working paper
department
of
economics
THE ECONOMICS
OFROTATING SAVINGS
AND CREDIT ASSOCIATIONS
Timothy Besley
Stephen Coate
Glenn Loury
No. 556May
1990
jT[5RH^iliM4Ta
institute
of
technology
50 memorial
drive
Cambridge, mass.
02139
THE
ECONOMICS OF
ROTATING SAVINGS
AND CREDIT ASSOCIATIONS
Timothy Besley
''Stephen Coate
Glenn Loury
The
Economics
of Rotating Savings and Credit Associations* byTimothy
Besley Princcion UniversityStephen Coate
Harvard
Universityand
Gltnn Loury
Harvard
UniversityMay
1990 AbstractThis
paper
examines
the roleand
performance
ofan
institution for allocating savingswhich
isobserved
world wide
- rotating savingsand
credit associations.We
develop
ageneral
equilibrium
model
ofan
economy
with an
indivisibledurable
consumption
good and
compare
and
contrast theseinformal
institutionswith
creditmarkets
and
autarkic saving in terms of the properties of their allocationsand
theexpected utility
which
they obtain.We
also characterize Pareto efficientand
expectedutility
maximizing
allocations for oureconomy, which
serve as usefulbenchmarks
for the analysis.Among
our results is the striking finding that rotating savingsand
creditassociations
which
allocatefunds
randomly
may
sometimes
yield a higher level of expected utility to prospective participants thanwould
a perfect credit market.*The
authors are grateful toAnne
Case,Richard
Zeckhauser,and
seminar participants atBoston, Cornell,
Harvard, Oxford,
Princetonand Queen's
Universities for their helpfulcomments. Financial support
from
the Center forEnergy
Policy studies, M.I.T.,and
theJapanese Corporate Associates
Program
at theKennedy
School ofGovernment,
Harvard
University is gratefully acknowledged.Digitized
by
the
Internet
Archive
in
2011
with
funding
from
Boston
Library
Consortium
IVIember
Libraries
I. Introduction
This paper
examines
the economic; roleand performance
of Rotating Savingsand
Credit Associations (Roscas).
These
informal credit institutions arefound
all over the world, particularly in developing countriesi.While
their prevalence and, tosome
degree,robustness has fascinated anthropologists^, there has
been
very littlework
devoted
tounderstanding their
economic
performance.Here
we
shall subjectthem
to scrutiny in ageneral equilibrium setting, contrasting
and
comparing
them
with other financial institutions.Roscas
come
intwo
main
forms.The
first type allocates funds using a process ofrandom
allocation. In aRandom
Rosea,members commit
to putting a fixedsum
ofmoney
into a "pot" for each period of the life of the Rosea. Lots aredrawn
and
the pot israndomly
allocated to one of themembers.
In the next period, the process repeats itself,except that the previous winner is excluded
from
thedraw
for the pot.The
process continues, with every past winner excluded, until eachmember
of theRosea
has received the pot once.At
this point, theRosea
is either disbanded or begins over again.Roscas
may
also allocate the pot using a bidding procedure.We
shall refer to thisinstitution as a Bidding Rosea.
The
individualwho
receives the pot in the present perioddoes so
by
bidding themost
in theform
of a pledge of higher future contributions to theRosea
orone time
sidepayments
to otherRosea members.
Under
a Bidding Rosea,individuals
may
still only receive the pot once—
the bidding process merely establishespriority^.
We
take the view, supported in the literature, that these institutions are primarily ameans
of savingup
tobuy
indivisible goods, such as bicycles, or to financemajor
events,such as weddings.
Random
Roscas are not particularly effective as institutions forbuffering against risk since the probability of obtaining the pot need not
be
related to one'sthe pot immediately, only permit individuals to deal with situations
which
cannot recur,since the pot
may
be obtainedno
more
t^^an once. Furthermore, sincemany
kinds of risksin
LDC's
are covariant,many
individuals willhave
high valuations at thesame
instant.Roscas do play a greater role in transferring resources to
meet
life—cycle needs such asfinancing a wedding.
However,
even in this context, theyseem
more
appropriate fordeal-ing with significant, idiosyncratic events, rather
than
thehump
saving required for old age.Despite its manifest importance, there has
been
relatively little analysis in the savings literatureon
the notion ofsavingup
tobuy
an
indivisible good. Yet, the existenceof indivisible goods does,
by
itself, provide anargument
for developing institutionswhich
intermediate funds. In a closed
economy
without access to external funds, individualshave
to save
from
currentincome
to financelumpy
expenditures,and
can gainfrom
trading with one another.The
savings ofsome
agents canbe
used to finance the purchases of others.By
contrast,when
all goods are divisible individuals canaccumulate
them
graduallyand
there need be
no
gainsfrom
intertemporal trade. Clearly autarky with indivisible goods isinefficient. Savings lie idle during the accumulation process
when
they could beemployed
to permit
some
individuals to enjoy the services ofthe indivisible durable good.As
we
explain inmore
detail in the next section, Roscas provide onemethod
ofmaking
these joint savings work*.Given
theworldwide
prevalence of these institutions, itis of interest to understand precisely
how
theydo
mobilize savings and, in particular, thedifferences
between
Random
and Bidding
Roscas. It is also interesting to considerhow
theallocations resulting
from
Roscas differfrom
thatwhich
would emerge
with a fullyfunc-tioning credit market.
Such markets
are the economist's natural solution to theproblem
ofintermediation.
Yet
there aremany
settings inwhich
they are not observed.We
hope,through our
comparative
analysis, to suggestsome
reasonswhy
this is so.Our
study, in the context of a simpletwo—
good model
with indivisibilities,charact-erizes the
path
ofconsumption
and
of the accumulation of indivisible goods through timepro-perties of Roscas
and
of a competitive credit market,employing
the criteria ofex—
postPareto efficiency,
and
of ei-aiie expected utility. Neither type of Rosea, asmodeled
here, is ex-post efficient.We
find that, withhomogeneous
agents, randomization is preferred tobidding as a
method
of allocating funds within Roscas. Moreover, while Bidding Roscasare always
dominated by
a perfect credit market,we
show
that, in terms of ex-ante expected utility, this is not generally the case forRandom
Roscas.We
use amodel which
makes
no
claim to generality, butwhich
we
believe capturesthe essential features of the
problem
described above.We
keep themodel
simple in theinterests of
making
the insights obtainedfrom
it as sharp as possible.Some
importantissues concerning Roscas are not dealt
with
here. In particular,we
do
not address the, question of
why,
oncean
individual has received the "pot", hemay
bepresumed
to keep hiscommitment
topay
into the Rosea.The
anthropological literature reveals that the incentive to defectfrom
aRosea
:s curbed quite effectivelyby
social constraintss. Roscasare typically
formed
among
individualswhose
circumstancesand
characteristics are wellknown
to each other. Defaulters are sanctioned socially as well as not being permitted totake part in
any
future Rosea. Thus, in contrast to theanonymity
of markets, Roscas appear tobe
institutions of financial resource allocationwhich
rely inan
essentialway
upon
the "social connectedness"' of thoseamong
whom
they operate. Itwould
beinterest-ing to try to formalize this idea in a
model
allowing default. In futurework we
willconsider the sustainability ofRoscas in greater detail.
