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DEWEY
5
Technology
Department of Economics
Working Paper Series
Massachusetts
Institute of
EFFICIENCY PRICING,
TENANCY RENTAL CONTROL
AND MONOPOLISTIC LANDLORDS
Kaushik Basu
Patrick
M. Emerson
Working Paper 01-40
October 2001
Room E52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://papers.ssrn.com/abstract id= 290831
^ViASSACHUpffsi^^
__OFTECHNOLOGY
DEC
3 2001
LIBRARIES
Technology
Department ot Economics
Working Paper Series
Massachusetts
Institute of
EFFICIENCY PRICING,
TENANCY RENTAL CONTROL
AND MONOPOLISTIC LANDLORDS
Kaushik Basu
Patrick
M. Emerson
Working Paper 01-40
October 2001
Room E52-251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://papers.ssrn.com/abstract id= xxxxx
)
Efficiency Pricing,
Tenancy Rent Control and Monopolistic Landlords
Patrick M. Emerson
Department of Economics
University of Colorado at Denver
Kaushik Basu
Department of Economics
Cornell University
pemerson@carbon.cudenver.edu
and
Department of Economics
Massachusetts Institute of Technology
kb40@comell.edu
October, 2001
Abstract
We consider a model
the rent
on
taking on a
sitting tenants
new
of 'tenancy rent control' where landlords are not allowed to raise
set the nominal rent when
economy, landlords prefer to take short-
nor to evict them, though they are free to
tenant. If there is
any
inflation in the
Assuming that there is no way for landlords to tell a tenant's type, an adverse
problem arises. If in this context, landlords have monopoly power which, as we argue,
is indeed pervasive— then the housing market equilibria can exhibit some unexpected properties.
Most strikingly, landlords may prefer not to raise the rent even when there is excess demand for
housing. Such rents are labeled "efficiency rents" in this paper and their existence shows that
tenancy rent control can give rise to equilibria which look as it there were traditional rent control
in which the rent of each unit has a flat ceiling. In other words, tenancy rent control may not
achieve the flexibility, which it was expected to impart, to the system of traditional rent control.
staying tenants.
—
selection
(JEL
classification
(Keywords: Rent
Monopoly)
numbers: D40,
K
1
0,
L
Control, Rent Regulation,
1
2,
L5
1 ,
R3
1
Asymmetric Information, Adverse Selection,
Acknowledgments: We thank two anonymous
comments and suggestions.
referees of this journal for their insightful
Efficiency Pricing,
Tenancy Rent Control and Monopolistic Landlords
Introduction
I.
Standard, or
first
generation, rent control places a ceiling
landlord can charge. Hence, under standard rent control, excess
on the
rents that a
demand
for
housing
is
common.
Arguably, more
which allows a landlord
common
than standard rent control
when
to set the rent freely
'tenancy rent control,'
is
leasing to a
new
course, the tenant's right not to accept), but thereafter the landlord
the rent, nor can he evict the tenant.
'
It
will
be shown
in this
tenant (subject
is
standard rent control. That
is,
in equilibrium
keep the rent for new tenants so low that there
The landlord
voluntarily behaves as if there
is
it
may be
excess
was a
paper that tenancy rent
Stiglitz,
which looks as
if
in the landlord's interest to
demand
legal ceiling
for
housing
on the
This surprising result has analogues in the theory of efficiency
1957; Shapiro and
of
not allowed to raise
is
control can, through the workings of the market, result in an outcome
there
to,
at that rent.
rental rate.
wage
1984) or the theory of efficiency interest rate
(Leibenstein,
(Stiglitz
and
Weiss, 1981).
It
should be clarified here that our model
is
one where there
control and landlords have monopolistic power. In fact
the case of a monopolist landlord
in
Basu and Emerson (2000).
reality
- which
is
is
tenancy rent
what we consider
the other polar extreme to the case
In reality, landlords are often only limited in their ability to raise rents on,
inflation,
which
is
it is
paper
is
modeled
We do not claim that this case is in any way closer to the
of housing markets than the perfect competition assumption. The
the present analysis
in this
and
reality is surely
evict, sitting tenants.
