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					                        Franz Hubert

    Contracting with Costly Tenants

                             May 1994
                        forthcoming in
        Regional Science and Urban Economics 1995 (5)

                   Freie Universitat Berlin
              Fachbereich Wirtschaftswissenschaft

        mail: Franz Hubert, FUB, Boltzmannstr 20, 14 195 Berlin
                Tel.: 030 - 838 6357, Fax.: 030 - 838 4142
            email: Franz

  Helpful comments by Elmar Wolfstetter, Horst Tomann, Choon Poh Tan,
the referees and the editor are gratefully acknowledged.
     Contracting with Costly Tenants
                         Franz Hubert

    This paper analyses rental contracts in the housing market assu-
ming asymmetric information about tenant related `service cost' and
imperfect mobility. On the positive side it explains why long standing
tenants tend to enjoy lower rents | the so called `tenure discount'. On
the normative side, it shows that the market equilibrium is not e cient
because contracts protecting the tenant against arbitrary eviction suf-
fer from adverse selection. Tenure security laws, therefore, have the
potential to improve welfare.
Keywords: rental contracts, tenure security, adverse selection.
JEL D82, R21, K12.
1 Introduction
The relationship between a landlord and a tenant often stretches over many
months or years. It may be governed by a series of short{term contracts
or by a long{term contract stipulating conditions for the termination of the
lease, rules for rent reviews etc. This paper analyses rental contracts in
the housing market assuming asymmetric information about tenant related
`service cost' and imperfect mobility. Post{contractual selection, taking place
after landlords obtained information on their tenants through experience, is
the basis for pre{contractual selection, in which di erent contracts are used as
a screening device. Within this framework (i) we explain the empirical feature
that long standing tenants tend to enjoy lower rents than new tenants, the
so called `tenure discount', and (ii) we show that tenure security laws could
improve e ciency.
It has repeatedly been observed that long{standing tenants pay less than
those who entered their contract more recently (Borsch{Supan (1986), Beh-
ring (1988)). In this paper the tenure discount is explained as an attempt
to reduce the turnover of `low cost' tenants and to increase the turnover of
`high cost' tenants. The basic idea is as follows. In order to keep cost of
maintenance and administration low, a landlord expects his tenant to mini-
mize wear and tear, avoid trouble with neighbours etc. Many aspects of the
tenant's conduct, however, are not contractible | being unobservable by a
third party acting as an arbitrator or too vague to be explicitly stipulated.
It is therefore the landlord who bears the tenant related service costs (Mi-
ron (1990)). Tenants di er with respect to their expected service cost, but
a landlord will not be able to observe the di erence when lling a vacancy.
Experience gained during the course of tenure, however, will improve his as-
sessment of his tenant's character. The scope to react then is determined by
the nature of the contract chosen in the rst place. In the case of a long{term
contract with tenure security, the landlord has to put up with high service
cost until the tenant leaves voluntarily. Being non{contractible service cost

will not amount to a clear cut breach of contract. If the contract entails an
option to be terminated at will, he will evict high cost tenants and if the rela-
tionship is based on a sequence of short{term contracts, he will raise the rent
of `bad' tenants upon renewal. Since higher rents increase the probability of
a move, the landlord has an incentive to keep rents down for tenants with a
record of low service cost. However, whether or not `good' tenants enjoy a
`tenure discount' in equilibrium, depends on parameters such as mobility cost
and the expected service cost of unknown tenants and experienced tenants.
The hypothesis, that landlords grant a `tenure discount' in order to pre-
vent good tenants from moving is common in the literature (Schlicht (1983),
Goodman & Kawai (1985), Guasch & Marshal (1987), Miron (1990)). The
formal models, however, are not explicitly built upon di erences in the quali-
ty of tenants but on unspeci ed `turnover cost' borne by the landlord. Given
these, a tenure discount may arise if the tenant's willingness to pay decreases
over time (Schlicht (1983)) or if the landlord's assessment of the tenants'
preferences improves (Guasch & Marshal (1987)). One di culty with this
approach is that a di erence in the quality of tenants does not imply that
turnover cost are positive on the average. As the move of a good tenant
increases service cost so would the move of a bad tenant decrease them. This
paper explicitly analyses selection at the post{contractual stage by taking
these e ects into account. But it is assumed that turnover cost, in the nar-
row sense of moving cost, are borne by the tenant. In equilibrium, low rents
and a low turnover of good tenants as well as high rents accompanied by a
high turnover of bad tenants contribute to the tenure discount.
In many countries tenure laws for residential leases protect the tenant against
eviction, provided the landlord has no `just cause' for doing so (e.g. breach
of contract, rent arrears). Historically, tenure security laws had often be-
en introduced to enforce rent controls and to slow down the rate of con-
version to other, less regulated, forms of tenure. More recently, however,
the protection against arbitrary eviction has become the prime objective in
some cases. Notably in Germany the tenure security act of 1971 (Wohn-

raumkundigungsschutzgesetz) brought to an end a period of complete dere-
gulation by introducing mandatory protection against eviction. While the
initial rent of a new lease is freely negotiable, later rent reviews are tied to
the rent level of recent lettings of comparable dwellings (Vergleichsmietenre-
gelung). Similarly, in France the `Mehaignerie act' of 1986 deregulated rents
for new leases while requiring a minimum term of four years, during which
rent increases were tied to the index of construction cost. The regulation
of rent updating is clearly necessary to prevent the loophole of `economic
eviction' | an arbitrary increase of rent intended to force the tenant to give
notice on his own.
Eekho (1987), Hirsch (1988), Engels & Stutzel & von Weizsacker et al
(1984), Homburg (1993) and with quali cations Miron (1990) suggest that
these regulations will lead to a welfare loss if the cost of providing tenure
security are higher than the tenant's valuation of it | and are not necessary
otherwise. Borsch{Supan (1986) shows that long{term contracts providing
tenants with tenure security and ex ante speci ed rent may be superior to
a sequence of short{term contracts if tenants are risk{averse and prone to
the exploitation of immobility. Nevertheless, his justi cation of the regula-
tions appears to be premature because he does not explain why the better
contracts are not voluntarily chosen in an unregulated market. This paper
provides such an explanation based on asymmetric information and adverse
selection. As in Borsch{Supan (1986) we assume that forced moving involves
cost. Tenants, therefore, value pre{determined rents and protection against
eviction. However, if a tenant knows himself to be a low{cost type, he would
expect a lower probability of being evicted, if this option is not precluded in
advance. Under a sequence of short{term contracts he would expect a lower
rent in the future. Hence, if tenants know their type beforehand they will dif-
fer with respect to their preference for tenure security and stable rents. This
in turn enables uninformed landlords to use the option to give notice at will
| or alternatively the right to set future rents at will | as a screening device
at the pre{contractual stage. Since short{term or terminable contracts tend

to attract low{cost tenants while long{term contracts with tenure security
su er from adverse selection, the supply of the latter will be ine ciently low.
Tenure security laws, therefore, have the potential to improve welfare.
Finally, the paper is related to research stressing the importance of incenti-
ves. Homburg (1993) claims that contracts with protection against eviction
are likely to be ine cient because they fail to provide the tenant with proper
incentives to keep service cost low. Kanemoto (1990) argues that an e -
cient contract will allow the party in charge of maintenance and upgrading
to reap the bene ts of the investment: either the tenant is in charge and
enjoys tenure security or the landlord carries the duty and is able to evict
the tenant. In these papers `service cost', respectively investment levels, are
endogeneously determined in response to the incentives set by the contract.
It is, however, assumed that all tenants react exactly in the same way to an
incentive scheme. In this case, landlords would not bother to nd and to
keep `good' tenants. These issues, which are the theme of this paper, arise
only if di erent tenants respond di erently to an incentive scheme.
The next section introduces the basic assumptions of the model. In section
3 the focus is on selection at the pre{contractual stage. In order to simplify
the analysis we consider only long{term contracts for which post{contractual
selection is particularly simple: either evict or keep the tenant. In section 4
we bypass the issue of pre{contractual selection to analyse post{contractual
selection in a sequence of short term contracts in more detail. The nal
section summarizes the main results and explores further welfare implications
of mandatory tenure security. To ease the exposition most of the proofs have
been relegated to the appendix. A second appendix contains the more tedious
calculations. It is available on request.

2 Basic Assumptions
Consider a competitive market for rental housing services. In each of in ni-
tely many periods a cohort of new tenants or `youngsters' enters the market.
It exits after two periods of consumption. The size of the cohorts is constant.
Hence, at any time there are as many `youngsters' as `oldsters' in the mar-
ket. Landlords live in nitely long. When entering the market youngsters mix
with moving oldsters to compete for rental contracts.1 These could be short{
term contracts which set a rent only for the rst period leaving everything
else open to negotiation at the beginning of the second period, or long{term
contracts which x the rent for both periods. In the latter case contract
termination might be excluded (non{terminable contracts) or be allowed for
(terminable contracts).

