LIBRARY
OF THEMASSACHUSETTS
INSTITUTE
OF
TECHNOLOGY
working
p
department
of
economics
An
Approach
to theModelling
ofUrban
Housing Markets
Part
1:Housing Demand
Jerome Rothenberg
No. 149February,
1975massachusetts
institute
of
technology
^50
memorial
drive
Cambridge, mass.
02139
MASS. INST. TECH.
DEV/EY LIBRARY
An Approach
to theModelling
ofUrban
Housing Markets
Part
1:Housing Demand
Jerome
Rothenberg
No, 1A9
February,
1975This
research
issupported by National Science Foundation Grants GS-A0769 and
GS-370
lOX.The views
expressed
hereinare theauthor's
responsibility and
donot
necessarily
reflect
those
of
theDepartment
ofEconomics,
M.I.T.,
or theNational
Science
Foundation.
I Disl-jnctive Characteristics of
Urban
HousingMarkets
Urban
housing markets are not agood
representative of the abstract general market ofconventional
economic
theory.They
possess anumber
of distinctive features that are,af-
one
and
thesame
time, integral to urban housingand
difficult to integrate withinconventional
market
theory. Inclusion of these features does considerably complicatethe analytic,
and even more
the econometric, treatment of the field; but omission inthe interests of tractability
may
seriously misrepresent thephenomena
beingmodeled.
The
purpose of this essay is to bring to attentionsome
of these distinctive features,and
to delineate a theoreticalapproach
v/hich attempts to grapple withthem
in aconstructive
manner.
The approach
underlies a particular set of currentlyongoing
econometric
studies inwhich
the author is involved, but is compatible with a broaderfamily of empirical efforts.
The two
most important features of urban housing are the localization inconsuming
housing services
and
the durability of housing capital.1. Localization. Housing is a consumption activity that for
an
overwhelming
proportion of households occurs at a single spatial location during
each
conventionaleconomic
period (sayone
year).Commodities
areproduced
in a great variety of placesand
are transported to-orby-consumers
for consumption atone
ormore
places, orconsumers selectively travel to various sites
where
different Instances of consumptionmay
occur. Housing does not possess either the portability of this typicalcommodity
or the variety of sites at
which
sequences of consumptionmay
take place. For mosthouseholds consumption of housing services takes place at a single site,
where
theDigitized
by
the
Internet
Archive
in
2011
with
funding
from
IVIIT
Libraries
selected combination of
components
of the housingpackage
exists. Housing thereforeserves as
an
anchoring function, "piecing" a household at a particular spatial locationwhich, because
of the real costsinovercoming
distance,becomes
a base for a varietyof other forms of consumption involving both private
and
public goods. It is thislocal-ization that creates the
concept
of residentialneighborhood
and
household accessibility,and
hence deeply
links theeconomic
dimension of urban activities with the sociologicaland
political dimensions.2. Durability
Housing capital is
by
far the most durable capital incorporated inconsumer
goods.Indeed, it is
much
more
durable thanany
form of business capital as well, if thelong-evity of housing units in older countries like Europe is considered
—
units at every quality levelwhich
areup
to fourand
fivehundred
years oldIDurability
means more
than simply the continued physical existence ofa structure.It
means
that a structure does not ineradicably lose its current marketability justbe-cause of
growing age.
An
older unit remains agood
substitute in the rrKirket for unitsjust built,
even
atadvanced age.
This continued relative serviceability isdue
partlyto the fact that technological
change
in residential construction has not somuch
changed
the nature of housing services as the materials used to producethem.
The
more
fundamental changes in the nature (or quality) of the serviceshave
generallybeen
associated with capacities installable in older units at modest cost or withequipment
obtainable independently of the housing unit proper.Not
only doeseconomic
viability remain withage,
it is not marginal.While
depreciation does
occur
with timeand
usage,maintenance,
repairand
largeradvancing age.
Thus, old houses are not necessarily poor houses.They
can existthroughout the spectrum ofquality.
Advancing
age
does Influence the cost atwhich
any
given qualitycan be
maintained: itbecomes more
expensive with time. Thus, viaan economic
calculation, not as physical necessity, older houses are often observedto
be
oflow
quality (or In poor condition); but in othermarket
circumstances thehigher
maintenance
costs oreworth
making,
and
units are observed at higher qualities(and better conditions).
We
have
stressed the persistence ofeconomic
viability In terms of the servicesthat older unfts render. But there is another side to durability. Housing structures
are very expensive to
do
away
with.Even where age makes
serviceability disappearIt does not
make
the structure disappear. Unitswhich
are desired to be retiredmay
simply be
allowed
to crumble. But the siteoccupied
by them
cannot be reclaimed forre-^ise unless the structures are physically demolished. This demolition is very
expen-sive,
amounting
to a not inconsiderable proportion of the original cost of the structureItself (at constant prices). Thus,
even
when
notwanted,
old structures tend to remainIn existence and, under
many
circumstances, this continued existencemakes
iteco-nomically worthwhile to continue to use
them
for housing.3, Multi-dimensional Heterogeneity
Housing units in
an
urban area are heterogeneous, nothomogeneous.
They
differmaHcedly
In anumber
of dimensions. Differences in the physical units themselves areThe
recentsubstantialgrowth
inabandonments
showsan
opposite configuration.Rapid
neighborhood
declinedue
todemographic
factors, tax considerations, publicregulations like rent control,
moke
for a sudden collapse of profitability not associatedwith
age.
These units arewithdrawn
from the housing market despite basic structural Integrity.well-known, embracing
size,number
of rooms, architectural layoutand
style, structuralamenities,
and
condition,among
others. But a housing unit comprisesmore
than simplythe structure,
because
of the localization function referred toabove.
