of OPL/Or is ambiguous, and ~'ill llsually change ever space. Although what happens to :he component terms of eq. (19) with spatial movement is an empirical question, several comments are in order. The bidrent surface will vary among households according to income class, because t,~. and VL will tend to increase with income while it is common to argue in favor of a high income ela.~ticity of demand for space. To preserve some independence in the terms it is sensible to assume that marginal changes in income with shifts in location do not push the household from one income class to the next (i.e., assuming that tae changes in vx., VL and s with respect to income are discrete rather ~han cer, tinuous). Another problem related to independence of each term is the po,~,fibility that spatial variations in neighborhood quality may be internalized i,, the land or labor markets [Lind (1973), Freeman (1974), :SAssuming that the workplace has not been pr,-determined, the margina~ change in income .s given by (I4/,+W,') (w~+w..')-W,w~, with .:~,~wheld constant. Similarly, the marginal change in land expenditure is (PL,+PL,~ (S,+ ~,')- ~t,S,.
H.W. Richardson, Residentiallocation theory
263
Polinsky and Rubinfeld (1974)1. Although this has no effect on the individual's bid-rent surface as given by (19), the impact on his locational choice may be different if exte ~L.', ;. "~ifferentials are reflected in wages compared with when they are internal zed in rents. The full results could be known only by solving a simultan~.ous I ,odel of general equilibrium for the metropolitan area as a whole. Other elements in (19) present less trouble. SimlDier versions might allow deletion of the Oc/gr and Oq'/Or terms on the assumption that consumption of goods and housing structure preferences remain constant over space. However, this s t e p - possibly justifiable in an empirical analysis - i s restrictive because a change in location alters accessibilit.~ to consumption (via substitution possibihties between travel costs, T M an~' Tn, and consumption) while housing structure preferences may, despite tl~ high stability of housing characteristics quantities in the individual's utility function, change slightly over space due to the extreme heterogeneity of housing and possii~ie trade-offs between housing structure chartcteristics and neighborhood quality. Although wage rates will be higher in tfe major employmen I centers (ahd strongly positively correlated with the size of the center), the wage rate surface will in most metropolitan areas offer each household some choice in workplaces (abstracting from the restrictions arising from heterogeneous skills and specialization of labor among centers). However, a household may not be willing to pay more tot a residence close to a high-order center if it values residential neighborhood quality highly; travel costs and neighborhood quality will be negatively correlated in the 'normal' case since increasing distance from a major employment center raises e~vironmental quality. Since the sirras ,,t t3W, Or, O~/Pr and c3Lfl~r cannot all be positive (or negative), the net impact of "he allocation of time on bid-reins will depend on the relationship between the wage rat~: and values of travel time and of leisure. Although environmental quality may be directly related to distance from the CBD aad other major subcenters, its variation over space will be irregular influenced by considerations such as topography, neighborhood densities and social stav,s in addition to proximity to non-residential activities. The net result of all these influences is that the household's bid-~ent surface may be very irregular due to the resolution of complex trade-oils between income and leisure, accessibility and neighborhood quality, time and ~o,dential space. What role does the search process constraints play in this moclel? As suggested b~ ~,14~~,(17) and the discussion of optimal stopping, these constrain|s imply that the ho~i,,~ choices facing the individual household are severely restricted by limitations of space, time, information and cost. The search process constr~.i~ts (14}-(17) can be most easily dealt with by treating them as an influence o~,, the aggregate rental surface. Viewed in this light, they do not affect the bid-rent surface but have drastic consequences for the a~:~regatc residential land rent surface. The relevant aggregate rent surface does not
264
H.W. Richardson, Residential location theory
cover the metropolitan area continuously but degenerates into a few small patches (that coincide with the spatial search field). Also, even within these patches the rental opportunities will nc,t be continuous but dependent on available vacant houses (an exception is perhaps where the household is looking for a new house, and where these are developed in large estates). The effect of introducing the optimal stopping rule is to lower the utility obtained from the residential choice, reflecting the fact that searching is costly. The implications of a discontinuous (and irregular) aggregate land value surface and the irregular bid-rent surface for locational equilibrium are farreaching. The aggregate rent surface may coincide with the bid-rent surface at more than one location (multiple solutions), or it may coincide only with a much higher bid-rent surface (implying lower utility) than might have been expected had all locations been available (unlimited time and space for the search), or it may not coincide with any feasible bid-rent function (obtained by introducing subsistence constraints on consumption). If the household finds the land rent it would have to pay unacceptably high, it may reverse the decision to locate in the city, or extend its search, or temporarily move into the house rental market (presumably at lower levels of housing consumption). A potential problem with this model is how to reconcile individual locational equilibrium with the market determination of house prices. Individual households not only adjust to a ~,~ven house price (and rent) surface, but considered as a whole they help to determine that surface. A virtue of the most elementary form of the standard model is that its simple assumptions, including reducing the demand for housing to a demand for land and assuming homogeneity of tastes, ailow the macrospatial equilibrium rent and density gradients to be determined at the same time as the conditioas for household equilibrium. The highly irregular and diverse bid-rent surfaces of heterogeneous households in the model ol:tlined here would have to be known in order to build up the aggregate rent surface as the outer envelope of all these bid-rents. it may be possible, however, to sidestep this apparent difficulty. First, there is no strong reason apart from analytical convenience why it should be assumed that housing and land markets are ever in equilibrium. The urban spatial structure is continually changing as the spatial area of the city extends, as land is converted from one use to another, and as demolition and replacement takes place. Since urban structures are durable, the concept of a stock adjustment model is more plausible than that of static long-run equilibrium. Similarly, the results of the Koopmans-Beckmann (!957) assignmeitt problem remain relevant [but see the ad hoc exceptions of Hartwick (19747]: that because of tocational interdependencies a stable optimizing spatial equili~rium assignment of locators may not be sustainable, and hence the concept of an equilibrium rent surface may be invalid. Second, even if a general equilibrium were possible, it could only be developed wi~.h a comprehensive rnedel including all types of spatial activities including
H.W. Richardson, Residential location theory
265
non-residential land uses. Building up an aggregate equilibrium out of a residential location model alone requires exogenous specification of mine sections (or points) on the aggregate rent surface. In fact, the standard model even does this in a limited way by exogenously locating all non-residential activities in the CBD and by using rent at the edge of the CBD as a constant of integration in determining hod rent at any location.
