JOITRNAL OF URISAS ECOKOhlICS 6, W-1 11 ( 19’Z))

The Dynamics of the Housing Market: A Stock Adjustment Model of Housing Consumption

JOHN RI. QUIGLEY

Institution for So&l and Policy Studies and Department of Econonu’c:s,

I’ule Tiniversify, :\-ew Haven, Comedicrrt 06.520

Rec,eived September 10, 1976; revised March I, 1977

The importance of transactions costs and conversion costs in the market

for residential housing suggests that the observed housing consumption of an

individual household will generaI1.v differ from its “rqrrilibrium” level of dc-

mand (i.e., the housing consumption freely chosen in a frictionless world, given prices, incomes, and tastes). This paper develops an csplirit model of this

process and provides estimates of alternative stork adjustment models for two samples of renter households. Desired or “rq~~ilibrium” housing demand models

are estimated from samples of recent mover hollseholds; these demand param-

eters are then used to estimate the rate of adjustmcnnt to household equilibrium. The empirical analysis is replicated for two sampics and for two time intervals.

The results strongly indicate that there are significant lags in adjustment,

even for low income renter households. Failure to account for this dynamic

structure may give misleading conclusions about the operation of the housing market in the short run and the efircts of public policies.

1. IXTRODUCTION

One of the distinguishing features of housing as an economic commodity is the substantial transactions cost. associat,c>d with changing the sc% of housing services consumed. Typically, the act of choosing a different dwelling unit is associated with significant search costs, with the costs of moving housahold possessions, and often with nontrivial contracting cost,s, including brokers’ fws, security paymwts, and the likr. Thew t,ransaction costIs would bc of little conscqwncc in undcrstjanding consumrr behavior if dwelling units wwc rclatiwly mall(~ablo and if t,ha housing sorviws emitted by any dw4ing unit could ho modified iwxpcnsiwly. Howcvcr, the scope for modification of existing d\vclling units in response to changes

90

0094-1190/79/010090-22$02.00/O Copyright 0 1979 by Academic Press, Ipc. All rights of reprodurtion in any form rrsrrvrd.

TIE 1)YNAhlIC:S OF THE IIOUSIKC: MARKET 91

in t 11fs housing dcmantls of t Ilcir ocwpanl~s is tlt~mons:lra.bly limited, and any surh modification is usually quite costly.

Thcsc factors imljly that households’ consumption ol housing in an3 period may dcviatc significantly from their “equilibrium” levels of consumption (i.e., the amounts of housing freely chosen in a frictionlcss world , given household incomes, prcfcrcnccs, and relative prices). This paler focuses on the adjustment process by which households modify t,hoir obscrvcd consumption of housing services over time to close the gap between their actual and preferred consumption levels.

The empirical analysis discussed below is based upon reinterview data gathered for rcntcr households as a part of the Housing Allowance Demand Experiment (HADE) currently underway in the Phoenix and Pittsburgh metropolitan areas. Specifically, the conceptual model is tested using households in the HADIG ‘Lcontrol groups” (i.e., households from which data arc’ gathered, but which arc othern-isc unaffected by the cxperimcnts).

While the main purpose of the housing allowance cxpcriment is to provide information about t,he rwponsw of low income households to direct housing subsidies, the unique data gat8hcwd in the experiment can

support much broader analyses. In part’icular, the wpcated observations doscribing individual households and their housing consumption within identified housing markets permit for the first time, serious analysis of the short-run dynamics of housin g adjustment. The usefulness of these data is enhanced by their availability in two rat’hcr diffcrcnt housing markets, thus making possible more c>xtc>nsivc tests of housing mark& dynamics. Sevcrthelcss, it should be noted at the outset that, the empirical analysis which is used hrrc to waluatc the concrptual model is limitc>d (hi t,he design of the oxpcrimcnt) to renters in lower income groups.

2. CO~CWTUAI, FRAMEWC)RK

We begin by assuming that households of given sociodcmographic characteristics A derive utility from t,hcir consumption of ‘(housing services” H and other goods X :

U,,l(H, is) = I’(A, H, A-), (1)

whc~ro I’( .) is an indicat,or of the utility dc>rivcd by households with characteristics fl from their consumption of H and ,Y. Maximization of utilitJy sllbjcct to the household budgot constraint :

Y = Z'IIH + X, (2)

(where Y is household income, PII is the price of housing, and the price of other goods is th(x numcraircl), \-Ads a conventional demand curve for housing sorvicw

H = .f(s, 1-, P,[). (3

I):! IIASITSI-IEK AND QIJIGLEY

lC(Iwtion (3) iutlicatc~s t,hc utility maximizing flow of Ilollsirlg scvrvircv frcc~ly chosen. \\:r drnot c this “desircci lcvcl” of llousirlg wnsumptiou at time t by H,” and note that, at time 7 t,hc: dcsircd consumption lwcl Hi,<’ may differ due to changes in household income 1’, drmographic charactcr- istics A, or changes in rclat,ive prices, 1’11.

Let H, rcprescnt the actual housing consumption of the household observed at time t. As noted previously, if mobility were costless or if housing capital could be modified cheaply at all residential sites, each household would continuously adjust its location or the characterist,ics of its dwelling unit so that Ht = Hen.

If Ht < H,+ld( > Ht+ld), households will have an incentive to modify their housing consumption by purchasing more (less) housing services. However, due to the monetary and psychic costs of adjustment, cqui- librium and observed housing consumption Cl1 generally differ. The strength of the economic forces causing households to modify their consumption of housing services is mcasurcd by the size of the “gap” bctwccn desired housing consumpt,ion and its initial level, [Ht+l” - H,]. If we consider a group of households with similar moving costs, we should expect a monotonic rt:lat,ionship bet,wecn the size of the disequilibrium gap and the incentive for its rlimination.

Households may- climinatc the gap b&vten actual and desired levels of housing consumption cithcr by changing the characteristics of their current dwelling units or by moving. Since the scope for modification of existing dwelling units in rrsponsc to changing housing demands is limited, particularly for rcntcr households, it is rcasonablc to expect’ that t#hc “gap” is eliminated largely by intra-metropolitan residential mobility.

