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A theoretical and empirical analysis of the length of residency discount in the rental housing market
JOURNAL
OF URBAN
ECONOMICS
22,291-311
(1987)
A Theoretical and Empirical Analysis of the Length of Residency Discount in the Rental Housing Market* J. LUIS GUASCH* *Department Lu Jolla, California
AND ROBERT C. ~~ARSHAU~
of Economks, University of California, San Diego, 92093, and ?Departmest of Economics, Duke Universiry, Durham, North Carolina 27706
Received August 30,1983; revised November $1985 We present an analysis of occupancy discounts and suggest a decomposition of the discount into two components, a “sit” discount and a length of residency discount. Data from the national longitudinal survey of the Annual Housing Survey in which 75,000 housing units from around the United States were followed from 1974 to 1977 are used to obtain consistent and efficient estimates of those discounts. The econometric models account for censoring in the data by endogenously treating the tenant’s staying decision. The estimation indicates that neither discount is significant. This result is contrary to the commonly accepted result in the urban literature that landlords offer discounts to their current tenants when contracts are renegotiated. 0 1987 Academic Pmss, Inc.
1. INTRODUCTION Several recent papers have noted a negative correlation between the length of time a tenant occupies a rental unit and the rent paid.’ This phenomenon has been referred to as the length of residency discount. Two explanations are offered for this discount. First, landlords incur costs when a tenant moves out of a unit, including the cost of reconditioning, the cost of marketing a vacancy, and the rental income foregone during the vacancy. By asking a rent of the current tenant less than the vacant rent for the unit, the landlord decreases the’probability of incurring vacancy costs. Second, landlords use their rent-setting power to screen tenants, asking low rents of current tenants who have shown themselves to be desirable and thereby inducing them to remain in the unit. *We are indebted to Robert F. Engle, David F. Hendry, David M. Lilien, and Gary Zarkin for very helpful suggestions. John B. Knight, Mark Machina, Hal White, ami an anonymous referee also provided useful comments. J. Luis Guasch is indebted to the National Science Foundation for partial research support under Grant SES-8408219. Robert C. Marshall gratefully acknowledges partial support from the Shell Oil Co. Foundation. ‘See Bamett [l], Follain and Malpezzi [4], Goodman and Kawai (51, and Noland [15]. 291 0094-1190/87 $3.00 Copyright 8 1987 by Academic Press, Inc. All ri&s of reproduction in any form reserved.
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AND MARSHALL
However, market forces exist which might induce the landlord to raise the rent on an occupant. The landlord knows that the tenant’s willingness to pay for the unit in the last period was at least the contract rent and that moving costs will be incurred by the tenant if he leaves the unit. Those effects put upward pressure on renegotiated contract rents. First, we provide a theoretical formulation of how landlords set and adjust rents for vacant and occupied units. Second, using AMU~ Housing Survey data, we estimate the effect of the tenant’s residency length on the landlord’s rent-setting behavior. The theoretical analysis identifies conditions under which discounts are offered to tenants who remain in a unit for more than one contract period. We find that if discounts are offered, they can be of two types: a “sit” and a length of residency discount. The sit discount is offered at the first contract renegotiation and does not increase with occupancy length while a length of residency discount increases with occupancy length. The central issue of the econometrics section is that in order to estimate the sit and/or length of residency discounts, the effect of the reset contract rent on the tenant’s decision to stay or leave must be modeled. If a tenant finds the reset contract rent unacceptable, he vacates the unit and the landlord sets an optimal vacant rent for the unit. However, the former rent is not observed since it is never paid. Since only those asking rents found acceptable by tenants are paid, the observed rent is a censored asking rent. The problem of censored data must be addressed in estimating the effect of a tenant’s length of residency on a landlord’s asking rent. Previous estimates of the length of residency discount, based ordinary least-squares (OLS) estimates of log rent on characteristics (both the tenant’s and unit’s) and residency length of the tenant, may be biased and inconsistent since they do not take into account the censoring in the data. We pose the landlord’s rent-setting decision within a switching regime model. An aking rent equation with a tenure discount is observed only if the tenant finds the reset contract rent acceptable. Otherwise, an asking rent equation is observed which does not involve a discount. The latter equation describes how a landlord with a vacancy sets his rent. Estimation indicates that the sit and length of residency discounts are small and insigniticant. This result is contrary to the commonly accepted result in the urban literature that landlords offer discounts to their current tenants when contracts are renegotiated. 2. DEPENDENCE OF THE LANDLORDS RENT-SETTING BEHAVIOR ON INCOMPLETE OCCUPANCY SPELLS: A THEORETICAL FORMULATION We model the landlord’s rent-setting behavior to determine whether the rent asked is a function of the occupancy state of the unit in the previous
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period and/or how long the unit has been occupied. We assume that each of several landlords owns a single rental unit. The households or potential tenants are heterogeneous in preferences, income, size, age of the head, etc. Each household has a willingness to pay or reservation rent for a certain unit. The normalized distribution of reservation rents for a unit of a given description is denoted by f(Rm). Units are heterogeneous in characteristics, condition, neighborhood quality, public services, etc. Each type of unit has associated a different f(RmS) distribution. Without loss of generality, analysis is restricted to a given type of unit, since the relevant rent comparisons are sensible only if units are identical. Landlords can reset rents every period. It is assumed that the landlord of a vacant unit receives one random inspection per period and incurs a marketing cost, c, each period the unit is vacant. When a tenant moves out, the landlord incurs a net cost K, which represents the cost of reconditioning a unit for a new occupant. * We assume that K 2 c. The landlord is assumed to know the distribution of reservation rents for his unit but not the reservation rent of a specific tenant. In each period the landlord sets an asking rent, R,, and if the inspecting searcher or current occupant has a reservation rent not less than the asking rent, the unit becomes or remains occupied, respectively. Otherwise, the unit is vacant. The probability in period t of renting a vacant unit at R, is r(R,) = /
?r(R,lR,el)
= j-~mf(Rres)dRms i ,
for R, 2 Rlel
= 1
for R,I
R,-l.
In a nonstationary regime, e.g., an inflationary environment, the distribution of reservation rents changes over time. We assume that the distribution is shifted by the inflation rate, namely, f,(RmS) = f,+,(Rres(l + i)) and thus q(R) = T~+~(R(l + i)). The interpretation is that the average willingness to pay increases with inflation so that it remains constant in real terms. Since the observations are in nominal terms, we carry out the analysis in nominal terms as well. If the rent paid at t is R, and the inflation rate is i, 2c might be resources spent showing the unit, while K might include painting, cleaning up, etc.
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what is the maximum rent the landlord is certain his tenant will pay at t + l? It should be at least R, and perhaps as much as R,(l + i). Although average income might have gone up by the inflation rate, some sectors and individuals may have received a wage increase smaller than the inflation rate, and thus the landlord might select a lower value than R,(l + i) as his conjecture of the maximum rent his tenant is willing to pay at t + 1. We denote that rent by Z,,, and assume that R, I; Z,,, s R,(l + i). Then the conditional probability of rerenting the unit used by the landlord at t + 1 is
for R r+l =
1
forRt+1
2
&+1
s Z,+,.
(1)
The landlord with an occupancy faces a truncated (at Z,,,) distribution of reservation rents, allowing him to disregard rents below Z,,, when choosing his asked rent at t + 1. Suppose two landlords own identical units in period t - 1 and that one unit becomes rented in period t while the other remains vacant. We want the relationship of the optimal asking rents for the two units in period t + 1. The landlord’s objective is to maximize the expected net discounted utility of rental income. The landlord’s problem is formulated in a dynamic programming framework. Let V,“( R,) and y;O(R,I R+ 1) be the expected discounted utility of net rental income for a vacant unit in period t with an asked rent of R, and for an occupied unit in period t with an asked rent of R, when the rent paid in period t - 1 was RIel, respectively. VP,., and V,:; are the maximum attainable expected discounted utility of net rental income at t + 2 for an occupied and a vacant unit, respectively. Also, by I+( .) we denote the utility of income generated at time t. We subindex the utility function since the argument is in nominal terms. Therefore, u,(R) = ~,+i(R(l + i)). The objective function of a landlord with a vacancy at t + 1 is G(R,+1)
= ~+~tRt+l)b,+ltRt+~
- c,+I) + PV;O,~~R,+AZ~+~)I
+ (1 - 17t+ltR,+l))[~V,:;tR,+2)
+ K+~(-c,+I)],
(2)
while the objective function of a landlord with an occupancy is VP+#LIZt+1)
If the landlord
(R1+1IZ,+1)[~,+1(Rt+1) + PJ%(R,+zIZ~+~~ +t1 - ‘“r+1 @,+,IZt+d x [PKXRt+2) + %+1(-K+1)1. (3)
= *,+I
with a vacancy rents the unit, he gets the utility of the rental
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income net of the marketing costs associated with a vacancy plus the discounted value of an occupancy next period; if he does not rent the unit, he gets the discounted value of a vacancy next period plus the utility of the marketing costs this period. Similarly, if the landlord rerents the unit to the current tenant, he gets the utility of the rental income plus the discounted value of an occupancy next period; if the current occupant leaves, the landlord gets the discounted value of a vacancy next period plus the utility of the reconditioning costs associated with the tenant leaving. Let R:+l and RT,, denote the optimal asking rents at t + 1 for a landlord with a vacancy and with an occupancy at t, respectively; they are the values that maximize (2) and (3), respectively. Also, let R, be the rent asked by both landlords at period t. Then: If Z,+l < R,(l + i), then Ry+l = Ry+,.
THEOREM.
R,(l
Proof.
+ i), then Ry,, -c R:+l,
while if Z,,, =
The proof is in Appendix A.
If the landlord of an occupied unit does not condition the reservation rent distribution on the real rent R,(l + i), the asking rent for a sitting tenant is lower than that for a vacant unit. We then say that the landlord offers a rent discount to the sitting tenant in the amount of d,+l = Ry+l K+1, the motivation being to avoid the transaction costs when the unit becomes vacant. Under the presumption that the landlord of an occupied unit conditions the reservation rent distribution on a value Z,,, < R,(l + i), we analyze how the asking rent for both types of units, vacant and occupied, and the offered discount changes over time. We know that in this case R:+l > Ry+l. Then it follows that the asking rents of a landlord with a sitting tenant and with a vacant unit in period t + 2 are RT+2 = RT+,(l + i) and R;+* = Ry+,(l + i), respectively. Those results are obtained by noting that ?+1(Rr+d = ~~+~(R,+dl + i)), u~+~(R,+J = u,+~(R~+~O + i>), and u,+~(R,+~ - cI+r) = u,+~((R~+~ - c,+r)(l + i)), and substituting those expressions in the first-order conditions of the maximization problem of a landlord with a sitting tenant and with a vacancy. Therefore, the discount will be d rt-2 = RL2 (4 - RT+z = @;+I - RT+,)(l + i) = d,+,(l + i). The nominal discount increases over time by the rate of inflation; the higher the inflation rate, the higher the nominal discount. Thus, while the nominal rent on vacant units increases by the inflation rate, the nominal rent on occupied units increases by less than the inflation rate (due to the discount). Therefore we have established that if the landlord conditions on a z,+, < R,(l + i), there is a sit discount whose size depends on the transaction costs of a vacancy.