The
remainder
of the paper is organized as follows. Section II providesan
informal overview of theeconomic
role of Roscasand
outlines themain
questions to be addressed in this paper. Section III describes 'themodel which
provides theframework
for our analysis.Section
IV
then formalizes the workings of Roscasand
a creditmarket
in the context ofthis model. Section
V
develops properties of efficientand
optimal allocations of credit.The
role of SectionVI
is tocompare
the different institutions. W^e describehow
thealso establish the
normative
properties ofthe allocations. Section VII concludes.n.
The
EconoinicRole
ofRoscas:An
InformalOverview
Consider a
community
populatedby
20 individuals, each ofwhom
would
like toown
some
durable, indivisibleconsumption
good
—
say, a bicycle.Suppose
that bicycles cost$100. If individuals, left to their
own
devices,were
to save $5 permonth,
then eachindividual
would
acquire a bicycle after 20months.
That
is,under
autarkyand
given the savings behavior assumed,nobody would have
thegood
before 20months,
atwhich
time evciyonewould
obtain M. Obviously this is inefficient since, wiihany
preference for the early receipt of their bicycles, itwould
be possible,by
using theaccumulating
savings tobuy
one bicycle permonth,
tomake
ever;yone except the final recipient strictly better off.i
Roscas represent one response to this inefficiency. Let our
community
form
aRandom
Rosea which meets
once amonth
for 20months,
with contributions set at $5 permonth.
Every
month
one individual in theRosea would
berandomly
selected to receive the pot of $100,which
would
allowhim
tobuy
a bicycle. It is clear that this results in aPareto superior allocation.
Each
individual saves $5 permonth
asunder
autarky butnow
gets a bicycleon
average 10months
sooner.Note
that risk aversion is notan
issue since,viewed ex ante, the
Random
Rosea
does as well as autarky in every state of the world,and
strictly better in all but one.
Forming
aRandom
Rosea which
lasts for 20months
is only one possibility.The
community
could, for example,have
aRosea which
lasts for 10months
withmembers
making
the $5 contribution,and
someone
getting a bicycle every half—month.
Or, theRosea
could last for 30months
withmembers
contributing $5,and
a bicycle being acquired, everymonth
and
a half.Given
theuniform
spacing ofmeeting
datesand
the constant contribution rate,common
features of Roscas as they areobserved in practice, the length of theRosea
will be inversely proportional to the rate at w^hich thecommunity
saveswith a length
which
maximizes
the(ex—
ante expected) utility of its "representative"member.
Supposing that this is the case, one can then characterize the optimal length ofthe
Rosea and
the savings ratewhich
it implies.If instead of determining the ord< r of receipt
randomly
individualswere
to bid forthe right to receive the pot early, the
community
would be
operating aBidding
Rosea.Suppose
thatRosea members,
at their initial meeting, bid for receipt datesby
promising tomake
future contributions at various (cor^stant) rates, with a higher bid entitling one to anearlier receipt date. In our 20
month
example, the individual receiving the pot firstmight
agree to contribute (say) S6 per
meeting
for the life ofthe Rosea, the individualwho
is tobe last
might
onlypay
$4 per meeting,and
othersmight pay something
in between.What-ever the exact
numbers,
two
things should be true in equilibrium: overall contributionsshould total $100 per meeting; and, individuals should not prefer the contribution/receipt
datepair of
any
otherRosea
member
to theirown.
While both
Biddingand
Random
iloscasaUow
thecommunity
to utilize its savingsmore
effectively than under autarky, they clearly will not result in identical allocations.Even
when
theyhave
thesame
length, theconsumption
paths ofmembers
differ since,under
aBidding
Rosea, thosewho
get thegood
earlymust
foregoconsumption
todo
so.Moreover, there is
no
reason to suppose that the optimal length will be thesame
for bothinstitutions.
Of
particular interest are the welfare comparisonsbetween
the institutions.Understanding
what
determines their relativeperformance
gives insights into thecircum-stances
under
which
we
might
expect to saeone
or the other type ofRosea
in practice.Of
course, Roscas are not the only institutions able to mobilize savings.Imagine
the introduction of (competitive)
banking
into thecommunity.
Bankers
would
attractsavings
by
offering topay
interest, while requiring that interest be paidon
loansmade
tofinance the early acquisition of the bicycle.
These
borrowerswould
enjoy the durable'sservices for a longer period of time at the cost of the interest
payments on
theborrowed
with
which
to finance greaternon—
durablegood consumption
over the course of their lives.Under
ideal competitive conditiors a perfect creditmarket
would
establish a patternof interest rates such that,
when
indivifiuals optimally formulate their intertemporalcon-sumption
plans, the supply of savings -vould equal thedemand
for loans in each period.Such
amarket
is evidently amuch
move
complex
mechanism
than
theRoscas
describedabove. It is natural, however, to
wonder
how
the intertemporal allocation of durableand
non—
durableconsumption which
it procucescompares
with those attainedunder
Roscas.Of
particular interest are the welfare comparisons:how
do
these simple institutionsperform relative to a fully functioning cr<^dit
market?
m.
The
Model
We
consideran
economy
populatedby
acontinuum
of identical individuals.Time
is continuous
and
consumers have
finite lives oflength T.These assumptions
allow us totreat the
time
individuals decideto purchase the indivisible good, the length ofRoscas,and
the fraction of the population
who
have
the indivisiblegood
atany
moment,
as continuousvariables. This avoids
some
technicallyawkward,
but inessential, integer problems.Each
individual in theeconomy
receivesan
exogenously given flow ofincome
overhis lifetime at a constant rate of
y>0.
At any
moment,
individuals can receive utilityfrom
the flow of aconsumption
good, denotedby
c,and
the services of a durable good.We
imagine
that the durablegood
is indivisibleand
can bepurchased
at a cost of B.^Once
purchased,we
assume
that it does not depreciate, yielding a constant flow of services for theremainder
ofan
individual's lifetime.We
assume
that the durablegood
isproduced
for a large (global)
market whose
price is unaffectedby
the actions of agents in oureconomy.
