working paper
department
of economics
THE ECONOMICS OF ROTATING SAVINGS
AND CREDIT ASSOCIATIONS
Timothy Besley
Stephen Coate
Glenn Loury
No.
556
May 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
No.
556
''
May 1990
*«^^
The Economics
of Rotating Savings and Credit Associations*
by
Timothy Besley
Princcion University
Stephen Coate
Harvard University
and
Gltnn Loury
Harvard University
May
1990
Abstract
This paper examines the role and performance of an institution for allocating
savings which is observed world wide - rotating savings and credit associations. We
develop a general equilibrium model of an economy with an indivisible durable
consumption good and compare and contrast these informal institutions with credit
markets and autarkic saving in terms of the properties of their allocations and the
expected utility which they obtain. We also characterize Pareto efficient and expected
utility maximizing allocations for our economy, which serve as useful benchmarks for the
analysis. Among our results is the striking finding that rotating savings and credit
associations which allocate funds randomly may sometimes yield a higher level of
expected utility to prospective participants than would a perfect credit market.
*The authors are grateful to Anne Case, Richard Zeckhauser, and seminar participants at
Boston, Cornell, Harvard, Oxford, Princeton and Queen's Universities for their helpful
comments. Financial support from the Center for Energy Policy studies, M.I.T., and the
Japanese Corporate Associates Program at the Kennedy School of Government, Harvard
University is gratefully acknowledged.
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium IVIember Libraries
http://www.archive.org/details/economicsofrotatOObesl
I.
Introduction
This paper examines the economic; role and performance of Rotating Savings and
These informal
Credit Associations (Roscas).
credit institutions are
While
world, particularly in developing countries i.
their prevalence and, to
robustness has fascinated anthropologists^, there has been very
understanding their economic performance.
equilibrium
general
setting,
Here we
and
contrasting
found
shall subject
comparing
little
them
all
over the
some
degree,
work devoted
to
to scrutiny in a
them with other
financial
institutions.
The
Roscas come in two main forms.
random
allocation.
In a
Random
of the
except that the previous winner
type allocates funds using a process of
Rosea, members commit to putting a fixed
into a "pot" for each period of the
randomly allocated to one
first
life
of the Rosea.
members.
is
Roscas
At
this point, the
may
Lots are drawn and the pot
is
member
The
is
itself,
process
of the Rosea has received
either disbanded or begins over again.
also allocate the pot using a bidding procedure.
institution as a Bidding Rosea.
The
excluded from the draw for the pot.
Rosea
money
of
In the next period, the process repeats
continues, with every past winner excluded, until each
the pot once.
sum
individual
who
We
shall refer to this
receives the pot in the present period
does so by bidding the most in the form of a pledge of higher future contributions to the
Rosea or one time side payments to other Rosea members.
individuals
may
still
only receive the pot once
—
Under a Bidding Rosea,
the bidding process merely establishes
priority^.
We take the
means
of saving
view, supported in the literature, that these institutions are primarily a
up to buy
such as weddings.
indivisible goods, such as bicycles, or to finance
Random Roscas
are
major events,
not particularly effective as institutions for
buffering against risk since the probability of obtaining the pot need not be related to one's
immediate circumstances.
Even Bidding Roscas, which may allow an individual
to obtain
the pot immediately, only permit individuals to deal with situations which cannot recur,
since the pot
may
be obtained no more t^^an once.
LDC's are covariant, many individuals
in
will
Furthermore, since
have high valuations
many
at the
kinds of risks
same
instant.
Roscas do play a greater role in transferring resources to meet life—cycle needs such as
However, even in
financing a wedding.
this context, they
ing with significant, idiosyncratic events, rather than the
Despite
its
seem more appropriate
hump
saving required for old age.
manifest importance, there has been relatively
savings literature on the notion of saving up to
of indivisible goods does,
by
itself,
buy an
for deal-
little
indivisible good.
analysis in the
Yet, the existence
provide an argument for developing institutions which
intermediate funds. In a closed economy without access to external funds, individuals have
to save
from current income to finance lumpy expenditures, and can gain from trading with
The
one another.