References Amson, J.C., 1974, Equilibrium and catastrophic modes of urban growth, in: E.L. Cripps, ed., Space-time concepts in urban and regional models (Pion, London) 108-! 28. Artle, R. and P. Varaiya, 1974, On the ex,stcnce of positive rent gradients in Thiinen models, Electronics Research Laboratory Memo. no. ERL-M459 (University o! California, Berkeley, CA). Barr, J.L., 1973, Tiebout models of community structure, Papers of the Regional Science Association 30, 113-139. Beckmann, M.J., 1973, Equilibrium models of residential location, Regional and Urban E~:onomics 3,361-368. Ellickson, B., 1971, Jurisdictional fragmentation and residential choice, American Economic Review (Papers) 61,334--339. Ellis, R.H., 1967. Modelling of household location: A statistical approach, Highway R e , a r c h Record 207, 42-51. Freeman, III, M.A., 1974, On the estimation of air pollution control benefits from land values studies, Journal of Environmental Economics and Management I, 74-83. Hart~ick, J. M., 1974, Price sustainability of location assignments, Journal of Urban Fconomics, !, 147-161. ingrain, G.K., i.F. Kain and J.R. Ginn, 1972, The Detroit prototype of the NBER urban simulation meal.el (Columbia University Press for the National Bureau of Economic Research, New York). Kirwan, R.S. and M.J. Ball, 1974. The microeconomic analysis of a local housing market, in: Centre for Environmental Studies, Papers from the Urban Economics Conference. vol. 1 (CES CP9, London). Koopmans, T.C. and M.J. Beckmann, 1957, Assignment problems in the location of economic activities, Econometrica 25, 53-76. Lind, R.C., 1973, Spatial equilibrium, the theory of rents and public program benefits, Quarterly Journal of Economics 87, 188-207. Lippman, S.A. and J.J. McCall, 1976a, Job ~arch in a dynamic economy, Journal of Economic Theory 12, 365-390. Lippman, S.A. and J.J. McCall, 1976b, The economics o! job ~ a ' .l,' . A survey, Economic Inquiry 14, 155-189 and 347-368. Little, J.T., 1974, Household preferences, reiocmtion and welfare: An evaluation of the filtering concept, Institute of Urban and Regional S.udies WP-HMS 2 (Washington University, St. Louis, WA). McCall, J.J., 1965, ~rhe economics of information and oplimal stopping rules, Journal of Business 38, 300--317. Moses, L.N., 1962, Towards a theory of intra-urban wage differentials and their infl'aence on travel patterns, Papers and Pr(vcecdings of ~he Regional Science Association o 53.63. Muth, R.F., 1969, Cities and housing (Chicago University Press, Chicago, IL~. Oates, W.E., 1969, The effects of prorerty tax~ and local public spending on property values An empirical study of the tax capitalization and the Tiebout hypothesL% Journal of Political Economy 77, 957-971. Papageorgiou, G.J., 197b, On spatial consumer equilibrium, in" G.J. Papageorgiou, ed,, Mathematical land use theory (Le~:ingmn Books, Le×ingtor,, M A) 145-176.
266
H. It: Kichardson, Residential 1,~catio, z theory
Polinsky, A.M. and $. Shavell, 1973, Amenities and pr :~perty values in a general equilibrium model of an urban area, Urban Institute WP-1207- 5. Richardson, H.W., 1975, Discontinuous densities, urba a spatial structure and growth: A new approach, Land Economics 51,305-315. Richardson, H.W., 1977, On the possibility of posit tve rent gradients, Journal of Urban Economics 4, 60-68. Senior, M.L. and A.G. Wilson, 1974, Disaggrvgated r~ski©ntial location models: Some tvsts and further theoretical developments, in: E.L. Cril;ps, ~,1. Space-time concepts in urban and regional models (Pion, London) 141-172. Stegman, M.A., 1969, Accessibility models and resi(lential location, Journal of American Institute of Planners 35, 22-29. Telser, L.G. 1973, Searching for the lowest price, Americar~ Economic Review (Papers) 63, 40-49. Yamada, H., 1972, On the theory of residential location: Accessibility, space, leisure and environmental quality, Papers of the Regional Science Association 29, 125-135.
A GENERALIZATION OF RESIDENTIAL LOCATION THEORY* Harry W. RICHARDSON University of Southern Cahfornia, los Angeles, CA 90007, U.S.A. Received December 1975, final version received September 1976 This paper offers a more ger'~ral theory of residential location as an alternative ~o the 'tradeoff' model. The theory takes acceant of m u h i # e employment centers, time as wel~ as income constraints, environmental quality, non-work trips, local jurisdictions, the restricted number of houses available for occupation at any time and the constraints on choice imposed by the search process. The result is a very irregular bid rent surface. Its implications for the aggregate rent surface are discussed.