If we assume that, on average, households adjust to their equilibrium positions by closing the gap between actual and desired housing at a constant rate, t’hc rclat,ionship betwern consumption lcvcls over t’imc is

H lil = a[Ht,.ld - Ht] + 4Ht (4)

where a: is the rate of adjustment to equilibrium and 4 is one plus t8ho relative rate of housing price inflation during the interval between 1 and 1 + 1.l This formulation need not imply that individual households

1 By analogy to models of earnings and human capital investment, Eq. (4) indicates

that observed housing consumption in any period depends upon the entire history of desired housing consumption, i.e.,

H, = ;: a(+ - a)l--iHzd + (+ - CU)~HO. (N-1) i=1

More generally, if the inflation rate differs between periods, the expression becomes

t-1

Ht = aH,d + a x Hid A (+j - 01) + HO 14 (+j - a). s-1 ;=*+1 j=1

(N-2)

cBliminat (2 IOOOL~,!~;, of t 1~1 gal) lwt \vwn ac*t ual and clwircrd housing during a giwn pwiod. 11 is mow rcwonablc to prrsumc t.hat, those households who

c~hoosc to mow c4iminatcl compltMy the gap bc4,ween t)hcir desired and

actual levels of housing consumption. Thus, the coefficient of adjustment measures the avcragc propensity to respond (either by moving or by t,ransforming the bundle of housing services at the same location) in rcsponsc to disequilibrium in housing consumption.

The gap betn-trm the desired consumption level at t + 1 and the ohwrved consumption lcvrl at t can be decomposed into two components :

[Hid - H,], which mc’asurcs the gap bet~ween initial consumption and the desired consumption lcvcl, and

CHt+l d - Htd], th c c an e h g in thr equilibrium level of housing consumption during t,hc period.

A moro gcncral modol of the stock adjustment process would distinguish between thcsc components of change and investigate diffcrcntial behavioral wsponsw. In a more general formulation,

Ht+l = P[Htd - H,] + Y[H,+I” - Hldl + M,, (5)

p mcasurw the average speed of adjustment of housing consumption from the initial disequilibrium position, and y measures the spwd of adjustment to current-period changes in rquilibrium lt~cls of housing consumption. We may oxpcct that, on average, households adjust more rapidly to changes in their equilibrium position than to the magnitude of their initial disequilibrium, i.c., y > /3. This would imply that an exogenous change in the factors affecting housing drmand (say, a change in income or family size) would cause households to adjust their housing consumption more rapidly in the current period and to adjust less rapidly in subsequent periods to a new equilibrium position. Interpreted in terms of mobility behavior, this formulation hypothesizes that households are more likely to move in response to an cxogctncous change in housing demand in the period in which it occurs ; in subsrquent periods they are successively less likely to move.

In addition, thcrcl is some reason to c>xprct a “rachct effrct” in housing consumption, leading to asymmetry in the adjustment process. WC anticipate that] househclds whose current, housing consumption is loss than their desired lovcl will adjust to equilibrium more rapidly than those

If the time dependency of desired housing ronsumption H”(L) were known with certainty,

Il;q. (4) could be reformulated to indicate the optimal pattern of observable lifetime housing consumption, given a time pattern of transaction costs. However, we view the

uncertainty about hollscholds’ futru-r hollsing demands as being the essence of this problem.

94 IIANUSIIEK AND QlJIGLEY

with the expectation that cz+ > CY-, A straightforward generalization of t.his asymmetry in adjustment

based upon Eq. (5), is :

H t+l = /3+61[Ht" - Ht] + P-&[Ht" - Ht]

+ y+&[Htg" - Htdl + ~-G.rCHttl~ - Hid] + 6Ht (7) where

61 = 1, 6% = 0 if Htd > H,; ~3~ = 1; h1 = 0 if H,+,” > Htd;

ttl = 0, & = -1 if Htd < H,; 83 = 0; A4 = - 1 if Ht+ld < Htd,

with the expectations that y +a- > p-; p+ > p-, y+ > y-. Unfortunately, Eqs. (4) through (7) cannot be estimated directly

since they include a term for equilibrium demand which is unobserved. The following section discusses the derivation of this variable.

3. EQUILIBRlUhf DEMAND

A single cross section will generally include some individuals who have not yet adjusted their housing consumption to its equilibrium position. Therefore, without some information about the dynamic adjustment process, the est,imation of housing demand models from cross-sectional data could give misleading impressions about the true demands for housing, at least, if the adjustments follow the mow complex patterns sketched above. The importance of dynamic clcmcnts can be inferred, in part, from the mobility level of households. Figure 1 displays the length of tenure at the current address of urban houwholds in the U. S. in 1973. For all households, only ‘2070 have made an adjustment in their housing consumption by moving within the last year. While the mobility level of renters is substantial, it still seems rcasonablc t,o infer t’hat a large fraction of urban renters have undcrgonc changes in circumstances which would vary their desired housin, 0‘ consumption but which have not yet been reflect,cd in their housing consumption at a given point. in timo.

Past analyses of housing demand have provided a body of somewhat conflicting results about the values of key paramctcrs. For the most part,

THE DYNAMICS OF THE HOUSISG MARKET

I I I I I 5 10 15 20 25

YEARS AT CURRENT RESIDENCE

FIG. 1. IXstributions of owner and renter households by years at current residence. Source: U. S. Department of Housing and Urban Development, “Annual Housing Survey, 1973” Government Printing Office, Washington, 1). C. (1975).

the results seem quite dependent’ upon the particular aamplc and tho level of aggregation employed.

Research based upon aggregate time writs or cross-s&ional data using &ire cities as units of observation has established the sensitivity of housing demand to its relative price and to the avcragc incomes of residents (see Reid [12] and Muth [Y]). Considerable rcwarch has focused on the mc~asuremc~nt and interpretation of the awragc’ income variable used in aggwgat.e studies, and thcrc sums to bc a consensus on

t\Yo points :

(I) that the average income variables arc to b(l intrrprchtcld as proxies for long run or “permanent.” income; and

(3) that the elasticity of housing demand with respect to pcrmatwnt~ income is quite large and may exceed 1.0 (SW d(~I,wun- [S]).

Norc rortntly, thaw have brcn several studios Mimating the demand for residential housing bawd upon micro data on individual households and thcxir dwelling units (we Straszheim [14], Carlinw [Q], and Kain and Quiglcy [7]). l’hcsc stud& suggest :

(1) that the elasticity of housing demand may 1~3 considerably lowr with rwpcct to annual household income and may bc on the: ord(lr of 0.3 to 0.6; and

(2) that both annual income and pwmawnt incomtb, (the lattcar mc~asurwl somewhat crud(bly) arc important in c>xplaining the pat,trrri of ho1Isirq d(~mand.’

