Micro Econ Questions

WHAT IS THE PRICE ELASTICITY OF HOUSING DEMAND?

Eric A. Hanushek and John M. Quigley*

I. Conceptual Framework

DISAGREEMENT about the responsiveness of housing demand to variations in relative

prices persists, despite extensive empirical anal- yses. Two factors underlie this disagreement: the multidimensional character of housing makes di- rect observation of prices (as distinct from ex- penditures) impossible; and, the significant search, transactions, and moving costs asso- ciated with changing dwellings imply that, at any instant, a given household's consumption may deviate significantly from its utility maximizing level in a static equilibrium. While a number of ingenious attempts have been made to circum- vent these problems, each is quite indirect and relies upon strong, and untestable, assumptions (cf. Mayo (1978)). This paper provides direct estimates of price elasticities, based upon an explicit model of housing consumption dynamics and utilizing the experimental manipulations of housing prices incorporated in the Housing Al- lowance Demand Experiments.1 While the data are limited to two years of longitudinal data and pertain only to low income renters, they never- theless permit the direct estimation of this key parameter of housing demand.

This analysis focuses on the price responsive- ness of households, but clearly other changes in household circumstances (such as in income or family size) affect desired housing consumption. In fact, given the limited longitudinal data, in- formation about other demand adjustments pro- vides valuable insights into consumption dynam- ics. Price changes can be viewed as but one of a

variety of exogenous influences on housing de- mand.

A complete structural model of housing de- mand would consider the joint influence of household preferences, relocation costs, and prices on search and moving behavior and, con- ditional on this, their subsequent influence on housing consumption. However, both household preferences and relocation costs are generally unobserved, and estimation of such a complete model is simply beyond our current capabilities. We concentrate upon the more modest goal of modelling the reduced form relationship between housing consumption and housing stock dis- equilibrium (defined below).

Consumption dynamics are represented by variants of a linearized stock adjustment process . This formulation is based on the simple observa- tion that adjustments will generally be a monotonic function of the magnitude of dis- equilibrium in housing consumption. As indi- cated by past work (Hanushek and Quigley, 1979), this is both a convenient and powerful characterization of short run dynamics.

Two basic formulations of consumption dynamics are considered. Let Htd represent the "desired," or static equilibrium, quantity of housing demanded by a given household, and let Ht be the actual (observed) housing consumption at time t. In the simplest form, households are assumed, on average, to close the gap between desired and equilibrium housing consumption at a constant rate a, so that

Ht+ = a[Ht+Id -Ht ] + 4Ht (1)

where + is one plus the rate of relative price increase during the interval.2 This formulation, Received for publication August 15, 1978. Revision ac-

cepted for publication February 22, 1980. * University of Rochester and University of California,

Berkeley, respectively. This is a substantially shortened version of "What is the

Price Elasticity of Housing Demand?", Discussion Paper No. 7803, Public Policy Analysis Program, University of Roches- ter, August 1978. Copies of the more complete paper are available from the authors. Support for this research was provided by Abt Associates and a grant from the Alfred Sloan Foundation. The research assistance of Robbe Burnstine is gratefully acknowledged.

1 Alternative approaches to this problem can also be found in Friedman and Weinberg (1978).

2 While the H's refer to quantities, only consumption ex- penditures measured in current prices can be measured. Unit prices in the initial period are arbitrarily normalized to one, and 0 indicates changes in relative prices over time.

Given the large fixed cost component to searching and moving, individual households would not be expected to make a series of marginal adjustments to equilibrium but instead would be more likely, given a decision to adjust, to close substantially the gap. Thus, this relationship is best thought of as the reduced form consumption relationship that incorporates both adjustment decisions (searching and mov-

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however, does not allow for differences in behav- ioral responses occasioned by different patterns of equilibrium changes. Since the principal method of changing housing consumption is through searching and moving, households may be more responsive in any period to contem- poraneous changes in equilibrium demand than to initial disequilibrium. This possibility is con- sidered by decomposing the level of disequilib- rium into its time components with the hypothe- sis that y > f3:3

Ht+ = f[Htd -H] + y[Ht+ld- Htd] + Ht- (2)

Now consider a household for which housing prices are exogenously reduced by some propor- tion -j (O ? -r < 1). The new equilibrium housing demand, Htt+d, resulting from the price change is, by definition,

Ht+ I = Ht+?d(1 + E6) (3)

where E is the price elasticity of demand for resi- dential housing. Substitution into (1) and (2) yields

Ht+j= a[Ht+ld - H] + aEqHt+Id + Ht, (4)

and

Ht+= /3[Htd- H] + y[Ht+ld- Htd] + 13E71Htd + yEq[Ht+ld - Htd]

+ OHt * (5)

Both formulations explicitly recognize lagged responses to exogenous stimuli. From equation (4), the long run response to a price change is the price elasticity times the price change, En-, but on average only 100a per cent of this response will be observed in the first period. In equation (5) the observed response to an experimental change in housing prices is yE-q after one period and (y + ,/ - 83Y) Eij after two periods, even though the long run response is again El).