It is apackage,
consisting of the structure, the land lot, the
neighborhood and
the configuration ofaccessibility to different desirable destinations within the urban area'
and
outside.Each
of these other dimensions displayswide
differences as well. Lots differ in size,topography,
view, placement on
the block, etc.Neighborhoods
differ indemographic
character, assortment of private goods available, quality
and
variety of public goodsprovided, density
and
land-use mix. Locations differ greatly in the real cost involvedin reaching different desirable destinations
because
of distance,and
availabilityand
quality of different transportation
modes.
Most commodities
can beshown
to bepackages
of several components,and some
orall of these
components
can beshown
to differ for different brands or sellers. Butheterogenity has
some
special features for urban housing. First, households differsubstantially in their tastes for housing.
They
differ in the importance they ascribeto alternatives on
each
dimension of thepackage, and
in the relative importance ofthe different dimensions
—
i.e., the preference tradeoffs withineach component
and
acrosscomponents.
Thismeans
that the differencestobe observed in actual units matterin the market,
and
are thus likely not tobe
accidentia! but at least to a degree intended.Second,
householdsbehave
as though these differences matter a great deal.They
engage
in very substantial (and thus costly) search in order to findan
especiallyappro-priate
package.
Also, theyshow
a willingness topay
very highpremiums
for notablyattractive combinations. Price differences in the
market
ore considerable forpackages
Third, while for most commodities the multiple dimensions that matter to consumers
are subject to the control of producers
and
sellers, this is significantly untrue for urbanhousing.
A
producer/owner
of a housing unitcan
substantially control the structurecharacteristics of the unit over time and, to a lesser extent, the land
component.
But the
neighbrohood and
accessibilitycomponents
are at most selectedby
thepro-ducer/owner
at the outset (andeven
this selection is likely tobe
severely constrained),and
all subsequent changes are outside his control. Since such subsequentchanges
canhave
large effectson
the overall desirability of thepackage,
and
since theyoccur
without the direct mediation of market transactions
between
theowner and
thoseresponsible for the changes, their existence
amounts
to a serious potential for pervasiveexternalities within urban housing markets.
4. Convertibility of Existing Units
Housing units are highly durable. But theirdurability is not
independent
ofhuman
action during the aging process.
Continued
viability in themarket
requires continuedFnaintenance
and
repair. Indeed,human
intervention during the lifetime ofa housingunit
can have more
radical impact than ensuring its sheereconomic
survival.A
unitconstructed to render services at a certain quality level can be converted during its
lifetime to render services at eithera higher or lower level.
Upward
conversion isaccomplished
by
making
additions to a unit, orimprovement
ofplumbing
or electricity,or of
upkeep, making
architectural changes, or consolidatingtwo
ormore
originallyseparate units into a single larger unit. In most of these, real resources are being
in-vested to create
an improvement.
an
investment of real resources to decrease the quality of individual units. Thisoccurs
where one
unit is split intotwo
ormore
units,each one
containing less spaceand
probablyfewer
rooms than its predecessor. Investment is required here to putup
additional partitions
and
provide extra utilities tomake
each
unit complete.The
second
involves decreasing the level ofmaintenance and
repair so that usageand
timewill gradually worsen the condition of the unit.
5. Capital
and
Current Transactions:The
CapitalMarket
In urban
America,
given conventional standards of livingand
their translation intotechnology, a housing unit is a very expensive
commodity.
Most
peopleoccupy
unitswhose market
value is at least three times as great as their annualincome and
greaterthan their total portfolio ofassets.
The
durability of housingmakes
it possible toaccomodate
to thebudget
and wealth
position of households bytwo
means: 1) highmortgages
on
ownership claims; 2) a very important rental market. Probably a largerproportion of housing services are rented than for
any
otherconsumer good.
Both of thesemarket
patterns lead the housing sector tomake
veryheavy
demands on
loanable funds,especially long-term funds. Housing is
one
of the Icrgest clients of the national capitalmarket.
6.
Moving
CostsTo
amuch
greater extent than with other types ofconsumer
goods, achange
inconsumption
of housing is costly. First, since heterogeneity in the available stock ofhousing is considerable,
and
tastes regarding these differences quite important,sub-stantial search
may
be required fora household to assure itself of a satisfactorychange.
legal
and
otherspecialized services are requiredwhen
a sales transaction occurs,and
may
even
be necessarywhen
long-term leaseagreements
are involved.The
situationis especially complicated
when
an owner-occupier
household wishes tobuy
a differenthouse,
because
then the household must not only incur the costs of findingand
buying,but must also sell its present unit. Third, a
move
tonew
dwelling entails the cost ofphysically
moving
a household's possessions,an
expensiveand
oftentime- consuming
process
.
One
last aspect ofmoving
costs stems from the localization function performedby
housing.
A
household will oftenembue
its presentneighborhood
withan
emotionalaura
because
of the presence there of close friendsand
relatives,and because
ofpast associations. This is lost toa greater or lesser extent
when
moving, depending
on which
move
is Involved. Forsome
families thiscan
bean
extremelyheavy
cost.It is not so invariably,
however,
since families differ In their attitudes toward theirneighborhoods.
Some
actually dislike theirs,and
it Is in fact the desire tomove
outof the
neighborhood
that inducesthem
tochange
housing. Depslte differences, thedisruption of a particular style of life
anchored around
housing is a potentiallyimpor-tant
element
in considering the cost ofchanging
housing.The above
items all refer to achange
in consumption broughtabout
by
physicallychanging
the site of housing consumption. Butan
alternativeway
tochange
withoutmoving
is to convert a household's present residence to provide the desiredchanges
in housing services. This is generally not possible with rental occupancies,
and even
with ownership
occupancies
it is sometimes constrainedby
the existing lotand
perhapseven by
zoning.Moreover,
it applies only to structure servicesand
not to other aspectsphysical
moves
that avoides the substantial costs ofmoving.