96 HANUSHEK AND QUIGLEY

In addition, these more recent analyses suggest that other factors besides price and income, however measured, arc important determinants of the demand for housing services. For example, these analyses indicate ceteris paribus that larger households demand more housing services, that black households consume substantially 1~~s housing wrviccts than “otherwise comparable” \vhit,c: houwholds, and t,hat housing demand varies systematically with age and cducat’ion (if only bwause thw> latter variables arc proxies for long-run incorn(

deleeuw [S], in his review of previous demand clstimates, attempts to reconcile thra conflict,ing cvidcnco wporttld in aggregate studies through careful consideration of tho sampling schemes and the measurement of housing consumption and exogenous variablw. More recently, Polinsky [11] has provided a theoretical discussion of model specification and ag- gregation which could rcconcil(b results rcportcd in aggregate and micro analyses of housing demand. This analysis builds upon t’hc observation that, if a housing price gradient exists and is ignored in an wonomotric formulation, it will have differing effcxcts upon the estimat,c>d income (and price) elasticities J\-hen diffcrcnt units of observation and sampling schemes are used.

Consideration of the dynamics of housing consumption and tho impact of disequilibrium observations on t#he parametc>r t&ma& provides an additional reason for differcnccs in paramct,or cstimatcs. The c>xistence of disequilibrium in tho housing market \\-ill have a larger impact on estimation using individual households as units of obwrvat8ion ; using aggrcgatc data, the disequilibrium lc~els of individual houwholds will tend to “rwt out.” Thus, using micro data, the parameters wtimatcid from a single cross-section will rc~prrwnt short run clasticitics, prwumabl\ lower than t’hc comparable aggregat,e c&mates which rcprcwnt long run behavior. However, the est’imatc>s bawd on aggregate data will be biased: if t’he aggregate level of disequilibrium in the market does not average out; or even if the average disequilibrium is zero, as long as t,ht: adjustment process follows tither of the more complex patterns, Eqs. 6 or 7, described abovc.

The adjustment model itself is framed in terms of the (unobserved) “equilibrium” demands. h common technique for estimating such adjustment models (particularly in macro economic applications, c‘.g., investment functions) is to salvo the adjustmctnt equation in tc>rms of observed variables, i.e., to substitute for tht: desired, or equilibrium, demands. For sctvwal reasons, w do not, do this, but instcaad tstimatc tho ctquilibrium demands dirwt ly for uw it1 t 1~ adj&mcwt models. First, this direct formulation allo\vs analysis of whc4her changes in housc~l~oltl circumstances op(krat,e through (*hangin g cc~nilibrium demands, changing the adjustment patStoln, or lwth. Swond, 1)~ wtimat,ing tlw c~cl~lilil)rilitn

THE DYNAMICS OF THE HOUSING MARKET 97

demands directly and using these in the adjustment models, it is possible to test for a variety of adjustment patterns, such as the extended models (Eqs. 5, 6, and 7) which allow for different adjustment speeds with particular componcxnt8s of disequilibrium. Third, the longitudinal informa- tion available for this particular analysis is relatively short (two 6 month periods) ; substitution to eliminate the equilibrium demands requires very strong assumptions about t’he time paths of adjustment,, and the data arc too thin to obtain very reliable &imatcs of this long run adjustment pat)h. (Contrast this to the estimation in macro models (e.g., Muth [lo], 1,~ [S]) rh 11 ere relatively long time scG2s allow many more degrees of frcLcdom in estimation of the adjustment path). Finally, the equilibrium demand models are of independent intercut..

Distinguishing that group of households consuming their equilibrium levels of housing service is problematical, and a review of past research provides little guidance. Muth’s original study of housing demand [lo], based upon time series data (19151941) for the nation as a whole, suggests an aggregate adjustment coefficient of 0.3. Although it is outside t’hc spirit of his model, he appears at one point to intrrprtrt his aggrcgatrx results as if each individual household removed 30v10 of the discrepant> between actual and desired housing consumption in clach ycxar. dr:I,ct:uw and Ekanem’s [4] simulation model also specifics a stock adjustment, model for an entire housing market, but they do not cvaluat,c their simulations in terms of individual behavior. The dynamic mod& abovcb suggest t,hat inertia and significant transactions costs prevent individual households, in general, from consuming that level of h0usin.g sc>rvicc+ which would be optimal on the basis of income, housing pric(‘s, and othrr demographic characteristics. 3 However, thcrc sclcms littlrh reason to presume that those households which are currently making rcllocation decisions (and are observed to have ovcrcomc t)hc intlrt8ia) would got choose to consume their most prefcrrcd lcvcls of housing sclrvirc>s. Our characterization of this dynamic process, at the micro clconomic level, thus assumes that moving households do choose> equilibrium consumption levels; having chosen their opt’imal 1~~~~1s of housing consumption, changes in family circumstances cause housc>holds to drift from cqui-

3 If transactions costs were known, this information might be llsed to reformul:&l the

model and to define the sample of individuals who wordd be clxprctrd to be near rqui- lihrium. Unfortunately, there is very little empirical rvidenc*c about the mo:letary and utility costs of intrametropolitan moves. There is crude evidrncc~ abo\lt cnlt-of-po(akt%

caosts for owner oc*cupants (see delct~uw and l?kancbnr [a], and Shcblton [ 131). Thrscs &imatcs, suggesting that tr:tns:tc%icms ctrsts Ill:ly 1W on the* o&r of IO to 20’ ; 111’ ;mnlnd horlsing cxpensrs, cc~rtainly overstate the monetary costs for rcntchrs sinc*(> they itlc*lud~~

Irgal and brokerage fees. It should 1~ noted, however, that in addit,ion to t.he out-of- po(*ket costs of moving, there are sllbstantial psychic costs sinc%c so&l networks, sc~ho~~l :ttt,r:nd;mcr, and ot,her habits of 11rban lift arc lik(4y t,o I)(\ altc~rr~tl 1)y relor:lt,icm.

9s HANUSHEK AND QUIGLEY

librium. The analysis t#hus nuggwts that equilibrium housing demands arc freely expressed by “recent movers” in any housing market.