For
only necessary that the allowable rental increases are insufficient to keep up with
generally the case (see Basu and
Emerson (2000)
for a discussion).
somewhere
in
both polar cases so that
The standard
systems.
we
can better comprehend the
result
above the market clearing
of its good,
level
surprising
where
is
that
this is also
is
a
many
rental
for example, notes that in the
all
units in
rental
housing firms
of the
owners
in
that in
in the city
there are
But
there
housing markets are
remains
we can have
may
But, what
not
is
the opposite case
demand
for housing.
is
on rent control (see Amort, 1995,
a fair
far
amount of empirical evidence
from competitive. Cronin (1983),
Washington, D.C. suburbs of Virginia, on average
is
slightly
Cambridge, Massachusetts,
70% of
over four. Mollenkopf and Pynoos
6% of the city's households controlled
In addition, they estimated that
units.
90%
of the apartment
belong to an association of property owners 700 strong, and
20 owners account for
many
landlords there
Appelbaum and Glasser (1982;
California,
in this case)
our model as well.
in
as to give rise to excess
literaUire).
each submarket
housing
rental
that organization
when
raised
each submarket are controlled by one owner and mat the average number of
(1973) noted
70%
low
is
a monopolist with a fixed supply
in the theoretical literature
of the
for a thoughtful review
of modem rent control
of perfectly competitive or monopolistically competitive housing
common one
that suggests that
Of course
standard and arises
the landlord sets the rent so
suppliers
reality
essential to understand
and some of the product (housing,
under certain parametric conditions,
Tire assumption
is
it
of monopoly (with fixed supply) where price
unsold (vacant) arises in our model.
sell all
We do believe that
between the two models.
is
40%
of the
rental
that within
housing stock. Hence even
scope for monopoly behavior through collusion.
as cited in Gilderbloom, 1989) found that, in Isla Vista,
27 owners controlled
50% of the
rental
housing stock. Close by in Santa
Barbara, Linson (1978; as cited in Gilderbloom, 1989) reported that over
50% of the
rental
20%
housing was
of the
owned by only 60 owners and
rental stock. Finally,
seven of them could account for
Gilderbloom and Keating (1982; as cited
Gilderbloom, 1989) found that in Orange,
one
that
in
New Jersey, just ten owners control
close to
of the rental housing and that in Thousand Oaks, California, just one owner
third
controls over
30% of the rentals.
In an empirical study of supply side concentration of
the rental housing market in Boston, Cherry and Ford (1975) find that housing prices are
significantly
determined by concentration of within market segments.
Thus, there are
many reasons
to believe that housing markets are less than
perfectly competitive; the concentration
of rental property
in the
hands of only a handful
of major owners and the collusive opportunities from the presence of landlord's
associations are two.
There are
example Amort (1989) hypothesizes
well, for
monopoly power
theoretical bases for believing in
that the indivisibility
as
and heterogeneity of
housing markets leads to monopoly power on the part of landlords.
There
shortages
is
also reason to believe that landlords
by increasing
the Ontario provincial
initial rents
government relaxed Toronto's
housing
in
1998
rent control laws giving landlords
whenever a tenant moved
landlords did not respond with an expected building
built
to
and the supply of apartments. For instance
the right to raise rents to market levels
were
do not always respond
out.
However,
boom, fewer than 600 new
units
over the subsequent two years while Toronto's population increased by more
than 100,000. Vacancy rates in Toronto remain under 1%.
Again,
any closer to
this
does not
reality than
mean
one
that
that a
model
assumes
that
2
assumes landlords are monopolists
that they are perfect competitors.
What we
is
are
Barry Brown, "Apartment Shortage has Toronto Renters Searching for Space," The Buffalo News, October
1,2000,
p.
4A.
attempting to do in the present paper
is
to
examine the other extreme
that has
neglected in the literature and point out the aspects that are particular to the
case so that
II.
we may better understand
The Algebra
monopoly
of rent controlled housing markets.
of Rent
In this paper
controlled housing
this.
the true nature
been
is
we
will consider the effects
limited.
New
of rent control when the supply of rent
York, inner-city
Rent controls can however be of many kinds.
Mumbai and
It
Delhi are examples of
can take the form of a rent fixed by
a rent control authority or government (see Olsen, 1998, for a discussion of the different
forms of rent control) or of a law that gives landlords some or
when
leasing out property to
(or adjusted
new
upward only within
the landlord having
no
freedom to adjust rents
full
tenants but then requiring the rent to be held constant
limits) as
long as a tenant remains the lessee (and with
right to arbitrarily evict
a
sitting tenant).
control, called 'tenancy rent control,' (see, e.g.,
Arnott, 1995; Borsch-Supan, 1986)
is
This
latter
form of rent
Basu and Emerson, 2000; Nagy, 1997;
quite pervasive
and
is
the subject matter of this
paper.
