Housing vacancies, thin markets, and idiosyncratic tastes

Journal of Real Estate Finance and Economics, 2:5-30 (1989) �9 1989 Kluwer Academic Publishers

Housing Vacancies, Thin Markets, and Idiosyncratic Tastes

RICHARD ARNOTI" Department of Economics, Boston College, Chestnut Hill, Massachusetts 02167

Abstract

This paper presents a model with rental housing vacancies in equilibrium. Because of the indivisibility and multi-dimensional heterogeneity of housing units, the housing market is thin. As a result, a typical household entering the market is willing to pay a premium for its most-preferred over its second most- preferred available (vacant) unit. This confers monopoly power on landlords, which they exploit by setting rents above costs. Free entry and exit force profits to zero, with vacancies as the equilibrating mechanism. A nice feature of the model is that housing vacancies are socially useful in expanding the choice set of entering households, though there is no presumption that the market vacancy rate is socially optimal. Thin markets are modeled by assuming an idiosyncratic component to households' tastes over housing units.

The positive and normative properties of the basic model, which assumes costless search, are investigated. Then the model is extended to treat costly search. Finally, directions in which the model could usefully be extended are discussed.

Over the past quarter-century, since the seminal work of Muth (1969), L the theory of perfectly competitive housing markets has been developed and refined. It can now be confidently stated that our understanding of the operation of such ideal markets, while not complete, is very good. Yet there are many features of actual housing markets that are inconsistent with the perfectly competitive theory; e.g., costly search, imperfect capital markets, moving costs, and housing vacancies, to name a few. Fortunately, recent developments in microeconomic theory, notably the theories of search, imperfect information, and games, provide the tools to model at least some of these features. 2

This paper provides a model of rental housing vacancies. Despite the fact that almost all housing experts consider vacancies to be important in the process of housing market adjustment, remarkably little theoretical or empirical work on the subject has been done, 3 and there is no fully specified, general equilibrium model of housing vacancies in the literature. 4

The basic idea of the paper is that housing vacancies can be derived from three intrinsic characteristics of housing: (1) multidimensional heterogeneity of the product; (2) indivisibility (households cannot purchase fractions of different units and combine these fractions to obtain their ideal unit); and (3) multidimensional heterogeneity of household tastes. These three characteristics of housing combine

6 RICHARD ARNOTT

to render the housing market thin; even though there are many housing units, the dimensionality of the characteristics space for housing is so high that the units are sparse in this space. The mechanism generating vacancies is as follows: A landlord knows that, because of the thinness of the housing market, a typical entering household which most prefers one of his units from among those vacant is willing to pay a premium to rent that unit rather than its second most-preferred vacant unit. This gives the landlord some monopoly power which he exploits by setting rent above costs. Free entry and exit, meanwhile, force profits to zero, and profits are dissipated through housing vacancies.

The mechanism is similar to that in Chamberlinian-type models of monopolis- tic competition, except that profits are dissipated through vacancies rather than excess capacity. An attractive feature of the model, in contrast to price-search based models of housing vacancies (Read, 1987), is that vacancies perform a socially useful role in expanding an entering household's choice set, though there is no presumption that the equilibrium vacancy rate is socially optimal.

The reduced form of the model is similar to that of the random-interaction matching models in the labor economics literature (reviewed in Hosios (1986), with Diamond and Maskin (1979) being perhaps the best known example). However, the microfoundations of the model (the structural model) capture characteristics peculiar to the housing market.

The model which follows pro,Ades one of many possible formalizations of the vacancy-generating mechanism outlined above. The basic model, in which search costs are ignored, is presented in section 1. Section 2 extends the model to allow for costly search. Directions for future research are provided in section 3, with section 4 concluding.

1, The basic model

1.1. Model description

Households enter the market according to a Poisson process with constant arrival rate, 5 and depart from the market according to another Poisson process with departure rate equal to the arrival rate. When a household enters the market, it becomes instantaneously and costlessly informed of the characteristics of all vacant units, including their advertised rents. It immediately chooses one of the vacant units, and remains in that unit, paying the advertised rent, for its entire length of stay in the market.

Landlords supply units at a fixed cost per period. Each decides on the rent to charge so as to maximize his profits, taking the rent charged on all other units as given (Bertrand-Nash). While landlords know the characteristics of market de- man& they cannot perceive the tastes of a particular household. Because the household arrival and departure processes are random, the market's population

HOUSING VACANCIES, THIN MARKETS, AND IDIOSYNCRATIC TASTES

will fluctuate slightly. To simplify the analysis, it will be assumed that each landlord sets a time-invariant rent so as to maximize expected per period profit, based on the average state of the market. In order to capture the durability of housing, it will be assumed that the number of units in the market, U, remains fixed over time. These assure ~tions imply that, in setting rent, the landlord treats the average number of vacanl units, V, and average population, N, as given. Entry and exit are free.

Though housing units all cost the same fixed amount to supply, they differ from one another in many ways: floor plan, architectural style, color of paint and car- peting, etc. Household tastes differ over these characteristics and are completely idiosyncratic in the following sense: Randomly select a household i and a housing unitj. Let x0. be the extra amount the selected household would be willing to pay in rent to live in its ideal unit (supplied at the fixed cost) compared to the selected unit. Then x U is termed the mismatch cost of housing un i t j to household i. The assumption of idiosyncratic tastes is thatx U is the realization of a random variable 2 with probability density functionf(x), independent of the identity of the household and the housing unit.

With this characterization of tastes, one may conceive the choice of housing unit by a household which enters the market when V vacant units are available, as follows: The household costlessly visits every vacant unit, in no particular order. The first unit he visits has rentpl and mismatch costxl which is obtained as a random draw from thep.d. f f(x) . The sum of the rent and mismatch cost is termed the full rent, r; thus, r~ = p~ + xv The household then visits a second unit and discovers P2 andx2, and so on. It chooses that unit with the lowest full rent. The content of the assumption of completely idiosyncratic tastes will be discussed later.

To simplify the discussion, it is assumed that each landlord owns only one housing unit. The analysis is unaffected if some landlords own more than one unit as long as each landlord's units exhibit as much heterogeneity as other units in the market and no landlord owns a significant fraction of the stock of housing in the market.

1.2. Symmetric market equilibrium

In this subsection, a symmetric market equilibrium is solved for, which is defined to obtain if and only if:

1. rents charged on all units are the same (symmetry), the rent being termed the "market rent;"

2. taking the market rent, as well as the number of vacant units, asgiven, an in- cumbent landlord has no incentive to either alter the rent on his unit by a small amount or to exit; and

3. taking the market rent, as well as the number of vacant units, as given, no land-

8 RICHARD ARNOTr

lord who is not in the market has an incentive to enter and offer a unit at the market rent.

Several comments are in order. First, asymmetric equilibria, characterized by a distribution of rents, shall be ignored. 6 Second, since landlords consider only whether small deviations from the market rent are profitable, the equilibrium concept is local. Third, a property of symmetric market equilibrium in the context of the paper is that freedom of entry and exit forces profits to zero. Fourth, the equilibrium concept is Nash in rents. Fifth, the equilibrium concept is similar to that employed in spatial competition theory.

