- Email: [email protected]
Moving behavior and the housing market
ELSEVIER Regional Science and Urban Economics 25 (1995) 21-39
Moving behavior and the housing market A n n a M . H a r d m a n a, Y a n n i s M. I o a n n i d e s b'* aDepartments of Urban Affairs and Planning and of Economics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA bDepartment of Economics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA
Received September 1992, final version received March 1994
Abstract
This paper develops a model of the relationship between residential mobility and housing demand, which it embeds in an overlapping-generations model of the housing market. Moves, needed to adjust housing consumption, are costly; adjusting other consumption is not. Comparative static results for the impact of moving costs and the interest rate on the optimal number of moves and housing consumption in each residence spell are presented. We demonstrate that moving costs and constraints on moving can serve as a form of rationing of housing consumption which can be used in analyzing certain policy interventions in housing markets. Key words: Moving; Residential mobility; Housing; Frictions JEL classification: R21, R31, D50
1. Introduction
Residential m o b i l i t y - m o v i n g - is an important and relatively neglected feature of housing markets. In competitive housing markets, households decide how much housing to consume. In order to adjust its housing consumption to match changes in income, prices, or preferences, a household must enter the market, find a new dwelling and then move. Because * Corresponding author. 0166-0462/95/$09.50 (~) 1995 Elsevier ~icience B.C. All rights reserved SSDI 0166-0462(94)02065-5
22
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
search and moving are costly, most households adjust their housing consumption relatively infrequently. Thus, at any given moment, only a small fraction of the owners of housing are in the market to arrange trades of housing units or housing services. Similarly, most consumers of housing services, whether they are renters or owners, enter the market infrequently. Housing markets allocate shelter to households when they enter the market; they also provide housing equity, the most important and most commonly held asset in household asset portfolios. When institutional restrictions impede residential mobility, the allocative role of housing markets is disrupted. A number of papers on housing markets have addressed important aspects of residential mobility. The main emphasis has been on empirical aspects of moving behavior. 1 Another strand of empirical research has modeled the duration of residence spells (the intervals between moves). 2 Several recent papers present evidence that renters in rent-controlled housing markets move less frequently and consume less than optimal quantities of housing. 3 The administrative allocation of publicly owned housing has been shown to have similar effects in both Western and Eastern Europe. 4 In Europe, too, a number of empirical studies have examined the consequences of housing policies for labor mobility and hence the efficiency of labor markets. 5 Amundsen (1985) and Englund (1985), have explored residential mobility using models of individual behavior. The empirical literature cited above highlights the importance of interactions between housing policy and residential mobility. It is clear that policies which affect residential mobility have potentially significant effects on labor and capital markets as well as the housing market. Yet this area has so far received little attention in the theoretical literature. This paper integrates the residential mobility process into a model of the housing market by assuming that the quantity of housing consumed can be adjusted only by moving, and that moving is costly in terms of utility. The model allows us to compare states in which residential mobility is freely
For example, Hanushek and Quigley (1978), Venti and Wise (1984), loannides (1987), Feinstein and McFadden (1988), Rosenthal (1988), Henderson and Ioannides (1989), and Edin and Englund (1991). 2For example, Ioannides (1987), Rosenthal (1988), Henderson and Ioannides (1989) and Edin and Englund (1991). 3See, for example, Clark and Heskin (1982) and Olsen (1990) for the United States, Hardman (1987, 1990) for Egypt, Willis et al. (1990) for Ghana, and Turner (1988) for Sweden. 4 For example, for Poland, see Mayo and Stein (1988), and for the United Kingdom, Coleman (1988). 5See, for example, Mayo and Stein (1988), and Hughes and McCormick (1981, 1987).
A . M . Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
23
chosen with states in which it is impeded and households are not free to select the optimal number of moves. The remainder of this paper is organized as follows. In Section 2 we solve the household's lifetime optimization problem and examine its properties. In Section 3 we use the same model to characterize housing market equilibrium. Section 4 concludes.
2. The behavioral model
To examine the consequences of moving costs we use an approach that is grounded in Romer's model of the transactions demand for money (Romer, 1986). 6 The basic behavioral model used here is related to the models in Amundsen (1985) and in Englund (1985). 7 Consumption decisions are made by households. Households enter the economy and die continuously; each has a finite lifetime. An infinite number of overlapping generations of households coexist at any point in time, and households differ only with respect to the time they enter the economy. Households adjust their housing consumption at a finite number of points in time. With a large number of households, this discreteness is smoothed in the aggregate, so that market demand can be examined using standard tools. The household's optimization problem in an unrestricted environment is as follows. Consider a household which is formed at time t o and dies at time t o + T. Let ( t o , t 1 . . . . . t n _ l } , t o <~ t 1 • • • <~ t , _ 1 <~ t o + T , be the points in time when adjustments in the quantity of housing take place, with the corresponding housing quantities being {H 0(t 0) . . . . , Hn- l(t0) )- That is, Hj(t 0) is the housing stock purchased at time tj by a household formed at time t 0, j = 0 , . . . , n - 1. For completeness, let t n -- t o + T. We assume that a unit of housing stock generates a unit of housing services per unit of time. Households may adjust the amount of housing services they consume only by moving. The quantity of housing services consumed during each residence spell (the interval between moves) is constant. 8 6 See also Blanchard and Fischer (1989, pp. 168-178), who popularized Romer's model. 7 Amundsen's model of the role of moving costs in long-term optimal moving strategies assumes that utility is not discounted, prices are constant, there is a positive rate of interest, and moving costs are monetary and grow exponentially over time. Englund incorporates moving into a model of the housing market which assumes that non-housing consumption remains constant over an infinite lifetime horizon, and housing consumption may be adjusted only by means of residential moves. He adopts an isoelastic form for utility per unit of time, which implies log utility as a special case. s This feature reflects the quasi-fixed nature of housing. Housing deteriorates slowly, and not much else, except analytical tractability, is affected if housing consumption is allowed to vary exogenously during a residence spell.
24
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
Let Pc(t) and PH(t) denote, respectively, the nominal prices of nonhousing consumption per unit of time and of a unit of housing stock. C(t) denotes non-housing consumption during the residence spell which begins at time tj.9 Let Wo(to) denote lifetime human wealth for a household formed at time to, and i denote the nominal rate of interest. For simplicity we consider only dynamics along steady-state paths of the economy. The nominal rate of interest i is therefore taken as time-invariant and equal to its steady state value (cf. Cass and Yaari, 1967). The lifetime budget constraint may be written as
Wo(to) =
e-i(t-%1)Pc(t)C(t ) dt + ffI(to),
e -i(tj-l-'°) j=l
(1)
j--1
where t n --- t o + T, and/~(t0) denotes the present value at the beginning of its life of the household's net expenditure on housing. For a household formed at t o which plans to purchase Hj_~(to) units of housing stock at tj_ 1, hold it during the residence spell [tj_~, tj) and sell it at time tj, j = 0, 1 . . . . . n - 1, /-/(to) is given by
f-l(to) = ~ Hj_I(to)[PH(tj_I) e -i(tj-l-'°) - PH(t~) e-i(tj-t°)]
.
(2)
j=l
We assume that households are owner-occupiers. This formulation subsumes the notion of housing wealth under H(t0). ~° We assume that households value housing and non-housing consumption during their own lifetime and leave no bequests. H A household's lifetime utility maximization problem is stated as follows. A household chooses (C(t);Ho(to) . . . . . nn_l, t I ..... tt_l} SO as to maximize 9 We do not subscript C(t) because under our assumption that utility per unit of time is separable in housing and non-housing consumption, the optimal non-housing consumption does not depend upon the spell. On this, see Amundsen (1985, p. 577, Proposition 2). 10 Everything carries through if we assume instead that housing is rented, the housing stock is owned by private entrepreneurs or the state, and property owners earn the market rate of return in their capacity as landlords. This is clear if we recognize that a renter household's expenditure on housing services must obey the arbitrage condition that the owner of a unit of housing stock should be indifferent between selling it at time tj 1, and holding on to it, renting it for O(t), and subsequently selling it at time 4:
Pl-i(tj 1) e-i(Q-x to) = Pu(tJ) e-i('J-'°) + f,i' , O(t) e i('-'°) dt. 11 We purposefully leave out time-discounting for utility in order to avail ourselves, like A m u n d s e n (1985) and Romer (1986), of a substantial simplification of an otherwise intractable optimization problem. Nonetheless, we argue that an important implication of this assumption, namely that housing consumption increases over time, is at least partially defensible in the light of the empirical evidence. We take this up further later in the section.
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
/20=
ac j=l
lnC(t) dt+aH(ti-tj_l)lnHj_
1 -Bn,
25
(3)
tj 1
where parameters a c and a H satisfy a c > 0, a H > 0, and a c + a H = 1, and B denotes the utility cost of adjusting housing per move, subject to budget constraint (1). The utility cost B n is the only cost associated with the n moves the household plans to make over its lifetime. 12 This assumption, which is borrowed from Romer, accounts for the simplicity of the results obtained from our model. Some of the results below also depend critically on the assumption that utility is not discounted. 13 We obtain the necessary conditions for lifetime utility maximization in two stages. First we characterize the optimal values for the rate of consumption of housing and non-housing goods per unit of time, assuming that the values for the tj's are given. At a second stage we optimize with respect to the tj's. Let A denote the Lagrange multiplier that adjoins constraint (1) to lifetime utility according to (3). The necessary conditions for the rate of consumption C ( t ) , t E [0, T], become 1 ac
C(t)
- A e-i('-'°)Pc(t ) ,
t o <<-t <~ t o + T .