1. I n t r o d u c t i o n
The 'trade-off' theory of residential location, namely the idea that households choose where to live by trading-off CBD accessibility against mere space, is the micro-behavioral analogue of the standard monocentric urban land use model. Although there have been some criticisms of the trade-off model, it has remained remarkably resilient. But it has served its purpose and is now an obstacle to progress in urban economics. Its underlying assumptions (a single employment center, no locationai externalities - or spatially invariant externalities, transport cost savings as the sole determinant of location rent, ~ the neglect of information and search costs in finding a house, and the fact that many urban residential sites are occupied) make the trade-off model a very special ease that is of little help ir~ l, lderstanding residential !ocation behavior in a modern city. The modest a~m of this paper is to begin the development of a more general, and hopefully ri, her, model. If peripheral locations are ~onsidered desirable for reasons external to the land market (i.e., regardles~ of rent leve~s and plot size,'O, the door is open to much more complex residendali location decisions than implied in the trade-~fl" model. Introducing public goods and e×ternalities into residential loc~tion ®Thanks are due to an anonymous refer~ for very helpful criticisms. 1In general, this paper u ~ s P e term 'rent', even though the model developed is a buyer mode|. This is appropriate since mo,~t of the discussion focuses on housing cxpendi~urc~ rather than h o u ~ prices.
252
1t".IV. Richardson, Reside:~tiallocation theory
theory is an important step in its generalization. In addition, the marginal utility of distance in the standard model refers to distance from the CBD of the metropolitan area. In a multicentric model a distant location may offer both closer accessibility to jobs (at one of the subcenters) and escape from central city congestion. From the perspective and needs of some households, a peripheral location may offer both more access and more space. A serious limitation of the standard model is that it implicitly assumes that the household is free to locate anywhere. All households are assumed to be locating simultaneously and inslantaneously. In fact, only a small minority of households (movers) choose a site at any one time and only a small proportion of sites (vacant-including new-houses and lots) are available at that time. Moreover, the search process is constrained by limit¢ 4 time, a restricted budget and imperfect information, hard facts which force the houseseeker to narrow down the spatial focus of his search area very quickly. These considerations may force the residential locater to cheese a second-bes~ site, a decision which is unlikely to be rever~_~-dquickly because of locational inertia and (especially for owner occupiers) high relocation costs. The fact that t],e standard model usually treats housing demand as a demand for land is not pa~-ficularly serious, since housing characteristics may easily be included within the c~,mposite commodity set [Papageorgiou (1976)], particularly if construction costs can be assumed invariant with distance, while most households have relatively fixed house type preferences in mind so that the location model, in effect, standardizes for hous',ag characteristics. On the other hand, ff certain house types exist only at a few locations, this may become a severe constraint on the residential location decision (for instance, if the demand for new houses is income-elastic and if new houses are built only on the urban periphery). 2. Precedents Degpite the near monopoly of the standard model, there have been some hints of how a less restrictive theory of residential location might be developed. Ellis (1967) emphasized the importance of environmental attd neighborhood characteristics in residential location decisions and viewed the housing market as a process in which the heterogeneous preferences of households are matched up with the characteristics of the available stock. Stegman (1969) showed that accessibility could no longer be considered in unidimensional terms as minimizing distance from the CBD. Senior and Wilson (1974) similarly demonstlated that the residence-workplace nexus need not depend on the relationship bctween the CBD and radial locations, but could assume a cris~;-cross pattera between any pair of zones. Also they pointed ol~t the obvi,)us but longneglected fact that for some households (i.e., retired househok,~s) access to a workplace does not matter al all. Beckmann (1973) made a few c.oneessions to the staadard mo~el by allowing
for d
&&ed emp~o~ent (though distributed in pr oportion to p~J~u~ati0~ density), non-work trips and differences in family size core $evastatiq to the chances of survival of this model wab the introduction of public gA)cds aad choice of fiscal jurisdiction into residattial location decisions [Ellicks%~!(197]), O&S (196% BAR (1973)f. When public goods and other extern,~!ities are introduced, the familiar predictions ef negative rental (land value) and wage gradients @ve way to much more irre lar spatial patterns [an implication of the models of Polinsky and ShatjeU (1 ), and Polinsky and Rubinfgld (1974)]. Yamada (1972) also gave explicit atten to the role of environmental externalities in the residential site choice, but unnecessarily restricted the scope of the amxlysis by ussunling negative rent and density gradients and by makmg environmental quality a function of density and distance. Once irregular spatA variations in rents and densitics are permited, the problem of reconciling household equilibrium and overall macrospatial equilibrium may become seriorls. A possible solution, suggested by Ingram et al. (1972), is to argue that the housing market is always in a disequilibrium state, moving towards but never reaching an equilibrium. This kind of modification relieves the pressure on models of individual household location decisions; in particular, they no longer have to be so simple as to make it feasible to derive aggregate equilibrium conditiors. As a result, the compromise between simplirity ana realism may be allowed to shift a little in favor of the latter. There have been several recent papers that have stressed the need for ;I wider concept of urban rent than merely treating it as 1ocationaP rent (or even more narrowly, as CBD-accessibility rent). I,ittle (1374) suggested four dimensions to residential location (structure characteristics, accessibility, neighborhood, and local government jurisdiction) and argues th,,t these should be handled simulraneously. Particular attention was given to the desire of most famil’es to live in homogeneous neighborhoods [a point echoed by Kirwan and Bal’ (1973)J. Richardson (1977) has shown how introducing environmental externa.ities into t:le sundard model may generate a positive rent gradient’ while Ams(+n (1974), with a subsequent extension by Richardson (1975), demonstrated that minor cdganges in assumptions (specifically, a preference for low densiti:s and a minimum space requirement constraints) can lead to discontinuous density nctions (and by implication disconti Papageorgiou (19’76)has moved fu om the assumptic ns of the
254
H.W. Richardson, Residential location ttzee "y
standard model. The most drastic of his revisions ate the introduction of multicentric spatial structures (in particular, an intraurb~m hierarchy of ~nters on Christaller-L~Ssch lines) and a residential attractiveness variable, embracing housing quality, social prestige of neighborhvods and other factors, l'he multicentric environment generates irregular rent gradients while residential attractiveness leads to further irregularities, especially as it is adversely affe.-ted by proximity to city centers. In many sets of circumstances the location deci fion may be indeterminate; for example, the bid rent function of an indivitual household may coincide with the aggregate rent surface at more than on~ p ~int. In such a case, the residential location decision will be discontinuous even under the assumption of continuous space. The choice between two or more spatially separated residential sites may be determined by random factors- such a.~ the subjective price flexibility of the seller.