Estimation of this model requires longitudinal information on the actual housing consumption of households (Ht), their equilibrium demands

for given incomes, preferences and initial prices (Htd), and the price reduction (-r).

II. Empirical Analysis

Data from the Housing Allowance Demand Experiment, which provided housing subsidies to a sample of low income, renter households in Phoenix and Pittsburgh, allow estimation of the housing adjustment models in equations (4) and (5). This analysis concentrates upon 586 house- holds in Phoenix and 799 households in Pittsburgh. Of these, 302 in Phoenix and 424 in Pittsburgh were randomly assigned to treatment groups receiving experimental reductions in rental payments (-q) that ranged from 0.2 to 0.6 (i.e., rent subsidies of 20% to 60%); the remain- der were assigned to control groups that received no subsidy.4 Detailed longitudinal information about household characteristics and housing consumption was collected at the beginning of the experiment and annually for two subsequent years.

The empirical implementation involves spec- ifying the equilibrium demands of individual households (HI) in each period and then estimat- ing the dynamic responses and price elasticity parameters. Direct estimation of equilibrium de- mands was used instead of more common sub- stitution of exogenous demand determinants into the dynamic relationships for two reasons: prac- tical limitations imposed by the complicated ad- justments in equation (5) (particularly given the limited longitudinal data); and the lack of sepa- rate identification of changes in relative prices (k) with substitution.5

ing) and actual consumption choices. For present purposes, the focus on expected aggregate behavioral responses is sufficient, although information about the distribution of out- comes across individual households requires more detailed analysis of structural moving relationships (see Hanushek and Quigley (1978)).

3 Note that [H,+ Id - H,] [Htd - H,] + [H,t+d - H,d].

4 Receipt of subsidy did not depend upon any specific hous- ing consumption choices. The experimental payments were actually received for one full year after the period examined here, and experimental households knew that they would receive assistance in obtaining other subsidies after comple- tion of the experiment-thus minimizing effects of the limited duration of the experiment. (Other treatment groups receiv- ing income subsidies similar to a negative income tax are not analyzed here.) Sample median incomes were 0.5-0.6 of each SMSA's median income.

Since the experimental group is small relative to the entire housing market, there would be no perceptible supply re- sponses, and the price elasticity is therefore identified.

I Errors in estimating equilibrium demands may affect the subsequent estimation of the demand relationships. How- ever, in the simplest model (equation (4)) the problems with using instruments for equilibrium demands are the same as those encountered in direct substitution when equilibrium demands are stochastically related to the exogenous vari- ables. Moreover, as discussed below, it is possible to assess

PRICE ELASTICITY OF HOUSING DEMAND 451

In competitive markets where households face the same prices, contract rents (standardized for landlord provision of utilities and appliances) provide an unambiguous measure of the quantity of housing services actually consumed.6 Assum- ing that households recently making relocation decisions consume their utility maximizing quan- tity of housing services (Hd), equilibrium de- mand functions at initial housing prices for each housing market are then estimated by regressing expenditures on income, assets, family size, po- sition in the life cycle, and other demographic characteristics for recent mover households.7 These estimates, presented and discussed else- where (Hanushek and Quigley, 1979), are consis- tent with expectations and prior research esti- mates, but, importantly, they indicate systematic differences between the two housing markets. All previous investigations of price elasticities that utilized price variations across housing mar- kets have assumed that the demand relationships are the same across areas-an assumption that does not appear warranted.

By substituting relevant household income and demographic characteristics for each time pe- riod, household and time specific instruments for desired consumption (i.e., Hod, H d, and H2d) are constructed from the equilibrium demand func- tions. These estimates, combined with actual rent expenditures in each period adjusted for tenure discounts, allow direct estimation of the

price elasticity of housing demand-estimation that is free from the most stringent assumptions and data problems encountered by indirect methods.8

HI. Empirical Results-Adjustment and Price Elasticity Estimates

Table 1 presents coefficient estimates for households in the Phoenix and Pittsburgh hous- ing markets, replicated in two successive one- year intervals for the same sample of households. Assuming that the errors are normally distrib- uted, the estimates are obtained by maximum likelihood techniques incorporating the non- linear constraints on parameters.