7. Public constraints
Because
of the localization function,and
the close association of physical housingconditions
on
healthand
safety, housing hasbecome
the target of a variety of publicinterventions, including zoning regulations, health
and
building codes,and rent control,omong
others. These can exercise substantial constraintson
the actions of suppliersand demanders
in the market,and on
the v/orking of the market itself.The
housingmarket is not a perfectly free market.
II
Some
OperationalConsequences
1. Pre-existing Stock
and
New
Unih
Because of the durability of housing, units of
many
ages will offer servicesat
any
time. Indeed, inany
yearmore
than ninetypercent of all units offering services will
have
been
built before the beginning of theyear.
The
pre-existing stock is theparamount
component
in the supply of housing inany
period.Newly
produced
units are a small part of the story, although theymay
have
influence disproportionate to theirnumbers
insome
respects.The
pre-existing stock is not,however,
a passivecomponent
of the supply of2
housing. Because of convertibility,
owners
of existing units can modifythem
indirections suggested
by changing
market opportunities. Thus there aretwo
forms ofactive supply response to market forces: the building of
new
unitsand
the conversionof older units.
While
thetwo have
similarities, theyhave
differences as well.The
2
Similar to
one
of the several, sometimes confusing,meanings
of "filtering" insheer
magnitude
of the preexisting stockmakes
the conversion form of supply eminentlydeserving of understanding.
2. Information
The
great variety of housingpackages
and
the Importance of tastes regardingdiff-erences In those
packages
lends corresponding Importance to Information.Active
participants In the market must
be adequately
Informedabout
alternatives toavoid
large opportunity losses. Households looking for a unit can
expect
awide
variety tobe available. Substantial search is called for, often lengthy face-to-face search
because
of the multiple dimensions to beexamined.
Costly information Is important to sellers as well.
The
verysame
diversity in themarket
makes
it difficult for a prospective seller toknow
the most favorable terms hecan
realize for his particular unit.He
knows
that prospective buyers will differ Intastes
and
information,and
therefore inwhat
they are willing to offer for his unit.He
must therefore "search"among
prospective buyers for the most suitable. His "search"consists In deciding
on
asking prices, willingness to bargain, length of time to waitbefore bargaining, etc. This search,
no
less than buyer search. Involves cost—
oppor-tunity costs here, as against the combination of opportunity costand
active searchcosts for buyers.
3.
Market Segmentation
The great diversity of housing
packages
In the marketmeans
that they are notall perfect or
even
near-perfect substitutes forone
another.They
are likely to exhibita
whole
spectrum of substitutive relationships, from very close to nearly non-existent.markedly
In the degree of substifutabillty theyaccord
within a given set of units.The
presence of this spectrum of substitutabilitymeans
that "the" urban housingmarket is not
one
market but acomplex
of differentially related sub-markets.Each
sub-market
is a cluster of unitswidely
considered close substitutes,and
related toother clusters in terms of the differing degrees of substitutability with them.
This kind of segmentation of the market is abetted
by
the high cost ofadequate
information for participants. Participants cannot afford to scan all or
even
a majorpart of the market.
They
delimit their search to the most "relevant" portion of themarket, a
demarcation
probably based on various rules ofthumb
that reflect prevailingknowledge about
highly substitutive clusters.The
deliberate decision to avoid beingwell-informed outside modest portions of the
market
would
tend to consolidate boththe inequality of substitutability
and
theunevenness
in the distribution of degree.One
consequence
of this segmentation is that the different submarkets can experiencean Independent
margin of variation relative toone
another:one can have
high excessdemand,
another excess supply, another bewell-balanced.
So
housing pricesmay move
disparately
among
them.
Another consequence
Is that variation initiated withinone
subsectorwill
have
veryuneven
repercussionson
the other subsectors: in general, thecloser the relatedness the more nearly parallel the repercussion in direction
and
intensity.4. Transactional Friction
The
high costs of active participation in the market—
search, transactionsand
moving
costs—
imply that the prospective gains frombecoming an
active participant (i.e., seeking tomake
a change)may
have
to be quite substantial to warrant suchparticipation. Prospective gains arise from
changes
within the household that affectpackage, and between
housingand
non-4iousing commodities),and
fromchanges
Inthe character
and
prices of differentpackages
availableon
the market. Both of theseare likely to
occur
gradually most of the time, althoughsome
suddenchanges
In bothdo
happen.
This suggests that a household that has just
made
achange
Is not likely tomake
another
change
very soonbecause
of the high participation costs.So
activepartici-pation Is not likely to
be
continuous for most households but sporadic, with long periodsof non-partlclpation not
uncommon.
This In turn Implies that In such period thepartic-ular lineup of units available for sale or rental
and
households seeking units Is veryimportant In determining the
outcomes
of current transactions.Some
adaptiveself-selection
may
be Involved in determining this lineup, butchance
factors are likelyto
be
present as well. The upshot of this raises real questionsabout
the overall efficiencyof the market in pairing wants with availabilities, both at
any one
timeand
when
adaptive responses are
made
over timeby
bothdemanders and
suppliers (in the latterbecause
they are adapting to signals that are distorted).5.