Assumption 1 preferences] The utility functions of landlords and tenants
depend only on rent and are additive in time. Landlords are risk{neutral.
Tenants are not risk{loving. The common discount factor is 2 (0 1). The
tenants' utility in each period is denoted U : R ! R with U 0 < 0, U 00 0.

Tenants di er in two aspects: (i) their `quality' e.g. the propensity to cause
service cost (ii) their `mobility' e.g. the cost (or gains) which accrue to the
tenant if the contract is dissolved after the rst period.

Assumption 2 quality] Tenants di er in the probability                    2 = 0 1] to
cause c units of service cost during a given period.               is distributed with the
cumulative probability H .

The average cost{probability of all tenants is p
                                                          dH . Later we will
consider subsets of all tenants, e.g those who select a particular contract or
   1The interpretation in terms of overlapping generations is not to be taken too literally.
An alternative would be to introduce exogenous moves as in Greenwald (1986). It must
only be ensured, that evicted tenants can `hide' in the market.

have a particular record of service cost. The average cost{probability of these
subgroups will be denoted by adding subscripts.
An important feature of housing tenure is that | in the absence of con-
tractual stipulations | landlords enjoy some latitute regarding the rent of
sitting tenants. Not all tenants will move when their present rent increases
slightly beyond the level which is charged for vacancies. Mobility{cost may
result from psychological attachment to a at and neighbourhood as well as
from sunk investments in complementary consumption goods, expenses for
moving and renovation etc. On the other hand some tenants will move even
if the present rent is lower than the equilibrium rent for vacancies. Moving
bene ts may be due to a better wage o er elsewhere, formation or splitting
up of a household etc.

Assumption 3 immobility] If the contract is terminated at the end of the
  rst period, a tenant of type experiences an additive utility loss 2 =
     ]. is stochastically independent of and distributed with twice di eren-
tiable cumulative probability F and nite density f .

Note, that < 0 indicates a gain from moving. Later on, some additional
technical restrictions on the distribution F will be required to hold. Now,
we turn to the information structure and the timing of events (see gure 1
for an illustration).

Assumption 4 hidden information] Tenants know their `quality' and the
distribution of mobility cost F ( ) at the outset. Landlords know only the
distributions H ( ) and F ( ).

Central to the analysis is the assumption of asymmetric information on the
tenants' character. The assumption that information on mobility cost is
initially the same on both sides of the market is adopted for analytical con-
venience. It appears adequate if mobility cost develop during the course
of consumption. If, on the contrary, mobility is considered as a matter of

                       Abbildung 1: Timing of Events

    tenants                           tenants learn       continuation
     know                            mobility cost          (possibly
      and F                             landlords           with new
   landlords   landlords tenants     observe service        terms) or
     know        o er      select        cost and         termination
   H and F     contracts contracts   calculate p p
                                               g    b      of contract       time
                 1. negotiation         1. period       2. negotiation   2. period

personal character each tenant would know his before entering the rental
contract. This would complicate the analysis slightly without changing the
main results.
The rst negotiation consists of two steps: (i) landlords o er contracts { being
well aware that di erent contracts will attract di erent types of tenants, (ii)
tenants choose the contract which provides them with the highest expected
utility. Since the landlord (i.e. the principal) takes the initiative, the model
belongs to the `screening{approach'. The alternative `signaling{approach'
would see the tenants (i.e. the agents) `signal' their type by announcing
their o ers rst. There is, however, no rst mover advantage at this stage
because negotiations take place in a perfectly competitive framework. The
asymmetry of information on the expected service cost of a tenant, provides
the basis for pre{contractual selection. Post{contractual selection is shaped
by the way landlords learn about their tenants.
Assumption 5 imperfect learning] Each landlord observes the occurrence
of service cost for his own tenant. He does not obtain information on the
mobility .
By observing the occurrence of service cost in the rst period a landlord
obtains an improved assessment of the tenant's character. Those with a bad
record, `bad tenants' for short, have a higher probability to cause service cost
in the future as tenants with a good record (pb > pg ). Hence, when negotia-
ting for contract renewal, the landlord will take a tougher stand towards the
former. The particular form of these negotiations depends on the contract
chosen in the rst place and will be discussed later.

Assumption 6 no public record] The landlord cannot obtain information
on previous tenures of an applicant.

Information accumulated within a contract is not revealed to the market.
Hence, a tenant who moves is cleared o his reputation | whether good or
bad. This is an important di erence to otherwise similar models of the labour
market, where employers face the problem of assessing the productivity of
prospective workers. Freeman (1977) assumes that competing employers have
the same information on productivity as the current employer. Even if such
information is private in the rst place, employers could make inferences
from previous job assignments (Waldman 1984) or quit{decision (Greenwald
1986). Failure to obtain good job{assignments within the original rm or
repeated quitting are taken as evidence of poor performance. In the housing
market it is, however, much more di cult to draw useful conclusions from
information on previous tenures. First, there exists nothing comparable to
the internal labour market of big rms. Second, most contract terminations
result from a change of family composition, income or job. Hence, quitting as
such does not provide much information on the landlord{tenant relationship.
Third, information given by the present landlord cannot be trusted. Since a
good recommendation may ease a separation, the present landlord is better
o when lying about a bad tenant. In contrast to the labour market, there
is no `reputation' to curb this incentive for opportunistic behaviour, in the
(private) rented housing market. For these and other reasons landlords often
do not bother to obtain information on previous tenures, eventhough they
are concerned about selecting `good' tenants.

Contracting is analyzed as a multi{stage non{cooperative game within the
framework of an overlapping generations model with free entry. Therefo-
re, the set of equilibrium contracts has to be (i) a subgame perfect Nash{
equilibrium of the contracting game sketched in gure 1 and (ii) for each
contract expected cost must equal expected revenue. The zero{pro t{
constraint merely substitutes the explicit modeling of the overall supply
of housing. Subgame{perfectness rules out contracts using empty promi-
ses or empty threats concerning the nal negotiation on contract renewal
(post{contractual selection). It also implies that landlords take into account
the impact which their o ers have on the tenants choice of contracts (pre{
contractual selection). Finally, the Nash{equilibrium requires that, given the
existing o ers | and the induced self{selection |, there must not exist ano-
ther contract which earns strictly positive pro ts. As usual the equilibrium
is to be found by backward induction. The analysis is restraint to stationary

3 Tenure Security in Long{term Contracts
In this section I consider two types of long{term contracts. Both contracts
stipulate constant rents (rt , rn) for the time of duration, protecting there-
by the tenant against ex post exploitation of his immobility. In the rst
one, indexed with t (terminable), the landlord retains the right to give no-
tice at will at the end of the rst period. The second one, indexed with
n (non{terminable), protects the tenant against being evicted. For exposi-
tional convenience it will be ruled out that tenants give notice to quit | a
restriction which will be dropped in the next section. Formally this requi-
res < 0 and trivially implies that the average tenant has a preference for
staying E
               R dF > 0.
The initial negotiation (see gure 1) consists of two steps. First, landlords set
rents rt rn 2 R for both contracts. Since no tenant would strictly prefer the
terminable contract if not for being cheaper we can restrict the set of rents
to R = f(rt rn) 2 R 2 jrt rng. Second, tenants choose among the o ers.
It will be seen below, that this choice can be described by a parameter
a(rt rn ) 2 , with the property that, given (rt rn ) all youngsters of type
   < a strictly prefer the terminable contract. H (a) is the market share
of terminable contracts. The cost probability of youngsters choosing the
terminable contract is pt (a) = E ( j
                                           a) = 0 dH=H (a). For later use
note, that lima!0 pt (a) = pt (0) = 0 and that
                      p = H (a)pt (a) + (1 ; H (a))pn (a)        8a 2                    (1)
where pn denotes the cost{probability of tenants choosing the non{terminable
contract. During the rst period of consumption landlords observe the oc-
currence of service cost. Hence each landlord knows whether his tenant has a
good or a bad record (indexed g and b respectively) when he decides whether
or not to keep the tenant for the second period.