The general procedure for solving for such an equilibrium should be familiar from spatial competition theory: start with an arbitrary market rentp charged by all landlords and an arbitrary number of vacant units V. Then calculate the profit- maximizing rent of a single landlord ("the landlord"), when all other landlords charge p and the number of vacant units, including his own, is V,

po = po(p, v). (i)

Symmetry in equilibrium requires that

Po = P. (ii)

Finally, tree entry and exit force expected per period profits on each unit to ZerO, 7 i.e.,

p(1 - v) - oq-= 0, (iii)

where o,~-is the cost of a unit per period and v is the vacancy- rate, v = V/U. Equations (i)-(iii) provide three equations in three unknowns, any solution of which is a symmetric market equilibrium, according to the definition.

The calculation of the profit-maximizing rentp0(p, V) proceeds as follows: Con- sider a landlord with a vacant unit. If he sets his rent atp, households which wish to rent his unit will arrive at his unit according to a Poisson process with arrival rate equal to the arrival rate of households to the market divided by the number of vacant units; i.e., where pN is the arrival rate of households at the market, the arrival rate "at the landlord's unit" (of households who wish to rent his unit) is pN/ V. If, instead, the landlord sets rentp0 >p , the Poisson arrival rate of households at his unit will fall, since there is some probability that a household which "most prefers" his unit (i.e., the mismatch cost is lowest) will choose its second most- preferred unit because that unit's full rent is lower. As a result, the landlord's expected vacancy duration between tenancies increases, as does his expected vacancy rate. This effect is offset by the higher rent received per unit of time while the unit is occupied.

HOUSING VACANCIES, THIN MARKETS, AND IDIOSYNCRATIC TASTES

More precisely, the expected profit per unit time on the landlord's unit is

rr = p0(1 - Vo(Po;p, V)) - ~,, (1)

where n is profits, and v0 the landlord's expected vacancy rate when he chargesp0, other landlords choose p, and the number of vacant units is V. The landlord's expected vacancy rate is the expected length of a vacancy period divided by the sum of the expected length of a vacancy period and the expected length of an occupancy period. Since the process of arrival at the landlord's unit is Poisson, the expected length of a vacancy period is 1/[ao(Po;p, V)], where a0(" ) is the arrival rate at the landlord's unit. Similarly, since the departure process is also Poisson, with departure rate ~t per resident, the expected length of an occupancy period is 1/~a. Hence,

vo(Po;p,v) = ao(Po;p, V) = p (2) 1 + 1 P + ao(Po;p,V)

ao(Po;P, V) p

Let x l(V;<f(. ))) be the min imum of V draws from f ( . ). x 1(.) is a random variable and is termed the first order statistic. It is well known that the p.d.f o f x ~ is s

f ( x ' ; V , ( f ( . ))) = Vf(xl)(1 - F(xl)) V-i. (3)

Similarly, letx2(x 1, V;(f( �9 ))) denote the second lowest number drawn with Vdraws fromf(x), given that the min imum is x ~. x 2 is also a random variable and is termed the conditional second order statistic. The p.d. f o f x 2, conditional on x ~, is

f ( x 2) [ 1 - F(x 2) l V< ~(x2lxl;V,(f(.))) = ( V - I) 1 2F(-x ' ) 1 F ( ~ J " (4)

The probability that the landlord will lose a household which most prefers his unit because he sets Po > P, P(Po;P, V), is the probability that the full rent on the household's second most-preferred unit is less than that on the landlord's unit, i.e., Pr(x 1 + P0 > x 2 + P) or equivalently PF(X 2 <~ X 1 Jv P0 - p ) (i.e., the probability that the difference between the first and second order statistic is less than P 0 - P). Thus,

f0 m x 1 +P0-P P(po;p,V) = f ( x ' ; V , ( f ( . ))) f~ g(xZ[xl ;V, ( f ( �9 )))dxZdxk (5)

Substitution of (3) and (4) into (5) gives

10 RICHARD ARNOTT

ro x 1 + P 0 - P

= V(V - 1) f (x l ) (Jx~ f(x2)(1 - F(x2))V-Zdxgdx" P(po;p , V) ( 6 )

The arrival rate of households at the landlord's unit equals the arrival rate of households which most prefer his unit times the probability that such a household will not be put off by its higher rent; i.e.,

ao(Po;p,V) = { ~ - } ( 1 - P(po;p,V)). (7)

Substitution of (7) into (2), and substitution of the resulting expression into (1), gives

~ = p o { 1 - ~ } - ~ + ao(Po;p, V)

U ( 1 - P(po;p, 9) _ ( 8 ) = P0 V + N(1 - P(po;p,V))J

Since cost is fixed, the landlord maximizes expected profits by maximizing expected revenue, which entails choosing a rent at which expected marginal revenue is zero.

The first order condition with respect to P0 is

N( 1 P) Po VN aP - apo

- = 0 , or V + N ( 1 - P ) ( V + N ( 1 - p ) ) 2

N ( - ~p~-po) = O. (9) ( V + N ( 1 - p ) ) 2 ( 1 - P ) ( V + N ( 1 - P ) ) poV OP

From (6), 9

a P _ s174 Opo V ( V - l) f(x~)f(x 1 +Po - p ) ( 1 - f ( x ~ + P o - P ) ) V-2dx~. (10)

Now apply symmetry, P0 = P, which implies P = 0, to (9) and (10). Finally, apply the zero profit condition Po (N/[V+ N]) -~r 0. One then obtains 1~

ff 1 - "~V2(V- 1) (f(x'))2(1 - F(x~))V-2dx ' = O. (11) N

Equation (11) gives an implicit equation for the equilibrium number of vacancies

HOUSING VACANCIES, THIN MARKETS, AND IDIOS~2qCRATIC TASTES 11

depending on (o~,N,(f( �9 ))). While closed form solution of(11) is possible only for a very restricted class of functions, solution by numerical methods is straight- ~brward.

The following (local) comparative static results can be proven: 11

1. As population increases, the number of vacant units increases and the vacancy rate decreases.

2. As the cost of housing increases, the number of vacant units and hence the vacancy rate decreases.

3. As the housing market becomes thinner (in a precise sense explained in note 11), whether due to increased diversity of housing or to increased intensity of preferences for differences between housing units, the number of vacant units and hence the vacancy rate increases.

The first part of result 1 is explained as follows: Assume for the sake of argument that the equilibrium with a higher population has the same number of vacancies as the original equilibrium. The arrival rate at a representative landlord's vacant unit increases in proportion to the increase in population, for whatever amount he sets his rent above the market rent (see (5)), and his expected vacancy duration decreases proportionally. Thus, the marginal cost of raising rent a certain amountmthe increase in rent foregone from a longer expected vacancy duration-- decreases proportionally, while the marginal benefit of raising rent by that amount rises somewhat due to the higher occupancy rate. Since the marginal benefit to each landlord of raising rent above the market level equalled the marginal cost in the original equilibrium, with a higher population and the same number of vacancies all landlords have an incentive to raise rents, which is inconsistent with equilibrium. The other comparative static results can be similarly explained. Result 3 confirms intuition: as the market becomes thinner (in the precise sense explained in note 11) landlords' market power increases; in response, they raise rents which causes the vacancy rate to rise.