Moreover,we
assume
that ti-e services of the durable are not fungible across agents—one
must
own
it to enjoy its services.''We
assume, for analytical convenience, that preferences are intertemporallyeach instant
an
individual receives a flow of utility at the rate v(c) ifhe does nothave
thedurable good,
and
v(c) 4- i^ if he does. Moreover,we
assume
that individualsdo
notdis-count the future.
Thus
there isno motive
for saving except to acquire theindivisible good.These
assumptionsmake
our analysis;much
lesscumbersome
without substantiallyaffecting the
main
insights ofthe paper.We
make
the followingassumption
concerning theutility function v(•):
Assumption
1: v: K -i IR is three times continuously differentiable,and
satisfies v'(-)>
0,
v"(-)
<
and
v"'(-)
>
0.As
well as being increasingand
strictly concave,we
require the utility function tohave
apositive third derivative. This assumption,
which
is not essential formuch
ofthe analysis,is satisfied
by most
plausible utility functions. It is sufficient for individuals tohave
ademand
for precautionary savings (see Leland (1968))and
also guarantees proper riskaversionin the sense of Pratt
and
Zeckbauser (1987).A
consumption
bundlefor an individual in thiseconomy
may
be
describedby
a pair<s,c(-)>,
where
s G^e
[0,T] U {u} denotes the date of receipt of the durable good,and
c:[0,T] -t K gives the rate of
consumption
ofthenon—
durable at each date.By
"s=
u"we
mean
the individual never receives the durable good.We
suppose, without loss ofgenerality, that the population is uniformly distributed over the unit interval
and
we
indexdifferent
consumers
withnumbers
a6[0,l].An
allocation is then a set ofconsumption
bundles, one for each
consumer
type,and
may
be representedby
a pair of functions <s(-),c(-,-)>, such that s:[0,l] -»^and
c:[0,l]x[0,T] - [RThe
function s(-), hereafter referred to as the assignment Junction, tells us the dates atwhich
different individuals receive the durable.By
relabelling individuals as required,we
may
assume
withno
loss of generality that individuals with lower indexnumbers
receive the durable earlier. Thus,that s(-) is non-decreasing
on
[0,q], <q
< 1,and
thats~
(u)=
(a,l]. Ifa
=
1, thens~
[u)=
and
everyone receives thegood
—
this being the caseon which most
of theanalysis will focus.
The
secondcomponent
ofan
allocation gives us theconsumption path
{c(a,r):re[0,T]} ofeach individual a.^
Under
the allocation <s,c>,an
individual oftypea
enjoys utility:(3.1)
u(a;<s,c>)
=v(c(a,r))dr
+
e(T-s(a)), s(a)6 [0,T]T
v(c(a,r))dr, s(a)=i/.
To
be feasible, the allocationmust
consume
no
more
resourcesthan
are available overany
timeinterval.
To make
this precise, forany
assignment function s(-)and any
date re[0,T],we
define N(r;s) to be the fraction of the populationwhich
has received the durableby
time rwith
assignment function s(-).9Then
an allocation<s,c>
is feasible if, for allr£[0,T]: (3.2) rT (y rl c(a,x)da)dx > N(r;s)B.
The
lefthand
side of (3.2) denotes aggregate saving at rand
the righthand
side denotesaggregate investment. Hence,
an
ailocation is feasible if aggregate investment neverexceeds aggregate saving. In the sequel
we
will letF
denote the set of feasible allocations.IV. AlternativeInstitutional
Arrangements
The
object of this paper is tocompare
different institutionalarrangements
forallocating savings
and
it is the task of this section to describe four possibleways
ofcredit market.
Each
institution thatwe
consider will result in a particular assignmentfunction
and
set ofconsumption
paths,and
thus a particular distribution ofutilityon
[0,1].We
can evaluate the institutional alternativesby
comparing
these utility distributions.IV.1
Autarky
The
simplest institution isAutarky
—
i.e.,no
financial intermediation at all. In thiscase saving occurs in isolation.
An
individual desiring to acquire the durable, given strictlyconcave utility
and no
discounting, will save the requiredsum
B
at the constant ratey—
c, oversome
period of accumulation [0,t]. Acquisition is desirable if the utilityfrom
such aprogram
exceeds T-v(y), i.e. the life—time
utility attainable without saving tobuy
the durable. Consider, therefore, an individual's life—time utilitymaximization problem
ofchoosing c
and
t to:Max{t.v(c)
+
(T-t).(v(y)+0}
(4.1)
subject to
t-(y-c)=B, 0<c<y and
0<t<T.Let (ca
j^
) denote the solution to thisproblem and
letW
. denote themaximal
value.To
developan
insightful expression forW
. , it is convenient to define(4.2)
Kfl
^o^c?y
r'^'y!i''''^l. i >-"
Assuming
that t .<
T
at theoptimum
in (4.1),lo then substitutionand
rearrangementleads to the following:
(4.3)
'
e]-B-KO-10
Expression (4.3) admits an appealing interpretation, anticipating a
more
general resultobtained in section V.
The
value of opiimal acquisition of the durableunder
Autarky
isthe difference
between
thetwo
termson
ihe righthand
side of(4.3).The
first equalswhat
lifetime utility
would
be if the durable v/ere a free good. Hence, the secondterm
may
beunderstood as the utility cost of acquiring the durable
imposed by
theneed
to save up.From
(4.2), it is easily seen that this cost is directly proportional to the price of thedurable
and
to the loss of utility per unittime
during the accumulation phase,and
isinversely proportional to the length of that phase.
The
optimal autarkicconsumption
(and hence savings) rate, c. , is chosen tominimize
this cost.It should be noted that the function /i(•) possesses all the usual properties of a cost
function
—
it isnon—
negative, increasingand
strictly concaveii. Moreover,from
theEnvelope
Theorem
the optimal acquisition date, t . ,may
be
obtainedby
differentiating thecost function, i.e. t .
=
B-n'{().These
facts will be used repeatedly below.It is apparent
from
(4.3) that a.^.quisition of the durable underAutarky
will bedesirable [i.e.
W
. >T-v(y)] ifand
only if B-/i(^) < ^-T.We
formalize this requirement inthe
assumption
that follows:Assumption
2: (Desirability) Let £(y,^)be
definedby
/i(£)/^=
J.