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We now investigate whether tenants with different lengths of tenure have different discounts. Consider two rental units, k and j, both vacant at t - 1 with the same asking rent at t, kR’; =jR:. In t, k becomes occupied while j remains vacant. Then we know that at t + 1, kRT+l < jR:+l provided that Z,, i < R:(t + i). Suppose now that at t + 1, unit j becomes occupied and unit k remains occupied Then at t + 2, kRy+2 = kRT+l(l + i). If kRT+l(l + i) 2 Z/+2, then jRT+, =kRT+2 since the value on which the j th landlord conditions the reservation rent distributions does not become binding; both tenants, although having different tenure lengths, pay the same rent and are offered the same discount and there is not a length of tenure discount. However, if kRT+ i(l + i) c Z{+*, then jRT+:2 = Z,‘,2 > k RT+ 1(l + i) and the tenant of unit k who has been in the unit for two periods pays a lower rent than the tenant of unit j who has lived in the unit only one period. The difference from the case above is that the value Z/+ 2 upon which the j th landlord conditions the reservation rent distribution becomes binding, and that value being greater than the rent the k th occupancy is paying implies that the rent charged to the k th occupant is higher than that charged to the jth occupant. Our results extend to nonconstant inflation rate environments. Then if the inflation rate is nondecreasing, tenure discounts, arising from the boundary problem discussed in the above paragraph, are not observed. If the inflation rate falls sufficiently fast, length of residency discounts may occur.3 3. EXTENSIONS Imagine a population of tenants who differ in their predisposition to maintain and care for the unit that they occupy. Once the tenant has been in the unit for a period, the landlord can infer his tenant’s predisposition, i.e., whether he is a good or a bad tenant. Bad tenants increase the maintenance costs of the unit and the cost to bring the unit to market specifications after the tenant has left. Consider a framework where there is a proportion, (Y, of good tenants in the population and, 1 - (Y,bad tenants inducing maintenance cost Qo and Qa, respectively. Then we conclude, by cost arguments similar to those above, that landlords drawing good tenants try to keep them by charging lower rents than the asking rents of vacant ‘An example may illustrate the point. Assume that if a tenant paid R, in period t; then the landlord conjectures that the tenant is willing to pay at least R, in period r + 1, or Z,,, = R,. Let R, be the rent charged when both units are vacant. Suppose that k becomes occupied and j remains vacant. Then at t + 1 the asked rents are’R:+ 1 =jRy(l + i,, 1) and kRy+ 1 < j R:, 1. Suppose that in t + 1 unit i becomes occupied and k remains occupied. Thus kRT+, = ‘Rr,,(l + i,+r). If it+a 2 i,+i, then kRT+z >jR;, kRT+, =kRT+i(l + i,+r), and’Ry+, = ‘R;(l + i ,+i). Therefore there is not a boundary solution and both tenants, while having different occupancy tenures, are given the same discount.
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units (i.e., offering a discount), the benefit to the landlord being his savings on maintenance costs. On the other hand, landlords who draw bad tenants compensate by asking higher rents than the asking rents of vacant units. Thus on average, more bad tenants than good tenants turn down the rent increase. Under symmetric costs and for (Y 2 l/2, we expect, on average, lower rents to be paid for occupied than for vacant units with RF < R; < RF, where Ry, RF, and R‘; stand for the rent asked to a revealed good tenant, the rent asked to a revealed bad tenant, and the asking rent on a vacant unit, respectively. Note that the discount offered in this scenario is again not a length of stay discount since the landlord learns everything he needs to know about his current tenant in one period of occupancy. The increase of the discount over time in an occupied unit is solely the product of inflation as it was in our previous story with homogeneous tenants. The size of the discount depends on the proportion of good and bad tenants in the population and on the nature of the replacement of them each period among the searching tenants, as well as on the values of the maintenance costs, Qa and Qo. Our formulation could easily be modified to introduce n period lease contracts with the results obtained being of the same nature as those reported above. In summary, it has been shown that a landlord who is behaving optimally offers a rent discount to a tenant who remains in a unit for at least one period if there are transaction costs associated with a vacancy and the landlord conditions the reservation rent distribution on a Z,+i -C R,(l + i). It has also been argued that if tenants are of two types, good and bad, and if landlords need one period to type a tenant, then good tenants (but not all tenants) will receive rent discounts after occupying a unit for at least one period. 4. THE DATA The data come from the national longitudinal survey of the Annual Housing Survey (AHS) in which 75,000 U.S. housing units were followed from 1974 to 1977. The survey consists of yearly snapshots of the same units for 4 consecutive years. Every October interviewers visited each housing unit and obtained information about the unit and its residents. A subsample of rental units from the AHS is used here which consists of 477 rental units from 11 cities across the United States. The variables used here along with the criterion used to construct the data set are discussed in Appendix B. Table 1 presents descriptive statistics. The most interesting are the average rents for tenants who have occupied a unit for at least 1 year (sitting tenants) versus the average rent for those who have been in a unit for less than 1 year (nonsitting tenants). Overall, and in each year of the
298
GUASCH AND MARSHALL TABLE 1 Descriptive Statistics from the Data (1) Frequency counts by cities City Cleveland Dallas Denver Houston Indianapolis Minneapolis-St. Paul New Orleans Philadelphia Phoenix San Francisco Seattle Total
No. of units 63 24 24 51 18 41 23 95 12 97 29 411
(2) Average size of a unit = 3.91 rooms Average number of units in a structure = 11.18 Percentage built: Before 1930 40.2 After 1930 and before 1965 36.1 After 1965 23.7 (3) Number of records where tenant in unit more than 1 year = 1383 Number of records where tenant in unit less than 1 year = 525 Mean tenure overall = 4.16 years Mean tenure for sitting tenants = 5.74 years Median tenure of an occupant = 2 years (4) Mean rent overall and by year Overall 1974 1975 1976 1977
Sitting tenant ($) 151.23 134.33 146.81 154.42 168.42
Nonsitting tenant ($) 171.81 153.99 160.88 172.13 201.84
panel, sitting tenants pay lower monthly rentals than do nonsitting tenants. With respect to the tenure averages, note that mean tenure appears somewhat high (4.61 years) but median tenure (2 years) is consistent with previous studies of household mobility. 5. ECONOMETRIC MODELS AND ESTIMATION The objective is to obtain a consistent estimate of the length of residency discount and of the sit discount which was introduced in the theory section. The statistical model of the landlord’s rent-setting decision accounts for the
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OF RESIDENCY
TABLE 2 Previous Estimates of the Length of Residency Discount Fokin and MaIpezzi [4] Goodman and Kawai [5] Bamett [l], Lowry [12], Noland [15]
- 0.010” -0.011” -0.038"
Note. The discounts are given as fractions of rent paid. “Significant at the 5% level.
censoring in the data by modeling the endogeneity of the tenant’s staying decision. A more detailed discussion of the models and parameter estimates can be found in Marshall and Guasch [14]. Econometric Models
Previous researchers typically cross-section data:
have estimated the following model using
ri = X,‘& + y tenure, + ei
(5)
where ri = log(Ri), and Ri is the contract rent per month; Xi = vector of observed characteristics of unit; tenure, = length of residency in years of the current tenant (i.e., tenure, =2 implies that the tenant has lived in the unit for at least 2 years but not yet 3 years).
Table 2 shows some estimates of the tenure, coefficient obtained by other researchers. In all these studies, the coefficient of tenure is negative and significant. However, these estimates probably are biased and inconsistent estimates of the effect of tenure on asking rents because they have not accounted for the censoring in the data. Rent paid is a poor proxy for rent asked when high reset contract rents, which induce sitting tenants to leave units, never are observed. To overcome the censoring problem, (5) is generalized to a switching regime model. Two alterations are made to the specification. First, the rent observed on a survey date was set in nominal dollars anywhere from 0 to 12 months prior to that date (assuming l-year contracts in the market). It is possible that time-varying unit-specific characteristics observed at the time of the survey have changed in response to the contract rent which was set some time before the survey; some of the Xi may be functions of the rent. Then, estimates of (5) would suffer from simultaneity
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bias. The panel nature of the data can be used to surmount this problem since it is reasonable to assume that rent is a function of unit-specific characteristics observed at the previous survey date whereas the converse is inadmissible. Hence we have r; = Xi’, _ 1& + y tenurei + ei.
(6)
Since observed nominal asking rents may be set at some arbitrary time between the last and current survey data, there may be differences in rents due solely to inflation.4 Second, by entering only tenure, as a regressor, the discount offered in the first year for staying in the unit is set equal to the discount offered by the landlord in each year thereafter. In Section 2 we postulated that a difference may exist between the discount offered by landlords for each additional year spent in the unit, the length of residency discount. Each of these different discounts can be estimated by separating tenure into two components. First we define an indicator variable, Zi = 1 if tenure, 2 1 and Zi = 0 otherwise. Note that tenure, = Zi - tenure, = Zi(l + tenure,, -r) = Z, + Z, tenurei, _ i. Therefore, entering Zi and Zi - tenure,, -r separately in (6) instead of just tenure, is equivalent to relaxing a coefficient restriction. Thus, ri = Xi: -J3i + ylZi + y2Zi . tenurei, _ i + ei.
(7)
We can obtain (6) from (7) by imposing the restriction that yi = y2. Equation (7) and the ensuing censoring-corrected generalizations of it are estimated separately for each of the 1975, 1976, and 1977 cross sections. OLS estimation of (7) is equivalent to taking the expectation of r, conditional on Xi, -i, Ii, and Zi . tenure,, -i. For OLS parameter estimates to be unbiased, it is necessary that the expectation of ei conditional on 4A variable D, a correction for the data collection procedure of the AHS, was also included in the X matrix. D is equal to the number of months between the contract negotiation date and the survey date. It varies between 0 and 12 in integer increments. A tenant who is observed to occupy a unit in 2 consecutive survey years will have experienced a rent renegotiation at some point between the surveys. This tenant would be equally likely to experience the renegotiation in any month between the survey dates. However, for units with occupants who have resided in the unit less than 1 year, it is far more probable that the rent was renegotiated closer to the current survey date than to the previous one. This is because a unit which became vacant between survey dates probably experiences a vacancy of some duration before a new occupant is found. Without correcting for this bias, a negative coefficient on tenure may be found simply because new tenants on average negotiate rents closer to the current survey date than do sitting tenants. In fact, for the entire data set, the average value of D for sitting tenants is 4.97 while for recent occupants the average value of D is 4.01.