Neighborhood
ExternalitiesThe
localization function of housingmakes
theneighborhood
component
of thehousing
package
Important. Since third party changes inneighborhood
can
affect thedesirability ofa housing unit Independently of the actions of its
owner
or tastechanges
by
its user, external effects In the housing market can be strongand
pervasive.The
usualmarket
distortions resulting from externalities are likely to result. In addition, less usualforms of resource immobility
can
result. Forexample,
adapatlon tonew
marketoppor-tunities might be optimal Ifa
group
undertook It, yetno one
member
ofthe group findsDecentralized behavior fails to secure such
an
expectationand
so results in inappropriateindividual—and hence
aggregate—
market performance.6. Vulnerability to Capital
Market
The
heavy dependence on
the national capital market gives the housing sector aspecial vulnerability.
Aggregate
forces can impingeon
the capital market for reasonsthat
have
little todo
with the worth-whileness of housing versus other production orconsumption
sectors; yet thesemay
have
a powerful positive or negative impacton
housing.Macroeconomic
conditionsmay
call forheavy
use ofmonetary
policy.This will
have
effects in housing for greater thanon
other productionand
consumptionsectors, yet unmotivated
by
any
realchange
in relative national prioritiesamong
thesectors.
Thus, vulnerability to the national capital market
makes
for rather special featuresin housing.
7.
Dynamics
Most
ofthe foregoing suggests that actual sequencesand
timings of actionsby
participants matter. Disequilibria maintained through frictions and,uninformedness
and
immobllities,
and
substantial adjustment lags,make
it important to understandongoing
processes,
even
more, perhaps, than hypothetical equilibrium destinations. Indeed,they
even
raise the questionwhether
equilibrium is ever tobe
observed underany
butvery special circumstances.
The
housing market's most distinctive featuresmay
wellbe dynamic, and
so efforts to understand it theoreticallymay
have
tohave
much
thecharacter of
dynamic
models.13
present paper includes the portion
devoted
to housingdemand.
Papers being currentlycompleted
deal with supplyand
market adjustment.The
overallmodel
explicitly attemptsto
embed
many
of the distinctive characteristics of urban housing,and
tocomprehend
many
of the asserted
consequences
describedabove.
On
the other hand, it isby
no
means
being
claimed
that thismodel
definitively handles most of the inherent complexitiesof the reality being
modeled.
Indeed, it is noteven
fullydeveloped
within itsown
presumed
scope. It is being presented as one of a series of theoretical studies dealingwith a
common
approach
to the understanding of urban housingphenomena.
Ill
A
Dynamic Model
ofUrban
HousingMarkets
A.
Individual Short-Run HousingDemand
1. Utility Function
We
begin withan
urban population containingN
households,some
similar but atleast
some
different.Each
household has a utility function,which
may
be similar toor different from, that of others.
(1) U'
=
U'(Z,H)
i=
1,...,N
where
Z
is the composite service flow ofall other commodities (the numeraire commodity);H
is theamount
or quality of housing services rendered in the present periodand
isin fact a vector of housing
package
components,each
itself a vector:(2)
H
=
(H3 , H|_,H^
,H^,
H^)where
H_
is the vector of housing structure characteristics likenumber
of rooms,interior size, architectural style, interior amenities
(number
of bcfh rooms, fireplace,14
H
is the vector of lot characteristics, like size, shape,placement
in block, topography, etc.;
Hj. is the vector of
neighborhood
characteristics, likedemography,
variety of private goodsavailable, character of public services rendered
sanitation, schools, parks, etc.), aesthetic character, land use mix, etc.
H*
is the vector of accessibility to the various desirable destinations withinand
outside of the urban area.
H-r is the tenure status; it reflects a style of
occupancy;
in addition, it proxies forcertain idiosyncratic features ofa unit, those reflecting the often
custom-made
character of ownership units as
opposed
to themere
usually standardized character ofrental units (appropriately designed for a larger
number
of occupants with a variety oftastes).The shape of
U
with respect to the housingcomponents
—
tradeoffsamong
the H.
and
H
and
Z
—
depends on socioeconom
ic characteristics of household i,F:
(3)
f'=C
Y.,N^j
,N21
,A.
, Ej, R. ,W.,...]
i =l,...,N
where
Y. is thepermanent income
level of household iN,.
is the household sizeN
2j is thenumber
of schoolage
childrenA.
is theage
of the householdhead
R. is a designation ofethnic group or race
Wj
is theamount
of liquid assetsowned
2. Housing
Package
ofComponents and
Component
PricesThe
five types ofcomponents
listed in equation (2) are not purchased separatelyby
the consumer, but as indissoluble parts of the overallpackage.
Relative tosome
starting15
a
different overallpackage
rather than simply transformingeach
while holding the3
rest
unchanged
in thesame package.
If the housing market is largeand
variedenough,
aconsumer
is faced with a very broad range of differentcomponent
combin-ations. In effect, he then can put together almost
whatever
combination hewants--as though he
were
in fact buyingeach component
separately.This suggests that the several housing
components
could be treated like separatecommodities,
and
the consumer'sbudget
allocation could then be treated asan
allo-cation over the numeraire
commodity
Z
and
the various housingcomponents
ratherthan over sonciething called housing.
Which
ismore
appropriate?The problem
issalient
because
while thecomponents
are parts of packages,we
have
not spoken yetabout
amagnitude
of housing packages. Housing is a vector. Strictly speaking,we
can
speakunambiguously
of size of thepackage
only ifallcomponents
increase ordecrease together, or less generally, if
some
components change
inone
direction(greater or lesser)
and none
in the other.But
we
considersome
pairs ofcomponents
tobe
substitutes,some
to becomplements and some
—
under
certaincircumstances—
to be sometimes one, sometimes the other.So
we
expect
to find consumers voluntarilyincreasing consumption of
one
component
while decreasing that of another,and
we
must be prepared to infer something
about
the "size" of differentpackages where
vectordominance
does not hold. This is quite difficult, but it turns out that bothapproaches
involve important difficulties, yet the
consumer
process can beviewed
In bothways.