Giving Notice to Quit. Having observed the occurrence of service{cost
in period one, the landlord calculates the conditional cost probability for
tenants with a good respectively bad record (pg pb). From the Bayes{rule
for the calculation of conditional expectations follows that pb > pg . Since the
probability of service cost is a xed character of the tenant it must be true
                                  pt   = ptpb + (1 ; pt )pg :                            (2)
If a contract is terminated both parties will turn to the market where evicted
oldsters mix with youngsters. The expected pro t with an old tenant staying
for a second period before exiting the market, i is
                                  t ; pi c) +             i 2 fg bg
                            i = (r                  t                                    (3)
  2   Note, that p is also the fraction of tenants with a bad record after period one.

where t, the expected pro t with a new terminable contract, can be written
     t   = (rt ; pt c) + (1 ; pt ) g (rt ; pg c) + pt b(rt ; pbc)         (4)
           + (1 ; pt )(1 ; g )(rt ; pg c) + pt (1 ; b)(rt ; pb c) = (1 ; )Z ]
   Z        1 + (1 ; pt) g + pt b + (1 ; pt )(1 ; g ) +       pt (1 ; b )

Let g b 2 0 1] denote the probability of evicting a tenant with a good
(respectively bad) record. The landlord chooses g , b in order to maximize
the expected pro t.
Program 1: Decision to Evict

                   max 2 =            t + (1 ;        i   i 2 fg bg
                                  i              i)

Note that g and b in the expression for t refer to the simultaneous decisions
of other landlords and are therefore no instruments of the program.
Proposition 1 In equilibrium a tenant is evicted if and only if he has got
a bad record   b   = 1,   g   = 0.
The terminable contract can be interpreted as an implicit agreement which
makes the eviction conditional on the occurrence of service cost in the rst
period. This is possible because it is in the landlord's best interest to follow
the agreement eventhough the exact type of his tenant is not revealed to him
and the service cost of an unknown new tenant depend on the behaviour of
the other landlords.

The Tenants' Choice of Contracts. Given the prospect to become evic-
ted the expected utilities for the contracts are (1+ )U (rn) and (1+ )U (rt) ;
  E respectively. Setting both equal de nes a marginal tenant a 2 .
                 0 = a ; minf (1 + ) (U (rt ) E U (rn)) 1g                 (5)
Any tenant with           a   weakly prefers the terminable contract.
The Landlords' Contractual O ers. As usual in selection models the
Nash{equilibrium for the contractual o ers is characterized by two conditi-
ons: First, pro ts are zero for every contract. Second, given the existing
o ers, there must not exist another contract which earns strictly positive
pro ts. For the non{terminable contract, rent{revenue and expected service
cost are the same for both periods, hence the zero{pro t condition simply
                               0 = rn ; pn(a)c                            (6)
The corresponding expression for the terminable contract is obtained by eva-
luating (4) at and using (2)
          0 = rt ; pt(a)c (a )           (a ) (1 + pb ) + (1 ; pb)
                                                  (1 + pt ) + (1 ; pt
                                                                      )    (7)
Note that (a ) > 1 since pb > pt and 2 (0 1), and that lim !1 = 1.
This reveals the rst externality in the market operating via post{contractual
selection. A landlord who evicts a bad tenant reduces his cost at the cost of
other landlords o ering vacancies. Clearly, in a stationary equilibrium, this
externality nets out when aggregated. Hence, if we abstract from discounting,
the rent for the terminable contract must be equal to the expected service
cost of youngsters choosing this contract. To enjoy the low cost of tenants
with a good reputation, landlords have to bear the higher cost of an unknown
tenant in the rst period. This `front{end loading' of the cost pro le requires
a rent which is higher than the expected cost, when discounting is taken into
A landlord will not be able to attract any tenant if he charges a rent hig-
her than his competitors. Lowering rt will certainly decrease pro ts since
it attracts tenants who are `worse' than those who have already chosen the
terminable contract. Lowering the rent of the non{terminable contract ho-
wever attracts tenants with lower average service cost which might o set the
reduction of revenue. Hence, a zero{pro t o er (rn rt ), compatible with the
self{selection constraint, is a Nash{equilibrium if no non{terminable contract
charging a lower rent is pro table.
                        n (r r )
                            n t
                                       n (~ r )
                                          rn t       8rn < rn
                                                      ~                            (8)

Equilibrium.       Since g , b have been already incorporated when deri-
ving (5) and (7), the Nash{equilibrium of the contracting game is given by
a rt rn which simultaneously ful l expressions (5), (6), (7) and (8).
As usual in screening models for particular parameters a Nash{equilibrium
may fail to exist. In response to the possible non{existence of a Nash{
equilibrium alternative equilibrium notion have been proposed. Referring to
Wilson (1977), a zero{pro t o er is called a `Wilson{equilibrium' if no other
contract exists which would yield a positive pro t even after all contracts
that make losses as a result of the entry are withdrawn. According to Riley
(1979) a `reaction{equilibrium' is a set of zero{pro t contracts for which no
defecting contract with positive pro ts exists which in itself cannot be made
unpro table through a reacting contract. Unfortunately the two concepts
appear to lead to di erent equilibrium outcomes (Riley 1979).3 Furthermo-
re, both equilibrium concepts make demanding assumptions on the agents'
awareness of the reactions of competitors. This might be appropriate for
markets in which a small number of sophisticated suppliers operate e.g the
insurance market. It is less appealing when applied to the rental housing
market with its large number of small suppliers. We, therefore, restrict our
attention to parameters for which a Nash{equilibrium exists.

Proposition 2 A situation in which only non{terminable contracts are tra-
ded a = 0 cannot be an Nash{equilibrium.

Even if both contracts are being traded in equilibrium (0 < a < 1), the
separation of tenants is incomplete. All tenants whose cost{probability
is below the threshold value a are `pooled' at the terminable contract, all
   3Since Wilson developed his notion for a nite set of agents while Riley considered a
continuum a direct comparison is di cult.

others choose the non{terminable contract. Partial pooling is possible be-
cause the screening device is discrete, the protection against eviction is either
being granted or denied, only the rent can be varied in a continuous man-
ner. Therefore, the ne tuning between two parameters of the contract (wage
& education, insurance premium & coverage etc), which usually erodes any
pooling constellation in sorting models, is not possible in the present con-
text.4 Partial pooling is even necessary for the credibility of the sanction. If
self{selection were perfect | as it would be, if there were only two types of
tenants in the market | the landlord would be unable to make any additio-
nal inferences from the observation of service cost. Hence, there would be
no incentive to evict tenants for bad performance which in turn makes the
threat of doing so incredible.

Comparative Statics.          Assuming that an equilibrium exists, we can
derive the comparative statics of the pre{contractual selection from equations
(5), (6) and (7) bearing in mind the condition (8). rt and rn are strictly
increasing in pt and pn which in turn are strictly increasing in a . Hence it
is su cient to consider a which determines the market share of terminable

Proposition 3 The market share of terminable contracts H (a ) decreases
(i) as mobility cost increase @a =@E < 0 and (ii) as the precision of learning
(the value of experience) increases @a =@pb < 0. (iii) Terminable contracts
become more competitive if service cost increase @a =@c > 0. (iv) The impact
of discounting is ambiguous @a =@ R 0.
   4 If stripped of its particular time structure the model is one of screening, as in Roth-
schild & Stiglitz' (1976) treatment of the insurance market. They consider two classes of
customers and a continuum of contracts. Here, in contrast, the customers are represented
by a continuum, but there are only a discrete number of contracts. In this respect the mo-
del is similar to the original selection{model of Akerlof (1970) who considers a continuum
in quality of cars and the decision to buy or not to buy, using a simple sale{contract.

These results are in line with intuition. Higher mobility cost E make the
tenants less inclined to forego tenure security in order to obtain a lower rent.
This result suggests, that lower mobility cost might be one of the reason
that long term contracts providing tenure security play only a minor role in
the USA while in Europe, they apparently had been well established even
before tenure laws made them obligatory.5 As the di erence between pb
and pg increases, with pt being kept constant, the private gains from evicting
tenants with a bad record become bigger. Since evicted tenants turn to other
landlords, total cost are not reduced. Instead cost for unknown applicants
in the rst period increase and cost for good tenants in the second period
decrease. The cost pro le becomes more `front{end loaded' which makes
terminable contracts less competitive. A general increase in service cost c
makes terminable contracts more attractive for landlords due to their lower
probability of such cost. The impact of an increase of the discount factor (or
a decrease of the discount rate) is ambiguous. Tenants will be deterred from
terminable contracts because they are more concerned about the expected
utility loss of forced moving. Landlords will be attracted because they have
less to worry about the front{end loading of service cost.

E ciency.      Holmstrom & Myerson (1983) suggest the distinction of ex
ante and interim e ciency depending on the time of the assessment. Ex
ante e ciency refers to the time when even the better informed side is
ignorant | in our case before tenants learn . Interim e ciency is eva-
luated when the asymmetry of information has developed | tenants know
their type but landlords are not informed. The e ciency of the equilibri-
um is assessed by comparison with the case of tenure security regulations
which make terminable contracts unfeasible (formally the latter corresponds
to a = 0 n = pnc = pc).