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Survey
Contract renegotiotion
Survey
I t-s t
t-1 FIGURE
1
xi, - 17 I,, and Ii . tenure,, -i be equal to zero. The censoring correction to be employed is based on evaluation of E(.ri 11,). If we can specify the asking and reservation rent equations at the time of a contract renegotiation, we can formulate the stochastic process describing the probabilities with which a tenant decides to stay in or leave a unit. Figure 1 illustrates the correspondence between the survey data and economic decisions made by agents. The survey records information about the unit and tenant at t - 1 and 1 year later at t. With l-year contracts in the market, the contract renegotiation for the tenant in the unit at t - 1 will occur somewhere between t - 1 and t, say at t - 6 (where 0 < S c 1). The tenant formulates a reservation rent at t - 6 which depends on the characteristics of his household and the unit and, in addition, may depend on the number of contract periods he has resided in the unit. The reservation rent is not observed. The landlord formulates an asking rent which is observed at t if and only if the tenant decides to stay in the unit. However, characteristics of the unit and of the tenant who occupied the unit at t - 1 are observed at t - 1. Therefore, it is possible to specify an asking rent and reservation rent equation at t - S. These are below:
asking rent: reservation rent:
r,* = Xl -i & + y1 + y2 tenurei, -i + &i rires = xi’,-lP2 + Z/,-I& f Y3 + Y4 tenure,,-,
(8) + E3, (9)
where
zi -I=
vector of tenant-specific characteristics (see Appendix B for definition of the variables in Z), reS= log of the reservation rent of the current tenant at t - 6, and ‘:.*I = log of the monthly contract rent asked by the landlord of the current occupant at t - 6.
Whereas (8) captures observed rent-setting behavior, (9) describes how landlords set rents at times of contract renegotiation, whether or not these
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rents are observed (ri = ri* if Ii = 1; ri* will be unobserved if Ii = 0). Whether or not the tenant stays in the unit depends on the relationship between ri* and rims. The tenant stays in the unit, or Ii = 1, if ry - ri* 2 0 or
xi:-l(& - L$>+ Zil,--lPj+ (Y3- Yl>+ (Y4- Y2)tenure,,-,2 &i- &3i* 00) Since both (8) and (9) are formulated conditional on the tenant having been in the unit tenure, years at the time of the contract renegotiation, (y3 - yi) is unidentifiable and we amalgamate it with the Xi,,i constant coefficient and maintain the j3 notation. Then the above condruon can be rewritten as X[-l(&
- PI) + Zil,...#3 + (y4 - y2) tenure,,-,
2 Ei - Eji.
Let Qi = (Xi:-1, Z( -1, tenurei, -I), P’ = (82 - &83, ~4 - YZ), and LJ = q - e3i, Then the above condition becomes QiP > vi. The tenant leaves the unit, or Ii = 0, if rfes - ri < 0 or QiP < vi. We alIow the conditional variance of log rent for sitting tenants to differ from that for new occupants. This leads to ri=x;-l
Instead .E(E&
,
j3 1 + ylIi + y21i - tenure,,-,
of calculating = 0). Assuming
and utilizing we have
E(qlli),
we need
results established by Heckman
E(rilXi,
+ Ii&ii + (1 - Ii)+.
(11)
lJ3(~i~l1~ = 1) + (1 - Ii)
[6,7] and Lee and Trost [ll],
-1, Ii, Ii * tenurei, -1)
= xi~-lpl
+ F(l
+ ylI, + y21i - fIJj ”
- li)hiF(Q,P/uv)/F(-Q,P/u,).
(12)
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where f(x) F(x)
= standard normal density evaluated at x; = cumulative distribution of the standard normal at x.
If we knew P/u,, then the parameters in (12) could be estimated by OLS. But we do not. However P/u, can be estimated by probit analysis. This is step 1 of Lee and Trost’s [ll] two-step estimation method. Once (P/u,) is obtained, OLS can be applied to the equation below. r;=x;-l ,
p 1 + ylIi + yzIi . tenure,, -i
+%(I - Ii)fiiF(Qi(~“))/‘(-Qi(~“)) ”
+ qi
(‘3)
where E(qi]li) = 0. OLS applied to (13) is step 2 of Lee and Trost’s [ll] two-step technique. Consistent estimates of the standard errors of the estimated parameters can be obtained by a White’s [16] technique as modified by Lee [9]. Since the two-step estimation method provides consistent estimates of the parameters, these can be used to start the maximum likelihood iteration.5 Then only one step of scoring need be taken to obtain estimates which are fully efficient (of the first order). Alternatively, we could have formulated the stochastic process described above as a switching regime model. In regime 1, the tenant stays in the unit and an asking rent is observed which possibly contains residency discounts. In regime 2, the tenant leaves and the vacant rental price is observed, regime 1: iff regime 2 : iff
ri=x/-l , /I 1 + y1 + y2 tenure,, -1 + &li Ii = 1 QiP 2 ui or pi = x~-,a, + &2i QiP < vi or Ii = 0.
‘In addition to 8, (G;,), q, and c, can be obtained from the equations
04 05)
consistent estimates of &T and 6et are needed. These
where N1 and N2 are the number of observations from regimes 1 and 2, respectively.
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GUASCH AND IviARSHALL
Since P is estimable only up to a scale factor, we impose the normalization here and throughout that u,’ = 1. Note that (14) of regime 1 and (15) of regime 2 can be stacked together by multiplication with Ii and (1 - Ii), respectively, to yield (11) where the error covariance for regimes 1 and 2 is assumed to be the same as that for (11). It is clear from (14) and (15) that across-regime coefficient restrictions have been imposed, namely, the coefficients of the Xi, -1 are assumed equal. This constraint implies that landlords evaluate the market worth of their unit-specific characteristics in the same way regardless of the occupancy status of their unit. The likelihood function for the switching regime model follows immediately. Let f(qi, vi) be the bivariate normal density of qi and vi, and let g(ezi, vi) be the bivariate normal density of ezi and vi. Then
Estimation In Table 3, estimates of the sit and length of residency discounts are reported. For expositional clarity, we have reported neither coefficients of the Xi, -i nor probit estimates.6 Columns (1) and (2) of Table 3 report OLS estimates without censoring corrections for the pooled data set. All remaining columns present estimates for models in a given year of the data set. Columns (3) (6) and (9) are uncorrected OLS estimates of (7) for 1975, 1976, and 1977, respectively. Columns (4), (7) and (10) present two-step censoring-corrected estimates of (13) for the corresponding years while columns (5), (8), and (11) contain maximum likelihood estimates of the switching regime model described by regime 1 and regime 2 where the likelihood function is given by (16) (again for the years 1975, 1976, and 1977, respectively). ‘The X matrix consists of variables (1) through (17), 10 distinct city dummies (Philadelphia is the excluded dummy) and, in the case of panel estimates, two time dummies for 1976 and 1977. The Z matrix consists of variables (18) through (23) and lagged tenure. Of course, estimates of the coefficients of Z in the offer equation are not possible (not identified), but certainly the coefficients from the probit estimates are possible. The two-step consistent estimates rely on probit estimates which are also not reported. All the unreported estimates and the data set are available from the authors upon request.
- 0.0100 (3.26)
Note. fraction “Ratio “Ratio “Ratio
- 0.0059
(1.63)
- 0.00724
(2.16)
0.591 38.115 0.2923 477
- 0.10458 (3.36)
- 0.09%5 (4.64)
0.564 131.074 0.3064 1431
OLS”
OLS”
(3)
- 0.1379 (1.37) + 0.0578 (0.84) 0.594 37.813 0.2918 477
(0.63)
- 0.0028
-0.1161 (0.97)
Censoring b correction
(4) 1975
477
- 0.1767 (1.94) + 0.0578 (0.70)
(0.87)
- 0.0024
- 0.1321 (1.15)
MLE’
(5)
0.568 42.934 0.3102 477
(2.08) - 0.1704 (1.76) +0.1019 (1.56) 0.574 42.296 0.3086 477
(0.95)
-0.0044
(0.57)
- 0.0078
- 0.0635
(2.46)
Censoring b correction
(7) 1976
- 0.0844
OLS”
(6)
477
-0.1707 (2.08) +0.1019 (1.15)
- 0.0637 (0.55) - 0.0044 (1.54)
MLE’
(8)
0.534 48.211 0.3287 477
- 0.1233 (3.68) - 0.0078 (2.09)
OLS”
(9)
Estimates for 1975,1976, and 1977 cross sections
-0mo6 (0.W + 0.2580 (2.45) 0.541 47.400 0.3268 477
l +0.1587 (0.70) - 0.0071 (1.32)
Censoring’ correction
(10) 1977
477
+ 0.0195 (0.10) + 0.2822 (2.30)
+0.1760 (0.83) - 0.0073 (1.81)
MLE’
(11)
yt measures the sit discount. y2 measures the discount per year of tenancy after the first year. Both discounts are given as a of rent asked in columns (4), (5), (7), (8), (lo), and (11) and as a fraction of rent paid in columns (2), (3), (6), and (9). of coefficient estimates to White standard errors for panel data reported in parentheses. of coefficient estimates to White standard errors for cross-section data reported in parentheses. of coefficient estimates to maximum likelihood estimates of standard error reported in parentheses.
R2 0.558 SSR 132.880 SEE 0.3083 N 1431
0Q”
0Cl”
Y
Y2
Yl
OLS”
Pooled estimates ~_..____ (2) (1)
TABLE 3 Estimates of the Discount
306
GUASCH
AND MARSHALL
Column (1) reports an OLS estimate of the panel equivalent of (6) for the pooled data set.’ The ratio of p to its White [16] standard error is reported in parentheses below 9. p is presented to establish comparability between these data and those used by other researchers. A 1% yearly significant discount is found which is within the range of discounts reported in Table 2. Column (2) presents an OLS estimate of the panel equivalent of (7). The sit discount is approximately 10% while the length of residency discount now is reduced to 0.7%. Both coefficients are significant. This initial result indicates that a substantial discount is being offered for the first year of occupancy and that the discount increases slightly each year thereafter. The yearly OLS estimates T1 and j$ in columns (3), (6), and (9) indicate the same result. Again, these results should not be interpreted as conclusive since they potentially suffer from censoring bias. However, for each year of the panel the censoring-corrected estimates of (13), reported in columns (4), (7), and (lo), yield insignificant estimates of the sit and length of residency discount. While many of the coefficients have not changed greatly from the estimates in columns (3), (6), and (9), they no longer are constant across samples with the mean of Ti = 0 and that of 92 = -0.005. Thus the evidence of large and significant discounts disappears once the anticipated censoring bias is corrected. Although these coefficient estimates are consistent, they are not fully efficient. The fully efficient maximum likelihood estimates in columns (5), (8), and (11) confirm this result. In no case are either T1 or j$ signifkantly different from zero. We are unable to reject either the null hypothesis that yi = 0 or the null hypothesis that y2 = 0 and, in addition to their variation between samples, the means of the estimated yi and y2 parameters are very small. Conversely, in each cross section one of the covariance parameters is playing an important role, again indicative of some censoring bias. Thus the significance of 7 in column (1) and also 9i and R in columns (2), (3), (6), and (9). Once the tenant’s staying decision is modeled as directly dependent on the landlord’s reset asking rent, we find no evidence that landlords are offering either sit or length of residency discounts to their occupants. Indeed, these results indicate that the negative significant estimates of the discounts found in columns (l), (2), (3), (6), and (9) appear to be due to the fact that tenants stay in units for which they pay low rents, rather than the result of systematic intentional discounting of rents by landlords.