Then
(>i{y,'^)A^
Desirability requires that
some
combination
of the utility ofdurable services (if), the lengthof the lifetime over
which
these servicesmay
be enjoyed (T),and
theincome
flow (y) belarge
enough
relative to the cost of the durable (B) tomake
its acquisition worthwhile.Even
if durablegood
acquisition is not desirableunder
Autarky, itmay
be sounder
otherinstitutional arrangements.
We
shall return to this point below.If the durable
good
is desirable, then livesunder Autarky
will be characterizedby
two
phases. In the first of these individuals save at a constant rate until theyhave
12
representative
member
of thisRosea
perceives his receipt date for the pot to be arandom
variable, f, with a uniform distribution
on
the set {t/n, 2t/n,...,t}.Given Assumption
1,each
member
saves at the constant rate equaJ toB/t
over the life of the Rosea.Thus
eachmember's
lifetime utility is therandom
variable:(4.4) W(t,T) =
t.v(y-B/t:
+
(T-t).(v(y)+0
+
(t-r)^.The
continuum
casemay
be understood as the limit of this finite case as n approaches infinity,i^As
n grows, theRosea meets
more
and
more
frequently.^^ In thelimit it is
meeting
at each instant of time,and
the receipt date of the pot, r,becomes
acontinuous
random
variablewhich
is uniformly distributedon
the interval [0,t].Thus
theex ante expected utility of a representative individual
who
joins theRosea
of length t isgiven
by
the expectation of (4.4), taking r to be uniformon
[0,t]. Hence,(4.5)
W(t)
=
t.v(y-B/t)
+
(T-t).(v(y)+e)
+
t^/2.In the
above
discussionwe
took the length of theRandom
Rosea, t, as given. Itseems natural, however, to
assume
that it will be chosen tomaximize members'
ex-ante expected utility—
W(t).Thus
consider than theproblem
of choosing t tomaximize
(4.5).We
denote the solutionby
t-pand
useW^
to represent themaximum
expected utility.Notice that this
problem
is almost identical to that in (4.1) (after solving for cfrom
thebudget constraint in (4.1)).
We
may
therefore proceedby
analogy with (4.1)through
(4.3)and,
assuming
tT%<
T, writei^(4.6)
Wj^=T.[v(y)+{]-B-M{/2).
13
tR=B-/i'(^/2).
Expression (4.6) reveals the elegant simplicity of our
model
of aRandom
Rosea.Comparing
(4.6) with (4.3), it isimmediately
apparenthow
aRandom
Rosea
improvesupon
Autarky.The
finaneial intermedia ion providedby
theRosea
lowers the utility eost of aequiring the durable, i.e.W-p
—
W
.=
B-[/i(^)—
/i(^)]>
0, sinee aswe
noted above,(jl{-) is increasing.
So far,
we
have
taken the question of desirabilityunder
aRandom
Rosea
forgranted. This does, after all,
seem
reasonable.The
savings in aRosea
are being put towork
and
eaehmember
of aRandom
Rosea
expects to enjoy the services of the durablet-n/2 units of time
more
than hewould by
choosing thesame
saving rateunder
Autarky.So even if acquisition of the durable
good
is undesirableunder Autarky,
itseems
reasonable to suppose that the agentswould
prefer toform
aRandom
Rosea and
acquire the durablegood
rather than go without it. This situation does indeed occurifWt^>T- v(y)>W
..
This belies, however,
two
more
subtle issues related to desirability. First, for themodel
of aRandom
Rosea
developed here to be consistent, itmust
be that eachmember
would
actually choose to invest in the durablegood
upon
winning
the pot, as isassumed
inthe foregoing derivations. This will be true only if the horizon remaining after the last
member
has received his winnings is longenough
to justify the investment,which
willdepend
upon
theconsumption
valueof the durable's services.Second, the finite formulation
whose
limit is characterized in (4.6)assumes
that thenumber
ofmeeting
dates of theRosea
is equal to thenumber
ofmembers.
Relaxation of this assumption allows for the possibility thatmembers
might
elect to face animproper
probability distribution for the
random
receipt date r. Suppose, for example, that with an an evennumber
ofmembers,
meetings occurredon
dates {2t/n,4t/n,6t/n,...,t}.Now,
tomake
his contribution ofB/n
at each meeting, an individualwould
save at the constant rate ofB/2t. In thecontinuum
case, conditionalon winning
the pot, each agentwould
face14
receiving the pot
would
only beone—
half.Could
it be that such a "partially funded"Rosea
would
be preferred to the "fully funded'' versionwhose
versionwhose
welfare is given in (4.5)?To
guarantee thatRosea member;
will alwayswant
to design aRosea
so that theyall eventually receive the pot
and
that bvying the durable is alwaysan
optimal response towinning,
we
require a stronger desirabili-.yassumption than
we
made
under
Autarky. Itturns out that if i^ is twice as big as the
minimal
valueneeded
to induce acquisition ofthedurable
good under
Autarky, then (4.6) does indeed characterize theperformance
of an optimalRandom
Rosea.Thus
we
replaceAssumption
2 with:T
Assumption
2': (Strong Desirability) ( > 2£{y,Tj).Then
we
have:Lemma
1:Under
Assumption
2'an
optimalRandom
Rosea
involves everymember
receiving the pot during the life of the Rosea. Moreover, every participant of the optimal
Random
Rosea
will desire to use the proceeds to acquire the durable good.Proof: See the
Appendix.
In fact, specifying the
Random
Rosea
so that everyone receives the pot once is notessential for
much
of our comparaiive welfare discussion in section VI.Our
analysis therecompares
Random
Roscas constrained to provide probability one ofeventual receipt ofthe pot to alternative arrangements.The
firding that aRandom
Rosea
is preferred to otherinstitutions is only strengthened
by
allowing the possibility that not everyone wins.On
the other hand, partially funded Roscas do not appear to
be
common
in practice.We
do needAssumption
2', however, for its implication that all winners will prefer to invest in15
the durable good. Otherwise the analysis culminating in (4.6) is inconsistent.
The
intuitive reason for the stronger desirability requirementcomes from
notingthat randomization is
tantamount
to introducing divisibility of the durable good, sincebeing offered a probability of receiving the durable
good
is like being offered a fraction of it tx ante.Assumption
2 assures that "all" is preferred to "nothing"—
the only choices facingan
Autarkic individual.Assumption
2'must
be
stronger since it has to guarantee that"all" is preferred to "anything less", i.e. intermediate levels of durable
consumption
must
also
be
dominated.Let <Srj,Crj> denote the allocation achieved with the optimal
Random
Rosea under
Strong Desirability. Recalling thatan
individual's type, a, is identifiedwith
his order of receipt of the durable, the assignment function will be s-n(a)=at-p.As
under
Autarky, allindividuals
have
identicalconsumption
pathswhich
fall intotwo
distinct phases.To
bemore
precise, CT^(Q:,r)=CTj, for Te[0,t-p]; and, CT^(a,r)=y, for re(tx^,T],where
CT%=y—
B/t^.