   5On the di erences of mobility see Schneider, Stahl, Struyk (1985) on early forms of
tenure see International Labour O ce (1924)).

Proposition 4 (i) if both contracts are traded in equilibrium (a             <
1) tenure security laws improve interim e ciency according to the
compensation{criterion and ex{ante e ciency according to the Pareto{
criterion. (ii) if only terminable contracts are traded (a = 1) interim e -
ciency is improved according to the Pareto{criterion.

The possible ine ciency of markets plagued by asymmetric information is
well known since Spence (1974). Here, terminable contracts decrease welfare
in two ways: (i) the time{pro le of service cost becomes front{end loaded
because high cost tenants move more often and (ii) forced moving implies
mobility cost. Foregoing protection against eviction serves as a signal for
being a good tenant. The private cost of signaling are the expected utility
loss in the case of contract termination and the compensation for the front{
end loading of the cost pro le. Private gains result from a redistribution of
rent{payments among tenants, and are therefore matched by social cost of
the same magnitude.

4 Short-term Contracts
In the preceding section the set of feasible contracts has been restricted to
long{term contracts with xed rents for both periods. This leaves the landl-
ord little choice but to evict tenants with a bad record | if the option
was not precluded in advance. Now we consider a sequence of short{term
contracts. Since any single contract xes the rent for only one period, the
landlord can adjust the rent according to previous experience. In order not
to clutter up the analysis of post{contractual selection, we skip the issue
of pre{contractual selection. Short{term contracts are of particular interest
since they trivially ful l the requirement of incentive compatibility. Nothing
is promised for the second period, hence no promises can be broken. They
entail the option of contract termination through `economic eviction', that is,
a rent high enough to induce even the most immobile tenant to give notice.
Moreover, short{term contracts are less costly to write and enforce. Lower
transaction cost will give them a speci c advantage when competing against
long{term contracts.
Since attention is restricted to those who choose short{term contracts in
equilibrium, the game (see gure 1) starts with the landlords setting the rst
period rent r1 and proceedes to the second negotiation when landlords and
tenants have to agree on a second term or to separate. Tenants know their
mobility cost at this stage. Landlords only know the distribution of this
characteristic F . Based on the experience gained in the rst period they set
two rents ri i 2 fb gg depending on whether the record is good or bad. The
tenants respond by accepting the o er or by giving notice to quit. In the
latter case they turn to the market and rent for r1 .
This sequence gives the landlord a rst mover advantage in the negotiation
for contract renewal which enables him to exploit the tenant's immobility. He
is, however, prevented from reaping all the gains from trade by his imperfect
knowledge of the mobility cost. Most of the following results do not depend
on this particular distribution of bargaining power. If the tenant would enjoy
the strategic advantage of a take{it{or{leave{it o er the landlord would be
left just indi erent between renting at ri i 2 fb gg and renting at r1 to an
unknown tenant. Second period rents as well as rates of turnover would be
lower | but otherwise the results would be similar.
In this section we assume that some tenants do bene t from moving, ( < 0).
To ensure that the problem of post{contractual selection is well behaved, it
is assumed that (1 ; F ( )), i.e. the probability that mobility cost are above
a given value, is log{concave.6
    6 This requires that ln 1 ; F (
                                    1 + (1 ; ) 2 )] >    ln 1 ; F ( 1 )] + (1 ; ) ln 1 ;
F ( 2 )] 8 1 2 2        2 0 1]. The class of log{concave distributions includes (in some
cases with restrictions on the parameters) the multivariate beta, Dirichlet, exponential,
gamma, Laplac, normal, uniform, Weibull, and Wishart. See Caplin & Nalebu 1991 and
the references cited there. Since 2 ] is the loss from moving, we can de ne          ;
as the gains from moving, with distribution G( ) 1 ; F ( ). In other word we assume

Decision to Move.       The tenants' decision to accept the rent o ered or
to move can be described by the marginal tenants in terms of mobility{cost
 i2     ] i 2 fb gg.
                        <                          i   U (r1 ) ; U (ri ) <

                        > U (r1) ; U (ri)              U (r1 ) ; U (ri ) 2
             i (r r ) =                            i                         ]   (9)
                 1 i
                        :                          i   U (r1 ) ; U (ri ) >

The fraction of tenants who move is F ( i) i 2 fb gg. We de ne r and r
as rents for which all tenants move, respectively stay, by (F ( i(r1 r)) 0)
(F ( i(r1 r)) 1).

Second Period Rent. On contract renewal, landlords propose the second
period rents which maximize their expected pro ts 2. Obviously ri < r can
never be optimal and pro ts are invariant for ri > r. Hence, we can restrict
the instrument to r r].
Program 2: Second Period Rents

                     max 2 =         F ( i ) s + (1 ; F ( i )) i
                      r  i

                  subject to         ri   2   r r]             i 2 fb g g

where    i= (ri ; pi c) + s is the expected pro t conditional on the record
i 2 fb g g and s is the expected pro t with a short term contract chosen by
an unknown type of tenant. To ease notation we denoted partial derivatives
of with subscripts and de ne Fi d=dri F ( i(r1 ri)) = ;f U 0(ri) > 0 and
Fii d2 =dri F ( i (r1 ri )) = f 0 U 02 ; f U 00 . The following lemma is crucial
for the characterization of post{contractual selection and the comparative

Lemma 1 If i2 = 0 at ri then ii = ;2Fi + Fii ( s ; i ) < ;Fi i 2 fg bg

that the distribution of moving gains is log{concave.

Lemma 1 is an implication of log{concavity of 1 ; F (and non{convexity of
U ). While stronger than necessary to ensure that program 2 is well behaved,
it is crucial for the comparative statics. In this sense the requirement of
log{concavity of 1 ; F is tight. Since lemma 1 rules out any local minima
the solution to program 2, denoted ri , is given as
               <      i    1 + ( s ; i)Fi 0            at r
          ri =
               > ri   i    1 ; F ( i) + ( s ; i)Fi = 0 at ri 2 (r r)
               : r    i    ( s ; i)Fi 0                at r              (10)

Contractual O ers. As in the previous section pro ts are zero in equi-
librium and there must exists another short{term contract earning positive
pro ts. The zero{pro t condition (11) can be obtained from (4) with obvious
      s    = (r1 ; pc) + (1 ; p)F ( g )(r1 ; pg c) + pF ( b)(r1 ; pb c)
              + (1 ; p)(1 ; F ( g ))(rg ; pg c) + p(1 ; F ( b))(rb ; pbc)
           = 0                                                            (11)
A contract (r1 rg rb ) ful lling (10) and (11), constitutes a Nash{equilibrium
if no contract charging a lower initial rent is pro table.
                       s (r r r )
                           1 g b
                                     s (~ r r )
                                        r g b      8r < r1
                                                    ~                    (12)
Such a contract creates lower revenue but it also induces more tenants to
quit their current contracts. (12) rules out that there are enough additional
movers with a very low cost{probability to o set the fall in revenue by a
decrease of expected cost.

Equilibrium.          The next proposition and its corollary characterize the
Proposition 5 In equilibrium landlords charge bad tenants strictly higher
rents than (i) good tenants (rg < rb ) and (ii) unknown tenants (r1 < rb ).
                            Tabelle 1: Types of equilibrium
                       type 1                      type 2
              0 = F ( g ) < F ( b) = 1    0 < F ( g) < F ( b) = 1
              0 ;( s ; g )Fg ; 1        0 = 1 ; F ( g ) + ( s ; g )Fg
                    0 s; b                      0 s; b
                       type 3                      type 4
              0 = F ( g ) < F ( b) < 1    0 < F ( g) < F ( b) < 1
              0 ;( s ; g )Fg ; 1        0 = 1 ; F ( g ) + ( s ; g )Fg
            0 = 1 ; F ( b) + ( s ; b)Fb 0 = 1 ; F ( b) + ( s ; b)Fb

Corollary 1 The most mobile (if not all) bad tenants move | some of
them do so in spite of positive moving cost: 1 F ( b) > F (0) > 0. Good
tenants have a strictly lower turnover than bad tenants: F ( b) > F ( g ) 0.

Post{contractual selection is characterized by the trade o which a landlords
faces at the beginning of the second period. On one hand he would like to
exploit the tenant's immobility at that stage | suggesting a `mark up' on
initial rent. In the case of a bad record, this inclination is even fostered
by the wish to get rid of high{cost tenants. In the case of a good record,
however, he wants to keep the tenant | suggesting a rent discount. If all
the bargaining power rests with the landlord, the result of this trade o
between exploiting switching{cost of sitting tenants and preventing low{cost
tenants from moving remains open. If the tenant should have the rst mover
advantage in the negotiation for renewal, a tenant with a good reputation
would always demand a tenure discount.
Table 1 displays the di erent types of equilibria.7 An equilibrium of type
1 resembles very much the market for long{term terminable contracts. All
  7   It can be shown by numerical examples that all these equilibria do in fact exist.