3. A generalized residential location model A more broadly defined theory of residential location needs to take account of the following considerations: (1) Housing tastes and preferences differ among household types, stratified by income, family size, stage of family life cycle, social class, race, etc. Here it is assumed that there are many household groups (i = 1). (2) Location r is an argument in the utility func',tion. Location is two dimensional so that the household may choose one cf several workplac,;s (2 = 1) in addition to the CBD. Also, the household interacts with ~ .weral other locations (j = 1) for shopping, school trips, leisvre, etc. However, locations are desirable not for their own sake but for their attributes, including accessibility and environmental qualities. 4 (3) Time is a scarce resource to be allocated, much the same way as income. Leisure is assumed to enter the utility function. Working time is also assumed to be variable, though the special (thovgh common) case of fixed working hours can be easily analyzed. (4) A distinction is drawn between the price of land and the price ofthe structure. If a house is a bundle of utility-generating characteristics, it can be broken down via use of a hedonic price index into a set of dwelling ch~:racteristics each with its own price. These prices are assr,med to be invar,:~nt over space [it seems reasonable to assume little variation in construction costs with a metropolitan region; but see Muth (1969 pp. 46-69)]. *if these attributes are correctly and fully specified, location per se makes no separate contribution to utility. Otherwise, it acts as a residual picking up unspecified locational characteristics.
H. 14". R~chardson, Residential location theory
2.55
(5) The location decision involves not only choosing a house but also an area in which to live. Although land prices and travel costs are elements in this choice, neighborhood and environmental characteristics are also important. These include public services (and taxes), air quality, the quality of schools, the average density of the neighborhood, and social or ethnic homogeneity (or, since preferences vary, heterogeneity). (6) The weights in the utility function differ" between buyers and renters. Buyers :ire also subject to an additional constraint, the availability of capital. The model explicitly discussed is a buyer model, though with ::ertain obvious differences it is applicable to renters. (7) The residential location decision is severely constrained by the relatively small proportion of houses vacant, ih: time available for the ~earch process, the relatively narrow spatial limi~ s to the search field and the costs of searching. The result is that the actual residential choice is much more restricted than if the total housing stock were axailable or ir transaction costs were zero. (8) Income-earning opportunities vary over space. Wage rates may vary among employment centers (even for the same occupational group), while wage income may also change becau.~e of variable working hours. A model reflecting these attributes is easy to develop, though it is not as readily manipulable mathematically as the standard trade-off model. The residential locator maximizes his utility function subject to constraints in the usual way. However, the utility function is more complicated than usual, and the constraints are of two different types, tone set refers to resource constraints (income, time and capital), while the other set refers to the searcil process. The resource constraints determine ,~here and 2: ";,hat type of housing the household would like to live given its utility function, income and access to capital. The search process constraints determine the housing opportunities available to the household, how it responds to these constraints, and hence what the household will settle for within a limited time period. Even if the chosei: location is less than ide~ I, it will tend to bc an equilibrium location because of the existence of relocatioa costs (buying and selling costs for owners, leasing restrictions and costs for renters). 'rhe r,:~idential site choice involves maximizing a utility function of the followinj type: .. L, {.~,, M a x U = U[c, r, s, .., ~ ~,,,.., 2 .~,,),(qt , q 2 , ..., q=}],
{l}
y ~ - ( l + g,}pJ, - C~, - T~Pdr >_ O,
{2)
subj{~*c t 0
6 i
(3}
H.W. Richardson, Residential location theory
D
D,,
(4)
Di + F i ~_ Ai-A N-Wi--T~-L
*i, ~ ~_ O,
t ~ ~_ (A i - [ A ' i + Dt])~ l,
(5) (6)
(7)
where c r s z
ql
,
composite consumption, distance from CBD, housing space, workplace (measured as distance from the CBD; if the workplace is the CBD z drops out of the utility function), y~ = income received by household i living at r and working at z, L l = leisure of household i, set of neighborhood and environmental characteristics as:~ociated with site r, set of dwelling characteristics, property tax rate (i.e., local property taxes as a fraction of house price), gr = c',= consumption expenditure of household i located at r, T~M - total money costs of travel of household i at r, e i = : price of house purchased by i, D i = housing deposit (downpayment) made available by i, Op downpayment required by lender at location r, according to the risks of the neighborhood, K i = capitalization factor, a variable determined by the rate of interest, the period of the mortgage loan and the credit rating of i, l = search costs per hour for household i, E F i = search costs made available by i, expenditure on housing of i at r, Ai= assets of i, A *i -_ minimum reserve assets of i, N = total time available (net of sleeping time), W i = working time of i, time spent on travel by household i.
....