Est’imation of t,hc equilibrium demands for housing is based upon

observations of individual houwholds in t\\-o metropolitan awas, Pittsburgh and Phocnis, in 1973. In c>ach mcbtropolit.an awa, a st,ratified random sample of households was drawn ; information about the initial (indicated subscqucntly by t = 0) housing consumpt,ion, income, and soriodt>mographic composition of clach houwhold was gathrrcd. Certain rctrospcctiw data \vcrc also gat8hcwd for clach houw- hold at t = 0. Subscqucntly, households ww assigned to a LLtreatm(~nt,” or a “control” group and similar information was gathc>wd after a (i month interval (t = 1) and again after a second G month interval (t = 2).

The equilibrium demand models wportcd lwlow arc’ ctstimat,c>d using the sample of “rwent movers” houwholds in each city, i.t’., those who mowd into their t = 0 dwelling units within the pwvious 12 months.4

We hypothesize that the equilibrium housing d(>mands for this group of recent movers is a function of annual incom(~, n-c%alth, houwhold size’, age, years of education, and varies by raw. Appcxndix Table Al and A2 comparc the avcragc’ of thaw sociodomographic c*haracbtcvistics for the group of rwcnt mowrs (collwtcd in 1973 71) \\-it It published wnsus information (for 1969) for the t\vo metropolitan arcas. Also included are the average sociodcmographic characteristics of t)hc control housc- holds, which are used below to tost the stock adjustment mod(als. The comparisons reveal t#hat the sampled households arc slightly large, are hcadod by older individuals with slightly 1~~s chducation, and are of about the same racial composition as th(a avcragc households in clach housing market.

Howtver, it should be noted that t.he sampled households arc, by design, not representative of the populations in t.he metropolitan awas. In particular, t’hey have substantially low-w incomes t,han the rcprcwnta- tivc household. Thcrefow, some caution must bc used in extrapolating the results to thr entire population, at least if one brlicvw that the param&rs of interest, arc different,, for some reason, for lo\\. income households.

The csquilibrium demand cyuations \vcr(’ cstimattrd in sweral altcrna- t,ivcs to investigate the specification of age or “life cy~‘le” c#&s. Dcamand models were estimated wit,h a linear and a quadratic age tclrm (see Appendix Table A3). In addition, each sample xvas stratified into two age groups (less t’han 45 y’ars, and 45 years and older) to trst for complete

Yormg _ .-__ ~_.. .~

Incomr: (thowmds) 4.020 (3.19)

Assets (thousands) -1.310

(0.9s) 15ducation (years) 4.950

(5.06)

Horxehold size 3.511

(1.27) PJack - 1 ;i.ss:l

(3.61)

Other nonwhite - 14.9% (O.SO)

Age 1 .40:1 (4.(U)

I<ofrigcrator 22.267 (2.89)

bS:tC)VC 1.074

(0.1(i) Air conditionclr

(‘onSt:Lnt 1.283 I? o.:i1s

d,r :i1:<

b:stimated income elasticity (at point of means) 0.144

Estimated income elasticity (at SLlSA medians) 0.51.’

_.- ~~ _____

n Head of ho~whold aged 45 or more.

Oldn --~~

4.92) (2.11)

1.118

(1.58)

1.31s

(0.9s)

8.479

(3.66)

- 13.159

(1.513)

-3.761 (0.11)

0.450

(1.09)

- 13.272

(O.S7)

25.063 (1.80)

THE DYNAMICS OF THE HOUSISG MARKET

TABLE 1

Hwxing I)emand Eyuations for llwent LIowr Households

Stratified by Age

99

Phoenix

Yo1rng Ol&

6.290 ti.NXl

(7.74) (3.14)

I .2:io 1.X30

(0.9::) (1 .X2)

3.647 0.600 (4.9 I ) (0.49)

1.144 s.795

(0.94) (1.49) - 25.54 1 -X1.44:1

(4.02) (1.56)

- lG.370 -4.466 (4.20) (0.41)

I .8 I‘2 - 0.5w (4.90) (1.47)

-0.470 2.295 (1.40) (0.3s)

22.926 -6.868

(4.79) (0.57) 15.434:

(i .,,:,;

2i.(ilH

(I .7(i)

7.41s 104.907 o.:i15 0.247

502 1%

O.‘L:IS 0.185

O..XO 0.5::1

interaction of the age variable. The rclcvant F-t& indicates t,hat the sample should hc stratified, and these models arc prcwntcd in Table 1.; Including the age Aratification, t’hc demand cstimatw reported in Tablr 1 explain 38% of the variation in contract rents in I’hocnix and 33% in Pittsburgh.

The dcpendcnt, variable in each of the rcgrcssions pwsont~c~tl is monthly contract writ,, \:-hick differs from t,hc index of monthly housing wrvicw

100 HANUSHEK AND QUIGLEY

(or gross rent) by the terms of individual landlord-tenant agreements. To adjust for this, thn demand equations include dummy variables if the landlord provides major appliances. The individual estimat,cs of the appliance premiums arc impwcisc~ ; howcwr, F-t&s for thr sets of coefficients indicate that t’hcy arc jointly important.G

The regrossion cshimatw prcscnted in Table 1 and in t’he Appendix provide considerable support for the qualitative resulk of previous analyses of the demand for housing bawd upon micro data. At the point of sample means, the income cocfficicnt suggests an elasticity of housing demand with rcspcct to annual income of about 0.15 in Pittsburgh and about 0.23 in Phoenix. The ion- avcragc income elasticity computed from the linear model reflects the over sampling of 101~ income households. At the SMSA median income and rent levcIs, t,hc elasticity of housing demand wit,h respect to annual income is cst.imatcd t.o bc 0.52 to 0.63 in Pittsburgh and 0.53 to 0.59 in Phoenix.