‘The model here is ‘;, = XLmI& gives the panel model
+ y tenure,, + q,. Breaking tenurei, into two components
q, = X,:- 1PI + yl I,, + y2 I,,
tenure,,-1 + Ed,.
LENGTH
OF RESIDENCY
307
6. CONCLUSION Using data from the Annual Housing Survey, we have consistently and efficiently estimated the sit and length of residency discounts, accounting for censoring in the data by modeling the endogeneity of the tenant’s staying decision. To account for the censoring bias, the landlord’s rent-setting decision has been posed within a switching regime model. We have been unable to reject either the null hypothesis that yi = 0 or that yz = 0 even with a sample of 477 units (see Learner [8]), but it is well known that this does not imply that discounts are nonexistent. It means that after correcting for censoring in the data, we have insufficient evidence to conclude definitively that landlords offer residency discounts. Nevertheless, this apparent nonresult has both statistical and economic content. The much smaller mean numerical values of estimated discounts cautions against accepting the values reported in potentially biased situations.8 In light of our theoretical formulation, the empirical results tend to indicate that landlords with occupied units fully correct rents by the inflation rate. Other factors not considered in our theoretical model might also help to account for the apparent nonexistence of discounts. We assumed that the transaction costs were state dependent, and that they did not occur in the state of occupancy. However, the marketing cost c which was modeled as a per period cost for the duration of a vacancy may, in fact, be a fixed cost to the landlord. With regard to the reconditioning costs (Y, most lease contracts require a security deposit which is refundable after a tenant leaves a unit conditional on the status of the unit. APPENDIX A Proof of theorem. To show the result, it suffices to demonstrate that the derivative of (3), the objective function of the landlord with an occupancy, with respect to R,+l evaluated at the optimal asking rent of the one with a vacancy, R;, 1, is negative. The first-order condition for (2) is
av,V,1= 4+dK+1)b,+l(K+1 -JR t+1 +rt+1
( R;,,)[
u;+~(R:+~
- 4 + PVP:*(Rt+*l~t+2)1 -
c~+~
- c,+l) + p avp:,!$;+tzt+2)]
-~;+,(R~+,)[pl/,~~(R,+,) + u,+k-c,+1)1= 0. Let R;+l
(‘w
be the solution of the above equation. Then the derivative of (3)
*Also note that our corrected models were capable of reproducing the results of other researchers as they embodied as specializations the caseswhen (1) Ok,”= uee,”= 0, (2) ucpz= q,, ad (3) K = YZ.
308
GUASCH
with respect to R,+l
AND MARSHALL
evaluated at R:+ 1 is
wL(Rt+llzt+l) 8R,+1
R’ r+1
n,:,(R:+l)[u,+,(R:+,)
+
PVP:I(R~+K+~] w%Rt+AZ,+d ,jR
+ ut+l C-K1+1
-~l+,(R:+,)[pV,‘,‘,(R,+,)
1
r+1
(A-2)
)I]
1 x
UJ
f,,
1 ( ROffer)
&Offer
’
JZ 1+1 We show, term by term, that (A-2) is smaller than (A-l) establishing the result. As the rent increases, the probability of renting the unit falls so rt’+,(R:+J < 0, and since u( R’;,,) > u( Ry,, - ct+& the first term of (A-2) is smaller than the first term of (A-l). By concavity, u;+ 1(RT+l c,+d ’ u:+,(R:+,) so th e second term of (A-2) is smaller than the second term of (A-l). Finally, since K,+l 2 c,+i and - 7~‘(Ry,,) > 0, the third term of (A-2) is also smaller than the third term of (4). Therefore, < 0. Now note that RJ,, = R:(l + i), namely, that the ~vp,1/~R,+,IK+1 nominal asked rent for a vacant unit at t + 1 equals the nominal rent at t adjusted by the inflation rate or that the real asked rent for vacant units remains constant. Recall that landlords conjecture that his tenant is willing to pay at least Z,,, at t + 1. Thus unless Zl+l = R;(l + i), we have &‘+I < R:+l; otherwise a boundary solution on the above maximization problem occurs and RT+l = Ry+l, with no occupancy discounts being offered. APPENDIX
B: VARIABLE
DEFINITIONS
AND CODINGS
Coding
Definition of coding
Rent Sit or I
In dollars 1
Tenure
0 T-T
Monthly contract rent Current tenant has been in unit at least 1 year In unit less than 1 year TT = No. of years tenant has lived in unit Unit located in central city Not in central city Unit built before 1939 Built between 1940 and 1950 Built between 1950 and 1960 Built between 1960 and 1964
Variable
(1) Metro
(2) Age
1 0 30 45 55 62
LENGTH
APPENDIX Variable
Coding 66 69 YY
(3) Nunits
(4) Rooms (5) Kitch (6) Air (7) Elec (8) Gas (9) Oil (10) St Noise (11) St Crime (12) Aban (13) Run-down (14) Odors (15) Tram (16) Sch (17) Police (18) Rstr (19) Rhse (20) Inc (21) Grade
(22) Age
(23) HHsize
1 2 3 6 12 25 60 N 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 DDDDD 1 0 YY NN
309
OF RESIDENCY
B: -Continued Definition of coding Built between 1965 and 1968 Built between 1969 and 1970 YY. = year built if built after March 1,197O 1 unit in the structure 2 units in the structure 3 to 4 units in the structure 5 to 9 units in the structure 10 to 19 units in the structure 20 to 49 units in the structure 50 + units in the structure N = number of rooms in the unit Exclusive use of a kitchen No exclusive use of a kitchen Air-conditioned unit No air conditioning Consumption of electricity included in the rent Not included Consumption of gas included in rent Not included Consumption of oil included in rent Not included Street noise is not bothersome Bothersome Street crime is not prevalent Prevalent No abandoned buildings in neighborhood Condition exists Run-down buildings in neighborhood They exist No bad odors in the neighborhood Condition exists Public transportation adequate No adequate Schools adequate Not adequate Police protection adequate Not adequate Resident likes street on which unit is located Does not like Resident likes house or apartment Does not like DDDDD = Gross family income in dollars Head of household is high school graduate Not high school graduate YY = Age of head in years NN = Number of people in household
310
GUASCH
AND MARSHALL
The tenure variable was constructed by subtracting the date the head of household moved in from the date of the survey and truncating the difference. Hence, if in 1974 the current head of household moved in, then in March 1974 tenure is coded zero. If in March 1972 the head moved in, then the tenure variable is coded 2. Our data set was constructed to meet five criteria: (a) none of the cities had any form of rent control legislation during the period 1974 to 1977; (b) a rental unit must be an apartment or house and not a mobile home or a unit in a transient hotel; (c) any unit which was not interviewed in a given year is excluded from the sample; (d) a unit must be occupied during all 4 years at the time of the survey; and (e) any unit which had a miscoding in a time-invariant variable or other crucial variable was excluded from the sample. At no time was the monthly contract rent used to determine whether a unit should be excluded from the sample. There are two main types of miscodings in the data. First, over the course of the panel for a given unit, time-invariant variables may vary. If for a given unit the year that the structure was built is coded 1972 for 3 years and in the fourth year the coding is 1930, then the unit was excluded from the data set. Second, if in a year the date that the head of household moved in is less than the date that the head of household moved in during an earlier year, then the unit was also excluded. For a general discussion of the inconsistencies in the codings of variables in the Annual Housing Survey, see Beveridge [2]. REFERENCES 1. L. Bamett, “Using Hedonic Indexes to Measure Housing Quality.” The Rand Corporation, Santa Monica, CA, R-2450-HUD (1979). 2. A. A. Beveridge, Reliability in large scale household surveys: The case of the Annual Housing Survey, Mimeo presented at the Annual Meeting of the Association of Public Data Users, Washington, DC (Oct. 14-15, 1982). 3. R. F. EngIe and R. C. Marshall A micro-econometric analysis of vacant rental housing units, in “The Urban Economy and Housing” (R. E. Grieson, Ed.), Heath, Lexington, MA (1982). 4. J. R. Follain and S. MaIpezzi, “Dissecting Housing Value and Rent,” The Urban Institute (1980). 5. A. C. Goodman and M. Kawai, Length of residency discounts and rental housing demand: Theory and evidence, Working Paper No. 108, Johns Hopkins University, (1982). 6. J. J. Heckman, The common structure of statistical models of truncation, sample selection, and limited dependent variables, and a simple estimator for such models, Ann. Econom. Social Measure., S(4), 475-492 (1976). 7. J. J. Heckman, Dummy endogenous variables in a simultaneous equations systems, Econometrica, 46(6), 931-959 (1978). 8. E. E. Learner, “Specification Searches: Ad Hoc Inference with Non-experimental data”, Wiley, New York (1978). 9. L-F. Lee, Identification and estimation in binary choice models with limited (censored) dependent variables, Econometrica, 47(4), 977-995 (1979). 10. L.-F. Lee, Some approaches to the correction of selectivity bias, Rev. Econom. Stud., 49(157), 355-372 (1982).
LENGTH
OF RESIDENCY
311
11. L.-F. Lee and R. P. Trost, Estimation of some limited dependent variable models with an application to housing demand. J. Econom., 8,357-382 (1978). 12. I. S. Lowry, Rental housing in the 1980s: Searching for the crisis, in “Rental Housing: Is There a Crisis?” (J. C. Weicher, Ed.), Urban Inst. Press, Washington, DC (1981). 13. G. S. MaddaIa, Disequilibrium, self-selection, and switching models, Cal Tech. Social Science Working Paper No. 303 (1980). 14. R. C. Marshall and J. L. Guasch, Occupancy discounts in the U.S. rental housing market, Oxford Bulletin of Economics and Statistics, 31, 41-62 (1983). 15. C. W. Noland, “Assessing Hedonic Indexes for Housing,” The Rand Corporation, Santa Monica, CA, N-1505-HUD (1980). 16. H. White, A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity, Econometrica, 48(3), 817-838 (1980).