                          Tabelle 2: Comparative Statics
                                        Equilibrium Type 4
                         r1                 rg        rb     F ( g)      F ( b)
                         |                 |         |         +          +
          E              |12               +1        +1       |1          |1
           pb            +                 |3        +        |1          +14
              c          +                 +         +        ??          ??
          1       ! 1,        2   rg   ! r1 ,    3 p ! 1,     4 p ! 0.

tenants with a bad record would be `economically' evicted by a su ciently
high rent F ( b) = 1, all good tenants would be prevented from moving
by a low rent F ( g ) = 0. Given the assumption that some tenants prefer
to move < 0 a tenure discount is required rg < r1 to achieve this. In
a type 4 equilibrium only the most mobile tenants of both groups move
(0 < F ( g ) < F ( b) < 1) and mix with new entrants after the rst period.
The other equilibria are mixed cases.

Comparative Statics. Table 2 summarizes the result of the comparative
statics for the equilibrium of type 4. Given the empirical evidence supporting
a tenure discount for average long standing tenants, we consider the case
r1 rg .
Due to the higher turnover of bad tenants, the expected cost in the rst period
of a contract is higher than in the second. At the same time, competition
among landlords is strong at this stage. In the second period, when costs
are already lower, the landlord is able to exploit the immobility of his old
tenants. Therefore the landlord faces a de cit in the rst period of a contract
and enjoys surplus in the second. The zero pro t condition requires that
the interest on this `investment' is covered by the rent{payments. As the
discount factor increases | the discount{rate decreases | competition

forces rents down. The fall in r1 induces a downward shift of rg and rb which
is, however, not strong enough to prevent the turnover from raising. To
determine the impact of mobility, the expected mobility cost E is considered
as a parameter, shifting the distribution F in the sense of rst order stochastic
dominance, (FE < 0). As moving cost increase, tenants would become more
prone to ex{post exploitation. Rents for oldsters increase, while turnover
rates decrease. At the same time the tenure discount gets smaller.
The impact of an improved assessment of the tenants at the post{contractual
stage (pg pb moving away from p), the comparative statics depend on whether
the landlords learn `more' about good tenants or bad tenants. From (2)
follows that for a given p, pg and pb are related according to
                              pg   =      (1 ; pb)
If the cost probability of youngsters p is high, then not observing cost during
the rst period is `good news' from which a very low pg can be inferred,
while `bad news' are hardly news at all. In this case a small increase in pb
must be accompanied by a large drop of pg , to keep p constant. If p is low,
pg is insensitive to a change of pb . Some results can only be obtained for
the limiting cases p ! 1, when landlords learn only about low cost of good
tenants and p ! 0, when landlords learn about high cost of bad tenants. In
the latter case, a marginal increase of pb raises rb so much that the turnover
of bad tenants rises. The resulting increase of expected rst period cost of
unknown tenants leads to a rise of r1 , which puts the landlords in a stronger
position vis{a{vis old tenants with a good record. Nevertheless, the change
of rg is weak in relation to the increase of r1 so that the turnover of good
tenants declines. As p approaches one, an increase of pb results in a much
reduced cost probability of tenants with good record. Landlords lower rg
reducing thereby the turnover of good tenants. The withdrawal of low{cost
tenants from the pool of movers requires an increase of r1 which makes the
overall impact on the turnover of bad tenants ambiguous.
An increase of c means that all tenants are becoming more costly to serve. It
has to be compensated by a general increase in rent. Since the cost increase
is in proportion to the cost probability, the absolute di erence between good
and bad tenants grows. It could therefore be expected, that the incentives to
keep good tenants and to deter bad tenants becomes stronger. For the type
4 equilibrium, however, no clear cut results could be derived.

E ciency.        E ciency requires that tenants move if and only if they be-
ne t from moving. With short{term contracts the time{pro les of expected
rents are upward{sloping for bad tenants and (possibly) downwards sloping
for good tenants. As with terminable long{term contracts, the turnover of
tenants with a bad record is, therefore, ine ciently high. The turnover of
good tenants, however, might be too low. If their rent is lower than the in-
itial rent of a new contract (the case of a tenure discount) some tenants will
forego the gains from a move in order cash in on their good reputation. Since
it will be di cult to prevent landlords from o ering good tenants favourable
rents, the scope to improve e ciency through regulation is more limited in
this case.

5 Concluding Remarks
This paper is based on the assumptions that tenants di er with respect to
non{contractable service cost and that landlords learn about these di erences
through experience. The analysis of the preceding sections addressed two
related questions: (i) how do landlords respond to information which they
gain during tenure, given the contract they agreed upon in the rst place,
(ii) what types of contracts will be chosen in equilibrium.
In section 4 the focus was on post{contractual selection. It was shown that
within a sequence of short term contracts the landlord would charge tenants
with a bad record a higher rent upon renewal than tenants with a good re-
cord. Correspondingly, the rate of turnover of bad tenants would be higher

than those of good tenants. In an extreme case, all tenants with a bad re-
cord would be economically evicted by a very high rent and turnover of good
tenants would be inhibited by a large enough tenure discount. The compa-
rative statics suggest that a tenure discount is more likely in equilibrium if
mobility cost are low and if a good (or a bad) record of service cost leads to
a substantial revision of expected service cost.
In section 3 the focus was on pre{contractual selection. To ease the exposi-
tion we restricted the class of feasible contracts to long{term contracts with
and without tenure security and ruled out that tenants want to move. Post{
contractual selection is absent in a non{terminable contract and takes the
simple form of evicting all tenants with a bad record in a terminable contract.
Since the probability of eviction is smaller for low cost tenants, terminable
contracts are able to `skim the cream' if tenants are better informed about
their type ex ante. Contracts providing tenure security, in turn, su er from
adverse selection. The equilibrium will always entail some trading of termi-
nable contracts. Their market share increases as mobility cost decreases or
as non{contractable service cost increases.
Terminable contracts enable low cost tenants to di erentiate themselves from
high cost tenants. Compared to a situation with equal treatment, the rent of
the former is reduced at the cost of latter. This redistribution among tenants
is achieved at some cost | the moving cost in case of eviction. But eviction
serves no social aim | service cost are xed and the evicted tenant will rent
from another landlord anyway. Hence, tenure security laws which force all
parties to enter non{terminable contracts increase welfare.
Tenure security regulations can be found in Japan and in most european
countries | e.g. Germany, France, the Netherlands, Italy and (until recent-
ly) in Great Britain. As for the USA and Canada tenure laws are not uniform.
Most jurisdictions, however, follow a free market approach towards contrac-
ting in the rental housing market. On its normative side the paper has, so
far, emphasized on the potential of tenure security laws to enhance welfare.
It is to be stressed that this result applies only to regulations which are not
combined with measures to depress rents. The German tenure security act
of 1971 or the French `Mehaignerie act' of 1986 can be regarded as examples
of this kind. Furthermore, the unambiguous impact of tenure security laws
on welfare depends on the fact that, given assumptions 1 and 2, society can
provide tenure security at zero cost. This feature is easily lost if we draw a
more realistic picture of contracting problem.

   1. If the tenant has some discretion over the probability or the size of the
      service{cost, the landlord's threat to give notice creates incentives to
      keep these cost low.8 The protection against eviction would entail costs
      in the form of a higher incidence of nuisance, negligence etc. which have
      to be traded o against the welfare gains of reduced contract disruptions
      and stable rents.
   2. The right to give notice prevents the landlord against becoming stuck
      to a `bad' tenant. If landlords are risk averse, security of tenure can no
      longer be provided without a premium for this risk.
   3. It has been assumed that the expected service cost of a tenant is inde-
      pendent from the landlord. In reality landlords di er in their valuation
      of service cost. An old lady subletting parts of her house may consi-
      der lady visitors after ten o'clock unacceptable, whereas a manager of
      a commercial building company would probably not mind. To redu-
      ce service cost, the more di cult tenants should be matched with the
      less sensitive landlords. The abolition of terminable (or short term)
      contracts through tenure security laws would result in an increase of

While each of these modi cations makes the protection against eviction cost-
ly, none of them restores necessarily the e ciency of an unregulated market.
Due to adverse selection, the private cost of providing tenure security still
   8See Stiglitz & Weiss (1983) for a general discussion of the incentive e ects of contract
termination and Homburg (1993) for an application to the housing market.