The models also include the education lcvc~l of the household head which is, in part, a proxy for permancnt income effects. The independent effects of education level indicate that t’hc elasticity with respect to current income understates the elasticity with rcspwt, to permanent, incomc.7

6 For the fonr columns of Table 1, the ZT-ratios are 2.1X, 0,.X, 4.30, and 3.51, respec-

tively. 7 It is possible to approximate the “income elasticity ” associated with the cd~lcation

term, at least crudely, by relying upon exogeneous information about the relationship between schooling and earnings in each metropolitan area. Note bhat crlrrent inc~omc

(Y) includes a permanent (Y,,) and a transitory (I’,) component ; assume E( Y,) = Cov- (Y,,Y,) = 0. The equilibrium demand equations (Tables 1, A3) are

H,’ = A,, + A,(I-, + I’,) + A&D + ., (N-3)

where ED is years of schoolillg. Then the elasticity of dernalld with respect to l’p,

Separate estimates of the earnings model.

log Yz2 = b, f b,ED + 1)tE.V + baES2, (N-5)

(where ES is experience) were made for malt workers in the Pittsburgh and Phoenix

S&ISA’s from the 1970 Public Use Sample (see Hanushek [6]). For Phoenix b, is estimated to be 0.060, and for Pittsburgh b( is estimated to be 0.085. Substitution of the values reported in Tables Al and A2 into (*y-4) yields estimates of elasticities of housing

demand with respect to permanent income (EYJ of:

erp rstimatrd at _ ~--.

Sample mems PR4Sh medians

Phoenix 0.X 1.02

1’itt.sbllrgh (I.;, I l,lG

THE UYNAMICS OF THE HOUSING MARKET 101

TBBLlq: ‘2

Basic Stock Adjustment hIode

H (+I = a[H,+,” - Ht] + $JH~

Phoenix Pittsburgh -~.

0 + 1 1 ---f 2 Pooled 0 ----t 1 I + 2 Pooled

CY 0.278

(6.10) P 1.022

(1.47)

IF 0.030 Sl“P J J 3 1 .fiti

(U 305 Stratification test :

(P-ratio)

0.265 0.270

(0.04) (8.53) I .046 1.004

(3.12) (0.35)

0.694 O.citil 29.08 30.44

278 si)7

0.73

0.124 0.159 0.143

(4.22) (5.19) (6.75)

1.016 1.001 1.008 (1.90) (0.16) (1.35)

0.7m 0.746 0.753

10.81 21.58 20.71 3% 38’2 7W

0.97

Note : t-statistics in parentheses.

‘I t-statistics refer to the null hypothesis that + = 1.0 (i.e., that the rate of housing price

increase equals the rate of overall price inflation).

The regression also indicates that black households in Pittsburgh (Phoenix) spend 12% (13Oj,) less on housing than otherwise comparable whites. There is no indication that the housing market behavior of those few other non-white households in Pittsburgh differs from that of whites. But in Phoenix, where other non-whites (Mexican-Americans) comprise 28% of the sample, the results suggest that they spctnd 1354 less on housing services than comparable white households. J,argcr households consistently demand more housing services.

One novelty of these demand equations is the inclusion of household

While these estimates are somewhat crude, they do suggest that a substantial part of the differences in income elasticities reported in micro and aggregate studies of housing

demand may be reconciled by considering the effect of education on long run income. Secondly, if we ronsidcr the effect of estimating the demand models for all individuals

in these two samples and not just recent movers (ix., including households whose observed consumption is not in equilibrium), we find substantially lowrr estimates of

elasticites with respect to clxrent income (9”;, lower in Phoenix, and 4X<,{: lower in Pittsbllrgh). The diBercnce between the two (*itics is consistent with the higher mobility

rate of Phoenix (whicah wo~dd suggest the amount of disrq~~ilibrium xross individnals w~Jrlld bc loss there).

Finally, the bias notrd by Polinsky [I 11, arising from ignoring a prire gradicallt trrm in t,he demand models, is probably unimportant for these two particular cities. Both SLISA’s have very tlisp~~rsod c~mployrnc~nt sitcts and rrsidcntial patt,crns (Pittsburgh has

the, flat&t tletlsity gradient of 40 c.itics :m:il~~~d t)y .\lllt,li [<I]) which sllggeat snu~ll lxic*c: gradients. III addition, tlircct inv&igation of pripc gradients for Pittsburgh

conducted in ~orljlulc%ion with Nl%la:l: hcrllsitl g 1n0&1 (rc~lx~rtc~d by Ingr:lnl ill priv:ltc:

c~(lrI~c~sl)o~~(lcrlc:C) was Frrlit,l(xs.

102 HANUSHEK AND QUIGLEY

assets as a crude proxy for houwhold wcalt8h. The inclusion of t,he asset measure stems to have little effect upon housing consumption, at, least the housing consumption of low income rcntcrs.

4. TESTI?;G THE STOCK AD,JUSTMEn’T MODEI,

Entimatcs of the equilibrium housing demand function permit a direct investigation of the stock adjustment model. In t5his section, these adjuxtmwt models arc estimated scparatcly for the two samples of control households in Phoenix and Pittsburgh.

For each sample household, its actual housing consumption at three points in time (Ho, HI, Nz) is obscrvcd. Estimates of its equilibrium housing demand (Nod, Hid, Np’l) are obtained from its income and sociodemographic composition at timw 0, 1, and 2 by using the demand functions reported in Table 1.

Table 2 presents estimates of the basic stock adjustment model (Eq. 4) for both groups estimated for two intervals; Table 3 presents estimates of the more general adjustment model which distinguishes between the initial Icvc~l of discyuilibrium and changes in equilibrium demands.

The estimated adjustment coefficient for Pittsburgh households is

TABLF: :3

Expanded Stock Adjustment Models

Ht+1 = P[H,” - Htl + y[Ht+1”

Phoenix

0 --t 1 1 --t 2 Pooled

B 0.272 0.266 0.269

(5.91) (6.04) (8.44)

Y 0.405 0.139 0.312 (2.07) (0.3) (‘2.01)

@ 1.021 1.046 1 .0x3

(1.35) (3.14) (3.13) IF 0.630 0.69.5 0.661

PI< I+: 3 1.69 29.12 30.37 Stratification test :

(F-ratio) 0.70

Test for equality of coefficients

/3 = y: (I-ratio) 0.67 0.M 0.2s

-

0 3 1 l-2 Pooled

0.122 0.157 0.141

(4.16) (2.13) (6.67)

0.471 0.497 0.491 (2.96) (5.12) (3.69) 1 .OlO 0.997 1 .003

(1.17) (0.35) (0.45) 0.765 0.747 0.75.5

19.71 21.55 20.6::

0.60

2.19 I .36 2.63

TIIE DYNAMICS OF THE IIOIJSING MARKET 10::

a+

I? SI+X

Stratification test: (F-ratio)

Test for equality of

coetficients a+ =a~:

(t-ratio)

0.364 0349 0.357

(5.51) (5.3) (7.67) 0.09s 0.074 0.079

(‘L.87) (3.42) (4.33)