= := = =
?It
In addition, tbr completeness there are the foUowing definitional equations: = PL.S
, Pil.,
H . W . Pdcluu,dson, R e s i d e n t i a l location t h e o r y
2.¢7
g, = a,b,,
~1~9
c; = t, oc;,
(1 ])
r~,, = X n,J,~d~,s ' ' + 2f,~d!~,
(12)
J yt/z =
i l + y~.w, W~w,
(] 3)
where rent of land at location r, P~n, = expenditure of household i on s~ructure, price of dwelling characteristic m converted to a rent payment per period, a , = tax rate per dollar of assessed value at r, b r --- assessment-value ratio at r, distance between i's location and destination j, d',,- distance between i's location and workplace z, i lit1 --'- number of non-work trips to destination j, /rS = transport cost per unit of distance between r and j, f , : = transport rate between home and workplace, ),,~= wage rate of i at workplace z, .)'Nw = non-wage income of h,,a.iehold i. p/.,. =
The time and space search process constraints migl~t take the following form:
where t~= dt = ht = H = Vt, = AH,, = H~'=
H,, < E , H + A H , , ,
(14)
t i ~_ dih i,
(15)
H~' ~ R'H,,,
(16)
HI, ~ It,~,' c H,, c tl,
(17~
time available for household i's search, days available, average search time (numb~~ ~,~ hour:.) per day, housing stock of metropoi~ap, area, vacancy rate over ~eriod #, net addition to housing stock in period t ~, number of houses av~lable in spatial search field of household i during period t~,
258
H. W. Ri~ardson, Residemial location theory
R ~ = parameter of the search field, determining the proportion of available houses (H,,) considered by household, H~ = selected house and site of household i. These equations summarize the model. Eq. (2) is the budget constraint, similar to that employed in the standard trade-off model. The definitional equations (8)-(13) show how it differs from the standard case. As stated in (8), housing expenditures are divided into expenditures on land and on the dwelling, while dwelling price is merely the aggregation of price x quantity of separable housing characteristics (9). The property tax rate (g,), as given by (10), is the result of local government decision. It is a necessary element in the budget constraint when the residential location decision includes the possibility of a choice of local tax jurisdictions. Eq. (11) shows that household consumption varies over space (in addition to the fact that consumption is a function of income), though the f.o.b, price of composite consumption varies with location, partly because of substitution between goods and housing as their relative prices change, partly because changes in location alter the costs of acquiring goods (the money and time costs involved in shopping trips). In general, the more decentralized the city and the larger the number of subcenters, the smaller the spatial variation in these shopping costs. Eq. (12) is the aggregate travel cost function, allowing fcr a flexible number of nonwork trips to a variety of locations and assuming that the household contains one worker making a trip to and from work each day. Since the workplace need not be constrained to the ' C B D while the non-work trips may radiate out in all directions from the home size, the travel cost equation (12) is one of the major consequences of relaxing tlae monocentric city assumption. Fq. (13) shows how income-earning opport~:nities vary with location. For a given workplace working hours (I4;) will tend to vary with the length of the worktrip. The wage rate (w.) changes from one workplace to another. For example, if the employment centers are arranged in a size hierarchy, the wage rate may be positively related to the order of the hierarchy due to scale economy effects. Generalizing the wage gradient concept [Moses (1962)] to two-dimensional space, there will be a wage surface due to the fact that commuting costs give employers in low-order centers a degree of protection against wage equalization. The house price constraint (3) is important for housebuyers, since a housing deposit is the entry price to the buyer segment of the housing market while access to mortgage capital determines how much a household can spend on housing. However, since the deposit is frequently the result of past savings, ~hile the size of the mortgage granted tends to be linearly related to income, the house price constraint is predominantly a wealth and income constraint in disguise. Constraint (4) reflects the restriction placed by leaders on the mortgage loan/value ratio due to perceived variation in the riskiness of neighborhoods
IT. IV. Richardson, Residential location theory
259
over space. Constraint (5) shows that both the downpaymer, t and money search costs will be limited by the household's wealth and reserve needs. The time constraint (6) is introduced because leisure confers utility and because working time (and hence income) is variable. The allocation of" time is an important ingredient in the housing decision because substitution of work for leisure increases the ability to spen~l more on housing while the substitution of travel time for leisure widens the spatial range of the site choice. Constraint (7) shows that search time is limited by the allocation from savings for the search and by search costs per unit of time. Constraints (14)--(16) refer to the limitations of time and space on the search process. A potential buyer has, from (7), onl) a limited time available (t ~) to find a house, if the search process fails, he will shift into the rental market. Potential renters usually have a much tighter time constraiL,, and will take second- or third.-best choices if necessary. The stock of hou, ing available for occupation in period t ~is limited to the number of vacant houses on the market during that period plus net additions (new houses minus demolitions) to the stock in t ~. Con,traint (16) reflects the fact tbat because of limited time and/or area preferences, houseseekers will limit the areal extent of ,heir search to a very few neighborhoods within the metropolitan ar~.'a that match up with their x" preferences (some of which will be functionally related to income). If the workplace z has already been chosen, the houses Hf; will ali fall within commuting limits from z; if the workplace is to be selected after the residet~tiai site r, the spatial search area may be much wider. It is important to not~. that the areas within the search field need not be continuous. As shown by (17), the effect of the search process constraints is that the choice available to a residential locator is much narrower than the urban housing stock. Given the narrowness of choice and the spatial discontinuities in available supply, the 'optimal' choice is likely to be a second-best choice. However, the search process constraints are different in kind from the resovrce constraints. The latter determine housing expenditure, house and neighborhood type, locationai choices, i.e., the main parameters of the residential location decision, whereas the search constraints suggest that the actual housing choice will deviate, though within a relatively small range, from the ideal ~ c a u s e of supply limitations. The closeness of the actual to the ideal housing choice will be : .~unction of the time available and the boundaries of the search field, though also subject to a strong stochastic influence. In addition, the theory of search models may easily oe integrated into the analysis allowing more empt~asis on search cost~ and (,n the imperfect knowledge of participants in the housing market. Although the house search was listed as one of the more ot ~ous applications for optim~d stopping rules in an early paper by McCall (1965;, the idea has never received serious attention from residential location theorists. One reason raay be that the a~sumptions of the standard stopping rule model (an infinite time horizor, no discounting, random
260
H. IV. Richardson, Residential location theory
sampling from a known distribution but with one sample drawn per tlme period, no recall, indifference towards risk, no adaptation to new information, and a static framework) are too restrictive for analysis of the house-search process. However, it is not difficult to relax these assumptions as shown in studien of labor market behavior [see Lippman and McCall (1976a, b)]. In an elementary search model, the housebuyer would compare the expected utility from the last house seen, U] with the expected utility gained from continued searching according to the optimal stopping rule, U~. The optimal search policy is that the searcher should: Buy the house if
U,l > Ud,
Continue searching if
U,i < U~.