surmount the true social cost. Hence, in equilibrium the provision of tenure
security would still be to low at the margin. Making protection against evic-
tion mandatory for all leases, however, would have an ambiguous impact
on welfare. Such a measure may force tenants to pay a premium for the
insurance, which surmounts their valuation of it.
It is interesting to note that the aforementioned regulations in Germany and
France account for the di erent cost of providing tenure security. In France
the minimum term during which eviction is not possible is four years when
the landlord is a natural person but six years for commercial landlords. In
Germany, protection against eviction is not granted if the dwelling is furnis-
hed or if the landlord shares a small house with his tenants. Presumably
private landlords, and in particular those sharing the house with the ten-
ant, are more risk averse and more sensitive to service cost than commercial
As many other interventions in the market, tenure security laws need com-
plementary measures which might cause additional problems. The most pro-
minent, is the need to regulate the updating of rent in on{going contracts, in
order to prevent tenure security from being loopholed by economic eviction.
In France rent reviews have been indexed on construction cost | a proxy for
the cost of living index which cannot be used for legal reasons. In Germany
the rent increase is linked to the rent of other `comparable' dwellings. At
least in principle, rents of old contracts can be adjusted to the overall con-
ditions of the housing market. Recently, indexation on the rate of in ation
has been introduced as an alternative option. Fixing the rent in real terms
isolates the contract against the development of the market for the duration
of the term. The schemes have di erent implications for the allocation of
housing as well as the equilibrium price distribution if the housing market is
subject to real shocks (Hubert 1990). The optimal indexation has to strike a
balance between the insurance provided by xed real rents and the e ciency
of allocation provided by adjustable rents.
However, real world housing policy is often more concerned with (re)dis-
tribution rather than e ciency. Presumably, politicians will nd it more
di cult to resist pressure to `keep rents down' if some form of `rent control' is
already in place. The recent experience in Germany provides a good example.
When growing disposable income and a wave of immigration pushed rents
up in the late eighties, politicians gradually toughened restrictions on rent
increases. Within a couple of years, minor legal modi cations regarding
the determination and applicability of `comparable rents' have profoundly
changed the operation of the market (Hubert 1993). What was meant to be
a complementary measure for providing a secure tenure has mutated into a
system of thinly disguised rent controls. Nowadays, in most big cities tenure
security appears `necessary' to support legally depressed rents | as it used
to be in traditional `rent controls'.

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5.1 Selected Proofs
Proof of proposition 1
De ne X so that t X (1 ; )Z ];1. X < Z (rt ; pg c) =) t ; g =
(1 ; ) t ; (rt ; pg c) < 0 which implies g = 0. From X > Z (rt ; pb c) follows
 b = 1. Whatever the other landlords do, the best response is to evict a
tenant if and only if he has got a bad record. Being a dominant strategy it
establishes a unique Nash{equilibrium in the nal subgame.

Proof of proposition 2
For a = 0 (6) and (7) require rt = 0, rn = pc > 0 for which (5) implies a > 0
| a contradiction.

Proof of proposition 4
Since landlords' pro ts are zero in both cases, we are only concerned with
the impact on the tenants' welfare. Under regulation the expected utility is
(1 + )U (pc) for every tenant. In market equilibrium expected utility before
learning is given as H (a ) (1 + )U (t ) ; E ] + (1 ; H (a ))(1 + )U (n ).
Substitution with the help of the zero pro t condition and division by (1+ )
        H U (pt c ) + (1 ; H ) U (pn c) ; H
                                                 () E <                  (13)
                                               (1 + )
                           H U (pt c) + (1 ; H ) U (pn c)      U (pc)

The last inequality follows from (1) and non{convexity of U . The rst part
of (i) is obtained by interpreting H as a probability when calculating the
expected utility the second by taking it as a weight when aggregating. Claim
(ii) follows obviously from setting a = 1.

Proof of lemma 1
Note, that log{concavity of 1 ; F implies:
                        ln(1 ; F ) = ;f (1 ; F ) 2; f
                                        0               2
                  d 2                   (1 ; F )             <   0
                                       ;f ; f 0 1 ; F
                                                             <   0

Since i2 = 0 implies ;(1 ; F )=Fi = ( s ; i), the claim is equivalent to
(recall U 0 < 0 U 00 < 0)
                                         ;Fi ; Fii 1 ; F < 0
                      f U 0 ; (f 0 U 02 ; f U 00 )
                                                   1;F < 0
                                                   ;f U 0
                 ;U 0 ;f ; f 0 1 ; F
                                            ; U 0 (1 ; F )   <       0
                        |      {zf
                                        }     U
                     ({) by log{concavity

Proof of proposition 5
If ri 2 (r r) then @ 2 2=@ri @pi = picFi > 0 implies dri =dpi > 0. Note, that
  s ; i = (1 ; ) s ; (r ; pi c) is increasing in pi . If the border condition
holds for rb = r it will also hold for rg = r, if the border condition holds for
rg = r then it holds for rb = r. Hence, we have to rule out that rg = r or
rb = r to prove (i).
Assume rg = r then (10) requires r ; pg c 0, (9) implies r1 < rg = r, hence
r1 ; pg c < 0. All tenants would move, but a landlord would make losses even
if he catches a good tenant which contradicts (11). Assume rb = r then (10)
requires r ; pbc 0, from (9) follows r1 rb . But then a landlord would
make pro ts on all groups in the rst period, again contradicting (11). The
  nal step of the previous argument also proves (ii).

6 Technical Appendix
6.1 Calculations
6.1.1 Pro ts
The expression (4) for the expected pro ts with a terminable contract can
be obtained as follows.
Denote the fraction of oldsters among moving tenants as o, and the cost
probability of an unknown applicant pt .
               p      (1 ; p g
        ^ 1 +t pb + + (1 ;t ) )
                                                    + (1 ; p
                                    ^ pt ++tp bpb + (1 ; p t)) g pg
                                    pt =
                  t b       pt g           1 t b            t g
 t,   the expected pro t with a new terminable contract, is
                t   =    (rt ; pt c)
                        + o t
                        + (1 ; o)(pt b + (1 ; pt) g ) t
                                ^                                           (14)
                        + (1 ; o)(pt(1 ; b) b + (1 ; pt )(1 ; g ) g )
The rst term in (14) gives the rst period pro t of a terminable contract.
The last term gives the second period pro t for youngsters who are retained
for their good record. The other terms stand for those who were oldsters
at the beginning of the rst period and drop out and for youngsters who
become evicted. Both have to be replaced by new contracts and unknown
tenants for which expected pro ts are t. Rewrite t as (add and subtract
 (1 ; o)(pt(1 ; b) + (1 ; pt )(1 ; g )) t)
        t   = (rt ; pt c) + t
               + (1 ; o)(pt (1 ; b)( b ; t ) + (1 ; pt )(1 ; g )( g ; t))
             ^      ^
Substituting pt and o and solving yields (4). The expected pro ts of a short
term contract s can be obtained from (4) by replacing i through F ( i).

6.1.2 Pre{contractual Selection: Tenants Know Their Mobility
      before Contracting.
Assume that tenants already learn before contracting. As far as the landl-
ord's nal decision on whom to evict is concerned, nothing is changed. At
stage two however the terminable contract now attracts tenants who are bet-
ter or more mobile. A tenant of type ( ) weakly prefers the terminable
contract if
           (1 + )U (rn)      U (rt ) +   (U (rt ) ; ) + (1 ; )U (rt )
                                1 + (U (r ) ; U (r ))
                                           t       n
Let b             denote the set of tenants who weakly prefer the termina-
ble contract, the fraction of which is given by (b) = 01 F ( m) dH with
  m ( ) minf1 (1 + )(U (rt ) ; U (rn ))= g. The cost probability of tenants
preferring the terminable contract is pt(b) = 01 F ( m) dH=v(b). The for-
mula in the previous section can be rewritten by substituting (b) for H (a)
and p(b) for p(a). The precondition for the claim of proposition 2, that all
tenants buying non{terminable contracts cannot be an equilibrium, is that
limt!n p(b(rt rn)) < p which will be true except when mobility is inversely
correlated with quality. The proof of the welfare properties in proposition 4
rests on equation (1) and the fact that all tenants weakly preferred the non{
terminable contract, if provided on equal rent with the terminable contract.
Both are still valid.