0.990 1.013 1 .ooo

(0.56) (0.79) (0.02) 0.640 0.705 0.670

31.29 28.64 30.03

3.43 3.81

0.69

5.27

0.158 0.095 0.126

(S.50) (1.96) (3.78)

0.057 0.273 0.176 (0.78) (3.69) (3.38)

1.006 1.021 1.01:;

(0.42) (1.45) (1.37)

0.762 0.747 0.753 19.81 21.53 20.72

1.82

1.04 1.69 0.69

Note : t-statistics in parentheses.

a l-statistics refer to the null hypothesis that 6 = 1 .O (i.e., that the rate of harming pricae increase cqrmls the rate of overall price inflation).

quite similar for the two replications of the model; the results suggest that 12 to 16% of the gap between init,ial housing consumption and the cyuilibrium level is closed in each 6 month period. For Phoenix housc- holds, the adjustment, coefficients imply that 27 to 28oj, of the gap is closed in a 6 month period. The cocfficicnts indicate that housing prices increased slightly mow rapidly than other prices at both locations during the 1973 to 1974 prriod.

An F-test permits an explicit test of the rcplicability of the model. The covariancc t,cst accepts the hypothesis of equality of corfficicnts in the t,wo time intervals for each m&ropolitan area.

Table 3 provides tests of the adjustment model when t,hc gap is dis- aggregat,cd into the initial disequilibrium in housing consumption and the change in the equilibrium consumption lcvcl. With one exception, the ostimatcd coefficients indicate that consumers adjust more rapidl) t,o changes in their equilibrium positions than to their initial levels of disequilibrium. For t)he pooled Phoenix (Pittsburgh) sample, the results indicate that, 31% (49%) of a change in equilibrium housing consumption is removed in the same period. In contrast, only 27y0 (14%) of the initial lcvcl of disrquilibrium is removed. It should be noted, hon-cvcr,

104

.-

p+

P-

Y+

Y-

R2 SEE Stratification

test:

(F-ratio) Tests for

equality of coefficients : p+ = p-

(t-ratio) yf = Y- (t-ratio)

Phocnis Pittsburgh

0.352 0.353

(5.50) (5.26)

0.086 0.074 (2.63) (X42)

0.295 0.261

(0.90) (0.55) 0.486 0.130

(1.49) (O.Sl) 0.989 1.010

(0.47) (0.47) 0.643 0.705

s 1 .29 28.70

Pooled

0.370

(7.66)

0.077 (4.26)

0.251

(0.96) 0.385

(1.51) 1 .ooo

(0.0s) 0.672

30.01

0.46

3.62 3.80 . 5.‘39 2.63 1.25

0.36 0.19 0.32 0.15 0.88

o--r1 142 -___

0.160 0.103

(3.74) (2.06) 0.027 0.234

(I 35) (3.23)

0.490 0.735

(‘2.18) (2.00) 0.426 0.199

(1.35) (0.49) 0.997 1.004

(0.27) (0.25) 0.766 0.749

19.71 21.54

Pooled

0.158

(4.99) 0.042

(‘2.12)

0.61 1 (3.14)

0.284

(1.14) 0.987

(1.40) 0.754

20.71

2.11

2.m

0.92

Note : t-statistics in parentheses. Q t-statistics refer to the null hypothesis that C#I = 1 .O (i.r., that the rate of housing

price increase equals the rate of owrall price inflation).

that, with a formal test’, this behavioral difference is not statistically significant in Phoenix. Again, the F-ratios reported in Table 3 indicate that the estimated coefficients are equivalent for bot’h replications in each metropolitan area.

Tables 4 and 5 present the results when these two models are cxpandcd to distinguish between households consuming “too much” housing and those consuming “too little” housing. In Table 4 we distinguish bt+wwn the adjustment rate of households t,o higher levels of housing consumption and the rate of adjustment to lower lcvc~ls. For the Phoenix sample, t’hfb estimates indicat,o more rapid adjustment of the stock of housing to its equilibrium level by those households whose current’ housing consumption is less than its equilibrium levc?l than by those whose current consumption exceeds its equilibrium 1~~~1. This evidcwe of the so called “Ducscnhrrr;\

THE 1)YiVAMICS OF THE HOUSING MARKET IO.?

Tab10 5 1~rcwnls 1 lw 1’ull~. disaggwgat (4 stock adjust mc!nt niod~~l whic~h distinguishw bc+\\crn the initial gap in housing consumption and any change in equilibrium dtxmancl, by algchbraic sign.

The rclcvant stat,istical t&s indicate that the spwd of adju&mcnt to changes in the equilibrium lrvcl of demand is the same whether the equilibrium d(kmand has incrcascd (dw, o.g., to incrclasrs in income) or has decreased (due, c.g., to dccrcascs in income).

The asymmetry in the adjustment process is more evident> for housc- holds’ rcsponscs to tlwir initial lrvels of disequilibrium. The results generally sugg& that houwholds initially consuming “too little” housing change their obswwd consumption more rapidly than those initially con- wming ‘(too much” housing.

Finally, the results rcportcd in Tables 4 and 5 again indicate t’hat the replication test is paswd in cxach case. Within each housing market, the coefficients of each variant of the adjustment model are identical when the model is rcplicatcd for two time periods.

5. coKcLusIon-

The research rcport,ed in this papc’r has two objectives. First, an effort is made t,o charactcrizc bct,tcr the housing demand function for renter households. Second, the analysis attempts t’o dcscribc the adjustment behavior of consumers in the housing market by (Axploring a simple model of market dynamics.

The analysis is made possible by the caxistcncc of longitudinal data on the behavior of households at scvcral points in timcl in a single housing mark&. Thaw data, collecttld for the Housing Allowance Demand Experi- ment in two metropolitan areas, permit an estimation of the “equilibrium” demand funct,ion for rental housing by capitalizing on information about the mobility behavior of houwholds. The cstimat#cd drmand models, which arc quite consistent for both metropolitan arras, indicate an elasticity of demand wit,h rcspwt to current income of about 0.5 for representative households ; they also provide est,imatcs of how other demographic factors influence the demand for rental housing.

These longitudinal data permit estimation of scvcral variants of a simple stock adjust,ment model of housing consumption. In the dynamic models, we llypothcsize that changes in housing stock arc a function of the diffcrcncc between current consumption and dcsiwd consumption (the latt’clr being obtained from the equilibrium dtlmand cqnations) and of the housing price inflaGon in the local market.