We
beginby
treatingconsumer
choice overcomponents
as separate commodities.A
class of exceptionswas
notedabove
with respect to structurecomponents.
Some
desired structure changes can be physicallyaccomplished by
conversion, thereby holding all non-structurecomponents
unchanged
in the originalpackage.
To develop
thisanalogy
we
need
to be able to designate separate prices for the separatecomponent
commodities. This hastwo
parts. First,we
must be able to quantifyeach
component
so as to be able to distinguish—at the veryleast—between more
or lessof the
component.
Second,
we
must be able to specify a unit of quantity, so thata price per unit
may
be specified; or, failing that,we
must be able to distinguishdifferent states of the
component
(a nominal scale) so that a different overall pricecan be ascribed to
each
state.The
first requirement is not trivial, since all thecomponents
are vectors, mostare vectors of vectors,
and whereas some
vector dimensions lend themselves easily tochoice-relevant quantification, others
do
not.We
startby
assuming that marketparticipants
can
agreeon
the direction ofmore
or less, ormore
or less preferred foreach
dimension.Even
this fails forsome
dimensionsand
sub-dimensions. Architecturalstyle, certain amenities, will
mean
quite different ordinal things to different people.With
such profoundly Judgmental dimensionswe
candevelop
at best nominal scales,abandoning
theconcept
ofan
intersubjective ordering scale.Since
each component
is a vector,we
must either treat the ultimate separatecommodities as the most proximate set of scalar dimensions, or attempt to collapse
each
constituent set of sub-dimensions to a scalar index representing
each
first levelcomponent
(i.e.
Hg/H
, H|sg,H»
, Hj). Since theproblem
of indexing these dimensions is formallyvery similar to that involved in the
concept
of the"amount
of housing"we
leave it tothe later discussion.
We
therefore treat anew
set ofcomponents
as the commoditiesentering the consumer's utility function,
namely,
the scalar constituents of the elementsof(2):
What
are the corresponding prices of the"commodities"?
Since they are not in factbought
separately,
no
separate prices arequoted
forthem.
Buteach package
ofcomponents
has a quoted price,
and
these prices are in effectan
aggregate of the implicit pricesof the several constituents of the
package.
Adopting
the assumption thatpackage
pricesare linear aggregates, the relation
between package and component
prices for a given4
package
is :M
(5)
Pu-
2:
P.H
"
k=l
^ ^wfiere P is the market price of the housing
package^
H
Pi is the implicit market price of
component
Hi;
H
is theamount
ofcomponent
kembodied
in thepackage.
Since the
consumer
in the market is faced with a large set of obtainable packages, hereceives the information reflected in the matrix equation:
(6) P
=
H
PH
where
Pj, is theQxl
column
vector of marketpackage
pricesH
is theQxM
matrix:"ll
^12
•••^IM
H
21H
2M
,HQM
so that
row
i is the constituentcharacteristics of
package
iP Is the
Mxl
column
vector of implicit constituent prices.This is a set ofsimultaneous equations that, assuming fulfillment of the formal requirements
foro solution is given by:
(7)
h"^
P=
PM
Since our commodities are annual service flows, prices here refer to rental costs,
whether
18
Thus, market observations of
package
prices containenough
information in principleto enable
one
to discover the implicit prices associated with thepackage
components—
ifall sellers are implicitly quoting the
same
price foreach
component.
We
assume
that this last condition ismet by
the competitive forces operating in the market. If inan
early iteration of asking pricesone
seller is implicitly asking a higher price fora given
component
than are other sellersan
early impressionistic set of calculationsby
buyers will disclose the overpricingand
the seller will fail to sell his unit unlesshe
adjusts the outlyingcomponent
price. Thus, market clearing processesand
cal-culation of
component
pricesoccur
effectively simultaneously.We
therefore define a set of implicitcomponent
prices ateach
time t.With
theseprices
we
specify thebudget
constraint facing household i:(8) Y'
=
^
P,^H^
+ P^Z
(withP^=l)
3. Individual
Consumer
EquilibriumThe
consumer's equilibrium at time t is thus givenby
the solution of the followingmaximization
problem:(9)
max
U'=
U'IT,
H^.H^,
...,H^|F't]
M
subject to Y'
=
7"
P. H,+
Z
where
F' is household i'ssocio-economic
state at tand
P., ••»Pii are givenin the market.
This solution, with familiar optimal conditions, shows the optimal combination
of housing
components
and
numerairecommodity
that should bejpurchased tomaximize
utility,The
solution in these terms lackssome
intgtitive thrust: it does not tell"how
much"
housing is being purchased. Is it possible to formulate the consumer's choice process
theory.
We
treat "housing" as the output that results fromcombining
anumber
ofproductive
inputs—the
housingpackage components.
The
inputs are related to theoutputs
by
way
of a housing production function. This function is in reality away
of speaking
about
the household's utility function, since there isno
objective entityrecognizable as "housing" but only a subjective valuation of the overall attractiveness
of the
complex
of services that is conventionally labeled "housing," Therefore thereare as
many
different housing production functions as there are different householdutility functions.
By treating housing as the output of a housing production process
we
can distinguishtwo
aspects or stages of the household's choice problem: (1) the optimal combinationof inputs to
produce
any
level of housing output, (2) the optimal level of housingout-put. Housing level is defined as a
given
level of utility, for specified consumptionof
Z
and
various combinations of housing inputs.We
con define housing isoquants as utility level contours,and
the family of Isoquantsmap
out preference tradeoffsbetween
inputsand between
inputsand
outputs.The
household's choiceproblem
canbe
reformulated as follows. Define a setof housing Isoquants as:
(10)
h^
=5g
H'l-^w
=
U'[z,
( gH') If']4
allJlw
In the range ofI
^)
uiThus,
each
Isoquant Is the setgH
of housingcomponents
that, for given F'and Z,
give household I utility level
Jcw.