The ,,alue of U~ is, therefore, critical. In price-searching models its equivalent would be called the reservation price [Teiser (1973)], but in the house search context U~ represents the expected utility from the minimum acceptable house. This will be a function of price (p), location (r), neighborhood characteristics ( ~ , x ) and dwelling characteristics (~mq); thus U~ is directly related to characteristics of the individual's utility function. There is also an obvious relationship between search costs, F ~, and U~o; the higher these costs, the lower the value of U~. This permits a link between the search process and constraint (5), while yet another connection could be made by introducing F ~(negatively) as an argument of the utility function. The value of U~ is also affected by the nature and conditions of the search process. In this standard model Uo~ is the critical value associated with the o?,!mal stopping rule and it is chosen so as to equate the additional search costs incurred with the expected marginal return from looking at one more house; this allows the housebuyer to behave myopically and optimally by a sequer.tizl search. Thus, the virtue of this simple model, and the critical value concept, is that the optimal solution is obtained not by comparing the net bene~.s of stopping with those of continuing the search for an indefinite time but merely with those derived by looking at the next house on the viewing list. Realism may be introduced into the model in several ways. These include: (1) Discounting: The expected utility gained from the selected house is discounted over a much longer time horizon than the search costs. Thus, discounting tends to curtail the search period. (2) The time horizon is finite. This reduces U~. (3) Sampling with recall (i.e., houses seen in the past remain among the options to be chosen) is more appropriate than the sampling without recall (i.e., take it or leave it now) assumption of the basic model. Recall tends to increase U~. However, recall in the housing market is less than perfect, as ~ome past opportunities are sold to other buyers.
It. W. Richardson, Residential location theory
261
(4) In a dynamic economy, Uo~ will vary over time under the influence of a myriad of factors, especially the relative rates of change in house prices and incomes. (5) Also, the number of houses viewed may be unevenly distributed ove- ~ime, partly because of fluctuations in the intensity of the search, partly because of variation in the rate of new listings. As a result, the costs per ~,using opportunity are variable. (6) The risk-averse buyer has a lower U~, and stops the search proce~, s,~oner. On the other hand, the sampling with recall model provides a tJ~-~.ree of insurance since the best house seen in the past provides some protection against an unsuccessful search. (7) As implied in constraint (5), savings or ~ealth influence the search proc.'ss. Assets decline as the search process continues, and this i~lduces the ~earcher to accept a lower U~. (8) The house-search process is far from the random search assumed in the standard model. It is an adaptive process where each house seen ! a new piece of information used to update knowledge of the available properties and to revise U~. More particularly, it is reasonable to assume that the search process is systematic, starting from the apparently best opportunities and working down. In consequence, U~ tends to decline as the search continues. The upshot of this analysis is clear. Knowledge ls imperfect, information is costly, the future is uncertain, time is limited and search funds are scarce. An optimal search policy under these conditions may lead a residential locator to buy a house that confers lower expected utility than would be derived from the house chosen in the traditional zero-transaction costs model.