Given tenancy
rise to
rent control, the presence
an adverse selection problem. Landlords
staying tenants (as long- stayers
real rents
of even a small positive
now
staying tenants
know
prefer short- staying tenants to long-
impose greater costs on landlords due
during a single tenancy) but they have no
their type but
have no
inflation gives
way of telling
to the erosion
types apart.
3
of
Long-
interest in revealing this information to
prospective landlords. Curiously, the relation between rent control and inflation remains
This, of course, ignores the possibility that landlords offer tenancy discounts to keep
those that impose relatively low costs on the landlord. In our model tenants are
length of stay. See Hubert (1995) for an excellent analysis of markets with
"good" tenants
homogenous except
more and
for
less costly tenants.
-
.
We tried to develop the building blocks of a model
a neglected subject.
in
for analyzing this
Basu and Emerson (2000).
In the present paper,
model and build
into
we
develop some of the basic theory in a continuous -time
our model some elements of reality - to
monopolistic power on the part of landlords
Each
we
(potential) tenant
say that a tenant
after
t
periods. There
duration,
is
t,
of "type
is
is
given by
- which
wit, limited
supply and
has not been modeled thus
has an exogenously given duration of tenure
t"
we mean that
the tenant will
When
(>0).
t
move from an
4
far.
apartment
a continuum of tenants and their density function on the tenure
with F(t) being the corresponding distribution function. All
f(t),
agents are supposed to have the
same discount
rate
5e
[0,1)
We denote the total number of tenants in the rental market by N.
F(oo)= Jf(t)dt
Hence:
=N.
t=0
Suppose a landlord leases out
after
every
value
is
$1
t
.
to tenants
of only type
and each time a new tenant comes he
periods),
t
(that
is,
gets a type
fixes the rent so that
t
its
tenant,
real
Thereafter the nominal value of the rent remains fixed so long as the tenant
does not leave. Let the inflation rate be such that the value of each dollar erodes in each
period
at the rate
of the landlord's
of
real
1
- (3 where
,
income
is
(3
£
(0,1
denoted by
t
/4.\
v(t)
4
Some monopoly
=
f
le
)
Under
.
v(t).
these circumstances the present value
Clearly then,
t
-(P+5)x
,
dx + e
,
-5t f
I
t
e
-(P+6)xj
dx + e
,
-28t f
J
e
-(p+8)x j
dx+AA
,
elements were considered in a model of tenancy by Basu (1989), but the focus of that
paper was entirely on innovations and the context was that of a backward agrarian economy.
JY
_
,p+5,
o
e
(P
and above
much
(1)
61
)
selection ensures that, for each rental rate, only tenants of
will seek
the fact that short- staying tenants
they do not see as
-(^-5)t
+ SXl-e"
The process of adverse
a certain type
Mx
housing
no longer
in the rent controlled market.
find
tliis
worthwhile to rent
benefit for the erosion of real rents than
technical description of this process, see Step
demonstrates
it
1
This
in this
market as
of the proof in Appendix A, which
(in real
terms) earned by a landlord
is
v(t)
-
equal to
1
erodes each period by
procedure. v(t)
is
understand
tenant, conditional
square bracket
is
1
-
(3 ),
to rent
x> t,
at a
with and thereafter kept fixed nominally (so
it
and each rime a tenant leaves the landlord repeats the above
given by the following expression:
V (t)
To
real dollar to start
the
who manages
out his house to a tenant selected randomly from a tenant pool with tenure time
is
to
do long- stayers. For a
central mathematical character in such an analysis
expected present value of rents
which
due
result mathematically.
Hence, the
rent
is
=?
f (x >
rfe-^^dk + e^vColdx,N-F(OL Jo
J
this,
observe that f(x)/(N-F(t))
on tenants of type
t
the present value of rents earned
the probability of picking a type x
is
and above being
(2)
available.
when
the
first
The expression
tenant
is
of type
in the
x.
Now we are ready to state and prove the one technical result on which we will
build our economic analysis.
.
Proposition
1
If
:
t
.
>
t
,
then v(t*)< v(t')
Proof See Appendix A.