Standard covariance tests suggest little differ- ence in the estimated parameters in the two time periods and that the basic adjustment behavior of those households receiving experimental housing price reductions is no different from the behavior of control households who are unaffected by the experiment.9 The simple adjustment coefficient (a) indicates that, on average, 19% of the gap between desired housing consumption and ob- served initial consumption is closed in each one-year period in Pittsburgh. In Phoenix, where the average mobility rate of households is higher, the results suggest that 35% of the gap is closed in any year. In both housing markets, (A is sig- nificantly greater than one, indicating modest in- creases in relative prices. The rate of inflation in housing prices is consistently greater in Phoenix than in Pittsburgh.

Based upon the simple adjustment model, equation (4), the estimated price elasticity of demand is -0.64 in Pittsburgh and -0.45 in Phoenix. These are estimates of the long run

the importance of such problems in the extended models (equation (5)). Some attempts at using self-reported "satis- faction" indexes as instruments for housing disequilibrium were also made, but these crude instruments yielded unsatis- factory results.

6 A significant housing price gradient would imply that households face differing prices, but this does not appear to be a problem in the cities analyzed here. For Phoenix and Pittsburgh, Merrill (1977) tests for equality of coefficients between central city and suburbs in hedonic price equations. In Pittsburgh, coefficient equality cannot be rejected; in Phoenix, the hypothesis can be rejected, although the esti- mated prices are very similar and the standard error of esti- mate changes only slightly with stratification. Independent analysis of Pittsburgh for the NBER Urban Simulation Model (reported in private correspondence by Gregory Ingram) provides no evidence of price gradients. Finally, Muth (1969) found estimated population gradients for Pittsburgh to be insignificantly different from zero at the 0.10 level (and to have the third smallest point estimate for the 46 areas he analyzed).

7 All households, regardless of subsequent assignments to treatment groups, that moved within 12 months of enrollment (t = 0) are assumed to be in equilibrium and included in the estimation. This is consistent with the assumed micro- behavior in footnote 2.

8 Landlord cost savings from long term occupancy (result- ing from both reduced redecorating costs and lower vacancy rates) would be expected to be reflected, at least partially, in reduced rents. (See, for example, previous estimates by Kain and Quigley (1975) and Schafer (1979).) Here, measured ex- penditures are normalized to new occupants (consistent with the estimated equilibrium demands) by using the estimated tenure coefficients from the Phoenix and Pittsburgh hedonic price equations of Merrill (1977). These tenure discounts grow to 10% in Pittsburgh and 19% in Phoenix for more than 10 years of occupancy.

9 For Phoenix, the hypothesis of coefficient equality across time periods is rejected in the simpler model at the 5% level. However, in the more complex model, time period equality of coefficient is not rejected. Since -1 = 0 for the control house- holds, the relevant test is for equality of a's, a's, y's and O's across samples in each time period. This simply confirms the random assignment of households to treatment groups.

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TABLE 1.-ESTIMATES OF STOCK ADJUSTMENT MODELS OF HOUSING CONSUMPTION

FOR EXPERIMENTAL AND CONTROL HOUSEHOLDSa

Pittsburgh Phoenix

Pooled Pooled Coefficient 01 1 >2 Time Periods 0l 1- 2 Time Periods

a ~~~~~0.223 0.155 0.185 0.407 0.273 0.352 (7.65) (6.23) (9.79) (10.01) (7.61) (13.02)

p8 0.213 0.145 0.181 0.402 0.259 0.330 (7.24) (5.80) (9.56) (9.02) (7.09) (11.87)

y 0.466 0.444 0.376 0.462 0.534 0.512 (4.23) (3.46) (4.86) (4.26) (3.77) 6.38)

-0.573 -0.278 -0.689 -0.693 -0.642 -0.359 -0.484 -0.427 -0.385 -0.374 -0.453 -0.409 (3.07) (2.75) (2.71) (2.68) (4.13) (3.36) (2.90) (2.49) (1.80) (1.71) (3.53) (3.65)