The
household'smaximization problem
isnow
given as:
(11)
max
U'=
W'
[Z,
h' I p'^ ]subject to
Y\
=
p (h') h'+
Z
where
h Is the level of housing from the point ofview
of household I»* • *
p(h')h' Is the least cost function for
consuming
each
h' level of housing, analogousto the firm's total cost function.
In
maximizing
utility, the household selects thesame
budgetary allocation asunder
equation (9). Buttwo
conditions for optimalitycan be
expressed in terms ofthe concepts of equation (11):
(12a)
h:/Pj=h^P,^=...='^hVa[p(h')h']
orhlA[, =
Pj/P,^ a\\\M
where
h!=^h'/^
H.and
hj=ihV6H.
r I K
K
02b)
C6u'Ah'
]/
[bu'/bZ
]=
Cdlp(h')h' 3 ]/b
h'(12a) refers to equimarginal net
advantage
as determining the optimal composition ofthe housing
package and
simultaneously the marginal cost of housing; (12b) refers tothe optimal level of housing as
determined by
the marginal utilityand
marginal costof housing
.
In operationalizing this theory for empirical estimation, the second form of (12a)
will be important, since this will contribute toward the actual
measurement
of housinglevels.
4.
The Action
Situation of theHousehold
At
time t, household i, givenY'and
Pw,
Po*/..., P.„, hasdetermined
an
optimalpattern ofexpenditure
on
housingand
all other commodities. Call these optimalmag-A
A
A
A
A;A
A
21
however,
household I is alreadyconsuming
a housingpackage
H|.,and
with it, Z^.So
we
can specify a degree of discrepancybetween
idealand
actualbudget
pattern:A
A
H
-H
,Z
-Z
. This carries with it a discrepancybetween
attainable (in principle)utility level
and
actual utility level.We
can refer to these as losses,and
measurethem
inconsumer
surplus terms.We
can
distinguishtwo
dimensions of loss, corresponding to our formulation ofthe consumer's
maximizing
activity as relating to optimal composition of housingand
optimal level. L, is the loss resulting from
an
improper internal composition;Lo
the loss from improper allocation
between
housingand
all bther commodities.The
totalloss is L]
+
L2.a. Loss from Improper
Composition
This is
measured
from the baseline establishedby
condition (12a):we
presume
that (12a) is violated,
and
measure the extent of the violation:(^3)
^
A
h
=
^/2f
(H.-H.p
(hlnylp(hi)h']Ah'lh'=h']-Pj)
This is a version of the welfare triangle surplus measure: 1/2
^?^^
Q
—
i.e., the multiple ofwrong
price discrepancyand
wrong
quantity discrepancy. In (13),A
Hj - H, measures the extent of
wrong
usage ofcomponent
j; the next term measures theextent of
wrong
price discrepancy, in that fulfillment of (12a)would
make
the valueof marginal product of
component
[^measured at the optimal level h', equal to theprice of j.
The
marginal physical product is h , the value ofeach
extra unit of housingi
^* ^
• As. 'Iequal the relative marginal utility of
an
extra unit of housing. In effect, (13) measuressomething like the extra cost of
consuming
housingwhen
the internal composition isnot the least cost composition as defined
by
an
optimal choice.b. Loss from Improper Level
This implies violation of (12b).
We
measure
the cost of violation as asum
ofwelfare triangles under
demand
(or valuation) curves. See Figure 1.Figure ]
\i
m m
V(h') a valuation function: p
=
f(h').The shaded
triangle is the loss from operatingi A. A
at (hj, P|) rather than at (h', p;. It is given by:
(14)
4
=
l/2[|hj45
|.|V(hJ^)-V^J)|]
+1/2
[|z^-z|.|^(Zj)-
V))3
The
terms referring toZ
indicate that the discrepancy inconsumption
of housing hascreated a corresponding discrepancy in the consumption of
Z,
so the losses in bothc. Total Loss
Both L^
and
L_ refer to flow losses ineach
time period. It isassumed
that theywill be repeated in
each
time period that conditions remainunchanged.
Since theselosses are to be
compared
with prospective costs of entering the market to decrease orwipe
out these losses,and
such costshave
a different time shape,we
must convert theirexpected sequence
Into present discounted value terms: Li*and
L * respectively.Then
total costs are given by:(15) L*
=L^*
+
L2*.When
faced with a welfare loss of L* the household has the option ofbecoming an
active participant in the housing market with the purpose of
moving
to amore
appropriateA
housing unit
—
one
that conformsmore
nearly toH.
There is,however,
a costasso-ciated with active participation. This cost has three components: a cost of search, a
cost of conducting a market transaction
and
a cost ofmoving
to thenewly
chosen unit.Since the last
two
are not a function of theamount
of search carried on, they can becollapsed together as
an "overhead"
type of cost of concluding a transactionand
changing
residence accordingly. Its variabilitydepends on
the size of the transaction,the cost of conducting capital transactions,
and
theamount
of property that must bemoved.
Thus,
we
define a transactionand moving
cost:06)Ct=Ct(K',
r^)where
the cost of conducting capital transactionand
the size of the transaction, aswell as the
amount
of property to bemoved
aremade
functions of the wealth positionof household i, K' ,
and
either a market rate of Interest or a borrowing minus lendingThe
other cost of active participation is the cost of search. Search costdepends
on
theamount
of search conducted,and
is a rising function of thatamount.