6.2 Comparative statics of pre{contractual selection
(Proof of theorem 3)

Note that (8) implies
                     = 1 ; cp0n(a)an = 1 + cp0n 1 E   U 0 (rn ) > 0     (15)
The Jacobian is of system (5), (6), (7) and is:
              0                                                                                1
              B                      1                   1+ U 0 (rn)
                                                          E                  ; 1+ U 0 (rt)
                                                                                E              C
              @                  ;p0nc                          1                 0            C
                        ;(p0t c + pt c             a)           0                 1
First it will be shown that
                                     p0t c        + pt c    a>0                                     (16)
De ne
                                      (a)(1 ; )
                           = 1 + + pb(a)(1 ; )                               N
                             1 + + pt                                        D
and note that p0t p0b > 0. Then

                             p0t c
                                     N                   ;  0
                                         + pt c(1 ; ) pbDD2ptN
rearrange to obtain
                    N               pt (1 ; )                                         0
              p0t c         1;                                  + ptc(1 ; ) pb            >0
              | {z } |
                    D            1 + + pt (1 ;
                                         {z                )}           | {z D}
                +                        +                                +

It follows that the determinant of the Jacobian is positive
         jJ j = 1 + p0 c 1 + U 0 (r ) ; 1 + U 0 (r )(p0 c + p c
                        n                 n                         t    t            t    a) > 0
               |            {z
                             E               }|     E                   {z                   }
                    (+) by (15)                             (+) by (16)
             sign   @a
                             = sign ; 1 2 1 + (U (rt ) ; U (rn )) < 0
                    @E               E
A rise in E means a decrease in mobility. Not surprisingly, this tends to
decrease the market share terminable contracts.
Note that
         = (1 ; pb) 1 + + pt (1 ; )] ; (1 ; pt )2 1 + + pb(1 ; )] < 0
                             1 + + pt (1 ; )]

        sign   @a
                 = ; E 1 2 (U (rt ) ; U (rn)) +ptc  1 + U 0 (r ) R 0
               @   |          |
                               {z           }|     {zE
The e ect of an increase of is ambiguous.
Since pt < pn and jU 0(rt )j < jU 0(rn)j
            sign @a = sign 1 E (pt U 0 (rt ) ; pnU 0 (rn)) > 0
An increase in service cost c will increase the market share of terminable
An increase in the precision of learning (pb goes up implying a decrease of
pg ) leads to a lower market share of terminable contracts.

             sign @p = sign pt c 1 + +; (1 ; ) 1 E U 0 (rt ) < 0
                  @a                  1

6.3 Comparative statics of short term contracts
An equilibrium of type 4 is characterized by the rst order conditions for
an interior maximum, the zero pro t constraint and the requirement that
expected pro ts s are nondecreasing in the initial rent s in equilibrium.9
Since s ; i = (1 ; ) s ; (i ; pi c) i 2 fg bg these can be rewritten as
  9 In order to ease the notation: s r1 g r b r . Partial derivatives of F and
                                                  g   b

are denoted with subscripts: F f ( )@ =@i i 2 fs g bg. As before the denominator
                                      i   i

(1 ; )Z with Z 1 + (1 ; p)F ( ) + pF ( ) + (1 ; p)(1 ; F ( )) + p(1 ; F ( )) will
                                  g           b               g             b

be supressed. De ne N by  s
                                 N=(1 ; )Z . In equilibrium
                                                            = 0, hence N = 0. Since
@ =@x = (@N=@x)=(1 ; )Z when evaluated at N = 0 we can treat Z as a constant.

    0 = g = 1 ; F ( g ) + ((1 ; ) s ; (g ; pg c))Fg
         2                                                            (17)
    0 = b2 = 1 ; F ( b) + ((1 ; ) s ; (b ; pbc))Fb                    (18)
    0 = s = (s ; pc) + (1 ; p)F ( g )(s ; pg c) + pF ( b)(s ; pb c)   (19)
             + (1 ; p)(1 ; F ( g ))(g ; pg c) + p(1 ; F ( b))(b ; pbc)

    0<      s
            s   = 1 + (1 ; p)F ( g ) + pF ( b )                              (20)
                  + Fs (1 ; p)(s ; pg c) + p(s ; pbc)]
                  ; Fs (1 ; p)(g ; pg c) + p(b ; pb c)] > 0
The comparative statics are derived from (17) to (19) in the usual manner.
Given the empirical evidence supporting a tenure discount for average long
standing tenants, we consider the case of s g. It follows that that:
           (s ; pg c) (g ; pg c)   >   0                                     (21)
                       (b ; pbc)   >   0                                     (22)
                       (s ; pbc)   <   0 by (19) and s   g                   (23)
           In addition we have:
                            Fb Fg >    0                                     (24)
                              Fs <     0                                     (25)
                             FE    <   0 by rst order stochastic dominance   (26)

6.3.1 The derivatives
To ease notation we introduce the following shorthand notation for the second
order condition of program 2:
                     SOCi     ;2Fi ; Fii(i ; pic) i 2 fg bg < 0

The partial derivatives of system (17) to (19) are:
        gs = ;Fs + (1 ; ) s Fg > 0

        gg = ;2Fg ; Fgg (g ; pg c} + Fg (1 ; )
        2                        )                     s
             |       {z                 | {z }         g
                           SOCg <0        +
        2  = Fg (1 ; ) bs < 0
        bs = ;Fs + (1 ; ) s Fb > 0
        2                    s

        bg = Fb (1 ; ) g > 0
        2               s

        bb = ;2Fb ; Fbb (b ; pb c} + Fb (1 {z )
                                 ) | ;
        2                                          s
             |        {z                     ;
                        SOCb <0
         s   = 1 + (1 ; p)F ( g ) + pF ( b )
               +Fs (1 ; p)(s ; pg c) + p(s ; pbc)]
               ;Fs (1 ; p)(g ; pg c) + p(b ; pb c)] > 0
         g   = (1 ; p)(s ; pg c)F} + (1 ; p) 1 ; F ( g ) ; (g ; pg c)F}]
               |       {z         g          |           {z           g
                           (+)                     (0) by (17) and (19)
             = (1 ; p)(s ; pg c)Fg > 0
         b   = |(s ;{zbc)F} + p 1 ; F ( b ) ; (b ; pb c)F}]
               p     p    b      |          {z           b
                     (;)             (0) by (18) and (19)
             = p(s ; pbc)Fb < 0

The partial derivatives of system (17) to (19) with respect to the parameters

are (recall, that dpg =dpb = ;p=(1 ; p))
              gc  = (1 ; )Fg cs + pg Fg
               bc = (1 ; )Fb c + pb Fb
               2              s

                c = ; p + (1 ; p)F ( )pg + pF ( )pb
                s                    g          b

                     + (1 ; p)(1 ; F ( g ))pg + p(1 ; F ( b))pb] < 0

              g   = (1 ; )Fg s > 0
               b = (1 ; )Fb > 0
               2               s
                s = (1 ; p)(1 ; F ( g ))(g ; p c)
                     + p(1 ; F ( b))(b ; pb c) > 0

          gpb      = (1 ; )Fg      s
                                   pb + Fg dp c
           bpb     =   (1 ; )Fb pb + Fb c

            pb     =   ;(1 ; p)c F ( g ) + (1 ; F ( g ))] dpg
                       ;pc F ( b ) + (1 ; F ( b ))]

                   =   pc(1 ; )(F ( g ) ; F ( b )) < 0

          2        = ;FE > 0
          bE       = ;FE > 0
           E       = FE (1 ; p} (s ; pg c) ; (g ; pg c}]
                     | {z ) |              {z         )
                            ;                  +
                       + FE p (s ; pb c) ; (b ; pbc}] R 0
                         |{z} |          {z        )
                           ;              ;

6.3.2 Additional restrictions
Rewrite   s   to obtain
                     s    = 1 + (1 ; p)F ( g ) + pF ( b)
                            + Fs(1 ; p} (s ; pg c) ; (g ; pg c}]
                              | {z ) |             {z         )
                                   ;               +
                             + |{z} (s ; pbc) ; (b ; pbc}]
                               Fs p
                                    |         {z        )
                                ;              -
Since an increase of s reduces the turnover for both groups, there is a negative
feedback through a reduced turnover of good tenants (the forth term) and
a positive feedback through a lower moving rate for bad tenants (the fth
term). The equilibrium conditions require s to be positive. However, in
order to derive unambigous signs in the comparative statics, we introduce a
slightly stronger assumption.
              1 + (1 ; p)F ( g ) + pF ( b)                                  (27)
              + minfFs ;Fg g (1 ; p) (s ; pg c) ; (g ; pg c)] > 0
The restriction requires that the direct impact of an increase of the rst
period rent on revenue alone is stronger than the impact of a decreased
turnover of good tenants | caused either by a marginal increase of s or a
marginal decrease of g. Since (s ; pg c) ; (g ; pg c) vanishes for (g ) ! (s 1),
(27) will hold true, provided that the tenure discount is not too big and that
discounting is not too strong.
Given (27) we establish two useful lemmas:
Lemma 2       ;Fb ss + Fs   s
                            b <0
;Fb ss + Fs    s
               b   = ;1 ; (1 ; p)F ( g ) ; pF ( b)
                     ;Fs (1 ; p)(s ; pg c)] + Fs (1 ; p)(g ; pg c) + p(b ; pbc)]
                   < ;1 ; (1 ; p)F ( g ) ; pF ( b )
                     ;Fs(1 ; p) (s ; pg c) ; (g ; pg c)]
                   < 0