With
model of rent
the background,
this technical result in
When tenants make the decision
control.
apartment the alternative they have to keep in mind
rent controlled area or to
arrangement costs
C
now easy to
is
it
is
dollars (in present- value terms).
independent of the tenant's
of whether the person
Likewise in renting an apartment
them
to find housing in a non-
that this alternative
For simplicity,
housing
we assume
that
C is
This seems reasonable as well. In buying a house the
'type.'
cost will clearly be independent
full
whether to lease a rent -controlled
for
buy a house. Let us assume
describe a
is
a long- stayer or a short- stayer.
in a non-rent controlled area, the tenant's type is
unlikely to matter because the rent can be inflation indexed or be
made
contingent
on
the
length of the tenant's stay.
Now suppose that the rent (per period) in the rent controlled housing is R.
life- time
rental cost to a tenant
of type
each time at
is set
Consider
now
a
that
1
t
of Proposition
>v
—
1
I
(~*
/
\WR
1
by
Rv(t).
a tenant of type
t
monopoly
we know
who
landlord,
given by:
is
the
if the real rent at the start
of
t,
sets the real (starting) rent equal to R.
will accept this for
v'(t)
<
,
for
all
t, it
whom Rv(t) <
C. Since from
follows that
tenants types
all
t
such
\
will accept the offer.
J
It
follows from the definition of v(-) that the
landlord's expected present-value of rental earned from each
is
Recall that v(t)
1
Clearly only those types of tenant,
Step
clearly given
made by
present-value of life-time payment
tenure
t is
The
apartment
that is leased out
V(R) = Rv(v _l fc
From
Step
Proposition
rises,
of Proposition
1
1
we know
V(R) may
Figure
rise
or
1
K
we know
that as
R rises
_1
that as
v[v
R rises,
yyC
v
falls.
jj
'
\~A>
It is
)
rises.
now
Hence, from
transparent that as
R
fall.
represents a possible picture of V(R).
1
Earnings
per apartment
i
\
V(R)
Rent per period.
p
Figure
Define
informally,
>
t
,
and
t
for
to
t
is
all
the
t
>
be the supremum of the
Now define R
no
further takers
set \t|f(t)
upper support of f(t). So,
0, there exists tenants
among
such that v
the tenants.
[
=
Hence V(R)
t
>
such
is
t
of type
^y~
1
t
.
is
e
0).
In other words, and
[t
that there are
- 1,
Then
t
more
no tenants of type
t
J.
if rent
goes above
not defined for
R
> R
.
R
At
,
there are
R
,
the
.
only takers are of type
see that for
all
R< R
,
t
Hence V(R )= Rv(t) = Rv(t) =
.
V(R) < V(R
V(R) can
also evident that
It is
).
transparent if tenant types are
finite.
is
It is
This explains the shape of V(R) in Figure
over some stretches. This
fall
Then over some
is
easy to
1
especially
increases in R, large
short- stayers can decline the rental offer, leaving the pool
the landlord's point of view. This
RC/R = C.
numbers of
of tenants suddenly worse from
the classic adverse selection
problem (Akerlof,
1970).
III.
Excess Supply, Excess
The
results are the
Demand and
outcome of the landlord's optimization problem when
confronted with an earnings curve, V(R).
down
to zero
Efficiency Rent
was analyzed
in
The case of many
landlords
who
drive profits
Basu and Emerson (2000). Here we take on
the other polar
end: the case of limited supply and monopoly. Rent control applied to a fixed stock of
housing, such as in
New
York, and the evidence supporting the contention that rental
housing markets are not competitive,
To
begin
this analysis, let
> v" [Qp
1
t
housing, D,
is
)
will
want
it
worthwhile investigating
us derive the
function of the (per- period) rent, R.
type
make
From
demand
section 2,
for rent- controlled
we know
to lease rent- controlled housing.
given by
D(R)
} f(t)dt
w
sN- f(v" c/r
]
(
this
,
that
polar case.
housing as a
given R,
all
tenants
Hence, the demand for
of
10
From Step
1
of Proposition
R rises.
Such a demand curve
panel
a reproduction of Figure
is
is
1,
v
'
(Qjl
rises as
j
Hence, D(R) declines, as
lower panel of Figure
illustrated in the
1
R rises.
The upper
2.
.