(4)
It follows from (4) that under our assumptions the intertemporal variation in non-housing consumption is independent of moving costs. Consumption grows over time at a rate that depends on the rate of interest and on the price of consumption. Its actual magnitude is determined through the Lagrange multiplier A, which reflects lifetime resources. In the remainder of the paper we assume that the price of consumption is constant. No loss of generality is involved here because we study steadystate dynamics only. From that and from (1), (2) and (4), we obtain the optimal rate of consumption as follows:
C(t)-
W o ( t o ) - ff-I e (i)(t_to) TPc(to ) .
(5)
The optimal rate of consumption grows at a constant rate equal to i. By substituting into the expression for the lifetime utility function in (3) we have 12B can be seen as a function of the economy's information structure: a small value of B is associated with market features which facilitate matching between buyers and sellers. 13Both Amundsen (1985) and Romer (1986) do not discount for time.
26
A.M. Hardman, Y.M. Ioannides / Reg. Sci. Urban Econ. 25 (1995)21-39
i a o = ~ a c T2 - a c T [ I n T + In Pc(to)] + a c T + a , ~ (tj - t j _ , ) l n H j _ l ( t o ) j=l
In(W0-/t) (6)
- Bn.
Just as they adjust non-housing consumption, households wish to adjust their consumption of housing over time. The necessary conditions for the quantities of housing stock purchased each time a move occurs, /-/j, are obtained by differentiating (6) with respect to Hi, j = 0 . . . . , n - 1, and by using (2) and are as follows: 1 OlH(t
j --
tj 1)-Hj_l
1 OH a c T W o - ffI OHi-~I - O.
(7)
By rearranging (7) and by summing up over all j's we obtain /4 = aHW0.
A household's total expenditure on housing is a constant fraction of its initial wealth. This result is responsible for the tractability of the model; it depends crucially upon the assumption of no time-discounting of utility. It follows that the rate of consumption, from (5), becomes C(t)
acWo TPc(to ) e i(t-t°) ,
(8)
t o <~ t <~ t o + T .
The demand for housing stock at the time of the jth move by a household which enters the economy at time to, is then Hj l(t0)
--
-
tj - tj_ I auWo[PH(tj T
1) e-i('J
l-t°)
--
PH(tj) e i(tF'°)] -1
(9)
-
We assume that the price of housing stock increases at a constant growth rate X, PH(t) = Pn(to) eX('-'°).~4 Eq. (9) j = 1 . . . . . n, now becomes Hj l(t0) --
tj - tj_ l 1 T aHWo PH(tO) e-(i-x)(t,-1-to)[1 _ e-(i-x)(,,-',
i)] " (10)
The optimal quantity of housing stock acquired at the time of the jth move depends on the time the respective spell starts and on its duration. Under our assumptions, Hi(to) depends upon t o only via Wo(to) and Pu(to). This homogeneity result allows us to aggregate. If moves occur at N equidistant 14In Hardman and loannides (1991) we verify that this is indeed the case in general equilibrium.
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995)21-39
27
points in time, which is indeed the case at the optimum as we show below, (9) becomes 1 1 Hi(to) = - ~ aHWo pH(to) e-(i-x)(Jr/N) {1 _ e-(i-x)(r/N)} •
(11)
For (9) to yield a positive value for the housing stock Hi, we must have X < i: that is, the rate of housing price increase is less than the nominal rate of interest; owning housing stock entails an opportunity cost. If X > i then housing becomes an asset superior to saving at an interest rate i. If X < i, then the amount of housing stock desired increases with every spell, as the present value of the price of housing stock goes down. The condition may be motivated as follows. The total return to housing reflects both the housing services it produces and a possible capital gain. If the rate of capital gain were greater than the rate of interest then the total return to housing would exceed the rate of interest. With a perfect capital market, as long as housing and non-housing assets are held at equilibrium, it must be the case that x
Another implication of (11) is that the present value of housing wealth held by a household at the beginning of each spell is independent of the particular spell and equal to N-lotHWo[1 -- e-(i-x)(T/N)] -1 . This yields that in its lifetime the typical household spends on housing purchases an amount equal to aHW0[1 - e-(i-x)(r/u)] -~ and receives from housing sales an amount equal to a H W o e-(i-x)(r/u)[1 e-(i-x)(T/N)]. Our assumption of no time-discounting in our model is principally responsible for the result that housing demand, from (11), grows over a household's lifetime. This conflicts with both the standard life-cycle model and empirical findings of researchers who use the residence spells model and who find that housing demand peaks in mid life. ~5 However, other recent research which has emphasized housing decisions by the elderly a6 has shown that most elderly households often do not choose to reduce their housing consumption and especially their housing equity as much as the strict lifecycle model predicts. -
2.1. O p t i m a l times o f m o v e s
It remains to optimize with respect to the times of moves and thus the lengths of spells. We note that under our assumptions the optimal value of the consumption component of lifetime utility has been expressed explicitly in terms of parameters. The optimal times of moves follow from the is See Rosenthal (1988), Hardman (1987) and Henderson and Ioannides (1989). 16See Feinstein and McFadden (1988).
28
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
m a x i m i z a t i o n with r e s p e c t to ( t l , . . . , (6), that is:
-Bn
tn_l}
of the h o u s i n g c o m p o n e n t of
+ ~ OtH(tj -- tj_l){ln(t j -- tj_l) -- In T + l n ( a H W o ( t o ) / T ) - In Pn(O) j=l
+ (i - X ) ( t j _ 1 - to) - ln[1 - e -~i-x)~t' tj_~)]} .
(12)
B y d i f f e r e n t i a t i n g (12) with r e s p e c t to tj we o b t a i n a difference e q u a t i o n i n v o l v i n g (t]_l, tj, ti+l)
t +1-tj - I n tj - tj_ 1 - (i -x)[(tj+ -
-
1 - tj) - (tj - tj_l)]
1 - e x p [ - ( i - x)(ti+l - tj)] +In
1 - e x p [ - ( i - x)(tj - tj_l) ]
- ( t j - tj_l)(i - X )
e x p [ - (i - x)(tj - t~_l)] 1 - e x p [ - ( i - x ) ( t j - tj 1)]
e x p [ - ( i - x)(tj+l - tj)] +(tj÷ 1 - tj)(i - X) 1 - e x p [ - ( i - x)(tj+l - tj)] - 0 .
(13)
It can be s h o w n that given (ti_l, tj+l), tj = (tj 1 + t j + l ) / 2 satisfies (13), a n d t h a t it is t h e u n i q u e m a x i m i z i n g solution. It follows that all spells h a v e e q u a l lengths. 17 This result does n o t d e p e n d on o u r a s s u m p t i o n that the utility cost of m o v e s is p r o p o r t i o n a l to the n u m b e r of m o v e s . W e n o w c o n s i d e r the n u m b e r of spells, n = N, as the u n k n o w n variable, w h e r e tj - t] 1 T / N , which we treat as a c o n t i n u o u s l y v a r y i n g variable. 18 In d o i n g so we follow a n u m b e r of p r e v i o u s a u t h o r s including, in particular, A m u n d s e n (1985) a n d R o m e r (1986). B y using (11) in (12) we h a v e =
-BN
a.W o T(N+ a H T In Pg(to-------~ - In N + (i - X) 2N
-ln[1-exp[-(i-x)T]]].
1) (14)
T h e o p t i m a l length of r e s i d e n c e spells follows by m a x i m i z i n g (14) with r e s p e c t to the n u m b e r of spells, which we treat as a real n u m b e r .
17This result is reminiscent of Amundsen (1985), Proposition 3, when the rate of growth of moving costs is equal to the rate of interest. Our result is identical to Romer (1986), except that he deals with a single good, consumption, and it is trips to the bank that are equidistant over time. 18A new symbol for the number of moves is used in order to underscore that it is a decision variable.
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
29
Thus by differentiating (14) with respect to n we obtain the following necessary condition for the optimal number of spells: +
T2 e x p [ - (i - x ) ( T / N ) ] Nz (i - X) 2 1 - e x p [ - ( i - x ) ( T / N ) ]
/3 _ ( i _ x ) (i - X) an
1 T2 + (i - X) 2 -~ N---7 = 0.
T 1 n (15)
T h e optimal number of spells is obtained implicitly from (15) as its unique positive root. By using the auxiliary unknown x = - ( i - x ) ( T / N ) and rewriting (15) we have -x X2
e
1-e
_x-L
+ x - y x1
2
,
where L = - ( i - x ) ( B / a n ) . Rewriting once again yields L +x+½x
Q(x; L ) -
L +x -ix
2
z = eX.
(16)
T h e RHS of the above is the exponential function, an increasing function CeX~, with well-known properties: e ° = 1, ~ ix=0 = 1. For the LHS, which we denote by Q(x; L ) we have by differentiating with respect to x:
OQ(x; L ) 0x
2Lx + x 2 ' ( L + x - ½ x 2 ) e"
T h e r e f o r e , the function Q ( x ; L ) has the following properties: it is increasing for all positive values of x; it tends to +0% when x--* 1 + ~/1 + 2L from below, and to -0% when x---~ 1 + V1 + 2L from above; and, it tends to _ 1 , when x---~ +o0. Its value for x = 0 is equal to 1, and its slope at the same point is equal to 0. Since e x assumes a finite value when x = 1 + x / i + 2L, we conclude that Eq. (15) has a unique strictly positive root, which we call x*. This is clarified further by Fig. 1, where it is shown that Q(x; L ) intersects the function e x at x = 0 and at x*, 0 < x * < 1 + X/I-+ 2L. From x* we obtain N = (i - x ) ( T / x * ) . We use N(i - X ; B/Xn) to denote the optimal number of spells:
2( i - x ;
=--(i-X) x . ( i _ x ( B / a l 4 ) ) ,
where x*(.) denotes the unique root of (16).