1.3. Social optimum

One may imagine an omnipotent (but not omniscient) social planner who sup- plies heterogenous housing of constant cost. When a household enters the market, the planner allows it to choose a unit from among those vacant at the time and requires that the household remain in that unit during its entire stay in the market. The household entry and exit processes are as before. The planner knows tastes in the aggregate, which are characterized by f(x), but not individual households' tastes. His decision variable is the number of housing units.

Alternatively, one may imagine a Lange-Lerner socialist planner who has only indirect control over the number of vacant units through his power to set rent.

12 RICHARD A R N O T r

Since rent and the vacancy rate are simply related according to the free entry condition, p(1 - v) - ~-= 0, control over rent gives him effective control over the vacancy rate.

The socially optimal number of vacancies minimizes the sum of the cost of unoccupied units and mismatch costs. The former cost is Co = f V ' , where V' is the actual number of vacant units. As noted before, because of the stochastic nature of the arrival and departure processes, with a fixed number of housing units, the number of vacancies will fluctuate slightly. However, since Co is linear in V,

Co = f fV (12)

provides an exact measure of the expected cost of unoccupied units, where Vis the average number of vacant units. The social mismatch costs are

fo fo N h(V') x'~(xl; V ' , ( f ( . )))dx'dV',

where h(V') is thep.d.f of the number of vacancies. Since f ( . ) is not linear in V' (see (3)),

f0 f0 f0 h ( V t) x I ~ x I V t xlf(xl; V,(f( ))) d x : . f ( ; , ( f ( . )))dx'dV' 4=

To make the analysis tractable, however, it will be assumed that the social planner computes social mismatch costs as

fo ~ C 1 = N x~f(x~;V,{f( .)))dx ~

fO c~ "~- N V x l f ( x l ) ( 1 - F ( x l ) ) V - l d x 1 (using (3)). (13)

The total social cost is S - Co + C1 and is at a local minimum when OS/OV = 0 and O:S/OV 2 > 0. The first order condition is 12

fO ~ .~-+ N x~f(xl)(1 + Vln (1 - F(x')))(1 - F(x'))V-~dx ' = O. (14)

The Vwhich solve (14) and satisfy the second order condition are local socially optimal (average) number of vacancies. As with the market equilibrium, closed form solution is possible for only a few functions, but solution by numerical methods is practical.

It is straightforward to show that in the neighborhood of a local social optimum,

HOUSING VACANCIES, THIN MARKETS, AND IDIOSYNCRATIC TASTES 13

0 V/O~,~< 0 and 0 V/ON > 0. It can also be shown that as the market becomes thinner (in a stricter sense than as defined in note 11; see note 13) the socially optimal number of vacancies increases. ~3 The sign of Or~ON, however, appears to be ambiguous.

It is interesting to note that both the market and socially optimal vacancy rates are independent of the Poisson arrival-departure rates. The explanation of this result is different under the two regimes. In the market equilibrium, an increase in the arrival-departure rate decreases the expected vacancy and tenancy periods in the same proportion, and so does not affect the market vacancy rate. In the social optimum, the planner chooses V to maximize the expected utility of each household that enters the market, and this is independent of the Poisson arrival- departure rate.

The market vacancy rate and the socially optimal vacancy rate are now compared. The market vacancy rate is (locally) higher than optimal if a small rise in the number of vacancies from the market equilibrium level increases social costs, i.e. if (OS/OV)[ e > 0, where] e denotes "evaluated at the market equilibrium". From (14) and (11), this inequality reduces to

[fo lifo ] V2(V- 1) xf(1 + Vln (1 - F))(1 - F)V-~dx f2(1 - F)V-2dx

> - 1 ; (15)

when this inequality is reversed, the market vacancy rate is locally suboptimal. I have tried unsuccessfully to establish general properties of the LHS of the inequality (15); ~4 it would appear that the inequality can go in either direction, and therefore that the market vacancy rate may be larger or smaller than the socially optimal vacancy rate. Even if it were possible to establish that the market vacancy rate were locally higher than the socially optimal rate, since there may be multiple market equilibria, it might nonetheless be the case that the market would come to rest at an equilibrium with a sub-optimal vacancy rate.

The reason the market vacancy rate may be either higher or lower than optimal is that there are two market distortions operating in different directions. On the one hand, by exercising their monopoly power, landlords set rents too high, which effect by itself results in a super-optimal vacancy rate. On the other hand, landlords collectively neglect that by setting a higher rent, the equilibrium number of vacancies increases, which expands an entering household's choice set; this effect by itself results in a sub-optimal vacancy rate.

Thus, there does not appear to be a strong a priori case for government intervention. If, however, the government were able to establish that the market vacancy rate were higher than optimal, it could, at least locally, improve welfare by taxing vacancies or landlords' costs.

Because of the difficulty in establishing general results concerning the rela-

14 RICHARD ARNOTT

tionship between the market and socially optimal vacancy rates, it is of some interest to examine a couple of special cases.

1.4. Special cases

In the first special case treated, f (x) is uniformly distributed.

A) Uniform distribution:

|

f ( x ) = ~" forx E (O,D). (16) D

D is the largest possible mismatch cost and D/2 is the average mismatch cost between an arbitrary household and an arbitrary vacant unit. Thus, the larger is D, the greater is the diversity of housing units in the market, or else the more idiosyncratic are tastes in the sense that households are willing to pay a greater premium for a preferred unit to a less preferred one, holding the characteristics of the units fixed. D is termed the diversity of housing units in a market.

Substituting (16) into (11) and solving gives

= (17)

An intuitive interpretation of (17) comes from treating Y-as the num6raire, in which case D/[Nd~] is the average premium (in terms of unit housing costs) a household would be willing to pay for its most-preferred housing unit compared to its second most-preferred if there were no vacancies; as this premium increases, the landlord has more market power, and the equilibrium vacancy rate is higher. If T - D/[No~] is defined to be the thinness of the market, then

oroe 1 + V@-"

(17')

From substitution of (16) into (14), the socially optimal vacancy rate is given by

= or v ~ = ( v ) , ( 1 8 )

where 0 denotes the social optimum. Thus, Ov~ > 0, which implies Ov~ < 0,


and Ov~ O. Recall that the socially optimal vacancy rate was determined by the tradeoff between the increased cost of vacant units and the decreased mismatch costs as the vacancy rate increases. As .~increases, the cost of vacant units increases, and the optimal vacancy rate fails. The average mismatch cost equals the expected value o fx ~, which is

-.V+l ~-V- = -N"

Holding the vacancy rate fixed, therefore, mismatch costs increase with D/N. Hence, the higher is D/N, the higher the vacancy rate.