-The
allocation with the optimalRandom
Rosea
is exhibited in Figure 1.While
consumption
is still a piecewise horizontal function for each agent, the fraction of thepopulation
who
have
received the durable at time t isnow
increasingand
linear (see Figure lb). Moreover, while exhibiting thesame
general pattern for consumption,we
shallshow
below
that the accumulation phasewill not last thesame
length oftime
asunder
Autarky,i"IV.3
Bidding Rosea
We
turn next to the Bidding Rosea,where
the orderinwhich
individuals receivetheaccumulated
savings is determinedby
bidding.As
discussed in section II, the simplestassumption
tomake
is that the bidding takes placewhen
theRosea
isformed
at time zero, ^and
involves individualscommitting
themselves to various contribution rates over the lifeof the Rosea.
We
do
notmodel
the auction explicitly, but instead simply characterize16
require that, for a Bidding
Rosea
of length t, an individual receiving the "pot" at dater6[0,t]
commits
to contributing into theRosea
at the constant rate b(r) over this interval.A
set of bids {b(r):T6[0,t]} constitutes an equilibrium if: (i)no
individual could do betterby
out bidding another for his place in the queue;and
(ii) contributions are sufficient toallow each participant to acquire the durable
upon
receiving the pot.The
first condition isan
obvious requirement ofany
bidding equilibrium.The
second precludes the desirabilityofsaving outside the Rosea.
Any
such ia>^ingwould
lie idleand
thusbe
inefficient.Now
in the equilibrium of aBidding
Rosea
of length t, the utility ofan
individual receiving the pot at time r6[0,t] is given by: u=
tv(y-b(r))+(T-t)(v(y)+(^)+(t-r)^\Condition (i) then implies that, for all r,r'G[0,t]:
(4.7) v(y-b(r))
+
(1 -?)>'=
v(y-b(r'))+
(1"f
)e;while condition (ii) implies that
rT
(4.8)
b(r)dr=B.
Jo
These two
equations uniquely determine the function b(-), given the Rosca's length t.An
optimal Bidding
Rosea
willhave
its length chosen so that the utility level of a representativemember
ismaximized
subject to (4.7)and
(4.8).To
characterize the optimal Bidding Rosea, consider theconsumption
rate ofan
individual
who
receives the pot at date r,which
we
denoteby
c(r) = y—
b(r) for r 6 [0,t].We
may
describe the BiddingRosea
just as well with the function c(-) aswe
couldby
considering the bids directly. Substituting this into (4.8)
and
making
thechange
of17
(4.9)
t.[y-
[ c(a)da]=
B.Jn
Now
consider theconsumption
level, c=
c(x), of an individualwho
receives the durable atxt, (xg[0,1])
and
note that (4.7) implies that c(a)=
y~^{v{c)-{x-a)i), for all a6[0,l].Hence
all other individuals'consumption
levels can bedetermined
from
the equal utilitiescondition once c is
known.
Moreover, defining A(c,x)e/^[v (v(c)—(x—
a)^)]da as theaverage
consumption
leveland
using (4.9),we
may
write t=
B/(y—
A(c,x)).Hence
theutility level ofthe
member
who
receives the pot at datext can be written as:(4.10)
W(c)
=
T.[v(y)+^] -B.[v(y)-v(c)+xe]/[y-A(c,x)]
Since, all
members
have
identical utility levels,by
construction,maximizing
the utility of aparticular
member
is thesame
asmaximizing
thecommon
utility level.Thus
we
can viewthe
problem
solvedby
the BiddingRosea
as choosing c tomaximize
(4.10). Let Cd(x) denote the optimalconsumption
level for typex
individualsand
letWp
denote the welfarelevel achieved at the
optimum.
Usingby
now
familiararguments
we
may
write:(4.11)
WB
=
T.[v(y)+^]-B.MB(0,
where
(^(0
=Min
0<c<y
vYvj^-v(c]+x^y-A(c,x)
e>o.
This admits the
same
interpretation thatwe
noted forboth
Autarky
and
theBidding Rosea, as the difference
between
utilitywere
the durable freeand
the cost of(EA(Cp(x),x)), are
both
independent of xby
construction.'BNote
also that the optimallength ofthe
Bidding
Rosea,which
we
denoteby
tg, will be givenby
tg=B/(y—
Ag).Once
again, theabove
formulationassumes
that individuals will alwaysbuy
thedurable
when
they receive the pot.Thus
if it turns out to be the case thatsome
individuals could actually
do
betterby
not purchasing the durable, the analysis will beinconsistent. Intuitively, it seems reasonable to suppose that the (weak) desirability
condition given in
Assumption
2 should rule this out. After all, the financialintermediation afforded
by
the BiddingRosea
mean
that the costs of savingup
are lessthan
under
Autarky
and, since the allocation of the pot is notrandom, none
of the issuesraised in the previous
sub—
section arise. Unfortunately, however,we
have been
unable toestablish the validity of this conjecture.
While
it is clear that i^must
be "sufficientlylarge" relative to y, B,
and
T, it is not cbvious thatAssumption
2 or evenAssumption
2'provide appropriate bounds.
Hence
the exact condition necessary to ensure desirability inthe optimal
Bidding Rosea
awaits determination.Let <SxD,c-p> denote the allocation generated
by
the optimal Bidding Rosea.As
with theRandom
Rosea, the assignment function is linear, i.e. s-p(a)=
crt^. Below,we
shall
compare
t-pwith
tr>, the length of the optimalRandom
Rosea. UnlikeAutarky and
the
Random
Rosea, each individual receives a differentconsumption path under
aBidding
Rosea.However
the general pattern is similar withan
accumulation phase followedby
a phase inwhich
agentsconsume
all of their incomes.Hence
the allocation ofnon—
durableconsumption
is describedby
c-n{a,r)=c-r,{a), for rG[0,tr>]and ep(a,r)=y,
for re(tT5,T].IV.4
The
Market
The
final institution thatwe
consider is a credit market. In the present context,we
specify the behavior of such a
market
in the followingway.