OF URBAN
ECONOMICS
22,291-311
(1987)
A Theoretical and Empirical Analysis of the Length of Residency Discount in the Rental Housing Market* J. LUIS GUASCH* *Department Lu Jolla, California
AND ROBERT C. ~~ARSHAU~
of Economks, University of California, San Diego, 92093, and ?Departmest of Economics, Duke Universiry, Durham, North Carolina 27706
Received August 30,1983; revised November $1985 We present an analysis of occupancy discounts and suggest a decomposition of the discount into two components, a “sit” discount and a length of residency discount. Data from the national longitudinal survey of the Annual Housing Survey in which 75,000 housing units from around the United States were followed from 1974 to 1977 are used to obtain consistent and efficient estimates of those discounts. The econometric models account for censoring in the data by endogenously treating the tenant’s staying decision. The estimation indicates that neither discount is significant. This result is contrary to the commonly accepted result in the urban literature that landlords offer discounts to their current tenants when contracts are renegotiated. 0 1987 Academic Pmss, Inc.
1. INTRODUCTION Several recent papers have noted a negative correlation between the length of time a tenant occupies a rental unit and the rent paid.’ This phenomenon has been referred to as the length of residency discount. Two explanations are offered for this discount. First, landlords incur costs when a tenant moves out of a unit, including the cost of reconditioning, the cost of marketing a vacancy, and the rental income foregone during the vacancy. By asking a rent of the current tenant less than the vacant rent for the unit, the landlord decreases the’probability of incurring vacancy costs. Second, landlords use their rent-setting power to screen tenants, asking low rents of current tenants who have shown themselves to be desirable and thereby inducing them to remain in the unit. *We are indebted to Robert F. Engle, David F. Hendry, David M. Lilien, and Gary Zarkin for very helpful suggestions. John B. Knight, Mark Machina, Hal White, ami an anonymous referee also provided useful comments. J. Luis Guasch is indebted to the National Science Foundation for partial research support under Grant SES-8408219. Robert C. Marshall gratefully acknowledges partial support from the Shell Oil Co. Foundation. ‘See Bamett [l], Follain and Malpezzi [4], Goodman and Kawai (51, and Noland [15]. 291 0094-1190/87 $3.00 Copyright 8 1987 by Academic Press, Inc. All ri&s of reproduction in any form reserved.
292
GUASCH
AND MARSHALL
However, market forces exist which might induce the landlord to raise the rent on an occupant. The landlord knows that the tenant’s willingness to pay for the unit in the last period was at least the contract rent and that moving costs will be incurred by the tenant if he leaves the unit. Those effects put upward pressure on renegotiated contract rents. First, we provide a theoretical formulation of how landlords set and adjust rents for vacant and occupied units. Second, using AMU~ Housing Survey data, we estimate the effect of the tenant’s residency length on the landlord’s rent-setting behavior. The theoretical analysis identifies conditions under which discounts are offered to tenants who remain in a unit for more than one contract period. We find that if discounts are offered, they can be of two types: a “sit” and a length of residency discount. The sit discount is offered at the first contract renegotiation and does not increase with occupancy length while a length of residency discount increases with occupancy length. The central issue of the econometrics section is that in order to estimate the sit and/or length of residency discounts, the effect of the reset contract rent on the tenant’s decision to stay or leave must be modeled. If a tenant finds the reset contract rent unacceptable, he vacates the unit and the landlord sets an optimal vacant rent for the unit. However, the former rent is not observed since it is never paid. Since only those asking rents found acceptable by tenants are paid, the observed rent is a censored asking rent. The problem of censored data must be addressed in estimating the effect of a tenant’s length of residency on a landlord’s asking rent. Previous estimates of the length of residency discount, based ordinary least-squares (OLS) estimates of log rent on characteristics (both the tenant’s and unit’s) and residency length of the tenant, may be biased and inconsistent since they do not take into account the censoring in the data. We pose the landlord’s rent-setting decision within a switching regime model. An aking rent equation with a tenure discount is observed only if the tenant finds the reset contract rent acceptable. Otherwise, an asking rent equation is observed which does not involve a discount. The latter equation describes how a landlord with a vacancy sets his rent. Estimation indicates that the sit and length of residency discounts are small and insigniticant. This result is contrary to the commonly accepted result in the urban literature that landlords offer discounts to their current tenants when contracts are renegotiated. 2. DEPENDENCE OF THE LANDLORDS RENT-SETTING BEHAVIOR ON INCOMPLETE OCCUPANCY SPELLS: A THEORETICAL FORMULATION We model the landlord’s rent-setting behavior to determine whether the rent asked is a function of the occupancy state of the unit in the previous
LENGTH
293
OF RESIDENCY
period and/or how long the unit has been occupied. We assume that each of several landlords owns a single rental unit. The households or potential tenants are heterogeneous in preferences, income, size, age of the head, etc. Each household has a willingness to pay or reservation rent for a certain unit. The normalized distribution of reservation rents for a unit of a given description is denoted by f(Rm). Units are heterogeneous in characteristics, condition, neighborhood quality, public services, etc. Each type of unit has associated a different f(RmS) distribution. Without loss of generality, analysis is restricted to a given type of unit, since the relevant rent comparisons are sensible only if units are identical. Landlords can reset rents every period. It is assumed that the landlord of a vacant unit receives one random inspection per period and incurs a marketing cost, c, each period the unit is vacant. When a tenant moves out, the landlord incurs a net cost K, which represents the cost of reconditioning a unit for a new occupant. * We assume that K 2 c. The landlord is assumed to know the distribution of reservation rents for his unit but not the reservation rent of a specific tenant. In each period the landlord sets an asking rent, R,, and if the inspecting searcher or current occupant has a reservation rent not less than the asking rent, the unit becomes or remains occupied, respectively. Otherwise, the unit is vacant. The probability in period t of renting a vacant unit at R, is r(R,) = /
?r(R,lR,el)
= j-~mf(Rres)dRms i ,
for R, 2 Rlel
= 1
for R,I
R,-l.
In a nonstationary regime, e.g., an inflationary environment, the distribution of reservation rents changes over time. We assume that the distribution is shifted by the inflation rate, namely, f,(RmS) = f,+,(Rres(l + i)) and thus q(R) = T~+~(R(l + i)). The interpretation is that the average willingness to pay increases with inflation so that it remains constant in real terms. Since the observations are in nominal terms, we carry out the analysis in nominal terms as well. If the rent paid at t is R, and the inflation rate is i, 2c might be resources spent showing the unit, while K might include painting, cleaning up, etc.
294
GUASCH AND MARSHALL
what is the maximum rent the landlord is certain his tenant will pay at t + l? It should be at least R, and perhaps as much as R,(l + i). Although average income might have gone up by the inflation rate, some sectors and individuals may have received a wage increase smaller than the inflation rate, and thus the landlord might select a lower value than R,(l + i) as his conjecture of the maximum rent his tenant is willing to pay at t + 1. We denote that rent by Z,,, and assume that R, I; Z,,, s R,(l + i). Then the conditional probability of rerenting the unit used by the landlord at t + 1 is
for R r+l =
1
forRt+1
2
&+1
s Z,+,.
(1)
The landlord with an occupancy faces a truncated (at Z,,,) distribution of reservation rents, allowing him to disregard rents below Z,,, when choosing his asked rent at t + 1. Suppose two landlords own identical units in period t - 1 and that one unit becomes rented in period t while the other remains vacant. We want the relationship of the optimal asking rents for the two units in period t + 1. The landlord’s objective is to maximize the expected net discounted utility of rental income. The landlord’s problem is formulated in a dynamic programming framework. Let V,“( R,) and y;O(R,I R+ 1) be the expected discounted utility of net rental income for a vacant unit in period t with an asked rent of R, and for an occupied unit in period t with an asked rent of R, when the rent paid in period t - 1 was RIel, respectively. VP,., and V,:; are the maximum attainable expected discounted utility of net rental income at t + 2 for an occupied and a vacant unit, respectively. Also, by I+( .) we denote the utility of income generated at time t. We subindex the utility function since the argument is in nominal terms. Therefore, u,(R) = ~,+i(R(l + i)). The objective function of a landlord with a vacancy at t + 1 is G(R,+1)
= ~+~tRt+l)b,+ltRt+~
- c,+I) + PV;O,~~R,+AZ~+~)I
+ (1 - 17t+ltR,+l))[~V,:;tR,+2)
+ K+~(-c,+I)],
(2)
while the objective function of a landlord with an occupancy is VP+#LIZt+1)
If the landlord
(R1+1IZ,+1)[~,+1(Rt+1) + PJ%(R,+zIZ~+~~ +t1 - ‘“r+1 @,+,IZt+d x [PKXRt+2) + %+1(-K+1)1. (3)
= *,+I
with a vacancy rents the unit, he gets the utility of the rental
LENGTH
OF
RESIDENCY
295
income net of the marketing costs associated with a vacancy plus the discounted value of an occupancy next period; if he does not rent the unit, he gets the discounted value of a vacancy next period plus the utility of the marketing costs this period. Similarly, if the landlord rerents the unit to the current tenant, he gets the utility of the rental income plus the discounted value of an occupancy next period; if the current occupant leaves, the landlord gets the discounted value of a vacancy next period plus the utility of the reconditioning costs associated with the tenant leaving. Let R:+l and RT,, denote the optimal asking rents at t + 1 for a landlord with a vacancy and with an occupancy at t, respectively; they are the values that maximize (2) and (3), respectively. Also, let R, be the rent asked by both landlords at period t. Then: If Z,+l < R,(l + i), then Ry+l = Ry+,.
THEOREM.
R,(l
Proof.
+ i), then Ry,, -c R:+l,
while if Z,,, =
The proof is in Appendix A.