(17)
30
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995)21-39
Q(x;L~)/
L2
ex
x - (i-z)T L2>L1
l÷d-i x
x2
x
-1/2
Fig. 1. The optimal number of spells. 2.2. Properties o f the residence spells model
We get comparative statics results for X(-), the optimal number of spells, by working with (16). Since a Q ( x ; L ) / a L = - x 2 / ( L + x - ½ x 2 ) e < O , an increase in L = - ( i - x ) ( B / a H ) shifts the graph of the LHS rightwards and downwards, and implies an intersection with the graph of the RHS at a larger value of x = (i - x ) ( T / N ) . Because the higher is B, the higher is L, it follows that 2¢" is a decreasing function of B: aw 0B
q<0.
It is straightforward to show from (16) that W---~oo as B--~0 ;
W--~0
as B-~oo.
A . M . Hardrnan, Y . M . loannides / Reg. Sci. Urban Econ. 25 ( 1 9 9 5 ) 2 1 - 3 9
31
A higher value of i - X implies a higher value for x*. This follows from the effect of a change of i - X on L. The effect of a higher value of i - X on N is more difficult to determine because we have to isolate the effect of a higher value of i - X upon N, which is one of the components of x. Thus summarizing, comparative statics results for JV are as follows. The greater the disutility of a move, the smaller the optimal number of moves in a lifetime. The impact of a change in i - X is ambiguous and depends upon actual parameter values. An increase in i - X may cause JV to increase or decrease, but any increase will be less than proportionate to the increase in i -X: ON O---B- < 0 ;
O((i - X) T/JV) O(i - X ) >0 ;
ON -X------~ O(i "~0"
(18)
2.3. The impact of moving costs It is interesting to examine the intuition of the residence spells model relative to the costless adjustment case, that is, B = 0. Eq. (10) above implies, as a special case, the evolution of housing consumption when its adjustment is costless. We demonstrate this as follows. Consider Eq. (10) and define At~tj+ 1 -- tj. When At--->0, both the numerator and the denominator of the RHS tend to 0. Applying L' H6pital's Rule yields lim At~O
oLuWo(to) e(i_x)~tj_to )
: H i ( t 0 ) - TPH(to )
1
i -x
(19)
This is equal, to housing consumption when housing can be adjusted continuously. Eq. (19) differs from (10), the optimal non-housing consumption, because the household is, in effect, paying a rental of ( i x)PH(to) e -~i-x)~t-~°) per unit of housing services. It is interesting to compare optimal housing consumption when moving is costly with what it would be if it were to be adjusted continuously. Ceteris paribus, this comparison rests entirely upon the relative magnitudes of At~[1- e -~i-x)at] and 1 / ( 1 - X ) . From the convexity of e -°-x)at we have that e (i-x) At> 1 - ( i - x ) A t , which yields At 1 1 - e -(i-x) At .~ i -- X " It follows that optimal housing consumption at the beginning of the jth spell is, in our case, always greater than what it would be at the same point in time if it were to be adjusted continuously. This comparison also helps in providing intuition for the consequences of moving costs. The more frequently moves occur the closer the optimal housing consumption path is
32
A.M. Hardman. Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
Frictionless N=4 N=2
:\
.-"/
6
S /
J
6
T/4
T/2
6
3T/4
T
Fig. 2. Individual housing demand over time. to that of the costless case. In Fig. 2, we have plotted against time the value of ~ for the limiting (frictionless) case which varies, from (19), exponentially with time. We have also drawn there two alternative paths for the housing demand according to (11). As we showed above, households always oversize at the beginning of each spell and consequently undersize at the end of the spell. This particular feature of the model agrees with empirical findings by H e n d e r s o n and Ioannides (1989) and Edin and Englund (1991).
2.4. E x t e n s i o n s o f the behavioral m o d e l
It is a source of great simplification in this model that the optimal number of moves • depends only on the excess of the rate of interest over the rate of housing price appreciation, i - X , and not on any other price. The above model of residential moves can be extended in a number of ways that express alternative sets of circumstances and policy environments. Analytically, such alternative settings are reflected as modifications of Eq. (12). For reasons of brevity we do not present here the full analytical detail. We can investigate the impact of the availability of borrowing. The easiest way to model that is to assume that the household may borrow a fraction of the purchase price at an interest rate less than i (but greater than X)- The household is supposed to pay back by means of a 'balloon' payment when it sells the house at the end of a residence spell. A suitably modified version of (15) can then be shown to hold. Borrowing on such terms causes a decrease
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
33
in the optimal number of moves and thus an increase in the optimal length of each spell. It is also straightforward to extend the model in order to allow for maintenance. The result that spells of equal length represent the unique maximizing solution still holds when maintenance at a constant proportional rate is incorporated. The provisions of the tax systems in different countries can affect the optimal number of spells in non-trivial ways. For example, in the United States the effective borrowing rate depends on a household's income, because mortgage interest is deductible from income liable to income tax. Thus the effective cost of borrowing may vary over time because of variations in income and associated variations in the marginal tax rate. In that case the homogeneity of the model breaks down and the optimal length of residence spells is no longer constant.
3. The housing market We follow Cass and Yaari (1967) and Romer (1986) and embed the behavioral model of Section 2 into a model of the housing market within an overlapping-generations model with an infinite number of overlapping generations. Households supply their labor inelastically and their incomes consist of both labor earnings and interest income. Households may borrow and lend freely in perfect capital markets. The housing stock is assumed not to depreciate. It follows that as long as the population is fixed the real price of housing stock is constant over time at the steady-state equilibrium: X = 0. In order to simplify the analysis in this partial equilibrium model, we assume that the housing stock is initially owned by absentee landlords.~9 The housing stock is, at the steady state, always 'on the market'. It is continuously changing hands as new households enter the economy, old households die 19 As individuals come into the economy they purchase housing, financing their purchases by means of loans which they pay back when they resell their dwellings. This continuous buying and selling implies a return to housing which is at equilibrium must be equal to the rate of interest. Even though the housing stock is privately owned, its asset value does not show up as individual household wealth in our model. The two components of /~, the present value of payments for purchasing housing stock and the present value of receipts from selling housing stock, cancel out for the aggregate of all households. That is so because we have set up the model in terms of households' present value of total net spending on housing, /-). This important feature of our model is intimately related to the overlapping-generations structure. Alternatively, we may assume that the housing stock belongs to the population, in which case it is bequeathed (continuously over time) from one generation to the next. In such a case, individuals' income includes income from housing wealth which is equal to the rate of interest times housing wealth. We thank a referee for forcing us to clarify this feature of the model.
34
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
when they reach age T, and all other households buy and sell in order to adjust their housing consumption according to their optimal plans. The population's wealth at birth consists only of its human capital 2°, that is, the discounted value of the stream of lifetime wage income receipts: Wo(i ) =
L
(20)
w e-i' dt ,
where w denotes the steady-state real wage rate. The assumption of an infinite number of overlapping generations in the Cass-Yaari-Romer model means that a new generation of households is formed at each instant in time. With no population growth, generations are of equal size and their members have identical preferences. Households differ only with respect to when they were born. Thus, the inherent discreteness at the household level is completely smoothed out in the aggregate. We normalize the size of each generation to be equal to 1/T so that aggregate population is equal to ( 1 / T ) T = 1. The steady-state housing market equilibrium is fully characterized by pn, the price of housing stock, given the equilibrium rate of interest i, which is treated as exogenous, and with the price of the non-housing consumption good being used as the numeraire: Pc(to) = 1. 3.1. Aggregate housing d e m a n d
The evaluation of aggregate housing demand at each point in time is substantially simplified by the steady-state assumption. Aggregate housing demand, paralleling the demand for non-housing consumption, is equal to the average housing demand over a household's lifetime: 1 N l H ° ( t ° ) = N ~0 /-/J(t°) '
(21)
where Hj(to) denotes the housing demand at the beginning of the jth spell by a household of generation to (that is, who has entered the economy at time t o along the steady state). From (9) at the steady state, where X = 0 and PH(to) = Pn, a constant, and under the assumption that each household moves N times during its lifetime, we have that H i is independent of t 0, the time the household entered the economy: 1 CtHWo(i ) 1 /-/J = N P/4 e-"~/u)J[1 - e -i(r/m] '
j = 0, 1 , . . . , N - 1.
(22)
20If the populationowned the housingstock, then interest income, given by the term iP.17ts, would have to be added to the flow of income (see footnote 19).
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
35
Aggregate housing demand as a function of the rate of interest, of the real price of housing and of the number of moves within a household's lifetime, H = Ho(i , PH; N), now follows by aggregation from (21) and (23): el(T/N)
no(i, Pn; N) - anW°(i) P---T- (e ~r - 1) N 2 ( e ~ -
- 1) 2 .