The simple models, which view consumption changes as a function of the aggregate disequilibrium in housing consumpt,ion, appear satisfactory.

TIII- nioclt~ls vsl)la,ill som( (ifi (0 75(‘;, (II’ 111~~ v:1ri:L1 i0l1 it1 c~lltl-~~(~riotl

wnsimpi~ion : mow import:bnI Iy. sitiw t 110 data iiic~lutl(~ snmplt~s di-a\~t~ l’rom tuw ceil icbs for 1 \VO 1 in,,, iul~w:ds, c7rrI1 v;lGnt, ()I t II{, acljrisi ii~~111

model can bc rqAicatc~t1 for t IN: same horwholds in a difl’cwnt time pcGd. The cxpandcd versions of thcl stock adjust~mc~nt~ mod(4 allow diffcwnt

adjust.ment pattclrns dcpcnding upon int~ortt~mporal changw in cqui- librium and init,ial disequilibrium lcvc~ls. ‘l’hcw is some cvidwcc, although not conclusive, that adjustmc>nts to changes in equilibrium demands arc more rapid than adjustments to initial disequilibrium positions. On the other hand, t&s of asymmetry in adjustment (diffrrcnt, adjustment, speeds depending upon wh(+hw a houwhold was abovc or bc~lo\\- thc4r equilibrium demand) provide only n-cak cvidcncc that adjustments to increases in demand arc mow rapid than adjustmclnts to dccrt~asw in equilibrium demand.

Finally, the choice of control households from t.hc Housing hllowancc Demand Exprrimcnts (;.(I., thosr individuals rccciving no cxpclrimental treatmcnt8s but from n-horn data arc wgularlly collected) for this analysis is not accidental. Indwd, a major advantage of viwing the housing consumption of individual houwholds as a dynamic adjust,mcwt process is in providing a framwork for virw-ing the rcsponscs owr time of houw- holds participating in experiments of limited duration.

Incorporating the housing allowance trc>atmthnts thc~mwlws int’o the adjustment framwvork dcpcnds upon thcl specific aspects of the cxpcri- ment. The demand c>xpcaAmclnt is actually t\\-o dist,inct programs: the first offrrs a rcant, subsidy t,hat is bawd upon act,ual housing consump- Gons; t,he second is an income t,ransfw bawd upon income and family composit’ion. Th(l first’ plan simply roduws t,lw price of housing for scl&ed households; thcl swond plan is similar to a nogatiw income tax plan rxwpt, that wrtain rwtrictions, such as spending a dwignatc>d minimum amount, on rcntJ or mc&ng wrtain quality standards, aw imposed.

Within the framework of thcb adjustmclnt mod(>l, th(l c4fcct of tht, prim subsidy can br analyzed directly as long as t,he price clast)icity of d(>mand is constant across individuals. This price elasticity can ho cstlmatcd direchly by using the cstimatcd adjust,m& model as a maintained hypothesis and separating out t,hc subsidy amount from the> othcbr dctclrminants of dcmand.s

8 In terms of the basic stock ndjllstment model (I’q. 4), price reductions can lx> incorporated hy estimating

fft+~ = a[(1 + As)ff,+~~ - H,] + $JH, (N-6)

where A is the subsidy (percent of rent forgiven) offered to any household, and 11 is the price elasticity of demand. Equation (N-6) ran he estimated by ordinary least squares with 7 as a parameter.

‘I’hc incwluc~ (,rulisl’c~r I)~o~I~:~III i s 11101‘l’ wln})lic’:ll (VI. \\‘lll~ll Ill1~l~l~ aI’1 110

rwtrictions on housilig c~sl)(~ndil lir(‘s or Ilollsilrg c*lioic*c~ impos~~cl on recipients, tJlicl inwmc: t,ransfw slio~~ltl act t 11~ same as an incwaw in current income from any othrr sourw. Ho\vwor, bwauw thcl oxprrimcwt is of limited duration (and known to bc so by the participants), rwipicnts of this transfer may diwount the value of the transfer (i.c,., not] t,rclat it as an cquivalcnt inrrcasc in income from othclr sourws). Since t,he c~xpc~ctcd increase in housing consumption from an incwasc in income is knonm, it is possihlc t,o cstimatc thcl amount of discounting dirccxtly.!’

Individuals who rcwivc the transfer conditional upon a minimum rental expenditure or conditional upon the type of d\\-cllling oc*cupic>d fact a different decision prohlcm. Thry must decide whct,hcr or not, to ace@ tho subsidy (by meeting some wstrictions) and what their I~~W consump- tion lcvcl should be. Thus, thr adjustmcwt problem involws simult’anroua solution of the housing consumpt,ion prohlcm and th(l program participa- tion prohlrlm. In other words, a simultaneous equations model is ncwssary \\-here orw of t’hc equations involves a hinary choiw.

In any event, analysis of any cxpwimc>nt of limited durat’ion requires an explicit rcprcsentation of the dynamics of adjustment. This is especially true in the market for housing where transactions costs (psychic costs as well as out-of-pock& costs) arc expected to be important’. Since t’hr housing allowance experimental program is of limited duration (2 years for analysis purposes), many of the adjustmtnt,s to experimental treat- mcnts will not bc completed by the end of the c>xpcrimcnt.

Consider, for cxamplc, a changr in circumstances (such as a change in income or family size or the rcccipt of a housing subsidy) which incrcaws housing demand. The simple adjustment models indicate that only 45% of the totma adjustmont in Pittsburgh is observed by t,hc end of 2 years; 72GY” of the total adjustment in Pho&x is obscrwd by the end of 2 Scars. Ewn the expanded models, which allow for a mow rapid rr’sponw to disequilibrium in the period in which it occurs, imply that only 68% (73oj,) of the total adjustmclnt is ohscrwd in Pittshurgh (Phoenix) after 24 months. Thus, an evaluat,ion of the cffccts of an t>xpcrimcntal program (oven one which concentrates on low income households) rcquiros explicit consideration of the adjustment pat’h to provide clstimatcs of tho long run rcsponws to a nat#ional program.

9 By srlbstitrttirrg the housing demand equation into P:q. (4) and rearranging

Ht+, = a[.f(I’, -4) + 44 A) - ff,] + Nt (N-7)

where A is the income transfer offered to any household, and K is the discount factor which equates expected demand from experimental payments to expected demands from ordinary income. Again, Eq. (N-7) can be estimated by ordinary least squares.