If the landlord of an occupied unit does not condition the reservation rent distribution on the real rent R,(l + i), the asking rent for a sitting tenant is lower than that for a vacant unit. We then say that the landlord offers a rent discount to the sitting tenant in the amount of d,+l = Ry+l K+1, the motivation being to avoid the transaction costs when the unit becomes vacant. Under the presumption that the landlord of an occupied unit conditions the reservation rent distribution on a value Z,,, < R,(l + i), we analyze how the asking rent for both types of units, vacant and occupied, and the offered discount changes over time. We know that in this case R:+l > Ry+l. Then it follows that the asking rents of a landlord with a sitting tenant and with a vacant unit in period t + 2 are RT+2 = RT+,(l + i) and R;+* = Ry+,(l + i), respectively. Those results are obtained by noting that ?+1(Rr+d = ~~+~(R,+dl + i)), u~+~(R,+J = u,+~(R~+~O + i>), and u,+~(R,+~ - cI+r) = u,+~((R~+~ - c,+r)(l + i)), and substituting those expressions in the first-order conditions of the maximization problem of a landlord with a sitting tenant and with a vacancy. Therefore, the discount will be d rt-2 = RL2 (4 - RT+z = @;+I - RT+,)(l + i) = d,+,(l + i). The nominal discount increases over time by the rate of inflation; the higher the inflation rate, the higher the nominal discount. Thus, while the nominal rent on vacant units increases by the inflation rate, the nominal rent on occupied units increases by less than the inflation rate (due to the discount). Therefore we have established that if the landlord conditions on a z,+, < R,(l + i), there is a sit discount whose size depends on the transaction costs of a vacancy.
296
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We now investigate whether tenants with different lengths of tenure have different discounts. Consider two rental units, k and j, both vacant at t - 1 with the same asking rent at t, kR’; =jR:. In t, k becomes occupied while j remains vacant. Then we know that at t + 1, kRT+l < jR:+l provided that Z,, i < R:(t + i). Suppose now that at t + 1, unit j becomes occupied and unit k remains occupied Then at t + 2, kRy+2 = kRT+l(l + i). If kRT+l(l + i) 2 Z/+2, then jRT+, =kRT+2 since the value on which the j th landlord conditions the reservation rent distributions does not become binding; both tenants, although having different tenure lengths, pay the same rent and are offered the same discount and there is not a length of tenure discount. However, if kRT+ i(l + i) c Z{+*, then jRT+:2 = Z,‘,2 > k RT+ 1(l + i) and the tenant of unit k who has been in the unit for two periods pays a lower rent than the tenant of unit j who has lived in the unit only one period. The difference from the case above is that the value Z/+ 2 upon which the j th landlord conditions the reservation rent distribution becomes binding, and that value being greater than the rent the k th occupancy is paying implies that the rent charged to the k th occupant is higher than that charged to the jth occupant. Our results extend to nonconstant inflation rate environments. Then if the inflation rate is nondecreasing, tenure discounts, arising from the boundary problem discussed in the above paragraph, are not observed. If the inflation rate falls sufficiently fast, length of residency discounts may occur.3 3. EXTENSIONS Imagine a population of tenants who differ in their predisposition to maintain and care for the unit that they occupy. Once the tenant has been in the unit for a period, the landlord can infer his tenant’s predisposition, i.e., whether he is a good or a bad tenant. Bad tenants increase the maintenance costs of the unit and the cost to bring the unit to market specifications after the tenant has left. Consider a framework where there is a proportion, (Y, of good tenants in the population and, 1 - (Y,bad tenants inducing maintenance cost Qo and Qa, respectively. Then we conclude, by cost arguments similar to those above, that landlords drawing good tenants try to keep them by charging lower rents than the asking rents of vacant ‘An example may illustrate the point. Assume that if a tenant paid R, in period t; then the landlord conjectures that the tenant is willing to pay at least R, in period r + 1, or Z,,, = R,. Let R, be the rent charged when both units are vacant. Suppose that k becomes occupied and j remains vacant. Then at t + 1 the asked rents are’R:+ 1 =jRy(l + i,, 1) and kRy+ 1 < j R:, 1. Suppose that in t + 1 unit i becomes occupied and k remains occupied. Thus kRT+, = ‘Rr,,(l + i,+r). If it+a 2 i,+i, then kRT+z >jR;, kRT+, =kRT+i(l + i,+r), and’Ry+, = ‘R;(l + i ,+i). Therefore there is not a boundary solution and both tenants, while having different occupancy tenures, are given the same discount.
LENGTH
OF
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units (i.e., offering a discount), the benefit to the landlord being his savings on maintenance costs. On the other hand, landlords who draw bad tenants compensate by asking higher rents than the asking rents of vacant units. Thus on average, more bad tenants than good tenants turn down the rent increase. Under symmetric costs and for (Y 2 l/2, we expect, on average, lower rents to be paid for occupied than for vacant units with RF < R; < RF, where Ry, RF, and R‘; stand for the rent asked to a revealed good tenant, the rent asked to a revealed bad tenant, and the asking rent on a vacant unit, respectively. Note that the discount offered in this scenario is again not a length of stay discount since the landlord learns everything he needs to know about his current tenant in one period of occupancy. The increase of the discount over time in an occupied unit is solely the product of inflation as it was in our previous story with homogeneous tenants. The size of the discount depends on the proportion of good and bad tenants in the population and on the nature of the replacement of them each period among the searching tenants, as well as on the values of the maintenance costs, Qa and Qo. Our formulation could easily be modified to introduce n period lease contracts with the results obtained being of the same nature as those reported above. In summary, it has been shown that a landlord who is behaving optimally offers a rent discount to a tenant who remains in a unit for at least one period if there are transaction costs associated with a vacancy and the landlord conditions the reservation rent distribution on a Z,+i -C R,(l + i). It has also been argued that if tenants are of two types, good and bad, and if landlords need one period to type a tenant, then good tenants (but not all tenants) will receive rent discounts after occupying a unit for at least one period. 4. THE DATA The data come from the national longitudinal survey of the Annual Housing Survey (AHS) in which 75,000 U.S. housing units were followed from 1974 to 1977. The survey consists of yearly snapshots of the same units for 4 consecutive years. Every October interviewers visited each housing unit and obtained information about the unit and its residents. A subsample of rental units from the AHS is used here which consists of 477 rental units from 11 cities across the United States. The variables used here along with the criterion used to construct the data set are discussed in Appendix B. Table 1 presents descriptive statistics. The most interesting are the average rents for tenants who have occupied a unit for at least 1 year (sitting tenants) versus the average rent for those who have been in a unit for less than 1 year (nonsitting tenants). Overall, and in each year of the
298
GUASCH AND MARSHALL TABLE 1 Descriptive Statistics from the Data (1) Frequency counts by cities City Cleveland Dallas Denver Houston Indianapolis Minneapolis-St. Paul New Orleans Philadelphia Phoenix San Francisco Seattle Total
No. of units 63 24 24 51 18 41 23 95 12 97 29 411
(2) Average size of a unit = 3.91 rooms Average number of units in a structure = 11.18 Percentage built: Before 1930 40.2 After 1930 and before 1965 36.1 After 1965 23.7 (3) Number of records where tenant in unit more than 1 year = 1383 Number of records where tenant in unit less than 1 year = 525 Mean tenure overall = 4.16 years Mean tenure for sitting tenants = 5.74 years Median tenure of an occupant = 2 years (4) Mean rent overall and by year Overall 1974 1975 1976 1977
Sitting tenant ($) 151.23 134.33 146.81 154.42 168.42
Nonsitting tenant ($) 171.81 153.99 160.88 172.13 201.84
panel, sitting tenants pay lower monthly rentals than do nonsitting tenants. With respect to the tenure averages, note that mean tenure appears somewhat high (4.61 years) but median tenure (2 years) is consistent with previous studies of household mobility. 5. ECONOMETRIC MODELS AND ESTIMATION The objective is to obtain a consistent estimate of the length of residency discount and of the sit discount which was introduced in the theory section. The statistical model of the landlord’s rent-setting decision accounts for the
LENGTH
299
OF RESIDENCY
TABLE 2 Previous Estimates of the Length of Residency Discount Fokin and MaIpezzi [4] Goodman and Kawai [5] Bamett [l], Lowry [12], Noland [15]
- 0.010” -0.011” -0.038"
Note. The discounts are given as fractions of rent paid. “Significant at the 5% level.
censoring in the data by modeling the endogeneity of the tenant’s staying decision. A more detailed discussion of the models and parameter estimates can be found in Marshall and Guasch [14]. Econometric Models
Previous researchers typically cross-section data:
have estimated the following model using
ri = X,‘& + y tenure, + ei
(5)
where ri = log(Ri), and Ri is the contract rent per month; Xi = vector of observed characteristics of unit; tenure, = length of residency in years of the current tenant (i.e., tenure, =2 implies that the tenant has lived in the unit for at least 2 years but not yet 3 years).
Table 2 shows some estimates of the tenure, coefficient obtained by other researchers. In all these studies, the coefficient of tenure is negative and significant. However, these estimates probably are biased and inconsistent estimates of the effect of tenure on asking rents because they have not accounted for the censoring in the data. Rent paid is a poor proxy for rent asked when high reset contract rents, which induce sitting tenants to leave units, never are observed. To overcome the censoring problem, (5) is generalized to a switching regime model. Two alterations are made to the specification. First, the rent observed on a survey date was set in nominal dollars anywhere from 0 to 12 months prior to that date (assuming l-year contracts in the market). It is possible that time-varying unit-specific characteristics observed at the time of the survey have changed in response to the contract rent which was set some time before the survey; some of the Xi may be functions of the rent. Then, estimates of (5) would suffer from simultaneity
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GUASCH AND MARSHALL
bias. The panel nature of the data can be used to surmount this problem since it is reasonable to assume that rent is a function of unit-specific characteristics observed at the previous survey date whereas the converse is inadmissible. Hence we have r; = Xi’, _ 1& + y tenurei + ei.
(6)
Since observed nominal asking rents may be set at some arbitrary time between the last and current survey data, there may be differences in rents due solely to inflation.4 Second, by entering only tenure, as a regressor, the discount offered in the first year for staying in the unit is set equal to the discount offered by the landlord in each year thereafter. In Section 2 we postulated that a difference may exist between the discount offered by landlords for each additional year spent in the unit, the length of residency discount. Each of these different discounts can be estimated by separating tenure into two components. First we define an indicator variable, Zi = 1 if tenure, 2 1 and Zi = 0 otherwise. Note that tenure, = Zi - tenure, = Zi(l + tenure,, -r) = Z, + Z, tenurei, _ i. Therefore, entering Zi and Zi - tenure,, -r separately in (6) instead of just tenure, is equivalent to relaxing a coefficient restriction. Thus, ri = Xi: -J3i + ylZi + y2Zi . tenurei, _ i + ei.