A comparison of (17') and (18) gives

5 - : V = x / f > T U f = 0e>00" (19)

Thus, the market vacancy rate exceeds the optimal vacancy rate. The difference is, however, trivially small, is and may be due to the approximations taken in ignoring the stochastic fluctuations generated by the Poisson arrival and departure process.

B) Dimensionality of the characteristics space or variegation: In predicting the quantitative effect of some change on the vacancy rate, expect-

ed household welfare, etc., it is necessary to know the functional form off(x). The uniform distribution, as a one parameter (D) class of distribution, is overly restric- tive. Here, a two parameter class of distribution is derived, based on an intuitive geometric characterization of idiosyncratic tastes.

A household's idiosyncratic tastes over vacant units are characterized in terms of an idiosyncratic characteristics space. The location of a particular household's ideal unit is characterized by a randomly selected point (each point has an equal probability of being selected) in this space, the location of the first vacant unit by another, independently, randomly selected point, etc. Different households' tastes are characterized by different sets of randomly selected points. The geometry of the characteristics space then determines the functional form off(x).

Since it has been assumed that different households have the same f(x), it is natural to assume that, in a one dimensional characteristic space, housing units are randomly distributed on a circle with circumference D; in this situation, f (x) is the uniform distribution on [0,D] (the case analyzed above). Similarly, it is assumed that in a two dimensional characteristics space, housing units are randomly distributed on the surface of a sphere with circumference D, and that in higher dimensional characteristics spaces housing units are randomly distributed on the surface o f a hypersphere with circumference D. In this way a two parameter class of distributions is generated, with parameters D and 8, where 8 is the dimensionality of the characteristics space.

16 RICHARD ARNOTT

It is straightforward to show that

f(x) = { k(8)x~-~k(8)(D f~ r [ 0 ' D ] 2

- x ) 5-' forx C [D~ ,D] L / 2

(20a)

and

F(x) = {

k(6) x~ [ ] - - f o r x r 0, D 8 2

( D - x ) a f o r x r I D a , D ] ' l - k(8) 8 1 2 1

(20b)

where

828-1 k ( 8 ) - D6 (20c)

It follows from notes 11 and 13 that an increase in the diversity of housing (D) increases (locally) the market and optimal vacancy rates. As 8 increases, holding D constant, housing is said to become more variegated. An increase in the variegation of housing increases the landlord's market power and therefore the vacancy rate if (OZP/O_poOS) lp=po < 0. It does not appear possible to sign this derivative. The effect of more variegated housing on the socially optimal vacancy rate also appears to be ambiguous; what matters is whether the addition of a vacant unit decreases mismatch costs by more or less as 8 increases (02C~/0V08 ~ 0).

An interesting application of the above characterization of tastes would be to at- tempt to estimate the increase in mismatch costs caused by rent controls due to their depressing effect on the vacancy rate.

1.5. Comments

This section has developed a simple model which permits the calculation of market and optimal vacancy rates in a stylized housing market. The basic idea is that because housing units are very heterogeneous, a typical household is willing to pay a premium for its most-preferred vacant unit over its second most preferred, even when the costs of supplying the two units are the same and the number of vacant units is large. This confers market power on the landlord, which he exploits


by pricing above unit cost. Freedom of entry and exit dissipate profits by generating vacancies. To simplify the argument, it was assumed that tastes are completely idiosyncratic, i.e., that there is zero expected correlation between any pair of households' rankings of housing units with the same costs. The multidimensional heterogeneity of housing causes housing units to be sparse in characteristics space, which was taken to be the defining characteristics of a thin market. Thus it may be said that the model of the paper develops a theory of housing vacancies based on thin markets and idiosyncratic tastes.

In what ways does the model of the paper differ from the model of perfect competition? First, the latter model assumes a complete set of markets. In the context of a housing market in which housing exhibits multidimensional heterogeneity, the assumption of complete markets would imply that a household could rent a house with any set of characteristics it wanted. But the model of the paper assumes that the sets of housing characteristics between which the household can choose is restricted, thereby implicitly assuming an incomplete set of markets. Second, the model of perfect competition assumes symmetric information. In the paper, there is an asymmetry of information in that landlords do not know households' tastes. Suppose they did and that housing were non-durable. Then an entering household would face a group of landlords, each of whom would offer to construct the household's ideal housing unit. Bertrand competition between the landlords would then drive rent down to unit cost. Thus, from the perspective of the Arrow- Debreu model of perfect competition, the results of the paper stem from particular relaxations of the assumptions of complete markets and symmetric information.

Another observation is that the model of the paper is consistent with housing being either durable or non-durable, though I think of the model as describing the stationary state of a housing market with durable housing. It should also be noted that since landlords do not jack up rents once a tenant household has moved in, even though by assumption the tenant will remain in the unit during his entire tenure in the market, there must be a long-term enforceable contract between landlord and tenant.

It is appropriate at this point to introduce search costs, before discussing other extensions to the model.

2. Search costs

Thus far, it has been assumed that an entering household is perfectly informed concerning the characteristics and rent of all vacant housing units. In this section, the alternative assumption is made that entering households are completely unin- formed concerning the idiosyncratic characteristics of vacant units and must un- dertake costly search to obtain this information. They do, however, know which units are vacant.

To simplify the analysis, it is assumed that each search is equally costly, c. Along

18 RICHARD ARNOTF

with the other assumptions made, this ensures that each household has a reservation full rent search strategy.

Assume that the housing market is in symmetric equilibrium. Consider an entering household which has already been searching and the best unit it has found has x = ~. Should it search again? The expected benefit from an additional search is

fo x (2 - x) f ( x )dx ;

if, for instance, it found a uni t with x ' < ~, its mismatch costs would be reduced by (Y - x ') during each period, and its expected length of stay in the market is 1/p. The household will stop searching if it finds a unit for which the expected benefit from an addit ional search is less than or equal to c, i.e., the stopping rule is stop searching i f x ~ 2, where 2 is defined implicitly by the equation

fo ~ (2 - x ) f ( x ) d x = c. (21)

From (21) it follows that d2/dc > 0; as search costs increase, households become less fussy, i.e., 2 increases.

T h e market equilibrium with costly search is now investigated. The probability that a household will find its first vacant unit with x ~< 2 on the

n 'h search is F(2)(1 - F(2)) "-1. Thus, the expected number of searches is

v �9 (2,V) = ~. nF(2)(1 - F(2)) "-1 + V(1 - F(2)) v, (22)

n=l

where (1 - F(2)) v is the probabili ty that the household will search all the vacant units in the market without finding one with x < 2.