The
Asearch decision
can
be looked at in the followingway.
H
representsan
ideal housingconfiguration for household i. But it has not
been
found concretely in the market;it is only a goal of the search process.
Any
amount
ofsearch will scan a variety ofdifferent available units, not randomly, but in
accordance
with a selection procedurethat has only inadequate information:
hence
theneed
for closer,often face-to-faceconfrontation.
The
greater thenumber
of unitsexamined
the closer to the ideal islikely to be the best unit in the sample
examined.
Thus,we
can
measure theamount
of search in terms of the
expected degree
of potentialimprovement
discovered—
i.e.,the
expected
decrease in L*by
moving
to the most appropriate unit discovered.So
the search (or information) cost function also expressed in present discounted value, is
as follows:
(17)
c*
=
c*(^alvl;)
where
^
L* is theexpected
decrease in disequilibrium cost from L* due tomore
suitable housing being discovered by market search*L* is the original disequilibrium cost.
Then
total market participation cost is:(18)
C*=
C*(K',
r^+C*
(^ALVL*)
h
Cya
K'<
0,^C*^
^r*
>
0,dCJ/dDL>0,
dC*/dDL^
>0
where
DL=iiL*A*.
Household
I will enter themarket
asan
active participant if theexpected
decrease in L*exceeds
theexpected
cost of that participation.The
conditions foran
optimaldecision are:
(19a) -yf(DL)
=
(DDL*-
C*(DL,
K\r*)y
forsome
DL
(19b)
dTT/dDL
=
0,d^TT/dDL^<0
within the rangewherelp-Q
Equation (19a) establishes the net gain of at least
one
possible degree of activeparti-cipation. (19b) determines the optimal
amount
of participation, in terms of the extentofsearch, as
where expected
marginal gain from additional search is just offsetby
the marginal cost of that extra search.
Figjre 2 shows the
dependence
of the optimal actionon
K
, r* ,and
Lf :Figure 2
V*=DL-L*3
.*k^=
DL- L* ^ i^AL7tDL
1 ^a. For a very low initial disequilibrium cost L* . no
expected
amount
of search Is*
1
worthwhile
—
all the cost curvesexceed^
L*,everywhere
» For higher L""I * *
there are ranges of
expected
search (DL)where
6L.
"7C (exceptwhere
C*
Is veryhigh—
e.g.C^*
). In effect,b ^C%^L*^L*^^0,
i.e.high L* lowers marginal search cost of unit
improvement.
b. For high K' or low r*, C't is generally
low
enough
tomake
some expected
searchlevels profitable (e.g.
C*^,
C*2);where
the K', r* combination is such thatC
j=Cj2/
expected
costsexceed expected
gainseven
where
there is a very high initial disequilibrium.c. Since market conditions gradually diverge
more
and more
over time fromwhat
theywere
atsome
Initial point(when
household Imoved
to its current residence),and
household i's
own
demographic
circumstances are likely to diverge cumulatively overtime from that
same
initial period, the size of the housing disequilibrium is likely togrow
over time—
L*>0
generally.Asa
result,an
initial disequilibrium at t,L^
,
insufficient to warrant active market participation, will gradually Increase over time
until
some
market transaction is prospectively profitable.Assuming
household ibecomes an
active participant in the market, its equilibriumtransaction will
occur where
(20)
^^A
L*/&DL
=.^CVaOL
I.e., the marginal
expected
reduction in loss equals the marginal cost of bringingabout
that
expected
reduction. This equilibrium generally occurs aton expected
reductionless than the original loss (
AL/L
<
1).Moreover,
the larger is L* the largerB.
Aggregate
Short-RunDemand
1 ,
Aggregation
in Terms of Housing PackagesAt
any
given timeeach
household formulatesan
optimal housingpackage,
relatesthis to its actual housing,
and
decideswhether
the discrepancybetween
thetwo
isgreat
enough
to warrant it going into the market. If it is, the household will bepre-pared to carry out a degree of search necessary to decrease its loss from the disequilibrium
by
a designatedexpected
amount
through approximating its ideal target. Since theout-A
come
ofan
actual course of search will diverge fromH
in only arandom
fashion, theA
expected
configuration of search termination can be given as equal toH.
The
varianceA
of the actual bestH
found will be a positive function of L* - L*.Thus^
when
aggregating thedemand
of thewhole
city population,each
householdcan
be treated asdemanding
housing as follows:a) If inactive in the market, household i
demands one
unit identical with itspresent
occupancy
—
i.e., H^b) Ifactive in the market, household i
demands one
unit equal to its calculatedA
optimal
package
—
i.e.,H,
So
we
obtain aggregatedemand
simplyby enumerating
thenumber
of units ofeach
type
demanded
by
the population as awhole.
Symbolically:(21)
D^
=
D
C
(F' ), (H' ), Pi-..,P^D
allH
where
Dj, is thenumber
ofunits of typeH
demanded,'(F' ) is the set ofdemographic
profiles F' of the city population* P], .
.,Pm
ore the housingcomponent
prices.At
any
time t, all three sets of variables will be available, so thedemand
foreach
kind of dwelling can be determined.
2. Housing
Quality
awkward
to attempt to understand market forces in terms of multi-dimensional packages.We
seeksome
way
to increase the comparability of different housingpackages
so asto be able to understand the
working
of the market at amore
general—
and
powerful—
level.
We
argued
in section II of this essay that heterogeneitymakes
for a segmentationof the
market
into virtual submarkets, with various degrees of substitutabilityamong
submarkets.
The
submarket, then, is the proper level atwhich
to aggregatedemand
and
supply forces,and
atwhich
adjustmentsand
repercussions should beexamined.