APPENDIX TABLE: .41

Means and Standard Deviations of Variables Used for Pittsburgh

Variable HAI>E survq-

(1973)

Census aggregates

(1969) - __. Recent movers

Income (thousands) 4.379

(1.68) Assets (thousands) 0.636

(2.83)

Education (years) 11.009

(2.34)

Household size 3.316 (1.56)

Age of head 34.981

(15.52) Black proportion 0.23 1

Other non-white-(Spanish) proportion 0.009

Contract rent 123.307 (37.8<5)

Rent includes (1 = yes)

Refrigerator 0.142 Stove 0.196

-

Notes : Standard deviations in parentheses ; a Median for all families. D Median for adults aged 25 and over.

- , data not available.

--~ Control

group

<5.042 (2.09) 0.91 a

(2.60) 10.801 (2.54)

3.364

(1.71) 4T2.024

(I 7.34) 0.196

- Central SMSA

rity _-

8.800” 9.737”

- -

11.5h 12.lh

2.82 3.10

- 48.13

0.202 0.071

0.012 0.006 0.003 117.852 79d 76” (37.68)

0.122 -

0.1x2

c Estimated from midpoints of class intervals.

d Median for all occupied dwelling units.

TllE DYNAhllCS OF TllE IlOlJSIN(: XIAHKE’I LO!)

APPKNDIX TABLE A2

Means and Standard Deviations of Variables Used for Phoenix

Variable HAlIF: survey

(1973) -

Census aggregates

(1969)

Recent movers

Control

group

.___- Central SMSA

csity

Income (thousands)

Assets (thousands)

JjIducation (years)

Household size

Age of head

Black proportion

Other non-white (Spanish) proportion

,5.‘L24

(2.21)

0.572

(a.2r;) 10.753

(2.69)

:1.:x3

(1.75) 34.080

(15.61) 0.078

0.261

Contract rent

Rent includes (1 = yvrs)

Refrigerator Stove

Air conditioning

142.097 (44.23)

0.721 0.854

0.946

5.356 9.95@ (‘L.43)

1.252

(5.14) 10.688 12.3h

(2.80)

3.471 3.09

(2.03) 40.702

(18.44)

0.096 0.048

0.253 O.l:iS

133.783 1 OY (49.02)

0.623 0.752

0.868

9.856”

12.3*

3.14

45.7F

0.034

1.156

105”

Notes : Standard deviations in parentheses ; -, data not availahlc,. u Median for all families.

* Median for adults aged 25 and over. c Estimated from midpoints of class intervals.

” P\Iedian for all occupied dwelling units.

Income (thousands)

Assets (thousands)

Education (years)

Household size

Black

Other nonwhite

Age-squared

ltefrigerator

Stove

Air conditioning

Constant 122 df Estimated income elasticity

(at point of means) Estimated income elasticity

(at SMSA median)

(1) (2) (3)

4.430 (4.03) 0.440

(0.7:<) 3.871

(4.93) 6.652

(5.54) - 16.998

(4.42) 1.147

(0.07) (1.237

(a.00) -

15.790 (2.33) 4.333

(0.72) -

31.503 0.286

414

0.157

0.567

6.230 (8.26) 0.240

(0.32) 2.Q7’L

(4.59) X61::

(%48) - 23.5’$7

(X9& - 16.834

(4.52) 0.046

(0.41)

-5.750 (1.34) 14.044 (2.68) 16.872 (2.43) 46.089

0.260 641

0.229

0.5x5

-

- (4)

6.150 (8.28) 0.102

(1.39) 2.75!) (4.32) 1.8Y2

(1.71) -2X626

(4.04) - 15.902

(4.34) 2.756 (4.79)

-0.031 (4.80)

-5.000 (1.19)

14.777 (2.86)

18.62i (2.72)

2.973 0.2%

640

O.‘L26

0.577

ACKNOWLEDGMIPXS

The research reported in this paper was originally sponsored by Abt Associates. Further support was provided through a grant from the Sloan Foundation. The authors are grateful to Dan Koditchek and Robbe Burstine for research assistance. A previous version of this paper was presented at the Conference of the Mathematical Social Science Board, Los Angeles (December lQ75).

REFERENCE:,‘3

1. H. Aaron, “Who Pays the Property Tax,” Brooking6 Institution (1975). 2. G. Carliner, Income elasticity of housing demand, Review ojEconomics Rand Stotistirs,

55, 525-531 (1973).

TIIE DYXAMICS OF THE HOUSING MARKET 111

3. F. deleeuw, The demand for housing: A review of the cross-s&ion evidence,

Review of Economics and Statistics, 53, l-10 (1971).

4. F. deLeeuw and N. Ekancm, The supply of rental housing, American Economic

Review, 61, 806-817 (1971).

5. F. deLeeuw and N. Ekanem, Time lags in the rental market, L’rban Studies, 10, 39-68 (1973).

6. I<. A. Hanushek, “Ethnic Income Variations: Magnitudes and Explanations,!’ in

“-4merican IiXhnics Gro\lps” (Thomas Sowrll, Ed.), The Urban Institute, Washing-

ton, D. c. (1978). 7. J. F. Kain and J. >I. Quigley, ‘LHousing l\Iarkets and Racial Discrimination,”

National Bllreau of Economic Research (1975).

8. H. T. Lee, The stock demand elasticities of non-farm housing, RrzGew oj Econo~tics and Statistics, 46, 82-89 (1964).

9. II. F. Muth, “Cities and Housing,” University of Chicago Press (1969).

10. 1~. F. Muth, The demand for non-farm housing, in “The Demand For Durable

Goods” (A. C. Harberger, Ed.), University of Chicago Press (1960). Il. A. 11. Polinsky, The demand for housing: A study in specification aud grouping,

Econowwtrica, 42, 1-17-461 (1977).

12. 1I. Reid, “Housing and Income,” University of Chicago Press (1962). 13. J. P. Shelton, The cost of renting vrrslls owning a home, Land Economics,

49-Z (1968). 14. Rf. Htraszheim, “ An F:conometric 8nalysis of the Urban Housing &Iarket,” NBER

(1975). 1.5. U. S. Ijepartment of Commerce, Bureau of the Census, “1970 Census of Poprdation

and Ilorlsing,” Tables P-4, H-l, and P-2, U. S. GPO (1972).