The probabili ty that the household will choose a unit with x I C (x' - dx' ,x ') where x ' < 2 is the probabili ty that it finds a n x < 2 in Vsearches times the probability tha tx 1 E (x' - dx',dx') conditional on finding a n x ' -<2 in Vsearches. Since the household stops searching i f x ~< 2, then

f ( x 1 ) f ( x ' ) = (1 - (1 - V(2)) v) ~ f o r x I < 2 . r t x )

(23a)

The probability that the household will choose a unit wi thx 1 C (x' - dx' ,x') where x ' > 2 is the probabili ty that it does not find a n x ' < 2 in Vsearches times the probability that x ~ C (x' - dx',dx') conditional on not finding an x ' < 2 in V searches,

f ( X 1) = V f ( x l ) ( 1 - F ( x I ) ) V-1 f o r x ~ > 2. ( 2 3 b )


For future reference, note that the expected mismatch costs are

f0 if( C1 = N x x l )dx I

[ 1 f ( x ) = N Jo x F ~ - ( 1 - (1 - F(2))V)dx ~ + N Vx~f(x')(1 - F(x l ) )V- tdx '.

(24)

The determination of g ( x 2 1 x 1) is somewhat more complicated. For X 1 > X, ~(X2]X 1) is given by (4). F o r x I <2: I fx 1 < 2 was obtained on the n th search, then all the previous n - 1 draws were greater than2. Thus, thep.d . f ofx 2 conditional onx ~ < 2 being obtained on the n th search is

~(x2ln,x I < 2 ) = (n - 1)f(x2)(1 - F(x2)) n-2

(1 - F ( 2 ) ) " - ' , forx 2 > 2 and n >/2 (25)

and the p.d.f o f x 2 conditional o n x I < 2 is 16

V ff(xZlx ' < 2 ) = Z fi(x2ln, x~ < 2)F(2)(1 - F ( 2 ) ) ' - ' forx 2 > 2.

n=2 (26)

I fx ~ 42 , the probability that a landlord who raises his rent top0 > p will lose a prospective tenant is again the probability t ha tp + x 2 < P0 + x 1. Hence

P(p0;p,V) = f~po+pf(x ~) f x'+eo-p ~(x2,x ~ < 2 ) dx2dxl

+ ) Jxl g(x21xl)dx2dxl' (27)

and

OP ip ~ p = V ( V - 1)J~(= (f(xl))2(1 - F ( x l ) ) V - 2 d x 1. (28) Opo =

The interpretation of (28) is that if a household finds a unit of quality X 1 < 2, then x 2 > 2 and the probability that its second most-preferred unit has only slightly higher mismatch cost is negligible. Thus, when a landlord raises his rent by a small amoun t above the market rent, he need only worry about losing households which have not found a vacant unit with x 4 2 in Vsearches. Hence, the higher are search costs and the higher therefore 2, the greater the landlord 's monopoly power, the higher his markup, and the higher the vacancy rate.

20 RICHARD ARNOTI"

The equation determining the market number of vacancies, the analog to (11), is

_ v 2 ( v - 1) f o~ 1 - ~ ~ f ( x ~ ) 2 ( 1 - F ( x l ) ) V - 2 d x ' = O. (29)

It is evident that the market equilibrium has the local comparative static property that dV/d.~ > 0; since from (21), dude > 0, then dV/dc > 0, as claimed above.

At the social optimum, the number of vacancies minimizes the costs of vacancies (Co), mismatching (C0, and search (C2). From previous derivations, Co = i V , C1 is given by (24), and C2 = Ncd#(.~, lO, where ~(~, V) is given in (22). The expression for the optimal number of vacancies is complex. When search costs are suf- ficiently high that households take the first unit they search, vacancies do not perform a socially useful role. On the basis of this observation, I conjecture that the optimal number of vacancies falls as search costs rise. Since the equilibrium number of vacancies rises with search costs, an implication of the conjecture is that for search costs above some level, the equilibrium number of vacancies exceeds the optimal number. In this circumstance, the social benefit from search exceeds the private benefit, since additional search decreases landlords' market power and hence the efficiency loss from excessive vacancies. This in turn implies that household search should be subsidized. How this could be implemented is moot. A related policy would be to require more informative advertising, which would reduce unnecessary search and effectively reduce search costs.

3. Extensions and directions for future research

This section looks at a number of useful extensions to the model. To simplify, search costs are ignored for most of the discussion.

3.1. Temporary accommodation

Realistically, when a household enters a housing market, it does not instantaneously have to find a housing unit in which it will reside for a long time; temporary accommodation is possible, either in a motel or with family or friends. Associated with such accommodation is a rent, either explicit or implicit (incon- venience cost), of say p per period. A household will choose temporary accommodation when it first enters if the expected gains from doing so for an extra period exceed the expected costs. If the household waits before getting long-term accommodation, a new vacant unit may become available that it prefers to any unit that is vacant when it enters the market.

Suppose that the best vacant unit when it first enters the market has x = x'. It


will take the unit ifx' < x and not otherwise. ~ is determined by the condition that the expected marginal benefit of living in temporary accommodation a period of time dt equals the marginal cost. The expected marginal benefit equals the probability that a new unit becomes vacant during this period, gNdt, times the expected gain conditional on a new unit becoming available. This gain equals the probability the household will want the unit divided by the expected number of households in the temporary accommodation pool who will want the unit times the expected gain conditional on receiving the unit. Thus, the expected marginal benefit equals marginal cost condition is

gN fo r (2- - x) f ( x ) d x = p _ P, r

(30)

where T is the number of households in the temporary accommodation pool. The rate at which households leave the temporary accommodation pool is gN/T, while the rate at which enter households join the pool is (1 - F(x))VgN. Hence,

1 T = (1 - F ( x ) ) v " (31)

The landlord's profit-maximization problem is altered, since he will realize that there is some probability that his unit will be snatched up as soon as it becomes available by someone in the temporary accommodation pool. The social op- timization is altered since no household will have a mismatch cost in excess of x , and since the social cost of having households living in temporary accommodation needs to be counted.

From the above discussion, it should be evident that a full analysis of this extension is relatively straightforward. To the extent that those in temporary accommodation can be regarded as homeless, the model can be employed to analyze the homeless rate.

3.2. Internal mobility

A related extension would be to allow mobility within the market. A household would move into permanent accommodation, but would keep an eye out for preferred units as they become vacated. In particular, if the mismatch cost of its current unit were X and the cost of moving were rn, the household would move if a vacant unit became available with x < ~ where )?is a function of m, 9~, ( f ( . ) ) , and gN; it would adopt a reservation moving rule.

An interesting aspect of internal mobility is that the social benefit from a household's moving exceeds the private benefit, since when a household moves it generates a vacancy chain which will result in some other households being better

22 RICHARD ARNOTI"

matched than they otherwise would have been. Thus, moving costs should be subsidized.

The empirical literature on mobility indicates that the majority of moves internal to a housing market are triggered by demographic change. 17 This needs to be treated in any model of internal mobility satisfactory for empirical purposes. Relatedly, account should be taken of the different search behaviors of different demographic groups.

3.3. Vertically and horizontally differentiated housing

Treating tastes as idiosyncratic is a useful simplification with respect to some housing unit characteristics, such as style, color scheme, and floor plan. However, there are other housing characteristics that are not appropriately described by idiosyncratic tastes. One such set is vertically differentiated or quality characteristics: i.e., those characteristics for which everyone agrees that "more is better," such as structural quality, good ventilation, and quality of accessories and appliances. If households more or less agree on the relative weights to attach to different quality attributes, then vertical differentiation can be captured by a scalar, and the housing market may be regarded as a set of submarkets differentiated by quality.