Since jSOCb j > Fb by lemma 1, lemma 2 implies that SOCb s + Fs
                                                         s                 s
                                                                           b <0

           s      s
Lemma 3 c + pg g < 0
Proof: It is shown that ; c ; pg g > 0
                          s      s
             s              p                               p
         ; pc ;     s
                    g   =   pg
                                 + (1 ; p)F ( g ) + pF ( b) pb
            g                                                  g
                             + (1 ; p)(1 ; F ( g )) + p(1 ; F ( b)) pb
                             ; Fg (1 ; p)(s ; pg c)
                            (Note that pg < p < pb)
                        >   1 + (1 ; p)F ( g ) + pF ( b)
                            + (1 ; p)(1 ; F ( g ))
                            ;Fg (1 ; p)(s ; pg c)
                            (FOC implies 1 ; F ( g ) = (g ; pg c)Fg
                        >   1 + (1 ; p)F ( g ) + pF ( b)
                            ;Fg (1 ; p) (s ; pg c) ; (g ; pg c)]
                        >   0

Since jSOCg j > Fg by lemma 1, lemma 3 implies SOCg cs ; Fg pg           s

6.3.3 Comparative Statics of Equilibrium{Rent
The Jacobian of the system (17) to (19) is
         0                                                                          1
         B ;Fs + (1 ;           s Fg SOC + Fg (1 ;                Fg (1 ; ) b
                            )   s       g              )   s                 s
J4       B ;Fs + (1 ;
                                                           g                        C
     =   @                  )   s
                                s Fb    Fb (1 ; ) g
                                                  s            SOCb + Fb (1 ; ) b
                        s                      s                         s
                        s                      g                         b
The determinant simpli es to
                 jJ 4j =              s           s           s
                            SOCg SOCb s + SOCb Fs g + SOCg Fs b
                                  b s
                        = SOCg SOC{z s + Fs bs} + SOCbFs
                                              ]                    s
                                  +by lemma 2
                                                      | {z } > 0

Since jJ 4j > 0 the signs of the comparative statics are given as follows:

service cost c and discounting
       ds=dc   : ;SOCg SOCb cs + Fb pbSOCg bs + Fg pg SOCb g

               = ;SOC}(SOCg cs ; Fg pg g ) + Fbpb SOCg } > 0
                 | {z b                 s
                                             | {z b
                             |         {z         }    +
                              (+) by lemma 3
               :   Fs b Fb pb ;Fg pg SOCb s + Fs b ] ;SOCb Fs c > 0
                      s                     s      s          s
                   | {z } |
                                         {z         } | {z }
                                 +by lemma 1 und 2
       db=dc   :   ;Fs {z b } | s
                   | +g b
                         sF p ;F SOCg s ; F p s] ;SOCg F p s > 0
                                           {z g g g} | {z b b }
                                            c                     s
                                   +by lemma 1 und 3        +

       ds=d    : ;SOCg SOCb s < 0
      dg=d     : ;Fs SOCb s < 0
       db=d    : ;Fs SOCg s < 0

precision of learning pb
               : ;SOCg SOCb           s           g s            b s
                                      pb + Fb cSOC b + Fg dp cSOC g > 0
                   |       {z
                                      } | {z } |
                                          +                {z
                                                            b       }
                                  dpg              dpg s
      dg=dpb   :   Fs b Fb c ; Fg
                      s               c] ; SOCb Fg              s
                                                       c s + Fs pb ]
                                  dpb              dpb
               =         | {z } | {z } | {z } R 0
                   ;Fg dp c Fs bs + SOCb ss] + FsFb bsc ;FsSOCb pb   s
                   | {z } (;) by lemma 2
                                         + +
      db=dpb   : ;Fs       g Fb c ; Fg dp c] ;SOC Fb c s + Fs pb ] > 0
                           s                     g     s      s
                   |             {z
                                         b   }|       {z

Note that:

          = ; 1 ; p lim pb = 0                   lim dpg s = ;(s ; pg )Fg < 0
                                                 p!1 dpb g
   lim dpg = 0
   p!0 dp
                     lim s = (s ; pg )Fg > 0
                     p!0 g
                                                 lim s = 0
                                                 p!0 b
   p!1 dpb
             = ;1    lim s = 0
                     p!1 g
                                                 lim s = (s ; pb )Fb > 0
                                                 p!1 b

                                lim(dg=dpb) < 0
                                lim(dg=dpb) > 0
                            lim p!0(dg=dpb) = 0

Now we turn to the rate of turnover. We assume = 1 (for which             s    = 0).
To ease notation c = 1:
                          d               ds         dg
                             F ( g ) = Fs       + Fg dp =                       (28)
                                        | {z } | {z }
     Fs Fb b SOCg + Fg ] +
           s                   Fs Fg SOCb g ; Fg2 (SOCb s + Fs b )] < 0
                                          s             s      s
     | {z } | {z } |{z} |
                           dpb                      {z                }
              ;                                     +
                          d               ds         db
                             F ( b ) = Fs       + Fb dp =                       (29)
                                        | {z } | {z }
                                          ;      +

              Fs Fg g SOCb + Fb ] ;SOCg Fb Fb s ; Fs b ] ;Fb2 Fs g
                | {z } | {z } | {z } | {z } R 0
                    s                         s      s           s
          | {z } ;
                         +     + by lemma 2 +
                            lim(dF ( b)=dpb) > 0
                            lim(dF ( b)=dpb) R 0

expected mobility cost E
ds=dE    : ;   ((1 ; )(Fb bsSOCg + Fg g SOCb ) + SOCb SOCg )

           ;FE ( bsSOCg + gsSOCb )
dg=dE    : s SOCbFE + E ((1 ; )(Fg (Fs bs + s SOCb) ; Fb Fs bs) ; FsSOCb )
            s           s                      s

db=dE    : s SOCg FE + E ((1 ; )(Fb(Fs g + s SOCg ) ; Fg Fs g ) ; FsSOCg )
            s            s               s     s             s

To simplify the assessment of decreasing the mobility (increase in E ) we
assume that the error commited by abstracting from discounting is not too
large and set = 1 to obtain:
                       dg=dE     : ;SOCb      E Fs ;
                                              s        s
                                                       s FE ] > 0
                       db=dE     : ;SOCg      E Fs ;
                                              s        s
                                                       s FE ] > 0

The last two inequalities follow from

            E Fs ; s FE        = ;FE 1 + (1 ; p)F ( g ) + pF ( b)] > 0
            s      s

ds=dE   simpli es to:
           ds=dE         : ; E SOCbSOCg ; FE ( bsSOCg + g SOCb)
                              s                         s

                         = ;SOC} | E SOC{z FE }] ;FE SOCb g
                           | {z g s b + bs                  s
                                 +            ?
                                                         |    {z

Since E = FE (1 ; p)(s ; g) + FE p(s ; b) the term in the brackets can be
rewritten as

               FE      (1 ; p)(s ; g)SOC} + FE p(s ; b)SOCb + bs}
                                            |        {z         ]
                             +                          ;
Negativity of the second term is easily established by lemma 1 and b;pb c > 0.
The rst term becomes arbitrarly small for g ! s. Hence with some caution
we might conclude that ds=dE < 0. If there were only a small tenure
discount initially, it would decrease even further as mobility cost increase.
Now we turn to the rates of turnover. It will be shown that:
                    d                      ds           dg
                      F ( g ) = FE
                   dE           |{z} + | s {z } + Fg {z } < 0
                                           dE     | dE
                                           +            +
Since FE   <0   (30) can be rewritten as
                               F    ds      F      dg
                          1 > ;F s         ;Fg
                                 E dE           E dE
De ne x 1 + (1 + p)F ( g ) + pF ( b) and y (1 ; p)(s ; g) + p(s ; b), so
that s = x + Fsy and E = FE y. Recall that the denominator jJ 4 j has
       s                 s
been suppressed in all the derivatives of the endogeneous variables. Hence,
(30) is equivalent to
  jJ 4j = SOCg SOCb ss + FsSOCb gs + FsSOCg     s

                         yFsSOCg SOCb + Fs SOCb g + Fs SOCg b ; SOCb Fg x
                                                 s          s

                   SOCg SOCb s > yFs SOCg SOCb ; SOCb Fg x

                           SOCg SOCb x > ;SOCb Fg x
                                 jSOCg j > Fg
By a similar argument it can be shown that
                                   F ( b) < 0


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