(23)
If, in (23), the number of moves N is equal to N, the optimal solution for the number of moves which is given by (16) and (17), then H e gives the unconstrained, 'Walrasian', aggregate housing demand function, H o (i, In; N(i; B/an) ). The fact that H o is decreasing in Pn and has a housing price elasticity equal to - 1 follows from our assumption of logarithmic preferences. We note that the expression for aggregate demand in (23) holds whenever households plan equidistant moves over their lifetimes, even if the number of moves is not the optimal one. It is interesting, therefore, to examine the properties of aggregate housing demand with respect to N. We obtain first aggregate housing demand when moves are costless. When N----~~ (23) yields
1 anWo(i ) e it-1 H ° - T iPn iT
(24)
In order to determine the effect of changes in N, when N is finite, we work as follows. Let ~b(N; i) denote the component of the RHS of Eq. (23) that depends upon N: el(T/N)
6 ( g ; i) -- g 2 ( e i ( T / N ) _ 1)2,
e~(r/m
04) (N; i) = N3(e~ ON
(25)
(
[ T
1) 3 e i(r/N) i N -
]
T
2 + iN+ 2
)
(26)
Since, by convexity, e i(r/n) > 1 + i(T/N), it follows that 04)~ON > 0. That is, the more moves a household makes over its lifetime, the larger the aggregate housing demand. This is an important finding whose intuition we can explain as follows. Consider the definition of aggregate housing demand in (21) in conjunction with Fig. 2. Aggregate housing demand in our steady-state equilibrium is equal to the average height of the path of individual household demand over time. Our model of household behavior implies that when adjustment is costly individual households oversize when they move relative to the frictionless case at the corresponding point in time. In spite of this oversizing, aggregate demand always falls short of what it would be in the frictionless case.
36
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995)21-39
The dependence of aggregate housing demand on the rate of interest is complicated by the effect of the rate of interest on the optimal number of moves, which is ambiguous. By treating the wage rate from (20) as exogenous we consider first the direct effect of the rate of interest for a given number of moves: 1
0
1
H o Oi H°(i' PH; N ) = Wo(i )
OWo(i)
Oi
T e ir
0
+ e ir - 1 t- -~ th(U; i).
We can show that the sum of the last two terms above is positive. We note that the first term in the RHS of the above equation may be written as the interest rate times the interest elasticity of lifetime wealth. We thus conclude that the direct effect of a change in the rate of interest on aggregate housing demand is positive, provided that the first term is numerically small. Unfortunately, the total effect of a change in the rate of interest cannot be signed because of the ambiguity, from (18), in the sign of ON/Oi, as well.
3.2. Housing market equilibrium Housing market equilibrium is ensured by HD(i, PH, X ) = ISis, where /4s is aggregate housing supply, 14o is defined in (23), 2¢" as defined in (16) and (17), and W0(i) is defined in (20), or
1 1 - e -i*T e i*(T/N) P" =--~s az4w i* (e i - r - 1) X2(ei.~r/x)_ 1) 2 .
(27)
We have shown that for a given number of moves, the price of housing stock which equilibrates the housing market is an increasing function of the interest rate, provided that the interest elasticity of lifetime wealth is very small.
3.3. Properties of housing market equilibrium We noted in Subsection 3,1 above that the expression for aggregate housing wealth in (23) holds, when households plan equidistant moves during their lifetime, even when the number of moves is not optimized. Our housing market equilibrium model is thus applicable to contexts where the number or cost of moves is subject to administrative control or in other ways affected by public policy. Two points follow. First, we can examine the impact on housing market equilibrium of alternative values for B, the key parameter of the utility function that expresses the cost of residential mobility. We show above that the higher the cost of moving, the less frequently households move. From (27) we see that increases in B (the utility cost of moving) shift the housing market demand
A . M . Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
37
downwards. From (26) it follows that the fewer moves a household makes in its lifetime, the lower will be its average housing demand. Hence, the higher the utility cost of mobility, the lower the equilibrium price of housing stock, a result that fully accords with intuition. 2~ Second, in a non-Walrasian setting, where households are prevented from moving as frequently as they desire, the number of permitted moves may ration housing demand. It also follows that, in this model, changes in the number of moves permitted can be represented by shifts in the aggregate housing demand curve, while the latter remains bounded upwards by (24). An institutional feature of Eastern European housing sectors, and of some Western and Third World urban housing markets, has been the allocation of much of the housing stock at rents well below the market equilibrium price. This happened either because housing was subject to rent control, or because the housing stock was publicly owned and administratively allocated with rents well below market levels. Rent control typically means that while households may be able to secure housing services at a below-market price, moving is difficult because search costs are high. When housing is publicly owned, housing 'administrators' typically ration available housing services by directly controlling occupants' access to different packages of housing services and hence their residence spells. Housing policies such as rent controls and the administrative allocation of publicly owned housing in many housing markets have the effect of raising the cost of moving or effectively limiting the number of moves which households can make. The approach used here provides a useful way to model both phenomena.
4. Summary and conclusions The central feature of this paper is a housing market which adjusts to reflect both prices and frequency of moves. Following Cass and Yaari (1967) we assume a continuous-time finite lifetime, life-cycle model and an infinite number of overlapping generations coexisting at any point in time. A key result, which accords with intuition, is that economies with higher moving costs are likely to be associated with lower housing prices. Since residential mobility is the process through which arbitrage of the housing stock takes place, it is natural that higher arbitrage costs are associated with 21 We have assumed that the housing stock is owned originally by absentee landlords. If, instead, the housing stock is owned by the population, then the definition of Wo(i ) must be modified. The flow of income then includes interest from housing wealth, iPnlSIs . A solution for PH follows. While it is more complicated, the derivative of Pn with respect to B remains negative.
38
A.M. Hardman, "Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
lower housing prices. We conjecture that the effects of higher moving costs may be quite different in a more general model. In our model the discreteness of adjustments in housing consumption at the household level- m o v e s - is smoothed at the aggregate. This feature of the model makes it possible to apply to settings like those cited above, where administrative intervention is used to ration housing demand. We clarify the manner in which impediments to moving serve to ratio housing consumption.
Acknowledgements This paper has benefited enormously from the comments of two referees and the editors. Comments from participants in seminars at VPI and the University of Virginia, and at the 1991 meeting of the North American Regional Science Association, New Orleans, and the 1991 ASSET Conference, Athens, Greece, are also gratefully acknowledged. Ioannides acknowledges financial support from National Science Foundation grants SES9000200 and SES-9211913. We are grateful to Joan Boyd for her impeccable typing.
References Amundsen, E.S., 1985, Moving costs and the microeconomics of intra-urban mobility, Regional Science and Urban Economics 15, 573-583. Blanchard, O.J. and S. Fischer, 1989, Lectures in macroeconomics (MIT Press, Cambridge, MA). Cass, D. and M.E. Yaari, 1967, Individual saving, aggregate capital accumulation, and efficient growth, in: K. Shell, ed., Essays on the theory of optimal economic growth, ch. XIII (MIT Press, Cambridge, MA), 233-268. Clark, W.A.V. and A.D. Heskin, 1982, The impact of rent control on tenure discounts and residential mobility, Land Economics 58, February, 109-117. Coleman, D., 1988, Rent control: The British experience and policy response, Journal of Real Estate Finance and Economics, 1, no. 3, November. Edin, P.-A. and P. Englund, 1991, Moving costs and housing demand: "Are Recent Movers Really in Equilibrium?", Special issue on international studies of housing demand: The effects of government policies, Journal of Public Economics 44, no. 3, 299-320. Englund, P., 1985, Taxation of capital gains on owner-occupied homes: Accrual vs. realization, European Economic Review 27, 311-334. Feinstein, J. and D. McFadden, 1988, Dynamics of housing demand by the elderly. I. wealth, cash flow and demographic effects, in: D. Wise, ed., Economics of aging (University of Chicago Press, Chicago, IL). Hanushek, E.A. and J.M. Quigley, 1978, An explicit model of intrametropolitan mobility, Land Economics 54, 411-429.
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
39
Hardman, A.M., 1987, The role of side payments in a market in disequilibrium: The market for rental housing in Cairo, Ph.D. Dissertation, Department of Urban Studies and Planning, MIT, October. Hardman, A.M., 1990, Informal rental housing markets in Third World cities: Evidence from Egypt, Department of Economics, Virginia Polytechnic Institute and State University, April. Hardman, A.M. and Y.M. Ioannides, 1991, Residential mobility, the housing market and economic growth, paper presented at the Regional Science Association Meeting, New Orleans, October, 1991, Mimeo. Henderson, J.V. and Y.M. Ioannides, 1989, Dynamic aspects of consumer decisions in housing markets, Journal of Urban Economics 26, 212-230. Hughes, G. and B. McCormick, 1981, Do council housing policies reduce migration between regions? Economic Journal 91, December, 919-937. Hughes, G. and B. McCormick, 1987, Housing markets, unemployment and labour market flexibility in the UK, European Economic Review 31, 615-645. Ioannides, Y.M., 1987, Residential mobility and housing tenure choice, Regional Science and Urban Economics 17, 265-288. Mayo, S.K. and J.l. Stein, 1988, Housing and labor market distortions in Poland: Linkages and policy implications, World Bank Policy Planning and Research Staff Discussion Paper, Report INU 25, World Bank, Washington D.C., June. Olsen, E.O., 1990, What is known about the effects of rent controls? Report prepared for the Office of Policy Development and Research, U.S. Department of Housing and Urban Development, September. Romer, D., 1986, A simple general equilibrium version of the Baumol-Tobin model, Quarterly Journal of Economics 101, no. 4, 663-686. Rosenthal, S.S., 1988, A residence time model of housing markets, Journal of Public Economics 36, 87-109. Turner, B., 1988, Economic and political aspects of negotiated rents in the Swedish housing market, Journal of Real Estate Finance and Economics 1, no. 3, November, 257-276. Venti, S.F. and D.A. Wise, 1984, Moving and housing expenditure: Transactions costs and disequilibrium, Journal of Public Economics 23, 207-243. Willis, K.G., S. Malpezzi and A.G. Tipple, 1990, An econometric and cultural analysis of rent control in Kumasi, Ghana, Urban Studies 27, no. 2, 241-258.