(7)
We can obtain (6) from (7) by imposing the restriction that yi = y2. Equation (7) and the ensuing censoring-corrected generalizations of it are estimated separately for each of the 1975, 1976, and 1977 cross sections. OLS estimation of (7) is equivalent to taking the expectation of r, conditional on Xi, -i, Ii, and Zi . tenure,, -i. For OLS parameter estimates to be unbiased, it is necessary that the expectation of ei conditional on 4A variable D, a correction for the data collection procedure of the AHS, was also included in the X matrix. D is equal to the number of months between the contract negotiation date and the survey date. It varies between 0 and 12 in integer increments. A tenant who is observed to occupy a unit in 2 consecutive survey years will have experienced a rent renegotiation at some point between the surveys. This tenant would be equally likely to experience the renegotiation in any month between the survey dates. However, for units with occupants who have resided in the unit less than 1 year, it is far more probable that the rent was renegotiated closer to the current survey date than to the previous one. This is because a unit which became vacant between survey dates probably experiences a vacancy of some duration before a new occupant is found. Without correcting for this bias, a negative coefficient on tenure may be found simply because new tenants on average negotiate rents closer to the current survey date than do sitting tenants. In fact, for the entire data set, the average value of D for sitting tenants is 4.97 while for recent occupants the average value of D is 4.01.
LENGTH
OF
301
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Survey
Contract renegotiotion
Survey
I t-s t
t-1 FIGURE
1
xi, - 17 I,, and Ii . tenure,, -i be equal to zero. The censoring correction to be employed is based on evaluation of E(.ri 11,). If we can specify the asking and reservation rent equations at the time of a contract renegotiation, we can formulate the stochastic process describing the probabilities with which a tenant decides to stay in or leave a unit. Figure 1 illustrates the correspondence between the survey data and economic decisions made by agents. The survey records information about the unit and tenant at t - 1 and 1 year later at t. With l-year contracts in the market, the contract renegotiation for the tenant in the unit at t - 1 will occur somewhere between t - 1 and t, say at t - 6 (where 0 < S c 1). The tenant formulates a reservation rent at t - 6 which depends on the characteristics of his household and the unit and, in addition, may depend on the number of contract periods he has resided in the unit. The reservation rent is not observed. The landlord formulates an asking rent which is observed at t if and only if the tenant decides to stay in the unit. However, characteristics of the unit and of the tenant who occupied the unit at t - 1 are observed at t - 1. Therefore, it is possible to specify an asking rent and reservation rent equation at t - S. These are below:
asking rent: reservation rent:
r,* = Xl -i & + y1 + y2 tenurei, -i + &i rires = xi’,-lP2 + Z/,-I& f Y3 + Y4 tenure,,-,
(8) + E3, (9)
where
zi -I=
vector of tenant-specific characteristics (see Appendix B for definition of the variables in Z), reS= log of the reservation rent of the current tenant at t - 6, and ‘:.*I = log of the monthly contract rent asked by the landlord of the current occupant at t - 6.
Whereas (8) captures observed rent-setting behavior, (9) describes how landlords set rents at times of contract renegotiation, whether or not these
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GUASCH AND MARSHALL
rents are observed (ri = ri* if Ii = 1; ri* will be unobserved if Ii = 0). Whether or not the tenant stays in the unit depends on the relationship between ri* and rims. The tenant stays in the unit, or Ii = 1, if ry - ri* 2 0 or
xi:-l(& - L$>+ Zil,--lPj+ (Y3- Yl>+ (Y4- Y2)tenure,,-,2 &i- &3i* 00) Since both (8) and (9) are formulated conditional on the tenant having been in the unit tenure, years at the time of the contract renegotiation, (y3 - yi) is unidentifiable and we amalgamate it with the Xi,,i constant coefficient and maintain the j3 notation. Then the above condruon can be rewritten as X[-l(&
- PI) + Zil,...#3 + (y4 - y2) tenure,,-,
2 Ei - Eji.
Let Qi = (Xi:-1, Z( -1, tenurei, -I), P’ = (82 - &83, ~4 - YZ), and LJ = q - e3i, Then the above condition becomes QiP > vi. The tenant leaves the unit, or Ii = 0, if rfes - ri < 0 or QiP < vi. We alIow the conditional variance of log rent for sitting tenants to differ from that for new occupants. This leads to ri=x;-l
Instead .E(E&
,
j3 1 + ylIi + y21i - tenure,,-,
of calculating = 0). Assuming
and utilizing we have
E(qlli),
we need
results established by Heckman
E(rilXi,
+ Ii&ii + (1 - Ii)+.
(11)
lJ3(~i~l1~ = 1) + (1 - Ii)
[6,7] and Lee and Trost [ll],
-1, Ii, Ii * tenurei, -1)
= xi~-lpl
+ F(l
+ ylI, + y21i - fIJj ”
- li)hiF(Q,P/uv)/F(-Q,P/u,).
(12)
LENGTH
303
OF RESIDENCY
where f(x) F(x)
= standard normal density evaluated at x; = cumulative distribution of the standard normal at x.
If we knew P/u,, then the parameters in (12) could be estimated by OLS. But we do not. However P/u, can be estimated by probit analysis. This is step 1 of Lee and Trost’s [ll] two-step estimation method. Once (P/u,) is obtained, OLS can be applied to the equation below. r;=x;-l ,
p 1 + ylIi + yzIi . tenure,, -i
+%(I - Ii)fiiF(Qi(~“))/‘(-Qi(~“)) ”
+ qi
(‘3)
where E(qi]li) = 0. OLS applied to (13) is step 2 of Lee and Trost’s [ll] two-step technique. Consistent estimates of the standard errors of the estimated parameters can be obtained by a White’s [16] technique as modified by Lee [9]. Since the two-step estimation method provides consistent estimates of the parameters, these can be used to start the maximum likelihood iteration.5 Then only one step of scoring need be taken to obtain estimates which are fully efficient (of the first order). Alternatively, we could have formulated the stochastic process described above as a switching regime model. In regime 1, the tenant stays in the unit and an asking rent is observed which possibly contains residency discounts. In regime 2, the tenant leaves and the vacant rental price is observed, regime 1: iff regime 2 : iff
ri=x/-l , /I 1 + y1 + y2 tenure,, -1 + &li Ii = 1 QiP 2 ui or pi = x~-,a, + &2i QiP < vi or Ii = 0.
‘In addition to 8, (G;,), q, and c, can be obtained from the equations
04 05)
consistent estimates of &T and 6et are needed. These
where N1 and N2 are the number of observations from regimes 1 and 2, respectively.
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GUASCH AND IviARSHALL
Since P is estimable only up to a scale factor, we impose the normalization here and throughout that u,’ = 1. Note that (14) of regime 1 and (15) of regime 2 can be stacked together by multiplication with Ii and (1 - Ii), respectively, to yield (11) where the error covariance for regimes 1 and 2 is assumed to be the same as that for (11). It is clear from (14) and (15) that across-regime coefficient restrictions have been imposed, namely, the coefficients of the Xi, -1 are assumed equal. This constraint implies that landlords evaluate the market worth of their unit-specific characteristics in the same way regardless of the occupancy status of their unit. The likelihood function for the switching regime model follows immediately. Let f(qi, vi) be the bivariate normal density of qi and vi, and let g(ezi, vi) be the bivariate normal density of ezi and vi. Then
Estimation In Table 3, estimates of the sit and length of residency discounts are reported. For expositional clarity, we have reported neither coefficients of the Xi, -i nor probit estimates.6 Columns (1) and (2) of Table 3 report OLS estimates without censoring corrections for the pooled data set. All remaining columns present estimates for models in a given year of the data set. Columns (3) (6) and (9) are uncorrected OLS estimates of (7) for 1975, 1976, and 1977, respectively. Columns (4), (7) and (10) present two-step censoring-corrected estimates of (13) for the corresponding years while columns (5), (8), and (11) contain maximum likelihood estimates of the switching regime model described by regime 1 and regime 2 where the likelihood function is given by (16) (again for the years 1975, 1976, and 1977, respectively). ‘The X matrix consists of variables (1) through (17), 10 distinct city dummies (Philadelphia is the excluded dummy) and, in the case of panel estimates, two time dummies for 1976 and 1977. The Z matrix consists of variables (18) through (23) and lagged tenure. Of course, estimates of the coefficients of Z in the offer equation are not possible (not identified), but certainly the coefficients from the probit estimates are possible. The two-step consistent estimates rely on probit estimates which are also not reported. All the unreported estimates and the data set are available from the authors upon request.
- 0.0100 (3.26)
Note. fraction “Ratio “Ratio “Ratio
- 0.0059
(1.63)
- 0.00724
(2.16)
0.591 38.115 0.2923 477
- 0.10458 (3.36)
- 0.09%5 (4.64)
0.564 131.074 0.3064 1431
OLS”
OLS”
(3)
- 0.1379 (1.37) + 0.0578 (0.84) 0.594 37.813 0.2918 477
(0.63)
- 0.0028
-0.1161 (0.97)
Censoring b correction
(4) 1975
477
- 0.1767 (1.94) + 0.0578 (0.70)
(0.87)
- 0.0024
- 0.1321 (1.15)
MLE’
(5)
0.568 42.934 0.3102 477
(2.08) - 0.1704 (1.76) +0.1019 (1.56) 0.574 42.296 0.3086 477
(0.95)
-0.0044
(0.57)
- 0.0078
- 0.0635
(2.46)
Censoring b correction
(7) 1976
- 0.0844
OLS”
(6)
477
-0.1707 (2.08) +0.1019 (1.15)
- 0.0637 (0.55) - 0.0044 (1.54)
MLE’
(8)
0.534 48.211 0.3287 477
- 0.1233 (3.68) - 0.0078 (2.09)
OLS”
(9)
Estimates for 1975,1976, and 1977 cross sections
-0mo6 (0.W + 0.2580 (2.45) 0.541 47.400 0.3268 477
l +0.1587 (0.70) - 0.0071 (1.32)
Censoring’ correction
(10) 1977
477
+ 0.0195 (0.10) + 0.2822 (2.30)
+0.1760 (0.83) - 0.0073 (1.81)
MLE’
(11)
yt measures the sit discount. y2 measures the discount per year of tenancy after the first year. Both discounts are given as a of rent asked in columns (4), (5), (7), (8), (lo), and (11) and as a fraction of rent paid in columns (2), (3), (6), and (9). of coefficient estimates to White standard errors for panel data reported in parentheses. of coefficient estimates to White standard errors for cross-section data reported in parentheses. of coefficient estimates to maximum likelihood estimates of standard error reported in parentheses.