Earnings
per apartment
V(R)
Rent.
j^
R
Supply
Demand
D(R), Demand
and Supply
Figure 2
Next, draw the supply curve in the lower panel. The landlord,
S
units
of property. For simplicity,
it is
assumed
it
is
that the opportunity cost
assumed, owns
of leasing out
11
property
Let
S.
is
is
,
locate the landlord's
R
)
optimum
rent,
consider
which maximizes V(R). This,
not necessarily monotonic, there
that, if the
is
is
was
landlord
she would choose R*. This
maximized. Hence, the
earnings are
Next consider
total
However even
in Figure 2,
and
at
all
R*
R <R
where
total
Let
The
the
maximized
if V(R) rises, the total
at
R
must eventually (weakly)
same height
as
such a
rent, the landlord will
If the
equilibrium.
earnings.
optimum
if
.
R*.
.
R is raised starting from R
As
rise (since
as
R > R
V(R
rise,
R
is
)
= C > V(R),
since
raised.
demand
Let
R'
R*
or
R"
If
.
R", per apartment
R*>R
'
,
total
be
at
R"
,
earnings
then this
The monopolist holds back supply
is
R*e
for
falls
be the
earning
may be
all
axis.
[R, R'j, clearly, her
is
smaller
smaller at
and
R*
since
her apartments.
a fairly typical
in order to
,
).
be unable to find tenants for
turns out to
R
with
A, and R' the projection of E on the horizontal
earnings are greater at R*, since at such an
at
will coincide
earnings need not
landlord's chosen rent will clearly be either
fewer apartments are taken. Even
given by R*. Since
the per- apartment earnings are
earnings are maximized (subject to
E be
and locate
she manages to lease out the
,
below S and so more and more apartments remain vacant
rent
is
,
marked
restricted to selecting a rent less than or equal to
rents greater than or equal to
the earnings of tire landlord
R
rents less than
all
no reason why R*
because for
to wit, S,
R).
perfectly inelastic tlirough the point
To
same number of apartments,
all
is
be the rent which equates demand and supply.
Observe
R
the supply curve
R
the rent (left of
V(R)
Hence
zero.
push up
monopoly
the price
and her
2
1
The
for
interesting case occurs
when R*
housing exceeds supply (see lower panel of Figure
prefers not to raise the rent. This
is
1981).
We shall
therefore call
Usually,
we would
no
is
on
It is
rents,
the
is
'quality'
(from the
models of efficiency wage
(e.g. Stiglitz
and Weiss,
the 'efficiency rent.''
expect this kind of a rent to prevail on the market
In such a case
rent.
What our model
if rent
demand exceeding
illustrates is that
even
if there is
tenancy rent control can result in behavior such that the market
ceiling.
interesting to note that the
increases. This
R"
rather like in
form of an exogenous ceiling on
equilibnum mimics a rent
at
R*
compatible with equilibnum.
ceiling
is
1974; Mirrlees, 1975) or efficiency interest rates
control took the
supply
Nevertheless the landlord
2).
because a higher rent worsens the
landlord's point of view) of the tenant. This
(e.g. Stiglitz,
optimum. Here demand
turns out to be the
R*
because as S increases,
number of tenants
is
constrained
equilibrium
at
R"
by
is
the
more
likely
the landlord's profit
the
is
demand (remember
outcome as S
unchanged, since
at this point there is
excess supply of housing) and so an increase in supply does nothing to the landlord's
On the
income.
(see Fig 2).
other hand, at R*, the landlord's profit
So an equilibrium with excess demand
large portion
of the
rental stock is
for
is
given by V(R*) multiplied by S
housing
is
more
likely
when
a
under rent control.
IV. Conclusion
Well meaning urban policy makers of the 70's and 80's, attempting to correct the
glaring problems
had been
of old-style rent controls that placed ceilings on rents (problems that
illustrated quite vividly
by economists), turned
to a
form of rent control
that
was
13
more of a
tenant's protection legislation than a unit-by-unit rent restriction. This type of
'tenancy rent control' simply restricted landlord's ability to raise rents on sitting tenants
and prohibited most
flexible
side
payments and
program and one
that
was
was seen
arbitrary eviction. This
less susceptible to the inefficiencies
to
be a more
of the old rent
control laws.