Moving behavior and the housing market A n n a M . H a r d m a n a, Y a n n i s M. I o a n n i d e s b'* aDepartments of Urban Affairs and Planning and of Economics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA bDepartment of Economics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA
Received September 1992, final version received March 1994
Abstract
This paper develops a model of the relationship between residential mobility and housing demand, which it embeds in an overlapping-generations model of the housing market. Moves, needed to adjust housing consumption, are costly; adjusting other consumption is not. Comparative static results for the impact of moving costs and the interest rate on the optimal number of moves and housing consumption in each residence spell are presented. We demonstrate that moving costs and constraints on moving can serve as a form of rationing of housing consumption which can be used in analyzing certain policy interventions in housing markets. Key words: Moving; Residential mobility; Housing; Frictions JEL classification: R21, R31, D50
1. Introduction
Residential m o b i l i t y - m o v i n g - is an important and relatively neglected feature of housing markets. In competitive housing markets, households decide how much housing to consume. In order to adjust its housing consumption to match changes in income, prices, or preferences, a household must enter the market, find a new dwelling and then move. Because * Corresponding author. 0166-0462/95/$09.50 (~) 1995 Elsevier ~icience B.C. All rights reserved SSDI 0166-0462(94)02065-5
22
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
search and moving are costly, most households adjust their housing consumption relatively infrequently. Thus, at any given moment, only a small fraction of the owners of housing are in the market to arrange trades of housing units or housing services. Similarly, most consumers of housing services, whether they are renters or owners, enter the market infrequently. Housing markets allocate shelter to households when they enter the market; they also provide housing equity, the most important and most commonly held asset in household asset portfolios. When institutional restrictions impede residential mobility, the allocative role of housing markets is disrupted. A number of papers on housing markets have addressed important aspects of residential mobility. The main emphasis has been on empirical aspects of moving behavior. 1 Another strand of empirical research has modeled the duration of residence spells (the intervals between moves). 2 Several recent papers present evidence that renters in rent-controlled housing markets move less frequently and consume less than optimal quantities of housing. 3 The administrative allocation of publicly owned housing has been shown to have similar effects in both Western and Eastern Europe. 4 In Europe, too, a number of empirical studies have examined the consequences of housing policies for labor mobility and hence the efficiency of labor markets. 5 Amundsen (1985) and Englund (1985), have explored residential mobility using models of individual behavior. The empirical literature cited above highlights the importance of interactions between housing policy and residential mobility. It is clear that policies which affect residential mobility have potentially significant effects on labor and capital markets as well as the housing market. Yet this area has so far received little attention in the theoretical literature. This paper integrates the residential mobility process into a model of the housing market by assuming that the quantity of housing consumed can be adjusted only by moving, and that moving is costly in terms of utility. The model allows us to compare states in which residential mobility is freely
For example, Hanushek and Quigley (1978), Venti and Wise (1984), loannides (1987), Feinstein and McFadden (1988), Rosenthal (1988), Henderson and Ioannides (1989), and Edin and Englund (1991). 2For example, Ioannides (1987), Rosenthal (1988), Henderson and Ioannides (1989) and Edin and Englund (1991). 3See, for example, Clark and Heskin (1982) and Olsen (1990) for the United States, Hardman (1987, 1990) for Egypt, Willis et al. (1990) for Ghana, and Turner (1988) for Sweden. 4 For example, for Poland, see Mayo and Stein (1988), and for the United Kingdom, Coleman (1988). 5See, for example, Mayo and Stein (1988), and Hughes and McCormick (1981, 1987).
A . M . Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
23
chosen with states in which it is impeded and households are not free to select the optimal number of moves. The remainder of this paper is organized as follows. In Section 2 we solve the household's lifetime optimization problem and examine its properties. In Section 3 we use the same model to characterize housing market equilibrium. Section 4 concludes.
2. The behavioral model
To examine the consequences of moving costs we use an approach that is grounded in Romer's model of the transactions demand for money (Romer, 1986). 6 The basic behavioral model used here is related to the models in Amundsen (1985) and in Englund (1985). 7 Consumption decisions are made by households. Households enter the economy and die continuously; each has a finite lifetime. An infinite number of overlapping generations of households coexist at any point in time, and households differ only with respect to the time they enter the economy. Households adjust their housing consumption at a finite number of points in time. With a large number of households, this discreteness is smoothed in the aggregate, so that market demand can be examined using standard tools. The household's optimization problem in an unrestricted environment is as follows. Consider a household which is formed at time t o and dies at time t o + T. Let ( t o , t 1 . . . . . t n _ l } , t o <~ t 1 • • • <~ t , _ 1 <~ t o + T , be the points in time when adjustments in the quantity of housing take place, with the corresponding housing quantities being {H 0(t 0) . . . . , Hn- l(t0) )- That is, Hj(t 0) is the housing stock purchased at time tj by a household formed at time t 0, j = 0 , . . . , n - 1. For completeness, let t n -- t o + T. We assume that a unit of housing stock generates a unit of housing services per unit of time. Households may adjust the amount of housing services they consume only by moving. The quantity of housing services consumed during each residence spell (the interval between moves) is constant. 8 6 See also Blanchard and Fischer (1989, pp. 168-178), who popularized Romer's model. 7 Amundsen's model of the role of moving costs in long-term optimal moving strategies assumes that utility is not discounted, prices are constant, there is a positive rate of interest, and moving costs are monetary and grow exponentially over time. Englund incorporates moving into a model of the housing market which assumes that non-housing consumption remains constant over an infinite lifetime horizon, and housing consumption may be adjusted only by means of residential moves. He adopts an isoelastic form for utility per unit of time, which implies log utility as a special case. s This feature reflects the quasi-fixed nature of housing. Housing deteriorates slowly, and not much else, except analytical tractability, is affected if housing consumption is allowed to vary exogenously during a residence spell.
24
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
Let Pc(t) and PH(t) denote, respectively, the nominal prices of nonhousing consumption per unit of time and of a unit of housing stock. C(t) denotes non-housing consumption during the residence spell which begins at time tj.9 Let Wo(to) denote lifetime human wealth for a household formed at time to, and i denote the nominal rate of interest. For simplicity we consider only dynamics along steady-state paths of the economy. The nominal rate of interest i is therefore taken as time-invariant and equal to its steady state value (cf. Cass and Yaari, 1967). The lifetime budget constraint may be written as
Wo(to) =
e-i(t-%1)Pc(t)C(t ) dt + ffI(to),
e -i(tj-l-'°) j=l
(1)
j--1
where t n --- t o + T, and/~(t0) denotes the present value at the beginning of its life of the household's net expenditure on housing. For a household formed at t o which plans to purchase Hj_~(to) units of housing stock at tj_ 1, hold it during the residence spell [tj_~, tj) and sell it at time tj, j = 0, 1 . . . . . n - 1, /-/(to) is given by
f-l(to) = ~ Hj_I(to)[PH(tj_I) e -i(tj-l-'°) - PH(t~) e-i(tj-t°)]
.
(2)
j=l
We assume that households are owner-occupiers. This formulation subsumes the notion of housing wealth under H(t0). ~° We assume that households value housing and non-housing consumption during their own lifetime and leave no bequests. H A household's lifetime utility maximization problem is stated as follows. A household chooses (C(t);Ho(to) . . . . . nn_l, t I ..... tt_l} SO as to maximize 9 We do not subscript C(t) because under our assumption that utility per unit of time is separable in housing and non-housing consumption, the optimal non-housing consumption does not depend upon the spell. On this, see Amundsen (1985, p. 577, Proposition 2). 10 Everything carries through if we assume instead that housing is rented, the housing stock is owned by private entrepreneurs or the state, and property owners earn the market rate of return in their capacity as landlords. This is clear if we recognize that a renter household's expenditure on housing services must obey the arbitrage condition that the owner of a unit of housing stock should be indifferent between selling it at time tj 1, and holding on to it, renting it for O(t), and subsequently selling it at time 4:
Pl-i(tj 1) e-i(Q-x to) = Pu(tJ) e-i('J-'°) + f,i' , O(t) e i('-'°) dt. 11 We purposefully leave out time-discounting for utility in order to avail ourselves, like A m u n d s e n (1985) and Romer (1986), of a substantial simplification of an otherwise intractable optimization problem. Nonetheless, we argue that an important implication of this assumption, namely that housing consumption increases over time, is at least partially defensible in the light of the empirical evidence. We take this up further later in the section.
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
/20=
ac j=l
lnC(t) dt+aH(ti-tj_l)lnHj_
1 -Bn,
25
(3)
tj 1
where parameters a c and a H satisfy a c > 0, a H > 0, and a c + a H = 1, and B denotes the utility cost of adjusting housing per move, subject to budget constraint (1). The utility cost B n is the only cost associated with the n moves the household plans to make over its lifetime. 12 This assumption, which is borrowed from Romer, accounts for the simplicity of the results obtained from our model. Some of the results below also depend critically on the assumption that utility is not discounted. 13 We obtain the necessary conditions for lifetime utility maximization in two stages. First we characterize the optimal values for the rate of consumption of housing and non-housing goods per unit of time, assuming that the values for the tj's are given. At a second stage we optimize with respect to the tj's. Let A denote the Lagrange multiplier that adjoins constraint (1) to lifetime utility according to (3). The necessary conditions for the rate of consumption C ( t ) , t E [0, T], become 1 ac
C(t)
- A e-i('-'°)Pc(t ) ,
t o <<-t <~ t o + T .