Another set of housing characteristics that are not appropriately described by idiosyncratic tastes are horizontally differentiated characteristics. These are characteristics that have a natural ordering, but are not vertically differentiated. An example is a set of locations that are "equally accessible" from the perspective of the market, though a particular household will order the locations according to dis- tance from its work. Since households' rank ordering of housing units at equally accessible locations is systematic, household tastes over horizontally differentiated characteristics are not idiosyncratic. Another horizontally differentiated characteristic is the number of bedrooms, given total bedroom area.

The economics of markets with horizontally and vertically differentiated prod- ucts has been extensively examined in the literatures on product differentiation and spatial competition. Before the model of this paper can be applied in empirical analysis, it will be necessary to integrate it with these literatures so that housing units are simultaneously differentiated horizontally, vertically, and idiosyn- cratically. This is an important but difficult task. One problem is that non- existence problems plague models with horizontal differentiation, though the noise introduced by idiosyncratic tastes may eliminate this difficulty (de Palma et al., 1985). Another problem is to satisfactorily model the household's search process; search in the housing market is strongly systematic, but systematic search is difficult to model persuasively and in a way that is analytically tractable. Relatedly, the landlord's advertising problem should be considered, not only his expenditure on advertising, but also its informational content.


3.4. Heterogeneity over households

It will also be necessary to extend the analysis to treat systematic, as well as idiosyncratic, heterogeneity across households with respect to tastes, search costs, and information. An interesting aspect of this is that housing markets appear to be segmented according to the information of the household. New entrants to a housing market appear to settle in larger apartment buildings with similar units, which have higher rents and vacancy rates than other units of comparable quality, t8 The tentative stylized fact that the simple correlation between the vacancy rate and the idiosyncratic variety of housing units across submarkets is negative appears to con- tradict the model of section 1. One explanation, consistent with both the theory and the stylized fact, is that the costs of a vacancy for owners of large buildings are relatively low because they are less likely to be liquidity-constrained than small landlords. Another explanation, in line with Read (1987), is that, because they are poorly informed and need to find accommodation quickly, new entrants tend to settle disproportionately in high-rent (standardized for quality) units; old resi- dents, meanwhile, who have had the opportunity to search informally and at low cost, tend to take idiosyncratic units that fit their needs particularly well and ex- press their individuality. 19

3.5. Two-sided matching problem

In the model presented, there was a one-sided matching problem: households were looking for appropriate housing. In fact, of course, the housing market entails a two-sided matching problem since landlords are also looking for good tenants. Since a good tenant tends to be a good tenant, independent of who his landlord is and of where he is housed, the idiosyncratic element of this side of the matching problem appears unimportant.

The analysis of the two-sided matching problem should be interesting. Segmen- tation according to expected tenancy cost is to be expected, with low cost tenants (those with good "signals" and good references) paying less for the same housing. The labor economics literature on the two-sided matching problem, reviewed in Hosios (1986), should prove useful in the analysis, even though it does not ex- plicitly treat idiosyncracy. 2~

3.6. Nonstationarity

The paper analysed a (stochastic) stationary state. It will be of interest to examine how the behavior of the market differs when it is either growing or contracting, or when it is subject to anticipated or unanticipated shocks and policy changes.

To begin, consider modifying the basic model to allow for a constant Poisson

24 RICHARD ARNOTF

arrival rate that is either larger or smaller than the constant Poisson departure rate. In the former case, the market's population is growing at a constant rate; in the latter, it is contracting at a constant rate. This extension exposes a couple of interesting points. First, does a housing market become less thin as it grows? Prob- ably, but the effect is diminished by the greater locational differentiation of a larger market. Second, one can simply derive that the vacancy rate is higher, the more rapid the growth rate of the city. The intuition for this is that the higher rate of arrivals, relative to the size of the market, gives landlords more market power.

Casual empiricism suggests, however, that vacancy rates tend to be high in de- pressed cities and low in booming cities. The apparent discrepancy between this tentative stylized fact and the theoretical result probably derives from the distinc- tion between anticipated and unanticipated growth. In response to unanticipated growth, the vacancy rate falls because of rent stickiness. In this context, the anal- ogy between vacant housing units and inventories merits note. In any event, it will be useful to derive the comparative static response of the market to both anticipated and unanticipated changes in exogenous parameters and government policy.

Relatedly, the cyclical properties of the housing market and the vacancy rate should be examined, particularly since the housing market is so sensitive to the business cycle. For example, is the vacancy rate pro- or counter-cyclical, and why?

3. 7. Econometric analysis

Probably because of the absence of a well-articulated theory of vacancy rates, there has been remarkably little empirical analysis of housing vacancies despite their acknowledged importance in the process of housing market adjustment. 21 Hopefully, this paper and those by Read, drawn from Read (1987), 22 will stimulate further econometric analysis of housing vacancies, which in turn will generate stylized facts that will encourage further developments in the theory of housing vacancies.

4. Concluding comments

This paper has presented a new model of housing vacancies. The basic ideas were that because of idiosyncratic tastes and the indivisibility and multidimensional heterogeneity of housing units, the housing market is thin. The thinness of the market confers monopoly power on landlords, which they exploit by setting rents above costs. Free entry and exit force profits to zero, with vacancies as the equilibrating mechanism. A nice feature of the model is that housing vacancies perform a


socially useful role by expanding the choice set of the households entering the market, though there is no presumption that the market vacancy rate is socially optimal.

The basic model was very simple. Households were considered identical except with respect to an idiosyncratic component in tastes. Housing units too were identical except for an idiosyncratic component. Households entered and exited from the market at the same Poisson rates. When a household entered the housing market, it instantaneously became costlessly informed concerning all rental op- portunities, and immediately moved into that vacant unit with the lowest full rent (market rent plus mismatch costs) where it stayed until exiting from the market. In this simple model, the equilibrium vacancy rate was higher, the thinner the market and the lower operating costs. It also appeared that the market vacancy rate could be either higher or lower than socially optimal. On the one hand, landlords have monopoly power, which by itself causes the rent level and the vacancy rate to be too high; on the other hand, landlords collectively neglect the fact that by raising rents they raise the vacancy rate, which expands the choice set of entering households.

The basic model was extended to treat search costs, and it was found that an increase in search costs increases the market vacancy rate but decreases the socially optimal rate. Directions for future theoretical research were also discussed: the in- corporation of temporary accommodation and internal mobility within the market; the treatment of vertical and horizontal, as well as idiosyncratic, differentiation between housing units; heterogeneity among households, including their quality as tenants; and the response of the market to anticipated and unanticipated exogenous shocks, policy changes, and the business cycle. This is an am- bitious but feasible research agenda. When completed, the extended model should provide a sound basis for econometric analysis of vacancy rates that will improve our understanding of the role of vacancies in the housing market adjustment process. Moreover, the normative properties of the elaborated model should provide insight into appropriate government intervention vis-h-vis the rental housing market.