What we showed
in
Basu and Emerson (2000)
is
that this
new
type of rent control
brings about different kinds of inefficiencies due to the adverse selection problem
brought about by the asymmetric nature of information
present paper illustrates
can cause landlords
wit, they hold
(i.e.
one
such a
to operate in
but with
all
way
that
mimics the old
with excess demand, to
that will not stay too long).
rental
We call this
norm
to suggest large
no more of an abstraction than models
style rent control.
To
a better 'quality' tenant
may be
an abstraction from
amounts of concentration and
that
many
rental
housing markets,
assume perfect competition
it is
(as is the
We believe that, through the study of both extremes, a clearer
and more comprehensive understanding of the
Given the pervasiveness of 'tenancy
fully the nature
attract
housing markets to be monopolistic
of the evidence
in the literature).
the
'efficiency rent.'
possible collusion (through landlord's associations) in
certainly
What
in the presence of monopolistic landlords, tenancy rent control
down price, even
Assuming
reality,
is;
in these markets.
of the inefficiencies
Basu and Emerson, 2000)
is that
other types are hurt. In addition,
it
reality
of rent control will
rent control,'
creates.
What
it is
important to understand
the present
work
certain types of tenants are helped
it
illustrates a
result.
illustrates (as
by
this policy
does
while
type of strategic behavior on the part of
monopolistic landlords that has not been previously explored in the
literature.
14
Appendix A: Proof of Proposition
To prove
Proposition
1, first
1
note that v(t) can be simplified using
( 1 )
and
(2) to
be
written as:
5x
J"f(x)(l-e~
)v(x)dx
i
v(t)
(3)
x
N-F(t)-J"f(x)e^ dx
From
v(x), as
set
(3)
it
is
clear that v(t)
x ranges from
equal to
1,
to °=
t
for all x)
1
:
We will
show
is
this
is
obvious from the
now proved
that v(t) rises as
In other words, v'(t)
To prove
This
the weighted average of the different values of
fact that if v(x) is
removed
(i.e.
from the right-hand side of (3), then the right-hand term equals
The proposition
Step
.
is
<
,
for
in three steps.
t falls.
all
t.
note that:
Je-^dxUje^e^dxV*
(t)
f
v°
+
(
Je^e^dx \* J +A
V°
J
l-e HP+S)l
((3+5)(l-e-
/(t)=v(
Then:
Note
that v(t)
is
1.
positive.
((3
t
;
)
+ 5)e- (p+5)
l_ e n^oj,
,-<^-8)t
'
5t
5e
j_
\
-o,
e
We must now show that the term in parentheses is negative,
rearranging and collecting terms under a
showing the numerator, or
81
common
denominator reduces our problem to
.
15
((3
o^ 6 - - oe (p+8)t =
+
(3
We need to prove this for all
It
if
t
This establishes Step
:
As t
0,
.
To
and X'(t) <
,
X(0) =
Finally, note that
>
((3
then X(t)
is
negative.
f?t
].
if
<
,
see this note that
+ 5)5e [l-e
((3
X(0) <
follows that if
Step 2
=
t
5t
=
X'(t)
Hence X'(t) =
X(t)
'
t
>
0.
Vt >
0,
.
+ 5)-|3-5 = 0.
1
falls
from
t" to t'
To prove
this,
suppose
,
the weight
x>
t"
.
on each
v(x), for
The weight on
x>
t"
,
in (3)
v(x) in (3), denoted
falls.
by
w(x,t), is
given by:
w(x,t)
f(x)(l-e-
=
5
*)
5x
N-F(t)-Jf(x)e" dx
Note
that:
w(x, t')<w(x, t")
(4)
iff
x
N-F(t')-jf(x)e^ dx > N-F(t')- J^xje^dx
or
j"f(x)e~
5x
dx < }f(x)dx
t'
(5) is obviously true.
(5)
t'
Hence
(4) is true,
which establishes Step
2.
.
16
Step 3
:
Since the weights on v(x) in (3) always
v(x), for all
from Step 2
where x>
x>
that as
t
falls
t" to values
x,y such that x
It
t" implies positive weights
£
[t',
t* to t'
,
and y
>
t"
v(0<v(t').
to
1
,
v(x), for all
a decline in the weights on
xe
[t',
t'J
.
Hence we know
the weights get transferred from values of v(x),
of v(x) where x £
t'J
follows that
from
on
sum
[t',
t'J
.
From
Step
1
we know
v(x)
>
v(y) for
all
17
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