By
contrast,
savings of
when
all
some agents can be used
goods are divisible individuals can accumulate them gradually and
Clearly autarky with indivisible goods
there need be no gains from intertemporal trade.
Savings
inefficient.
to permit
lie idle
some individuals
As we explain
making these
is
in
to enjoy the services of the indivisible durable good.
more
detail in the next section, Roscas provide one
joint savings work*.
Given the worldwide prevalence of these
how they do
between Random and Bidding Roscas.
allocations resulting
from Roscas
tioning credit market.
intermediation.
differ
erizes the
method
institutions,
of
it
mobilize savings and, in particular, the
It is also
interesting to consider
from that which would emerge with a
how
the
fully func-
Such markets are the economist's natural solution to the problem of
Yet there are many settings
in
which they are not observed.
through our comparative analysis, to suggest some reasons
Our
is
during the accumulation process when they could be employed
of interest to understand precisely
differences
to finance the purchases of others.
study, in the context of a simple
why
We
hope,
this is so.
two—good model with
indivisibilities, charact-
path of consumption and of the accumulation of indivisible goods through time
under these alternative institutional arrangements.
We
also investigate the welfare pro-
Roscas and of a competitive credit market, employing the
perties of
Pareto efficiency, and of ei-aiie expected
is
ex-post
efficient.
We
find that, with
Neither type of Rosea, as modeled here,
utility.
homogeneous agents, randomization
bidding as a method of allocating funds within Roscas.
not generally the case for
utility, this is
We
Random
interests of
making the
issues concerning
,
insights obtained from
it
Roscas.
We
to
pay into the Rosea.
incentive to defect from a Rosea
among
are typically formed
known
to each other.
:s
believe captures
keep the model simple in the
In particular,
question of why, once an individual has received the "pot", he
commitment
we
Some important
as sharp as possible.
Roscas are not dealt with here.
preferred to
that, in terms of ex-ante
use a model which makes no claim to generality, but which
the essential features of the problem described above.
is
Moreover, while Bidding Roscas
we show
are always dominated by a perfect credit market,
expected
ex—post
criteria of
The anthropological
may
we do
not address the
be presumed to keep
literature reveals
curbed quite effectively by social constraints s.
that
his
the
Roscas
individuals whose circumstances and characteristics are well
Defaulters are sanctioned socially as well as not being permitted to
take part in any future Rosea.
Thus, in contrast to the anonymity of markets, Roscas
appear to be institutions of financial resource allocation which rely in an essential way
upon the
"social connectedness"' of those
among whom they
operate.
ing to try to formalize this idea in a model allowing default.
It
would be
In future
interest-
work we
will
consider the sustainability of Roscas in greater detail.
The remainder
of the paper
is
organized as follows.
Section
II
provides an informal
overview of the economic role of Roscas and outlines the main questions to be addressed in
this paper.
Section
Section III describes
'the
model which provides the framework
IV then formalizes the workings
this
model.
The
role of Section
Section
V
VI
of Roscas
and a
credit
for
our analysis.
market in the context of
develops properties of efficient and optimal allocations of credit.
is
to
compare the
accumulation and consumption time paths
different institutions.
differ
W^e describe how the
under the alternative arrangements.
We
also establish the
n.
normative properties of the allocations. Section VII concludes.
The Econoinic Role
An
of Roscas:
Informal Overview
Consider a community populated by 20 individuals, each of
some durable,
$100.
indivisible
If individuals,
individual
—
consumption good
own
to their
left
would acquire a bicycle
after 20
like to
own
Suppose that bicycles cost
say, a bicycle.
devices,
whom would
were to save $5 per month, then each
months.
That
is,
under autarky and given the
savings behavior assumed, nobody would have the good before 20 months, at which time
evciyone would obtain M.