Apendix: Notational glossary

(Variables presented in order of appearance in text)

U V N x

f(x) P

number of housing units in the market number of vacant housing units in the market population in the market mismatch cost p.d.f o fx (F the corresponding c.d.f) market rent

26 RI CHARD ARNOTT

r

o

~v yr

a

x 1

Y(x~; �9 ) x 2

~(XqX 1; ") P(')

Co h(.) C1 S e

0 D T k( . ) 8

r

~ ( . )

/7(x21n,x ~ <~)

~,(xqx' <2) c~ 2 T m s z(.) ~(')

full rent ( - p + x) fixed cost of a housing unit vacancy rate (-= V/U) Poisson arrival rate at, and departure rate from, the market expected profit per housing unit per unit time arrival rate lowest x (first order statistic) p.d.f o f x I (/~ the corresponding c.d.f) second lowest x (second order statistic) p . d f ofx 2 conditional on x ~ probability that the most-preferred unit does not have the lowest full rent social cost of unoccupied units p.d.f of the number of vacancies social mismatch costs total social cost evaluated at market equilibrium evaluated at social optimum diversity of housing units in the market thinness of the housing market intermediate function dimensionality of the idiosyncratic characteristics space or the variegation of housing units in the market search cost reservation mismatch cost with search expected number of searches p.d.f of x 2 conditional on x 1 < 2 being obtained on the n th search p.d.f o f x 2 conditional on x ~ < 2 social search costs reservation mismatch cost with temporal- accommodation size of temporary accommodation pool cost of moving reservation mismatch cost with internal mobility function used in definition of thinner market intermediate function

Acknowledgments

The research on this paper was undertaken in the summer of 1986 while I was a Visiting Scholar in the Faculty of Commerce and Business Administration at the University of British Columbia. I would like to thank the faculty there, in par-

H O U S I N G VACANCIES, THIN MARKETS, AND IDIOSYNCRATIC TASTES 27

ticular members of the Urban Land Division, for their hospitality. I would also like to thank participants at the McMaster Workshop on Recent Developments in Spatial Analysis and at seminars at U.B.C., Ohio State University, Universit6 Libre de Bruxelles, Carleton University, and Boston College for helpful comments, as well as Salman Wakil for excellent research assistance and Steffen Ziss for helpful comments.

Notes

1. In Muth (1969), Muth builds on papers he wrote earlier. 2. This theme is developed in Arnott (1987a). 3. A lengthy discussion of alternative explanations of housing vacancies is provided in Arnott

(1987b). 4. However, in a recently completed dissertation, Read (1987) develops several g.e. models of hous-

ing vacancies based on costly search by households to find the lowest rent. 5. A Poisson arrival rate of A means that the probability of a household arriving in an interval dt is

Adt, independent of history. 6. Asymmetric equilibria, characterized by a distribution of rents, may exist. 7. Strictly speaking, it is expected profits discounted to the time of entry that should equal zero. Since

a unit will remain vacant for some period of time after it has been introduced into the market, the two equil ibrium conditions coincide only with a zero discount rate. Thus, the maximizat ion of per period profits is strictly correct only when the discount rate is zero, in which case .~-has the interpretation as ongoing operating costs plus depreciation (since with a zero discount rate, amortized capital costs are zero).

8. The probability that x I C [x',x' + dx'] equals the probabili ty that the first number drawn lies in [x',x' + dx'] (f(x')dx') times the probabili ty that the remaining V - 1 numbers drawn exceed x ' , (1 - F(x')) v - l , plus the probability that the second number drawn lies in [x',x' + dx'] times the probability that the other V - 1 numbers drawn exceed x' , etc.

9. The second order condit ion is

OP O2P - 2 ( V + N(1 - P)) - - - P0 V ~ - < 0, where

OPo oPo (i)

02P fo ~ Op~ - V ( V - 1) f ( x l ) { f ' ( x 1 + p 0 - p ) ( 1 - F ( x I + p o - p ) ) v-2

- ( V - 2) ( f (x I + P0 - P))2( 1 - F( xl + Po - P))V-3} dxl. ( i i )

Applying (ii), along with symmetry and zero profits, to (i) gives

fff - 2 2(1 - F)V-2dx 1

~r foo _ F) V-2 N 3o {if '(1 - ( V - 2)/3(1 - r )V-3}dx 1 < O. ( i i i )

It appears that (iii) can be positive, in which case there may be multiple symmetric market equilibria.

xP~-A((X#),.4- l)((x~/)~d- I)UlA + l)(X~/)~f~/x.. =

xl) I-A((x)z.4 - l)((x)Zet- I)Ul A + l)(x)zfx 9f

uaql "0 < xaoj (x~/)i. q = (x)r=l leq~ tlans I > -~/mmsuoa aA!l -[sod e sls!xa aJaqlJ! `{lUO puej! (x)l .q ueql la>lJetu aauu!qz-~J e az!aalametla ol p!es s! (x)e.q "a.[a H 'fl

"0 < ixPl-.4((ix)d- I)z((,x)d- l)UI)( A +

l-.+((ix).4 - l)((ix)d - l)UlCtX)flxo~ N

s! uo!lipuoa JapJo puooas aq.l. [I • lleJ ol alea `{aup,aeA aql

asnea plnoa~ qa!wa 7uaa la>lJp,tu aql taola q sluaJ aaeao I ol aA!luaau! ue aaeq pinata spaolpue I ii , 'tunp -q!l!nba pasoddns s!ql u! "aaua H 'atup,s aq] su!eu,aa Jo Slle j luauaq leU!~Jetu aql "pasp,.aJau! lou seq alto ,{auednaao aql aau!s al!qta "sasp,aa.'m! `{isnon~!qtueun laâl la~petu aql aôqe luaa s!t I ,~u!s!p,J jo tu!q ol lso3 leU!~'aetu aql 'snq& "asp,aJ.'~u! uo!lelndod aql aao.la q uaaq aêq pinata aq t[p,ql l!un s!q saajaJd lsotu qa!qta ploqasnoq p, aso I ol `{la~l![ aJotu s! aq "luaa lO~laetu aql aAoqp, lunotup, uaA!~ e lua.I s!q flU!haS u[ "(pagueqaun s! (x)fpup,) sl!un lup,aeA aaom ~aou aap, aaaql asneaa~l 'luaa la)laetu aql aôqe lunotup, u!el -Jaa p, luaa s!q sas!eJ oq,~ pJolpue I aA!leluasaadaa aql aap!suoa tao N "atues aql su!muaa ~o saseaaaap l!un lueaeA qaea le alea leA!aJe aqI "uo!lp,lndod u] aseaaau! uP. ql.tax atues aql su!etuaJ ~o sas[J ale2 ~aueop,^ aql asoddn S :luatunffJe ailspnaq ffu.ttaOllO j aql ~u]sn paqs!lqelsa aq ue.~ I 1I nsaa jo lap,d puoaas aql