Let r(r) denote themarket
interest rate at time r. It is convenient to define 6{t) as the present value of a dollar at
19
present value prices {(^(r):rG[0,T]} at
which
the supply ofand
demand
for loanable fundsare equated. Hence, an individual
who
buys the durablegood
at time s pays 6{s)B for it.Given
the price path,an
individual chooses a purchase time s(a)and
aconsumption
path{c(a,r):rG[0,T]} to
maximize
utility.Hence, the optimization
problem
solvedby
each individual is givenby
Max
'^
{c(-),s}
(4.12) v(c(r))dr+
^(T-s) subject to <5(rk(r)dr+
5{s)B < y (5(T)dr, s G [0,T].Jo
'Jo
This
assumes
that each individual decides to purchase the durable atsome
time.Once
again, this will require that ( be big enough, given T, y
and
B.We
shallassume
that thisis true in
what
follows.20 Hence,we
define amarket
equilibnum to bean
allocation<Si^,C|^> and
a price path 6{-) satisfyingtwo
conditions: first,<S|^(a),c^(a,r)>
must
be a solution to (4.12) for all aG[0,l],
and
second, at each date tG[0,T],rt rl
(4.13)
[y-
Cj^(a,r)da]dr=N(t;Sj^).B.
"'0 "^0
Since individuals are identical, the first condition implies that, in equilibrium, all
individuals are indifferent
between
durable purchase times.The
second condition just says that savings equals investment at all points in time.We
useW,,
to denote theequilibrium level of utility enjoyed
by
agents in amarket
setting. Below,we
show
thatthis can
be
written in aform
analogous to (4.3), (4.6)and
(4.11).Direct
computation
of competitive equilibrium pricesand
the associated allocation20
we
are able to infer the existenceand
seme
of the properties of competitive equilibrium inour
model
by
using the fact that, with identical agents, itmust
coincide with a Paretoefficient allocation
which
gives equal utility to every agent. Hence, explaining properties of theMarket
allocationmust
await considiration ofefficient allocationsmore
generally.This completes our description
and
preliminary analysis ofthe different institutionalframeworks
which
are dealt with in this paper. In sectionVI
we
shallmake
detailedcom-parisons of
them.
Our
next task, however, is to discuss the implications of Paretoeffic-iency for the
model
described so far. This will aid the study of competitive allocations,while also providing insight into the baoic resource allocation
problem
posed hereby
the existence ofan
indivisible durable good.V. Efficient
and
Ex
Ante
Optimal
AIloCc.tionsWe
have
now
shown
how
vai'ious institutions for mobilizing savings produceparticular feasible allocations
and
associated distributions of utility in the population.To
evaluate the allocative consequences of alternative institutions for financial intermediation
we
requiresome
welfare criteria.Two
such criteria are natural here.The
first is ex postPareto efficiency, or
more
simply, efficiency.We
consideran
allocation tohave
beenefficient if there is
no
alternative feasible allocationwhich
makes
anon—
negligible set ofindividuals strictly better off, while leaving all but a negligible set ofindividuals at least as well off.
The
second criterion is defined in terms of ex ante expected utility.The
allocation<s,c>
is betterthan
<s',c'>, in this sense if/^u(a;<s,c>)da
>
/^ u(Q;;<s',c'>)da.The
preferred allocation gives a higher expected value of the lifetime utility of a type
a
indi-vidual, with
a
regarded as arandom
variable uniformly distributedon
the unit interval.This is a "representative
man"
criterionwhich
asks:which
allocationwould
be preferable21
particularly interested, therefore, in that allocation
which
yields the highest level ofexpected utility.
We
call this the ex ante optimal allocation. Notice that, since allindividuals enjoy the
same
level of utilityunder
Autarky, the Bidding Rosea,and
theMarket, the
two
criteria coincidewhen
applied to allocationsemerging
from any one
of these institutional forms. This is not the case for theRandom
Rosea. It is possible thatan
allocation generated
by
aRandom
Rosea might
be Pareto dominated,and
yet itself bepreferred to
some
Pareto efficient allocation in terms of the ex ante optimality criterion.Indeed,
we
will present anexample
inwhich
precisely this reversal occurs.In
what
follows,we
characterize efficient allocations for oureconomy
and
the exante optimal allocation.
While
this section is the analytic cornerstone of the paper, the readerwhose primary
interest lies in the resultsmay
wish just toskim
the nextfew
pages, focusing particularlyon
Corollary 1which
describes the essential properties of efficientallocations
and
the discussion ofthe optimal allocation followingTheorem
2.V.l Efficient Allocations
To
characterize efficient allocations,we
introduce weights ^(a)>
for each agentQe[0,l], normalized so that lr.6{a)da
=
1. Define the set of all such weights:Q={0
:[0,1]-t K
I
6 integrable
and
jr.6{a)da=
1}.By
analogy with a standardargument
in welfareeconomics,
an
efficient allocation shouldmaximize
a weightedsum
of individuals' utilities.Hence, if
<s',c'>
is efficient then theremust
be a set of weights in B such that/Q^(a)u(a;<s',c'>)da
> /^ ^(a)u(a;<s,c>)da, for all feasible allocations, <s,c>. In orderto investigate the properties of efficient allocations
we
therefore study, for fixed ^£0, theproblem:
rl
/r .\
max
W(^;<s,c>)
= ^(a)u(a;<s,c>)da,^'^^^
22
for
u(a;<s,c>)
asdeCned
in (3.1).21 Let<s«,Ca>
denote an allocationwhich
solves thisproblem,
and
W^
=W(^;<s^c^>)
denote themaximized
value ofthe objective function.Although
(5.1)may
appear to be a"non—
standard" optimization problem, it can besolved explicitly
under
our simplifying assumptions on preferences.The
solution isexhib-ited in
Theorem
1 below.We
develop theTheorem
and
discuss its implications atsome
length in
what
follows.The
proof of this result, presented in theAppendix,
exposes thenature ofthe resource allocation
problem which any
institution for financial intermediationmust
confront in oureconomy.
The
first point to note about the efficiencyproblem
is that it can be solved intwo
stages, loosely corresponding to static
and
dynamic
efficiency.The
first requires thatany
level ofaggregate
consumption
be optimally allocated across individuals, i.e.maximizes
theweighte.l
sum
of instantaneous utility at each date, while the second determines theoptimal acquisition path for the
durabk
good.We
shall focus, once again,on
the case inwhich
everyone receives the durable at theoptimum.