4 1.028 1.022 1.036 1.030 1.033 1.032 1.085 1.084 1.058 1.053 1.074 1.067 (27 I1)b (2.06)b (3.77)b (3.12)b (4.70)b (4.79)b (5.02)b (4.95)b (3.75)b (3.43)b (6.39)b (5.94)b

R 2 0.582 0.585 0.700 0.702 0.648 0.649 0.535 0.535 0.695 0.697 0.610 0.613

Sample Size Control 375 375 361 361 736 736 284 284 243 243 527 527 Experimental 424 424 407 407 831 831 302 302 265 265 567 567

Parameter Equality Experimental

& Control 0.292t 0.893c 0.043c l.796c 0.863c 0.571c 1.039c 0.977c Time Periods 1.278c 2.031c 4.l14c 2.399c

/ = y 5.240d 5.493d 6.434d 0.297d 3.770d 6.214d

a t-statistics are in parentheses. bt-statistics are calculated on null hypothesis that parameter equals one; i.e., that there were no changes in the relative prices of housing and other goods. c F-statistics. d t-statistics.

responsiveness of housing consumption to exog- enous price changes; i.e., the responses ob- served after all households had fully adjusted to the changed housing prices. The short run elasticities-those which would be observed after one year of altered prices-are, however, considerably less. While a 10% reduction in housing prices would lead eventually to a 6.4% increase in housing consumption in Pittsburgh, only a 1.2% increase is actually observed after one year. Similarly, in Phoenix only a 1.6% in- crease in housing consumption (from a 10% rent reduction) is observed after the first year, even though a 4.5% increase is expected in the long run.

Table 1 also presents the maximum likelihood estimates of the expanded adjustment model, equation (5), which distinguishes between initial levels of disequilibrium in housing consumption and changes in equilibrium demands. Again the model is replicated in two successive one-year intervals for the same sample of households in each market.10 Covariance tests indicate no dif-

ferences in adjustment behavior for experimental and control households and no differences be- tween time intervals.

The estimated price elasticities in the more general formulation of equation (5) are -0.36 in Pittsburgh and -0.41 in Phoenix. These esti- mates imply a somewhat less elastic response than found in the simpler models (and implicitly that achieving housing consumption goals through rent subsidies could be quite expensive). The estimates of equation (5) also strongly sug- gest significant differences in the short run re- sponsiveness of housing consumption to current changes in equilibrium demands and to initial levels of disequilibrium.

One concern throughout this estimation is the potential for biases arising from the inaccuracy in estimation of equilibrium demands and problems in distinguishing adjustment lags from serial cor- relation in housing demands. Measured house- hold characteristics explain roughly 40% of the variation in equilibrium demands. If unmeasured factors determining demand are uncorrelated with the observed exogenous variables, the pa- rameters of the equilibrium demand functions are, of course, unbiased, but the estimated equilibrium demand for individual households will contain errors. The expanded adjustment model (equation (5)) provides some insight into the importance of this problem. A portion of the residual variation undoubtedly reflects system- atic, but unmeasured, differences in household

10 In estimating equation (5), it must be recognized that 'q = 0 for all households at the beginning of the experiment. Thus, for the first interval (O -+1) the estimated model is

(N- 1) H1 = , [Hod - Ho] + y [Hid-H0d] + 7yciH1d + bHo

and for the second interval (1 -- 2), the estimated model is

(N - 2) H2 = 8 [Hld - H1] + y [H2d - Hld] + yE71 [H2d - Hld] + .CP1 Hld + 4H1.

PRICE ELASTICITY OF HOUSING DEMAND 453

tastes and other factors that remain stable over time. When changes in equilibrium demands are considered, any time invariant household factors will be eliminated, and, in the extreme, the error variance for this term will be zero. Thus, it is logical to expect biases in y to be less severe than those in p, and this may partially explain why estimates of /8 are smaller than those for y. How- ever, since the price discount was not in effect at enrollment, Xhe form of the estimation (as ex- plained in footnote 10) differs across the two periods, and any biases from inaccurate mea- surement and serial correlation of demands would differ across the two intervals. The finding of coefficient equality for the two periods thus offers considerable support for this explicit model of household dynamics.

For the simple models of stock adjustment, the 95% confidence interval for the price elasticity of housing demand is (-0.33 to -0.95) for Pittsburgh households and is (-0.20 to -0.71) for Phoenix households. For the expanded mod- els the confidence intervals are (-0.22 to -0.54) and (-0. 19 to -0.63) in Pittsburgh and Phoenix, respectively. Although the long run elasticity es- timates obtained from the expanded models are smaller, the latter models suggest a more rapid temporal response to price variation.