R2 0.558 SSR 132.880 SEE 0.3083 N 1431
0Q”
0Cl”
Y
Y2
Yl
OLS”
Pooled estimates ~_..____ (2) (1)
TABLE 3 Estimates of the Discount
306
GUASCH
AND MARSHALL
Column (1) reports an OLS estimate of the panel equivalent of (6) for the pooled data set.’ The ratio of p to its White [16] standard error is reported in parentheses below 9. p is presented to establish comparability between these data and those used by other researchers. A 1% yearly significant discount is found which is within the range of discounts reported in Table 2. Column (2) presents an OLS estimate of the panel equivalent of (7). The sit discount is approximately 10% while the length of residency discount now is reduced to 0.7%. Both coefficients are significant. This initial result indicates that a substantial discount is being offered for the first year of occupancy and that the discount increases slightly each year thereafter. The yearly OLS estimates T1 and j$ in columns (3), (6), and (9) indicate the same result. Again, these results should not be interpreted as conclusive since they potentially suffer from censoring bias. However, for each year of the panel the censoring-corrected estimates of (13), reported in columns (4), (7), and (lo), yield insignificant estimates of the sit and length of residency discount. While many of the coefficients have not changed greatly from the estimates in columns (3), (6), and (9), they no longer are constant across samples with the mean of Ti = 0 and that of 92 = -0.005. Thus the evidence of large and significant discounts disappears once the anticipated censoring bias is corrected. Although these coefficient estimates are consistent, they are not fully efficient. The fully efficient maximum likelihood estimates in columns (5), (8), and (11) confirm this result. In no case are either T1 or j$ signifkantly different from zero. We are unable to reject either the null hypothesis that yi = 0 or the null hypothesis that y2 = 0 and, in addition to their variation between samples, the means of the estimated yi and y2 parameters are very small. Conversely, in each cross section one of the covariance parameters is playing an important role, again indicative of some censoring bias. Thus the significance of 7 in column (1) and also 9i and R in columns (2), (3), (6), and (9). Once the tenant’s staying decision is modeled as directly dependent on the landlord’s reset asking rent, we find no evidence that landlords are offering either sit or length of residency discounts to their occupants. Indeed, these results indicate that the negative significant estimates of the discounts found in columns (l), (2), (3), (6), and (9) appear to be due to the fact that tenants stay in units for which they pay low rents, rather than the result of systematic intentional discounting of rents by landlords.
‘The model here is ‘;, = XLmI& gives the panel model
+ y tenure,, + q,. Breaking tenurei, into two components
q, = X,:- 1PI + yl I,, + y2 I,,
tenure,,-1 + Ed,.
LENGTH
OF RESIDENCY
307
6. CONCLUSION Using data from the Annual Housing Survey, we have consistently and efficiently estimated the sit and length of residency discounts, accounting for censoring in the data by modeling the endogeneity of the tenant’s staying decision. To account for the censoring bias, the landlord’s rent-setting decision has been posed within a switching regime model. We have been unable to reject either the null hypothesis that yi = 0 or that yz = 0 even with a sample of 477 units (see Learner [8]), but it is well known that this does not imply that discounts are nonexistent. It means that after correcting for censoring in the data, we have insufficient evidence to conclude definitively that landlords offer residency discounts. Nevertheless, this apparent nonresult has both statistical and economic content. The much smaller mean numerical values of estimated discounts cautions against accepting the values reported in potentially biased situations.8 In light of our theoretical formulation, the empirical results tend to indicate that landlords with occupied units fully correct rents by the inflation rate. Other factors not considered in our theoretical model might also help to account for the apparent nonexistence of discounts. We assumed that the transaction costs were state dependent, and that they did not occur in the state of occupancy. However, the marketing cost c which was modeled as a per period cost for the duration of a vacancy may, in fact, be a fixed cost to the landlord. With regard to the reconditioning costs (Y, most lease contracts require a security deposit which is refundable after a tenant leaves a unit conditional on the status of the unit. APPENDIX A Proof of theorem. To show the result, it suffices to demonstrate that the derivative of (3), the objective function of the landlord with an occupancy, with respect to R,+l evaluated at the optimal asking rent of the one with a vacancy, R;, 1, is negative. The first-order condition for (2) is
av,V,1= 4+dK+1)b,+l(K+1 -JR t+1 +rt+1
( R;,,)[
u;+~(R:+~
- 4 + PVP:*(Rt+*l~t+2)1 -
c~+~
- c,+l) + p avp:,!$;+tzt+2)]
-~;+,(R~+,)[pl/,~~(R,+,) + u,+k-c,+1)1= 0. Let R;+l
(‘w
be the solution of the above equation. Then the derivative of (3)
*Also note that our corrected models were capable of reproducing the results of other researchers as they embodied as specializations the caseswhen (1) Ok,”= uee,”= 0, (2) ucpz= q,, ad (3) K = YZ.
308
GUASCH
with respect to R,+l
AND MARSHALL
evaluated at R:+ 1 is
wL(Rt+llzt+l) 8R,+1
R’ r+1
n,:,(R:+l)[u,+,(R:+,)
+
PVP:I(R~+K+~] w%Rt+AZ,+d ,jR
+ ut+l C-K1+1
-~l+,(R:+,)[pV,‘,‘,(R,+,)
1
r+1
(A-2)
)I]
1 x
UJ
f,,
1 ( ROffer)
&Offer
’
JZ 1+1 We show, term by term, that (A-2) is smaller than (A-l) establishing the result. As the rent increases, the probability of renting the unit falls so rt’+,(R:+J < 0, and since u( R’;,,) > u( Ry,, - ct+& the first term of (A-2) is smaller than the first term of (A-l). By concavity, u;+ 1(RT+l c,+d ’ u:+,(R:+,) so th e second term of (A-2) is smaller than the second term of (A-l). Finally, since K,+l 2 c,+i and - 7~‘(Ry,,) > 0, the third term of (A-2) is also smaller than the third term of (4). Therefore, < 0. Now note that RJ,, = R:(l + i), namely, that the ~vp,1/~R,+,IK+1 nominal asked rent for a vacant unit at t + 1 equals the nominal rent at t adjusted by the inflation rate or that the real asked rent for vacant units remains constant. Recall that landlords conjecture that his tenant is willing to pay at least Z,,, at t + 1. Thus unless Zl+l = R;(l + i), we have &‘+I < R:+l; otherwise a boundary solution on the above maximization problem occurs and RT+l = Ry+l, with no occupancy discounts being offered. APPENDIX
B: VARIABLE
DEFINITIONS
AND CODINGS
Coding
Definition of coding
Rent Sit or I
In dollars 1
Tenure
0 T-T
Monthly contract rent Current tenant has been in unit at least 1 year In unit less than 1 year TT = No. of years tenant has lived in unit Unit located in central city Not in central city Unit built before 1939 Built between 1940 and 1950 Built between 1950 and 1960 Built between 1960 and 1964
Variable
(1) Metro
(2) Age
1 0 30 45 55 62
LENGTH
APPENDIX Variable
Coding 66 69 YY
(3) Nunits
(4) Rooms (5) Kitch (6) Air (7) Elec (8) Gas (9) Oil (10) St Noise (11) St Crime (12) Aban (13) Run-down (14) Odors (15) Tram (16) Sch (17) Police (18) Rstr (19) Rhse (20) Inc (21) Grade
(22) Age
(23) HHsize
1 2 3 6 12 25 60 N 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 DDDDD 1 0 YY NN
309
OF RESIDENCY
B: -Continued Definition of coding Built between 1965 and 1968 Built between 1969 and 1970 YY. = year built if built after March 1,197O 1 unit in the structure 2 units in the structure 3 to 4 units in the structure 5 to 9 units in the structure 10 to 19 units in the structure 20 to 49 units in the structure 50 + units in the structure N = number of rooms in the unit Exclusive use of a kitchen No exclusive use of a kitchen Air-conditioned unit No air conditioning Consumption of electricity included in the rent Not included Consumption of gas included in rent Not included Consumption of oil included in rent Not included Street noise is not bothersome Bothersome Street crime is not prevalent Prevalent No abandoned buildings in neighborhood Condition exists Run-down buildings in neighborhood They exist No bad odors in the neighborhood Condition exists Public transportation adequate No adequate Schools adequate Not adequate Police protection adequate Not adequate Resident likes street on which unit is located Does not like Resident likes house or apartment Does not like DDDDD = Gross family income in dollars Head of household is high school graduate Not high school graduate YY = Age of head in years NN = Number of people in household
310
GUASCH
AND MARSHALL
The tenure variable was constructed by subtracting the date the head of household moved in from the date of the survey and truncating the difference. Hence, if in 1974 the current head of household moved in, then in March 1974 tenure is coded zero. If in March 1972 the head moved in, then the tenure variable is coded 2. Our data set was constructed to meet five criteria: (a) none of the cities had any form of rent control legislation during the period 1974 to 1977; (b) a rental unit must be an apartment or house and not a mobile home or a unit in a transient hotel; (c) any unit which was not interviewed in a given year is excluded from the sample; (d) a unit must be occupied during all 4 years at the time of the survey; and (e) any unit which had a miscoding in a time-invariant variable or other crucial variable was excluded from the sample. At no time was the monthly contract rent used to determine whether a unit should be excluded from the sample. There are two main types of miscodings in the data. First, over the course of the panel for a given unit, time-invariant variables may vary. If for a given unit the year that the structure was built is coded 1972 for 3 years and in the fourth year the coding is 1930, then the unit was excluded from the data set. Second, if in a year the date that the head of household moved in is less than the date that the head of household moved in during an earlier year, then the unit was also excluded. For a general discussion of the inconsistencies in the codings of variables in the Annual Housing Survey, see Beveridge [2]. REFERENCES 1. L. Bamett, “Using Hedonic Indexes to Measure Housing Quality.” The Rand Corporation, Santa Monica, CA, R-2450-HUD (1979). 2. A. A. Beveridge, Reliability in large scale household surveys: The case of the Annual Housing Survey, Mimeo presented at the Annual Meeting of the Association of Public Data Users, Washington, DC (Oct. 14-15, 1982). 3. R. F. EngIe and R. C. Marshall A micro-econometric analysis of vacant rental housing units, in “The Urban Economy and Housing” (R. E. Grieson, Ed.), Heath, Lexington, MA (1982). 4. J. R. Follain and S. MaIpezzi, “Dissecting Housing Value and Rent,” The Urban Institute (1980). 5. A. C. Goodman and M. Kawai, Length of residency discounts and rental housing demand: Theory and evidence, Working Paper No. 108, Johns Hopkins University, (1982). 6. J. J. Heckman, The common structure of statistical models of truncation, sample selection, and limited dependent variables, and a simple estimator for such models, Ann. Econom. Social Measure., S(4), 475-492 (1976). 7. J. J. Heckman, Dummy endogenous variables in a simultaneous equations systems, Econometrica, 46(6), 931-959 (1978). 8. E. E. Learner, “Specification Searches: Ad Hoc Inference with Non-experimental data”, Wiley, New York (1978). 9. L-F. Lee, Identification and estimation in binary choice models with limited (censored) dependent variables, Econometrica, 47(4), 977-995 (1979). 10. L.-F. Lee, Some approaches to the correction of selectivity bias, Rev. Econom. Stud., 49(157), 355-372 (1982).
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OF RESIDENCY
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