Let c«(r) denote aggregate consuinptionin period r, i.e. c
Jt)£J
^cp{a,T)da.A
briefinspection of the efficiency
problem
should convince the reader that this should bedistributed
among
the different types ofindividuals so as tomaximize
the weightedsum
ofutilities
from consumption
in period r.To
state thismore
precisely, define theproblem
1
Max
(5.2) rl 0{a)y{x{ct))da subject to x{ck)da—
w
for all
w
>
0. Let Xo(*)W) denote the solutionand
letVa(w)
denote the value of the objective function.Then
a typea
indi\'idual'sconsumption
at time Te[0,T] is givenby
23
remains, therefore, to determine
sJa)
and
cJr).Note
first that since preferences are strictly monotonic, the feasibility constraint that<s,c>
eF
[see (3.2)]may
without 1jss ofgenerality be written as:rT rl
(5.3)
[y-
c(a,r)da]dy=
N(r;s)B, Vre[0,T].JQ
Jo
That
is, all savingsmust
be putimmediately
to use. Moreover, in the absence ofdiscounting, if the flow of aggregate savings
/^[y—
c(a,r)]da equals zero atsome
date r,then efficiency
demands
that itmust
also be zero atany
later datet'>t.
Otherwise,by
simply
moving
the later savings forward in timeone
could assignsome
agentsan
earlierreceipt date for the durable ' without reducing anyone's utility
from
non—
durableconsumption.
The
foregoing implies thatany
assignment function solving (5.1)must
becontinuous, increasing,
and
satisfy s(0)=0. This in turn implies that suchan
assignmentfunction will
be
invertibleand
differential)le almost everywhere.We
can use these facts towrite N(r;s)
=
s~^{t), for rG[0,s(l)],and
N(r;s)=
1, for re(s(l),T]. Substituting thisexpression for N(t;s) in (5.3)
and
differentiating with respect to r,we
find that for allre[0,s(l))
we
have
y-c(r)=B/s'(s
(r)),and
for all Te(s(l),T]we
have
y-c(r)=0.
Therefore,
we
may
conclude that the following conditionmust
be satisfiedby any
solutionto (5.1):
(5.4) s'(a)
=
B/[y-c(s(a))], for all Qe[0,l)i and, c(r)=y, for all rG(s(l),T].Equations (5.4) are the analogue of "production efficiency" in our model, i.e. there
can be
no
outright waste of resources. Hence, for parttwo
of the solution,we
are interested in theproblem
of choosing functionssJa)
and
cJr)
tomaximize
24 (5.5) rT
V^(c(r))dr+eT-^
rl 0{a)s{a)dasubject to (5.4).
By
analogy, with our earlier analysiswe
define(5.6) fiJ-y) E
min
0<(T<y
rV^(y)-V^(a)+7.
y-a
, 7>0,and
denote the solutionby
crAi).We
now
have:Theorem
1: Let<s,c>
be an efficient allocation withs~
(j^)=0,and
let^£0
be the weightsfor
which <s,c>
provides a solution in (^.l).Then
themaximized
value canbe
written inthe
form
W^=T.[V^y)
+
{]-B.j
/.^{/^^(z)dz)dx
and
the assignment function satisfies:a
s(a)
=
B.
[y-a
m]0{z)^)]
^dx, a6[0,l].Moreover, for all ae[0,l],
non—
durableconsumption
obeysc(a,s(x))
=
x^(Q!,^^(ax^(z)dz)), forxe[0,l];and
c(a,r)=
x^a,y),
for re(s(l),T].Proof: See the
Appendix.
As
noted, Pareto efficiency requirestwo
conditionsbeyond
the absence of physicalwaste of resources. First,
any
given aggregate level ofnon—
durableconsumption
should beallocated efficiently
among
individualsand
second, itmust
optimallymanage
theintertemporal
trade—
offbetween
aggregateconsumption
of thenon—
durableand
faster25
these stages.
More
needs to be said about thedynamic
efficiency considerations—
inparticular, the relevance of the minimization
conducted
in (5.6).To
see this consider the expressi;;n forW«
inTheorem
1. This welfaremeasure
isthe difference of
two
terms.The
Crsl,T-[V^y)+,^], would
be themaximal
weightedutility
sum
if the durablewere
a free ijood.The
secondterm
is, therefore, the (utilityequivalent) cost of acquiring the durable It is this cost
which
isminimized
in (5.6). It hastwo competing components:
non—
durableconsumption
foregone in the process of acquiring the durable (since c(s(a))<
y)and
durable services foregone in allowingsome
non—
durableconsumption
(since s'(a)<
B/y). Consider a small interval of timea6(a,a+da),
then thesum
of thesetwo components
is approximately [YJy)—V
Jc{s{a))+^j
0{z)dz], while theduration of the time interval is
s'(a)da=Bda/[y—
c(s(a))]. Efficient accumulationtherefore
means
minimizing the product of theseterms
at each aG[0,l]. This is preciselythe
problem
describedby
(5.6).A
geometric treatment of the minimizationproblem
(5.6)may
also be helpful (seeFigure 2).
The
functionV
J-)
is smooth, increasingand
strictly concave becausewe
have
assumed
that v(-) has these properties. Therefore, choosinga
tominimize
the ratio[Vfl(y)—V«((7)+7]/[y—a]
means
finding that point{a,VJa))
on
the graph ofV^(-)
suchthat the straight line containing it,
and
containing the point (y,V^(y)+7), is tangent to thegraph of V^(-). Notice
from
thediagram
that o'fl'f)must
be
decreasing, rising to y as7
falls to 0.
This observation, together with
Theorem
1, permits us to deducesome
usefulproperties ofefficient allocations:
Corollary 1: Let
<s,c>
be an efficient allocation withs"
(i/)=0.Then
(i) the assignment function s(-) is increasing, strictly
convex
and
satisfies 1im
s'(a)=
+0D,and
26
(ii) for all ae[0,l], c(a,•) is increasing
on
the interval [0,s(l)],and
constantthereafter.
Proof: (i) In view of (5.4)
and
Theorem
1,we
know
thatany
efficient allocation<s,c>
satisfies s'{a)
=
B/[y-CTi,^/ 6{z)dz)], forsome
Oe&.As
notedabove
(^Al) is decreasingand
approaches y as 7falls to 0. Hence, theresult.(ii) This follows
immediately
from
Theorem
1 after noting that Xni<^r) is increasingand
that crJ'y) is decreasing, d
The
properties ofthe assignment functionimply
that, in an efficient allocation, the fractionof the population
who
have received the durableby
time r is increasingand
strictlyconcave. In addition, the rate of
accumulation
(thetime
derivative of N(r;s)) approacheszero as rgoes to s(l).