By way of comparison, Muth (1971) estimates the price elasticity from the production function for new housing and reports three estimates of the price elasticity with 95% confidence intervals of (-0.51 to -0.99). Polinsky and Ellwood (1977), using an identical methodology, report two estimates, with a confidence interval of (-0.56 to -0.86). Again, these studies are for national samples of households and are confined to the purchasers of new single detached, FHA insured housing. This more direct analysis sug- gests that renters are somewhat less responsive to price variation and that responses do vary across housing markets.

The evolution of consumption responses to price changes depends not only upon the demand elasticity and adjustment parameters but also upon the supply elasticity. As shown in table 2, which describes the interaction of these factors, the response to a price reduction will be smaller with less elastic supply. For example, with a supply elasticity of 1.0, a 10% price reduction will elicit only a 3% to 4% increase in housing consumption over the long run, and, at the end of five years, only 60% to 90% of this eventual increase will be realized. The estimates thus sug- gest inelastic responses to price reductions and ones that evolve rather slowly over time.

TAbLE 2.-PERCENTAGE CHANGE IN HOUSING CONSUMPTION FROM A 10% REDUCTION IN HOUSING PRICES AS A FUNCTION OF DEMAND AND SUPPLY PRICE ELASTICITIESa

Demand Elasticity

Pittsburgh Phoenix Simple Expanded Simple Expanded

Adjustment Period/ Adjustment Adjustment Adjustment Adjustment Supply Elasticity (8) E=-.642 e=-.359 e=-.453 e=-.409

A. After one year supply elasticity 0.2 0.75% 0.81% 0.89% 1.02%

0.4 0.92% 1.01% 1.14% 1.38% 0.6 0.99% 1.10%10 1.26% 1.55% 1.0 1.06% 1.19% 1.33% 1.73% 00 1.19% 1.35% 1.60% 2.09%

B. After five years supply elasticity 0.2 1.35% 1.13% 1.34% 1.30%

0.4 2.03% 1.57% 2.00% 1.92% 0.6 2.44% 1.81% 2.40% 2.28% 1.0 2.91% 2.05% 2.86% 2.69% 00 4.11% 2.58% 4.01% 3.69%

C. After full adjustment supply elasticity 0.2 1.53% 1.28% 1.39% 1.34%

0.4 2.46% 1.89% 2.12% 2.02% 0.6 3.10% 2.25% 2.58% 2.43% 1.0 3.91% 2.64% 3.12% 2.90% 00 6.42% 3.59% 4.53% 4.09%

Calculations based upon estimated adjustment models of demand in table 1, assuming constant supply elasticities. Entries are obtained by substitution along the uncompensated demand curve, i.e., entries in the table are U,/8 - f, where 8 is the supply elasticity and f, is the responsiveness of demand to price change after period T (i, = ae, io. = e).

454 THE REVIEW OF ECONOMICS AND STATISTICS

REFERENCES

Friedman, Joseph, and Daniel Weinberg, Draft Report on the Demand for Rental Housing: Evidence from a Percent of Rent Housing Allowance, Abt Associates, Sept. 1978 (processed).

Hanushek, Eric A., and John M. Quigley, "The Dynamics of the Housing Market: A Stock Adjustment Model of Housing Consumption," Journal of Urban Economics 6 (Jan. 1979), 90-1i1. ,"An Explicit Model of Residential Mobility," Land Economics 54 (Nov. 1978), 411-429.

Kain, John F., and John M. Quigley, Housing Markets and Racial Discrimination, National Bureau of Economic Research, 1975.

Mayo, Stephen K., "Theory and Estimation in the Econom- ics of Housing Demand," paper presented at the North

American Meetings of the Econometric Society, Chicago, IL, Aug. 1978.

Merrill, Sally R., Draft Report on Hedonic Indices as a Measure of Housing Quality, Abt Associates, Dec. 1977 (processed).

Muth, Richard F., "The Derived Demand for Urban Residen- tial Land," Urban Studies 8 (Oct. 1971), 243-254. , Cities and Housing (Chicago: University of Chicago Press, 1969).

Polinsky, A. Mitchell, and David T. Ellwood, "An Empirical Reconciliation of Micro and Grouped Estimates of the Demand for Housing," this REVIEW 61 (May 1979), 199-205.

Schafer, Robert, "Racial Discrimination in the Boston Hous- ing Market," Journal of Urban Economics 6 (Apr. 1979), 176-196.