(4)
It follows from (4) that under our assumptions the intertemporal variation in non-housing consumption is independent of moving costs. Consumption grows over time at a rate that depends on the rate of interest and on the price of consumption. Its actual magnitude is determined through the Lagrange multiplier A, which reflects lifetime resources. In the remainder of the paper we assume that the price of consumption is constant. No loss of generality is involved here because we study steadystate dynamics only. From that and from (1), (2) and (4), we obtain the optimal rate of consumption as follows:
C(t)-
W o ( t o ) - ff-I e (i)(t_to) TPc(to ) .
(5)
The optimal rate of consumption grows at a constant rate equal to i. By substituting into the expression for the lifetime utility function in (3) we have 12B can be seen as a function of the economy's information structure: a small value of B is associated with market features which facilitate matching between buyers and sellers. 13Both Amundsen (1985) and Romer (1986) do not discount for time.
26
A.M. Hardman, Y.M. Ioannides / Reg. Sci. Urban Econ. 25 (1995)21-39
i a o = ~ a c T2 - a c T [ I n T + In Pc(to)] + a c T + a , ~ (tj - t j _ , ) l n H j _ l ( t o ) j=l
In(W0-/t) (6)
- Bn.
Just as they adjust non-housing consumption, households wish to adjust their consumption of housing over time. The necessary conditions for the quantities of housing stock purchased each time a move occurs, /-/j, are obtained by differentiating (6) with respect to Hi, j = 0 . . . . , n - 1, and by using (2) and are as follows: 1 OlH(t
j --
tj 1)-Hj_l
1 OH a c T W o - ffI OHi-~I - O.
(7)
By rearranging (7) and by summing up over all j's we obtain /4 = aHW0.
A household's total expenditure on housing is a constant fraction of its initial wealth. This result is responsible for the tractability of the model; it depends crucially upon the assumption of no time-discounting of utility. It follows that the rate of consumption, from (5), becomes C(t)
acWo TPc(to ) e i(t-t°) ,
(8)
t o <~ t <~ t o + T .
The demand for housing stock at the time of the jth move by a household which enters the economy at time to, is then Hj l(t0)
--
-
tj - tj_ I auWo[PH(tj T
1) e-i('J
l-t°)
--
PH(tj) e i(tF'°)] -1
(9)
-
We assume that the price of housing stock increases at a constant growth rate X, PH(t) = Pn(to) eX('-'°).~4 Eq. (9) j = 1 . . . . . n, now becomes Hj l(t0) --
tj - tj_ l 1 T aHWo PH(tO) e-(i-x)(t,-1-to)[1 _ e-(i-x)(,,-',
i)] " (10)
The optimal quantity of housing stock acquired at the time of the jth move depends on the time the respective spell starts and on its duration. Under our assumptions, Hi(to) depends upon t o only via Wo(to) and Pu(to). This homogeneity result allows us to aggregate. If moves occur at N equidistant 14In Hardman and loannides (1991) we verify that this is indeed the case in general equilibrium.
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995)21-39
27
points in time, which is indeed the case at the optimum as we show below, (9) becomes 1 1 Hi(to) = - ~ aHWo pH(to) e-(i-x)(Jr/N) {1 _ e-(i-x)(r/N)} •
(11)
For (9) to yield a positive value for the housing stock Hi, we must have X < i: that is, the rate of housing price increase is less than the nominal rate of interest; owning housing stock entails an opportunity cost. If X > i then housing becomes an asset superior to saving at an interest rate i. If X < i, then the amount of housing stock desired increases with every spell, as the present value of the price of housing stock goes down. The condition may be motivated as follows. The total return to housing reflects both the housing services it produces and a possible capital gain. If the rate of capital gain were greater than the rate of interest then the total return to housing would exceed the rate of interest. With a perfect capital market, as long as housing and non-housing assets are held at equilibrium, it must be the case that x
Another implication of (11) is that the present value of housing wealth held by a household at the beginning of each spell is independent of the particular spell and equal to N-lotHWo[1 -- e-(i-x)(T/N)] -1 . This yields that in its lifetime the typical household spends on housing purchases an amount equal to aHW0[1 - e-(i-x)(r/u)] -~ and receives from housing sales an amount equal to a H W o e-(i-x)(r/u)[1 e-(i-x)(T/N)]. Our assumption of no time-discounting in our model is principally responsible for the result that housing demand, from (11), grows over a household's lifetime. This conflicts with both the standard life-cycle model and empirical findings of researchers who use the residence spells model and who find that housing demand peaks in mid life. ~5 However, other recent research which has emphasized housing decisions by the elderly a6 has shown that most elderly households often do not choose to reduce their housing consumption and especially their housing equity as much as the strict lifecycle model predicts. -
2.1. O p t i m a l times o f m o v e s
It remains to optimize with respect to the times of moves and thus the lengths of spells. We note that under our assumptions the optimal value of the consumption component of lifetime utility has been expressed explicitly in terms of parameters. The optimal times of moves follow from the is See Rosenthal (1988), Hardman (1987) and Henderson and Ioannides (1989). 16See Feinstein and McFadden (1988).
28
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
m a x i m i z a t i o n with r e s p e c t to ( t l , . . . , (6), that is:
-Bn
tn_l}
of the h o u s i n g c o m p o n e n t of
+ ~ OtH(tj -- tj_l){ln(t j -- tj_l) -- In T + l n ( a H W o ( t o ) / T ) - In Pn(O) j=l
+ (i - X ) ( t j _ 1 - to) - ln[1 - e -~i-x)~t' tj_~)]} .
(12)
B y d i f f e r e n t i a t i n g (12) with r e s p e c t to tj we o b t a i n a difference e q u a t i o n i n v o l v i n g (t]_l, tj, ti+l)
t +1-tj - I n tj - tj_ 1 - (i -x)[(tj+ -
-
1 - tj) - (tj - tj_l)]
1 - e x p [ - ( i - x)(ti+l - tj)] +In
1 - e x p [ - ( i - x)(tj - tj_l) ]
- ( t j - tj_l)(i - X )
e x p [ - (i - x)(tj - t~_l)] 1 - e x p [ - ( i - x ) ( t j - tj 1)]
e x p [ - ( i - x)(tj+l - tj)] +(tj÷ 1 - tj)(i - X) 1 - e x p [ - ( i - x)(tj+l - tj)] - 0 .
(13)
It can be s h o w n that given (ti_l, tj+l), tj = (tj 1 + t j + l ) / 2 satisfies (13), a n d t h a t it is t h e u n i q u e m a x i m i z i n g solution. It follows that all spells h a v e e q u a l lengths. 17 This result does n o t d e p e n d on o u r a s s u m p t i o n that the utility cost of m o v e s is p r o p o r t i o n a l to the n u m b e r of m o v e s . W e n o w c o n s i d e r the n u m b e r of spells, n = N, as the u n k n o w n variable, w h e r e tj - t] 1 T / N , which we treat as a c o n t i n u o u s l y v a r y i n g variable. 18 In d o i n g so we follow a n u m b e r of p r e v i o u s a u t h o r s including, in particular, A m u n d s e n (1985) a n d R o m e r (1986). B y using (11) in (12) we h a v e =
-BN
a.W o T(N+ a H T In Pg(to-------~ - In N + (i - X) 2N
-ln[1-exp[-(i-x)T]]].
1) (14)
T h e o p t i m a l length of r e s i d e n c e spells follows by m a x i m i z i n g (14) with r e s p e c t to the n u m b e r of spells, which we treat as a real n u m b e r .
17This result is reminiscent of Amundsen (1985), Proposition 3, when the rate of growth of moving costs is equal to the rate of interest. Our result is identical to Romer (1986), except that he deals with a single good, consumption, and it is trips to the bank that are equidistant over time. 18A new symbol for the number of moves is used in order to underscore that it is a decision variable.
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
29
Thus by differentiating (14) with respect to n we obtain the following necessary condition for the optimal number of spells: +
T2 e x p [ - (i - x ) ( T / N ) ] Nz (i - X) 2 1 - e x p [ - ( i - x ) ( T / N ) ]
/3 _ ( i _ x ) (i - X) an
1 T2 + (i - X) 2 -~ N---7 = 0.
T 1 n (15)
T h e optimal number of spells is obtained implicitly from (15) as its unique positive root. By using the auxiliary unknown x = - ( i - x ) ( T / N ) and rewriting (15) we have -x X2
e
1-e
_x-L
+ x - y x1
2
,
where L = - ( i - x ) ( B / a n ) . Rewriting once again yields L +x+½x
Q(x; L ) -
L +x -ix
2
z = eX.
(16)
T h e RHS of the above is the exponential function, an increasing function CeX~, with well-known properties: e ° = 1, ~ ix=0 = 1. For the LHS, which we denote by Q(x; L ) we have by differentiating with respect to x:
OQ(x; L ) 0x
2Lx + x 2 ' ( L + x - ½ x 2 ) e"
T h e r e f o r e , the function Q ( x ; L ) has the following properties: it is increasing for all positive values of x; it tends to +0% when x--* 1 + ~/1 + 2L from below, and to -0% when x---~ 1 + V1 + 2L from above; and, it tends to _ 1 , when x---~ +o0. Its value for x = 0 is equal to 1, and its slope at the same point is equal to 0. Since e x assumes a finite value when x = 1 + x / i + 2L, we conclude that Eq. (15) has a unique strictly positive root, which we call x*. This is clarified further by Fig. 1, where it is shown that Q(x; L ) intersects the function e x at x = 0 and at x*, 0 < x * < 1 + X/I-+ 2L. From x* we obtain N = (i - x ) ( T / x * ) . We use N(i - X ; B/Xn) to denote the optimal number of spells:
2( i - x ;
=--(i-X) x . ( i _ x ( B / a l 4 ) ) ,
where x*(.) denotes the unique root of (16).