Finally, it is
worth drawing
attention to the relationshipbetween
thecharacteriza-tion in
Theorem
1and
the welfare expressionsfound
forAutarky
in (4.3), for theRandom
Rosea
in (4.6)and
for theBidding
Rosea
in (4.11).These
expressions all take thesame
general form, allowing the intuitive interpretation that welfare is the hypothetical utility
achieved
when
the durable is a free good, net of the implicit utility cost of acquiring the durable's services. This observation is the basis formost
of the results in section VI.V.2
The
Optimal
AllocationThe
optimal allocation is that efficient allocation inwhich
individuals are equally weighted. Since individuals are identicaland
are assigned types randomly, this solutionmaximizes
the ex ante expected utility of a representative agent. Hence, noting that 1G0,we
can write ex ante expected utility asW(l;<s,c>), and
the optimal allocation<s
,c>
must
satisfy:W(l;<s
,c>)
>W(l;<3,c>),
V<s,c>eF.
27
agents
have
equal weightsand
utility is strictly concave, then aggregatenon—
durableconsumption
should be allocated equallyamong
all agents. It is also useful to note that/i,(q)E//(q),
where
/i(-) is defined in (4.2),and
that ^i'{'y)=[y—(Tjj)]~ , using theEnvelope
Theorem.
These
facts, together withTheorem
1, yield:Theorem
2: Let<s
,c>
be the optimal allocation.Then,
ex ante expected utility can bewritten in the
form
rl
W(l;<s^,c^>)
EW^
-
T-[v(y)+e]
-B.
/i(e(l-a))da,and
the optimal assignment function satisfiess (a)
=
3-[
/i'(^(l-x))dx.Moreover, for all aefO,!],
non—
durableconsumption
obeysCo(^'SoW)
=
y-l/M'(^(l-x))
for xe[0,l],and
c^(a,r)=
y
for re(sQ(l),T].Proof: Provided that s (i^)=0, the
Theorem
follows at oncefrom
Theorem
1and
the foregoing observations, after noting that / 0{z)dz=ll(z)dz=l—
a
To
prove thats (i')=0,
we
show
that welfare under theoptimum
is increasing in the fraction of thepopulation
who
ever get the durable, a. Hence, letW
(a) denotemaximal
expected utilitywhen
a fractiona
of the population get the durable.Arguing
as in the proofofTheorem
1,we
obtain:W
fa)=
T.[v(y)+Q^]
-
B-[
//(e(^a))da.
Differentiating with respect to a, yields
(a
_
W'(a)
=
T^
+
B-
/i'(^(«-a)){da-B/x(0),Jq
which, after using the
Fundamental
Theorem
of Calculus, simplifies to28
By
Assumption
2,we
know
thatT(
> Bfj(^). Since fi(,-) is increasing, it follows thatW'(a)
>
0.0^ '
In addition to the properties outlined in Corollary 1, therefore, the optimal
allocation also gives individuals identical
consumption
paths.Theorem
2 also tells us thatc (0) equals c. , the
consumption
level in theaccumulation
phaseunder
Autarky22.The
pattern of
consumption
and
the fraction of the populationwhich
has acquired the durablegood
in the optimal allocation are illustrated, as a function of time, in Figure 1.Each
individual has
an
identicalconsumption
path beginning at c .and
risingsmoothly
to y atthe
end
of the accumulation phase.The
fraction of the population with the durable isincreasing
and
concave over the interval ofaccumulation.It is helpful to note the relationship
between
theproblem
solvedby
the optimalallocation
and
that of accumulating a perfectly divisiblegood under
autarky.As
notedearlier, ifthe durable
good
were
perfectly divisible, then therewould
beno
gainsfrom
tradeand
Autarky would
be optimal.23 It is precisely the indivisibility of the durablewhich
creates the problem. Nonetheless, the
economy
may
approximately replicate the situationunder perfect divisibility, even in the presence of indivisibilities,
by randomly
assigning individuals to positions in thequeue
at the initial date. This istantamount
to granting each individual a "share" of the aggregateamount
of the durablegood
available atany
subsequent date.The
exact linkbetween
thetwo problems
is spelled out in the followinglemma.
Here
the functionK(r)
is to be interpreted as the stock ofthe divisible asset the individual holds at time r.29
Lemma
2:The
optimal aggregateconsumption
path {c (T):re[0,T]} solves the problem:T
Max
[ [v(:i'r))
+
^(T-r)K'(r)]
dr
subject to B-K'(r)::=y-c(r);
K(0)=0;
K(T)=1;
0<c(r)<y.Proof: See the
Appendix.
Thus
theconsumption
path ofnon—
durableconsumption
is precisely that attainedby
anindividual
accumulating
a perfectly divisible durable good. In the absence of trade indurable services, financial intermediation provides the only
means
toovercome
the limitationsimposed
on
the agentsby
the constraint of indivisibility. In the optimalallocation it is completely
overcome
in :,he sense that ex ante expected utility is just thesame
as itwould
beunder
perfect divisibdity.While
we
have assumed
that preferences are quasi—linear in deriving our results,there will be gains
from
randomization {on the ex ante expected utility criterion) inany
equilibrium with indivisibilities. This is z. general point,
and
it is key tomany
of the resultsof section VI. It is important therefore to understand
why
it is true.Assumptions
aboutpreferences affect the extent to
which
randomization provides a perfect substitute fordivisibility, but not
whether
it improves over deterministic, equilibriumoutcomes
in thepresence of indivisibility.
From
an ex ante viewpoint, the representative agentwho
facesthe deterministic assignment function s''•) without
knowing
his typesa
is already, ineffect, bearing risk.
He
sees a probability N(r;s) of enjoying the durable's servicesbeginning
on
or before r,and
evaluates his expected utility accordingly.By
contrast, adopting an expostviewpointon
any
equilibrium, an agentknowing
histype
a
and
facing assignment function s(-),must
be willing to accept his assignment—
30
utility conditions of the Bidding
Rosea and
themarket
equilibrium.Thus
in the lattertwo
institutional settings, only those allocations
which
satisfy feasibilityand
equal utility canbe considered. This
amounts
to an additional constrainton
theproblem
ofmaximizing
exante expected utility.
This completes our characterization of efficient
and
optimal allocations.Having
understood their properties,we
are ready toexamine
how
well the various institutions thatwe
have
discussed perform relative to these criteria.VI.
The
Performance
ofRoscasThe
purpose of this section is to analyze the allocativeperformance
of Roscas indetail.
We
shall be interested in understandinghow
theyperform
relative to the efficiencyand
expected utility criteria, as well as in establishing additional features of the resource allocationwhich
they produce. In order to do this, it is helpful to set out conciselysome
ofthe results of the previous
two
sections. Thiswe
do
inTable
1.To
keep the notationconsistent
we
let t denote the date a^which
allhave
acquired the durableunder
theoptimum,
i.e. t =s (1).TABLE
1wpif-
pDate
by
which
allhave
J ,.,
,• acqui