'1 11 nsa-[ jo Bed ls2 U aql saqs!lqe}sa N/~,' sp, (11) Jalua N pup,j' lp'q& "ale2 `{aup,aeA 2a~ao 1 e u! sllnsaa "aiojaq tuoaj "tp]qta "la:,laetu ~u[snoq aqlJo ,flu!ua~la!q L e sp, laat.la ames aql ,{laSpaId seq "l!un ~elaUOtu aqlJO anleA aql ul. a~up,q3 palea[pu! aql qll?a ~UOle 's,,u.lal ip,aa u!x' u! aseaJau} s!t[l "snql affueqa aql aalje ~ pup,.~ u! aSueqa aql aaojaq salouap q aaaqta '(Iz + I] /X)q..q = (x)v.J lp'ql ,{eta P. qans u! saSup,qa (x).q uaql 'stu~a| [eu[tuou u[ pax~ su!etuaz.~, leql as anle^ u! sasea~au! l[un Aaelauotu aql J[ pup, z uo[Bodoad e `{q stuaa| [p,aa u[ saseaaau!ATj! leql alou "lxa N

• ~ llnsaJ saqs!lqelsa s!ql "tA < rA ~ (I - ,A)r(,A) < (I - zA)z(zA) "`{l!lenbau[ aAoqe aql pue (11) tuoaj "uaqi '(x), ..:/ol ~u!puodsaJaoa sa[aup,.aeAjO aaqtunu tun!aq!l!nba aql aq ~ la" 1

zp 7p r-A((Z)"4 -- l)r((z)J)

pup, "xp/zp (z)j= (x)zfuaq] "0 < x ii , ~oj I > xp/zp ql!ta ((x)z)lal = (x)z.q aau!s • stxe-x aql jo uopp,'fi'UOla 'leUOpaodoJd `{[!ap,ssaaau lou tl~noql 'luals[suoa e `{q (X)l _q tuoaj pau[el

-qo aq up,3 (x)z d lp,tll s! uo!l!puoa s!ql 'spJOA~ U l "0 < x IIe JOJ I > xp/zp pue 0 = ((0)Z)l.:l = (O)zJ ql!m ((x)Z)l ./= (x)z J lpmll qans (.)z uoIDunj alqe!luaaajj!p/[isnonu!luo3 e sls!xa a~aql j! ~lUO pue j! (x), .4 ueql la'~lJetu ~tqsnoq .tauut. qj-'o ue aZ!.la]ap,Jp,43 ol p!p,s s! (x)z._ q" "(x)U I pt, p, (.V)1.4 s.J~l r,~ o~1 a~led, q l

• lsaaalu! 3!tuouoaa jo lou s[ 0 = ([iV + A]/N) uo.qn[os art1 '0l

1LON~iV Q~IVHDI~I ~


lf0 = ~ zf l(z)(1 + Vln(1 - F l ( z ) ) ( 1 - F l ( z ) ) V - l d z (using z =- kx)

= ~ x f l (x)(1 + Vln (1 - Fl(x))(1 - Fl(x ) )V- ldx (replacing z with x).

From (14) it follows that a thinning of the market in this sense has the same effect on Vas an increase in population.

Defining 0(.~,N, V) = ,~-+ N f ~ x f ( 1 + Vln (1 -F) ) (1 -F)V- ldx , one obtains

OV ON

ON Cv "

where ON = f~ xf(1 + Vln (1 - F))(1 - F) v- ldx < 0 from (14), and 0v, given in note 11, is positive from the second order conditions for a minimum. Thus OV/ON > 0 and similarly OV/O.~-< O. Since

v

ON (N + V) 2 \ V ON "

the sign of do~ON depends on whether the elasticity of the number of vacancies with respect to population is greater or less than one, which appears ambiguous.

14. None of the inequalities in Hardy, Littlewood, and Polya [1934] appears applicable. 15. If, for example, the market vacancy rate is 4%, and the number of vacant units in the market is

100, the optimal vacancy rate is 3.96%. 16. Ifn = 1, there is no second most-preferred unit. This can be treated by giving$( - ) a mass point at

x 2 = oc with probability mass F(~). 17. The causality running in the other direction merits further investigation; i.e., the extent to which

demographic change is influenced by local housing market conditions (e.g., Smith et al., 1985). 18. I would like to thank Marion Steele and Andrew Muller for bringing this to my attention, and

discussing its possible causes. 19. A third explanation, which anticipates discussion later in the paper, is that tenants differ in their

cost to landlords and in their mobility characteristics. By whatever mechanism, high-cost, high- mobility tenants tend to gravitate to the larger buildings.

20. Obversely, the model presented in this paper may generate insight into the two-sided matching problem between employer and prospective employee in the labor market, which generates both job vacancies and unemployment. That is, it may prove useful in that context to examine idiosyncratic tastes on the worker's side, which affects his job satisfaction, as well as the idiosyncratic element in a worker's skill, i.e., how well his skills mesh with those needed in the particular job for which he is applying.

21. Two good, recent empirical papers on the subject are Rosen and Smith (1983) and Guasch and Marshall (1985). I discuss them, as well as the earlier theoretical and empirical literature on housing vacancies, in Arnott (1987b).

22. I understand that there is to be a special issue of the Journal of the Ameriean Real Estate and Urban Economics Association on housing vacancies.

References

Arnott, R. "Economic Theory and Housing." In E.S. Mills, ed., Handbook of Urban and Regional Economics. Amsterdam: North-Holland, 1987a.

30 RICHARD ARNOTr

Arnott, R. "Alternative Explanations of Housing Vacancies." Mimeo, 1987b. de Palma, A.; Ginsburgh, V.; Papageorgiou, Y.; and Thisse, J. "The Principle of Minimum Differentia-

tion Holds under Sufficient Heterogeneity." Econometrica 53 (1985), 767-781. Diamond, P. and Maskin, E. "An Equilibrium Analysis of Search and Breach of Contract. I. Steady

States." Bell Journal of Economics 10 (1979), 282-316. Guasch, J. and Marshall, R. "An Analysis of Vacancy Patterns in Rental Housing Markets."Journal of

Urban Economics 17 (1985), 208-229. Hardy, G.; Littlewood, J.; and Polya, G. Inequalities. Cambridge, England: Cambridge University Press,

1934. Hosios, A. "On the Efficiency of Matching and Related Models of Search and Unemployment."

Mimeo, 1986. Muth, R. Cities and Housing Chicago: University of Chicago Press, 1969. Read, C. "Three Essays on Vacancies in Rental Housing Markets." Unpublished Ph.D. dissertation,

Queen's University, 1987. Rosen, K~ and Smith, L. "The Price Adjustment Process for Rental Housing and the Natural Vacancy

Rate." American Economic Revi'ew (1983), 779-786. Smith, L., et al. "The Demand for Housing, Household Headship Rates, and Household Formation."

Urban Studies 21 (1985), 407-414.

Documents

Housing vacancies, thin markets, and idiosyncratic tastes