(17)
30
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995)21-39
Q(x;L~)/
L2
ex
x - (i-z)T L2>L1
l÷d-i x
x2
x
-1/2
Fig. 1. The optimal number of spells. 2.2. Properties o f the residence spells model
We get comparative statics results for X(-), the optimal number of spells, by working with (16). Since a Q ( x ; L ) / a L = - x 2 / ( L + x - ½ x 2 ) e < O , an increase in L = - ( i - x ) ( B / a H ) shifts the graph of the LHS rightwards and downwards, and implies an intersection with the graph of the RHS at a larger value of x = (i - x ) ( T / N ) . Because the higher is B, the higher is L, it follows that 2¢" is a decreasing function of B: aw 0B
q<0.
It is straightforward to show from (16) that W---~oo as B--~0 ;
W--~0
as B-~oo.
A . M . Hardrnan, Y . M . loannides / Reg. Sci. Urban Econ. 25 ( 1 9 9 5 ) 2 1 - 3 9
31
A higher value of i - X implies a higher value for x*. This follows from the effect of a change of i - X on L. The effect of a higher value of i - X on N is more difficult to determine because we have to isolate the effect of a higher value of i - X upon N, which is one of the components of x. Thus summarizing, comparative statics results for JV are as follows. The greater the disutility of a move, the smaller the optimal number of moves in a lifetime. The impact of a change in i - X is ambiguous and depends upon actual parameter values. An increase in i - X may cause JV to increase or decrease, but any increase will be less than proportionate to the increase in i -X: ON O---B- < 0 ;
O((i - X) T/JV) O(i - X ) >0 ;
ON -X------~ O(i "~0"
(18)
2.3. The impact of moving costs It is interesting to examine the intuition of the residence spells model relative to the costless adjustment case, that is, B = 0. Eq. (10) above implies, as a special case, the evolution of housing consumption when its adjustment is costless. We demonstrate this as follows. Consider Eq. (10) and define At~tj+ 1 -- tj. When At--->0, both the numerator and the denominator of the RHS tend to 0. Applying L' H6pital's Rule yields lim At~O
oLuWo(to) e(i_x)~tj_to )
: H i ( t 0 ) - TPH(to )
1
i -x
(19)
This is equal, to housing consumption when housing can be adjusted continuously. Eq. (19) differs from (10), the optimal non-housing consumption, because the household is, in effect, paying a rental of ( i x)PH(to) e -~i-x)~t-~°) per unit of housing services. It is interesting to compare optimal housing consumption when moving is costly with what it would be if it were to be adjusted continuously. Ceteris paribus, this comparison rests entirely upon the relative magnitudes of At~[1- e -~i-x)at] and 1 / ( 1 - X ) . From the convexity of e -°-x)at we have that e (i-x) At> 1 - ( i - x ) A t , which yields At 1 1 - e -(i-x) At .~ i -- X " It follows that optimal housing consumption at the beginning of the jth spell is, in our case, always greater than what it would be at the same point in time if it were to be adjusted continuously. This comparison also helps in providing intuition for the consequences of moving costs. The more frequently moves occur the closer the optimal housing consumption path is
32
A.M. Hardman. Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
Frictionless N=4 N=2
:\
.-"/
6
S /
J
6
T/4
T/2
6
3T/4
T
Fig. 2. Individual housing demand over time. to that of the costless case. In Fig. 2, we have plotted against time the value of ~ for the limiting (frictionless) case which varies, from (19), exponentially with time. We have also drawn there two alternative paths for the housing demand according to (11). As we showed above, households always oversize at the beginning of each spell and consequently undersize at the end of the spell. This particular feature of the model agrees with empirical findings by H e n d e r s o n and Ioannides (1989) and Edin and Englund (1991).
2.4. E x t e n s i o n s o f the behavioral m o d e l
It is a source of great simplification in this model that the optimal number of moves • depends only on the excess of the rate of interest over the rate of housing price appreciation, i - X , and not on any other price. The above model of residential moves can be extended in a number of ways that express alternative sets of circumstances and policy environments. Analytically, such alternative settings are reflected as modifications of Eq. (12). For reasons of brevity we do not present here the full analytical detail. We can investigate the impact of the availability of borrowing. The easiest way to model that is to assume that the household may borrow a fraction of the purchase price at an interest rate less than i (but greater than X)- The household is supposed to pay back by means of a 'balloon' payment when it sells the house at the end of a residence spell. A suitably modified version of (15) can then be shown to hold. Borrowing on such terms causes a decrease
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
33
in the optimal number of moves and thus an increase in the optimal length of each spell. It is also straightforward to extend the model in order to allow for maintenance. The result that spells of equal length represent the unique maximizing solution still holds when maintenance at a constant proportional rate is incorporated. The provisions of the tax systems in different countries can affect the optimal number of spells in non-trivial ways. For example, in the United States the effective borrowing rate depends on a household's income, because mortgage interest is deductible from income liable to income tax. Thus the effective cost of borrowing may vary over time because of variations in income and associated variations in the marginal tax rate. In that case the homogeneity of the model breaks down and the optimal length of residence spells is no longer constant.
3. The housing market We follow Cass and Yaari (1967) and Romer (1986) and embed the behavioral model of Section 2 into a model of the housing market within an overlapping-generations model with an infinite number of overlapping generations. Households supply their labor inelastically and their incomes consist of both labor earnings and interest income. Households may borrow and lend freely in perfect capital markets. The housing stock is assumed not to depreciate. It follows that as long as the population is fixed the real price of housing stock is constant over time at the steady-state equilibrium: X = 0. In order to simplify the analysis in this partial equilibrium model, we assume that the housing stock is initially owned by absentee landlords.~9 The housing stock is, at the steady state, always 'on the market'. It is continuously changing hands as new households enter the economy, old households die 19 As individuals come into the economy they purchase housing, financing their purchases by means of loans which they pay back when they resell their dwellings. This continuous buying and selling implies a return to housing which is at equilibrium must be equal to the rate of interest. Even though the housing stock is privately owned, its asset value does not show up as individual household wealth in our model. The two components of /~, the present value of payments for purchasing housing stock and the present value of receipts from selling housing stock, cancel out for the aggregate of all households. That is so because we have set up the model in terms of households' present value of total net spending on housing, /-). This important feature of our model is intimately related to the overlapping-generations structure. Alternatively, we may assume that the housing stock belongs to the population, in which case it is bequeathed (continuously over time) from one generation to the next. In such a case, individuals' income includes income from housing wealth which is equal to the rate of interest times housing wealth. We thank a referee for forcing us to clarify this feature of the model.
34
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
when they reach age T, and all other households buy and sell in order to adjust their housing consumption according to their optimal plans. The population's wealth at birth consists only of its human capital 2°, that is, the discounted value of the stream of lifetime wage income receipts: Wo(i ) =
L
(20)
w e-i' dt ,
where w denotes the steady-state real wage rate. The assumption of an infinite number of overlapping generations in the Cass-Yaari-Romer model means that a new generation of households is formed at each instant in time. With no population growth, generations are of equal size and their members have identical preferences. Households differ only with respect to when they were born. Thus, the inherent discreteness at the household level is completely smoothed out in the aggregate. We normalize the size of each generation to be equal to 1/T so that aggregate population is equal to ( 1 / T ) T = 1. The steady-state housing market equilibrium is fully characterized by pn, the price of housing stock, given the equilibrium rate of interest i, which is treated as exogenous, and with the price of the non-housing consumption good being used as the numeraire: Pc(to) = 1. 3.1. Aggregate housing d e m a n d
The evaluation of aggregate housing demand at each point in time is substantially simplified by the steady-state assumption. Aggregate housing demand, paralleling the demand for non-housing consumption, is equal to the average housing demand over a household's lifetime: 1 N l H ° ( t ° ) = N ~0 /-/J(t°) '
(21)
where Hj(to) denotes the housing demand at the beginning of the jth spell by a household of generation to (that is, who has entered the economy at time t o along the steady state). From (9) at the steady state, where X = 0 and PH(to) = Pn, a constant, and under the assumption that each household moves N times during its lifetime, we have that H i is independent of t 0, the time the household entered the economy: 1 CtHWo(i ) 1 /-/J = N P/4 e-"~/u)J[1 - e -i(r/m] '
j = 0, 1 , . . . , N - 1.
(22)
20If the populationowned the housingstock, then interest income, given by the term iP.17ts, would have to be added to the flow of income (see footnote 19).
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
35
Aggregate housing demand as a function of the rate of interest, of the real price of housing and of the number of moves within a household's lifetime, H = Ho(i , PH; N), now follows by aggregation from (21) and (23): el(T/N)
no(i, Pn; N) - anW°(i) P---T- (e ~r - 1) N 2 ( e ~ -
- 1) 2 .