The problem
ishow
to aggregate the individual units into these submarkets, so that:1) the groupings will conform to market aggregates that really
do
reflect a functionalhomogeneity
In market operations; 2) the relationshipsbetween and
among
groupingscan
becompactly
and
efficiently understoodand
expressed.The
clusterings of close substitutes,and
the patterns of graduations of substitutabilityamong
unitsand
clusters cannot confidently be asserted to be perfect in the sense ofmetric scalability (so that
everybody
agreeson
the degree of substitutabilitybetween
any two
units,and
degree of substitution forms a perfectly transitive ordering). Butthis
problem
is not unique with housing.Most commodities show
aberrationswhen
closely studied.
Even
so conventionaland
seemingly assured classifications as"In-5
dustrles" exhibit serious imperfections
when
actual market substitution Is Involved.Nonetheless
we
make
use of classifications likecommodity and
industry for mostanalytic
and
statisticalwork
ineconomics.
We
shall assume that In terms ofactualAnti-trust court proceedings on the
aluminum
Industry isone
of severalwhich
showed
significant slippage In conventional classifications of Industrieswhen
real market behaviorwas
studied.market processes the units of the standing housing stock
can
be tolerably arranged interms of degree ofsubstitutability (referring to disagreements
about
the position ofany
unit in the array)
and
moreover, that this array is approximately linear (i.e., thatdegree ofsubstitutabilityis a transitive relationship).
Insofar as the real
world
obeys these assumptions,and
thatwe
canmake
actualassignments to this array,
we
possess a strong analytic instrument for understandingthe
complex
of market activities. Becausewe
shallknow
thewhole
structure ofmarket
repercussions resulting from a
change
originating inany
part of the system.Each
change
will radiate from the point of origin typical substitutive types of adjustment
which
weaken
monotonically withsubstitutional distance from that origin.Given
the salientmultidimensionality of housing, this enables
me
to avoid the highly misleadingtreat-ment
of housing as ahomogeneous commodity which
varies onlyby
quantity, whileavoiding the opposite pitfall of having to deal with a market of thousands of different
kinds of commodities
whose
relations withone
another ore idiosyncraticand
requirepair-by-pair elucidation.
Thus,
we
assumean
orderly abstract space of substitutional distance (degree ofsubstitutability) along a single organizing dimension. This organizing dimension
we
label "housing quality."
A
number
of differentunits, units differing fromone
anotherfn various dimensions of the housing
package,
will typically fall ateach
level ofhousing quality. Thus, in effect the multi-dimensionality of the housing
package
isbeing collapsed for this purpose to a single dimension.
How
is thismapping
from M-space to a single dimensionaccomplished?
The
utility function ofeach
householdmakes
utility level a positive function ofthe housing
components.
So
foreach
household,utility differencesmark
degrees ofthere is
no
consensusamong
the population with respect to a direct array of unitsalong the quality scale.
An
alternativewould
be forsome
external source—
e.g.a
group of experts of housing—
to judge the relative quality of different houses. This leavesmuch
to bedesired, since "quality" is not
an
"objective" characteristic observable by inspection.It is
on
inferred property,stemming
from actual market transactions. It is themarket
that must indicate the appropriate
mapping.
The
market can perform the task,even
in the face of considerable differences intastes, in the following
way.
At each
equilibrium position of the market, everyconsumer
has adjusted his budgetary allocation so that the marginal rate of substitution
between
any
pair ofcommodities
equals the price ratiobetween them.
In highly competitivemarkets the prices facing all consumers are equal. So at equilibrium all consumers
have
thesame
marginal rote of substitutionbetween
the pair. This equality holds forsellers (producers) also. Their marginal rate of transformation (or ratio of marginal
costs) is
equated
to thecommon
price ratio—
and
thus, to+hecommon
marginal rateof substitution. Thus, a
common
tradeoffexists,and
ismeasured
by the observable priceratio, of consumers. This is
an
important form of consensus.In the case of housing
what
we
require in order toapply
thisapproach
is that theseveral housing
components
be subject to substitutionby
households. Householdsdo
in fact
show
a willingness to substituteone
component
for another. There are links,because
withinsome
rangescomponents
arecomplementary
withone
another, ratherthan substitutive (few families
would want
a unit with twentybedrooms and
only 1bathroom). But the ranges
where components
are treated as substitutes (land for interiorinterior space for condition, etc.) are substantia!.
So
preference tradeoffsamong
the
components
canbe
establishedand
we
canmake
use of themarket
equilibrium consensus or marginal tradeoffs to help constructan
index of housing quality.Assume
that all households,despite differences in taste,have
thesame
functionalform for the effect of housing on utility.
Only
theparameter
values differ.We
can
construct a representative functionwhich
relates housingcomponents
to "representativegratification" levels by using the
common
ratios of marginal utility discoverable atmarket equilibria. Expressing
each
marginal rate of substitution as thatbetween
ahousing
component and
the numerairecommodity
Z
,we
obtain:(22)
U^.
=
P^. all iwhere
U^-js the partial derivative of the "representative gratification" function;Hi '
Pjj. is the implicit price of housing
component
iSince
we
define the consensusas thatwhich
exists in the particular equilibrium,each
Ujj. is a constant, not a variable. Thus, the "representative gratification" function
is linear.
We
define itas follows:M
(23)
G
=
X
°- ^^•1=1 ' '
This is, of course, a form of
hedonic
index.From
(23),we
solve for a. ,a^,..,,aj^:(24) a.
=
P.We
can use this function asan
index of housing quality.Each
unit willhave
aG
value corresponding to its composition of
components.
Then
the standing stock canbe
arrayedby
G
value as a distribution of quality levels,and
we
thereby approximatethe degree of substitutability distribution in the market.