4. Loc~tioml equilibrium This model is much more complicated than the standard model, both because of its more complex ,.ttility function and lhe additional constraints. It is doubtful whether a general e, tuilibrium version ,:f this model can be solved, but in any event there are som,: sound reasons why a general equilibrium approach should be broadened to include non-residential location behavior, while in this model s,~erai important, ariables such as prices and wage rates have been assumed exogenous. In oth,:r words, this model would need considerable extension to make it a truly ge~leral equilibrium model, while it requires simplific; ~ion and r~stric, ion to obt:~ia a mathematical solution. Accordingly, it is wvser t~ a~sume that the individual residential locator faces a given - but imperfectly perceived lent (and land valt~e) surface, subject to the qualification that this rent.~l surface ~eflects, in part, past locational behavior of households with simi1~r utility ';unctions to thos~ outlined in the model. This means that the aggreg:~te renta!i
H. IV. Richardson,Residentiallocation theory
262
surfac~ may have irregularities and discontinuities of a kind not encountered in the standard monocentric model (tLe latter either generates or assumes a nicely smooth, differentiable and cc.ntinuous negative rent gradient in onedimen~gional space). Despite the complications of the model, it is possible to derive certain marginal conditions, e.g., substitutions between housing, other goods and land or between work and leisure. However, the critical utility maximizatign condition refers to locational equilibrium, and this is the only one to be discussed here. This condition may be stated (with superscripts i omitted) as t
¢
I
t
¢
~
+
f
B
F
[W;w.+ W,w'.+ W;wA+p c,+ 2 p , . q " +[p,. . pL.S,+PL, ,I m
+ ~ v~,.d"+ T/a+ vrT~, + VLL', = O, ~ n
where v~, vr and vL are the values (shadow prices) of neighborhood characteristic n, travel timeand leisure, respectively. From (18), the slf,p~ of the individual's bid-rent function (d/3,./dr) is given by the equation
~PL
I
1
Or "
t'wO W Wa~, OWOw\ +T"
Os +PLT,+ E
Oxn f~TM TT,+ -fTr
Oc+Ep,- o¢' nl
OTN
OL1 T;r "
(19)
Assuming that s > 0, the slope of the bid-rent function will be negative only if the sum of the terms in the square brackets is positive. In fact, the sign of OPL/Or is ambiguous, and ~'ill llsually change ever space. Although what happens to :he component terms of eq. (19) with spatial movement is an empirical question, several comments are in order. The bidrent surface will vary among households according to income class, because t,~. and VL will tend to increase with income while it is common to argue in favor of a high income ela.~ticity of demand for space. To preserve some independence in the terms it is sensible to assume that marginal changes in income with shifts in location do not push the household from one income class to the next (i.e., assuming that tae changes in vx., VL and s with respect to income are discrete rather ~han cer, tinuous). Another problem related to independence of each term is the po,~,fibility that spatial variations in neighborhood quality may be internalized i,, the land or labor markets [Lind (1973), Freeman (1974), :SAssuming that the workplace has not been pr,-determined, the margina~ change in income .s given by (I4/,+W,') (w~+w..')-W,w~, with .:~,~wheld constant. Similarly, the marginal change in land expenditure is (PL,+PL,~ (S,+ ~,')- ~t,S,.
H.W. Richardson, Residentiallocation theory
263
Polinsky and Rubinfeld (1974)1. Although this has no effect on the individual's bid-rent surface as given by (19), the impact on his locational choice may be different if exte ~L.', ;. "~ifferentials are reflected in wages compared with when they are internal zed in rents. The full results could be known only by solving a simultan~.ous I ,odel of general equilibrium for the metropolitan area as a whole. Other elements in (19) present less trouble. SimlDier versions might allow deletion of the Oc/gr and Oq'/Or terms on the assumption that consumption of goods and housing structure preferences remain constant over space. However, this s t e p - possibly justifiable in an empirical analysis - i s restrictive because a change in location alters accessibilit.~ to consumption (via substitution possibihties between travel costs, T M an~' Tn, and consumption) while housing structure preferences may, despite tl~ high stability of housing characteristics quantities in the individual's utility function, change slightly over space due to the extreme heterogeneity of housing and possii~ie trade-offs between housing structure chartcteristics and neighborhood quality. Although wage rates will be higher in tfe major employmen I centers (ahd strongly positively correlated with the size of the center), the wage rate surface will in most metropolitan areas offer each household some choice in workplaces (abstracting from the restrictions arising from heterogeneous skills and specialization of labor among centers). However, a household may not be willing to pay more tot a residence close to a high-order center if it values residential neighborhood quality highly; travel costs and neighborhood quality will be negatively correlated in the 'normal' case since increasing distance from a major employment center raises e~vironmental quality. Since the sirras ,,t t3W, Or, O~/Pr and c3Lfl~r cannot all be positive (or negative), the net impact of "he allocation of time on bid-reins will depend on the relationship between the wage rat~: and values of travel time and of leisure. Although environmental quality may be directly related to distance from the CBD aad other major subcenters, its variation over space will be irregular influenced by considerations such as topography, neighborhood densities and social stav,s in addition to proximity to non-residential activities. The net result of all these influences is that the household's bid-~ent surface may be very irregular due to the resolution of complex trade-oils between income and leisure, accessibility and neighborhood quality, time and ~o,dential space. What role does the search process constraints play in this moclel? As suggested b~ ~,14~~,(17) and the discussion of optimal stopping, these constrain|s imply that the ho~i,,~ choices facing the individual household are severely restricted by limitations of space, time, information and cost. The search process constr~.i~ts (14}-(17) can be most easily dealt with by treating them as an influence o~,, the aggregate rental surface. Viewed in this light, they do not affect the bid-rent surface but have drastic consequences for the a~:~regatc residential land rent surface. The relevant aggregate rent surface does not
264
H.W. Richardson, Residential location theory
cover the metropolitan area continuously but degenerates into a few small patches (that coincide with the spatial search field). Also, even within these patches the rental opportunities will nc,t be continuous but dependent on available vacant houses (an exception is perhaps where the household is looking for a new house, and where these are developed in large estates). The effect of introducing the optimal stopping rule is to lower the utility obtained from the residential choice, reflecting the fact that searching is costly. The implications of a discontinuous (and irregular) aggregate land value surface and the irregular bid-rent surface for locational equilibrium are farreaching. The aggregate rent surface may coincide with the bid-rent surface at more than one location (multiple solutions), or it may coincide only with a much higher bid-rent surface (implying lower utility) than might have been expected had all locations been available (unlimited time and space for the search), or it may not coincide with any feasible bid-rent function (obtained by introducing subsistence constraints on consumption). If the household finds the land rent it would have to pay unacceptably high, it may reverse the decision to locate in the city, or extend its search, or temporarily move into the house rental market (presumably at lower levels of housing consumption). A potential problem with this model is how to reconcile individual locational equilibrium with the market determination of house prices. Individual households not only adjust to a ~,~ven house price (and rent) surface, but considered as a whole they help to determine that surface. A virtue of the most elementary form of the standard model is that its simple assumptions, including reducing the demand for housing to a demand for land and assuming homogeneity of tastes, ailow the macrospatial equilibrium rent and density gradients to be determined at the same time as the conditioas for household equilibrium. The highly irregular and diverse bid-rent surfaces of heterogeneous households in the model ol:tlined here would have to be known in order to build up the aggregate rent surface as the outer envelope of all these bid-rents. it may be possible, however, to sidestep this apparent difficulty. First, there is no strong reason apart from analytical convenience why it should be assumed that housing and land markets are ever in equilibrium. The urban spatial structure is continually changing as the spatial area of the city extends, as land is converted from one use to another, and as demolition and replacement takes place. Since urban structures are durable, the concept of a stock adjustment model is more plausible than that of static long-run equilibrium. Similarly, the results of the Koopmans-Beckmann (!957) assignmeitt problem remain relevant [but see the ad hoc exceptions of Hartwick (19747]: that because of tocational interdependencies a stable optimizing spatial equili~rium assignment of locators may not be sustainable, and hence the concept of an equilibrium rent surface may be invalid. Second, even if a general equilibrium were possible, it could only be developed wi~.h a comprehensive rnedel including all types of spatial activities including
H.W. Richardson, Residential location theory
265
non-residential land uses. Building up an aggregate equilibrium out of a residential location model alone requires exogenous specification of mine sections (or points) on the aggregate rent surface. In fact, the standard model even does this in a limited way by exogenously locating all non-residential activities in the CBD and by using rent at the edge of the CBD as a constant of integration in determining hod rent at any location.