(23)
If, in (23), the number of moves N is equal to N, the optimal solution for the number of moves which is given by (16) and (17), then H e gives the unconstrained, 'Walrasian', aggregate housing demand function, H o (i, In; N(i; B/an) ). The fact that H o is decreasing in Pn and has a housing price elasticity equal to - 1 follows from our assumption of logarithmic preferences. We note that the expression for aggregate demand in (23) holds whenever households plan equidistant moves over their lifetimes, even if the number of moves is not the optimal one. It is interesting, therefore, to examine the properties of aggregate housing demand with respect to N. We obtain first aggregate housing demand when moves are costless. When N----~~ (23) yields
1 anWo(i ) e it-1 H ° - T iPn iT
(24)
In order to determine the effect of changes in N, when N is finite, we work as follows. Let ~b(N; i) denote the component of the RHS of Eq. (23) that depends upon N: el(T/N)
6 ( g ; i) -- g 2 ( e i ( T / N ) _ 1)2,
e~(r/m
04) (N; i) = N3(e~ ON
(25)
(
[ T
1) 3 e i(r/N) i N -
]
T
2 + iN+ 2
)
(26)
Since, by convexity, e i(r/n) > 1 + i(T/N), it follows that 04)~ON > 0. That is, the more moves a household makes over its lifetime, the larger the aggregate housing demand. This is an important finding whose intuition we can explain as follows. Consider the definition of aggregate housing demand in (21) in conjunction with Fig. 2. Aggregate housing demand in our steady-state equilibrium is equal to the average height of the path of individual household demand over time. Our model of household behavior implies that when adjustment is costly individual households oversize when they move relative to the frictionless case at the corresponding point in time. In spite of this oversizing, aggregate demand always falls short of what it would be in the frictionless case.
36
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995)21-39
The dependence of aggregate housing demand on the rate of interest is complicated by the effect of the rate of interest on the optimal number of moves, which is ambiguous. By treating the wage rate from (20) as exogenous we consider first the direct effect of the rate of interest for a given number of moves: 1
0
1
H o Oi H°(i' PH; N ) = Wo(i )
OWo(i)
Oi
T e ir
0
+ e ir - 1 t- -~ th(U; i).
We can show that the sum of the last two terms above is positive. We note that the first term in the RHS of the above equation may be written as the interest rate times the interest elasticity of lifetime wealth. We thus conclude that the direct effect of a change in the rate of interest on aggregate housing demand is positive, provided that the first term is numerically small. Unfortunately, the total effect of a change in the rate of interest cannot be signed because of the ambiguity, from (18), in the sign of ON/Oi, as well.
3.2. Housing market equilibrium Housing market equilibrium is ensured by HD(i, PH, X ) = ISis, where /4s is aggregate housing supply, 14o is defined in (23), 2¢" as defined in (16) and (17), and W0(i) is defined in (20), or
1 1 - e -i*T e i*(T/N) P" =--~s az4w i* (e i - r - 1) X2(ei.~r/x)_ 1) 2 .
(27)
We have shown that for a given number of moves, the price of housing stock which equilibrates the housing market is an increasing function of the interest rate, provided that the interest elasticity of lifetime wealth is very small.
3.3. Properties of housing market equilibrium We noted in Subsection 3,1 above that the expression for aggregate housing wealth in (23) holds, when households plan equidistant moves during their lifetime, even when the number of moves is not optimized. Our housing market equilibrium model is thus applicable to contexts where the number or cost of moves is subject to administrative control or in other ways affected by public policy. Two points follow. First, we can examine the impact on housing market equilibrium of alternative values for B, the key parameter of the utility function that expresses the cost of residential mobility. We show above that the higher the cost of moving, the less frequently households move. From (27) we see that increases in B (the utility cost of moving) shift the housing market demand
A . M . Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
37
downwards. From (26) it follows that the fewer moves a household makes in its lifetime, the lower will be its average housing demand. Hence, the higher the utility cost of mobility, the lower the equilibrium price of housing stock, a result that fully accords with intuition. 2~ Second, in a non-Walrasian setting, where households are prevented from moving as frequently as they desire, the number of permitted moves may ration housing demand. It also follows that, in this model, changes in the number of moves permitted can be represented by shifts in the aggregate housing demand curve, while the latter remains bounded upwards by (24). An institutional feature of Eastern European housing sectors, and of some Western and Third World urban housing markets, has been the allocation of much of the housing stock at rents well below the market equilibrium price. This happened either because housing was subject to rent control, or because the housing stock was publicly owned and administratively allocated with rents well below market levels. Rent control typically means that while households may be able to secure housing services at a below-market price, moving is difficult because search costs are high. When housing is publicly owned, housing 'administrators' typically ration available housing services by directly controlling occupants' access to different packages of housing services and hence their residence spells. Housing policies such as rent controls and the administrative allocation of publicly owned housing in many housing markets have the effect of raising the cost of moving or effectively limiting the number of moves which households can make. The approach used here provides a useful way to model both phenomena.
4. Summary and conclusions The central feature of this paper is a housing market which adjusts to reflect both prices and frequency of moves. Following Cass and Yaari (1967) we assume a continuous-time finite lifetime, life-cycle model and an infinite number of overlapping generations coexisting at any point in time. A key result, which accords with intuition, is that economies with higher moving costs are likely to be associated with lower housing prices. Since residential mobility is the process through which arbitrage of the housing stock takes place, it is natural that higher arbitrage costs are associated with 21 We have assumed that the housing stock is owned originally by absentee landlords. If, instead, the housing stock is owned by the population, then the definition of Wo(i ) must be modified. The flow of income then includes interest from housing wealth, iPnlSIs . A solution for PH follows. While it is more complicated, the derivative of Pn with respect to B remains negative.
38
A.M. Hardman, "Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
lower housing prices. We conjecture that the effects of higher moving costs may be quite different in a more general model. In our model the discreteness of adjustments in housing consumption at the household level- m o v e s - is smoothed at the aggregate. This feature of the model makes it possible to apply to settings like those cited above, where administrative intervention is used to ration housing demand. We clarify the manner in which impediments to moving serve to ratio housing consumption.
Acknowledgements This paper has benefited enormously from the comments of two referees and the editors. Comments from participants in seminars at VPI and the University of Virginia, and at the 1991 meeting of the North American Regional Science Association, New Orleans, and the 1991 ASSET Conference, Athens, Greece, are also gratefully acknowledged. Ioannides acknowledges financial support from National Science Foundation grants SES9000200 and SES-9211913. We are grateful to Joan Boyd for her impeccable typing.
References Amundsen, E.S., 1985, Moving costs and the microeconomics of intra-urban mobility, Regional Science and Urban Economics 15, 573-583. Blanchard, O.J. and S. Fischer, 1989, Lectures in macroeconomics (MIT Press, Cambridge, MA). Cass, D. and M.E. Yaari, 1967, Individual saving, aggregate capital accumulation, and efficient growth, in: K. Shell, ed., Essays on the theory of optimal economic growth, ch. XIII (MIT Press, Cambridge, MA), 233-268. Clark, W.A.V. and A.D. Heskin, 1982, The impact of rent control on tenure discounts and residential mobility, Land Economics 58, February, 109-117. Coleman, D., 1988, Rent control: The British experience and policy response, Journal of Real Estate Finance and Economics, 1, no. 3, November. Edin, P.-A. and P. Englund, 1991, Moving costs and housing demand: "Are Recent Movers Really in Equilibrium?", Special issue on international studies of housing demand: The effects of government policies, Journal of Public Economics 44, no. 3, 299-320. Englund, P., 1985, Taxation of capital gains on owner-occupied homes: Accrual vs. realization, European Economic Review 27, 311-334. Feinstein, J. and D. McFadden, 1988, Dynamics of housing demand by the elderly. I. wealth, cash flow and demographic effects, in: D. Wise, ed., Economics of aging (University of Chicago Press, Chicago, IL). Hanushek, E.A. and J.M. Quigley, 1978, An explicit model of intrametropolitan mobility, Land Economics 54, 411-429.
A.M. Hardman, Y.M. loannides / Reg. Sci. Urban Econ. 25 (1995) 21-39
39
Hardman, A.M., 1987, The role of side payments in a market in disequilibrium: The market for rental housing in Cairo, Ph.D. Dissertation, Department of Urban Studies and Planning, MIT, October. Hardman, A.M., 1990, Informal rental housing markets in Third World cities: Evidence from Egypt, Department of Economics, Virginia Polytechnic Institute and State University, April. Hardman, A.M. and Y.M. Ioannides, 1991, Residential mobility, the housing market and economic growth, paper presented at the Regional Science Association Meeting, New Orleans, October, 1991, Mimeo. Henderson, J.V. and Y.M. Ioannides, 1989, Dynamic aspects of consumer decisions in housing markets, Journal of Urban Economics 26, 212-230. Hughes, G. and B. McCormick, 1981, Do council housing policies reduce migration between regions? Economic Journal 91, December, 919-937. Hughes, G. and B. McCormick, 1987, Housing markets, unemployment and labour market flexibility in the UK, European Economic Review 31, 615-645. Ioannides, Y.M., 1987, Residential mobility and housing tenure choice, Regional Science and Urban Economics 17, 265-288. Mayo, S.K. and J.l. Stein, 1988, Housing and labor market distortions in Poland: Linkages and policy implications, World Bank Policy Planning and Research Staff Discussion Paper, Report INU 25, World Bank, Washington D.C., June. Olsen, E.O., 1990, What is known about the effects of rent controls? Report prepared for the Office of Policy Development and Research, U.S. Department of Housing and Urban Development, September. Romer, D., 1986, A simple general equilibrium version of the Baumol-Tobin model, Quarterly Journal of Economics 101, no. 4, 663-686. Rosenthal, S.S., 1988, A residence time model of housing markets, Journal of Public Economics 36, 87-109. Turner, B., 1988, Economic and political aspects of negotiated rents in the Swedish housing market, Journal of Real Estate Finance and Economics 1, no. 3, November, 257-276. Venti, S.F. and D.A. Wise, 1984, Moving and housing expenditure: Transactions costs and disequilibrium, Journal of Public Economics 23, 207-243. Willis, K.G., S. Malpezzi and A.G. Tipple, 1990, An econometric and cultural analysis of rent control in Kumasi, Ghana, Urban Studies 27, no. 2, 241-258.