References Amson, J.C., 1974, Equilibrium and catastrophic modes of urban growth, in: E.L. Cripps, ed., Space-time concepts in urban and regional models (Pion, London) 108-! 28. Artle, R. and P. Varaiya, 1974, On the ex,stcnce of positive rent gradients in Thiinen models, Electronics Research Laboratory Memo. no. ERL-M459 (University o! California, Berkeley, CA). Barr, J.L., 1973, Tiebout models of community structure, Papers of the Regional Science Association 30, 113-139. Beckmann, M.J., 1973, Equilibrium models of residential location, Regional and Urban E~:onomics 3,361-368. Ellickson, B., 1971, Jurisdictional fragmentation and residential choice, American Economic Review (Papers) 61,334--339. Ellis, R.H., 1967. Modelling of household location: A statistical approach, Highway R e , a r c h Record 207, 42-51. Freeman, III, M.A., 1974, On the estimation of air pollution control benefits from land values studies, Journal of Environmental Economics and Management I, 74-83. Hart~ick, J. M., 1974, Price sustainability of location assignments, Journal of Urban Fconomics, !, 147-161. ingrain, G.K., i.F. Kain and J.R. Ginn, 1972, The Detroit prototype of the NBER urban simulation meal.el (Columbia University Press for the National Bureau of Economic Research, New York). Kirwan, R.S. and M.J. Ball, 1974. The microeconomic analysis of a local housing market, in: Centre for Environmental Studies, Papers from the Urban Economics Conference. vol. 1 (CES CP9, London). Koopmans, T.C. and M.J. Beckmann, 1957, Assignment problems in the location of economic activities, Econometrica 25, 53-76. Lind, R.C., 1973, Spatial equilibrium, the theory of rents and public program benefits, Quarterly Journal of Economics 87, 188-207. Lippman, S.A. and J.J. McCall, 1976a, Job ~arch in a dynamic economy, Journal of Economic Theory 12, 365-390. Lippman, S.A. and J.J. McCall, 1976b, The economics o! job ~ a ' .l,' . A survey, Economic Inquiry 14, 155-189 and 347-368. Little, J.T., 1974, Household preferences, reiocmtion and welfare: An evaluation of the filtering concept, Institute of Urban and Regional S.udies WP-HMS 2 (Washington University, St. Louis, WA). McCall, J.J., 1965, ~rhe economics of information and oplimal stopping rules, Journal of Business 38, 300--317. Moses, L.N., 1962, Towards a theory of intra-urban wage differentials and their infl'aence on travel patterns, Papers and Pr(vcecdings of ~he Regional Science Association o 53.63. Muth, R.F., 1969, Cities and housing (Chicago University Press, Chicago, IL~. Oates, W.E., 1969, The effects of prorerty tax~ and local public spending on property values An empirical study of the tax capitalization and the Tiebout hypothesL% Journal of Political Economy 77, 957-971. Papageorgiou, G.J., 197b, On spatial consumer equilibrium, in" G.J. Papageorgiou, ed,, Mathematical land use theory (Le~:ingmn Books, Le×ingtor,, M A) 145-176.
266
H. It: Kichardson, Residential 1,~catio, z theory
Polinsky, A.M. and $. Shavell, 1973, Amenities and pr :~perty values in a general equilibrium model of an urban area, Urban Institute WP-1207- 5. Richardson, H.W., 1975, Discontinuous densities, urba a spatial structure and growth: A new approach, Land Economics 51,305-315. Richardson, H.W., 1977, On the possibility of posit tve rent gradients, Journal of Urban Economics 4, 60-68. Senior, M.L. and A.G. Wilson, 1974, Disaggrvgated r~ski©ntial location models: Some tvsts and further theoretical developments, in: E.L. Cril;ps, ~,1. Space-time concepts in urban and regional models (Pion, London) 141-172. Stegman, M.A., 1969, Accessibility models and resi(lential location, Journal of American Institute of Planners 35, 22-29. Telser, L.G. 1973, Searching for the lowest price, Americar~ Economic Review (Papers) 63, 40-49. Yamada, H., 1972, On the theory of residential location: Accessibility, space, leisure and environmental quality, Papers of the Regional Science Association 29, 125-135.