Obviously this
early receipt of their bicycles,
buy one bicycle per month,
to
it
would be
make
is
inefficient since,
possible,
wiih any preference for the
by using the accumulating savings
ever;yone except the final recipient strictly better
to
off.
i
Roscas represent one response to this inefficiency.
Random Rosea which meets
month.
Let our
community form a
once a month for 20 months, with contributions
set at $5 per
Every month one individual in the Rosea would be randomly selected to receive
the pot of $100, which would allow
buy a
on average 10 months sooner.
viewed ex ante, the
Random Rosea
strictly better in all
but one.
bicycle.
It is clear
Note that
that this results in a
as under autarky but
risk aversion
is
lasts for 20
months
is
not an issue since,
only one possibility.
could, for example, have a Rosea which lasts for 10
last
acquired, every
for
30
months with members contributing
month and
constant contribution rate,
$5,
bicycles.
Or, the
and a bicycle being
a half.
Given the uniform spacing of meeting dates and the
common
features of Roscas as they are observed in practice, the
length of the Rosea will be inversely proportional to the rate at w^hich the
and accumulates
The
months with members
making the $5 contribution, and someone getting a bicycle every half—month.
Rosea could
now
does as well as autarky in every state of the world, and
Forming a Random Rosea which
community
to
Each individual saves $5 per month
Pareto superior allocation.
gets a bicycle
him
It
seems reasonable to suppose that the Rosea
community
will
saves
be designed
with a length which maximizes the (ex—ante expected) utility of
member.
Supposing that
this is the case,
the Rosea and the savings rate which
If
it
the right to receive the pot early, the
make
"representative"
one can then characterize the optimal length of
implies.
instead of determining the ord<
Suppose that Rosea members,
its
r
of receipt
randomly individuals were to bid
for
community would be operating a Bidding Rosea.
at their initial meeting, bid for receipt dates
by promising to
future contributions at various (cor^stant) rates, with a higher bid entitling one to an
earlier receipt date.
In our 20
month example, the
agree to contribute (say) S6 per meeting for the
individual receiving the pot
life
first
who
of the Rosea, the individual
be last might only pay $4 per meeting, and others might pay something in between.
ever the exact numbers, two things should be true in equilibrium:
might
is
to
What-
overall contributions
should total $100 per meeting; and, individuals should not prefer the contribution/receipt
date pair of any other Rosea
member
to their own.
While both Bidding and Random
more
iloscas
aUow the community
to utilize its savings
effectively than under autarky, they clearly will not result in identical allocations.
Even when they have the same
length, the consumption paths of
members
differ since,
under a Bidding Rosea, those who get the good early must forego consumption to do
Moreover, there
institutions.
Of
is
no reason to suppose that the optimal length
will
so.
be the same for both
particular interest are the welfare comparisons between the institutions.
Understanding what determines their relative performance gives insights into the circumstances under which
Of
we might expect
to sae
one or the other type of Rosea in practice.
course, Roscas are not the only institutions able to mobilize savings.
the introduction of (competitive) banking into the community.
Bankers would attract
savings by offering to pay interest, while requiring that interest be paid on loans
finance the early acquisition of the bicycle.
Those deferring
made
to
These borrowers would enjoy the durable's
services for a longer period of time at the cost of the interest
funds.
Imagine
their purchase of the bicycle
payments on the borrowed
would receive
interest
on their savings
with which to finance greater non—durable good consumption over the course of their
Under
ideal competitive conditior s a perfect credit
of interest rates such that,
when
market would establish a pattern
indivifiuals optimally formulate their intertemporal con-
sumption plans, the supply of savings -vould equal the demand
Such a market
above.
It is
is
evidently a
Of
each period.
the Roscas described
wonder how the intertemporal allocation of durable and
natural, however, to
particular interest
for loans in
much move complex mechanism than
non—durable consumption which
lives.
procuces compares with those attained under Roscas.
it
how do
are the welfare comparisons:
perform relative to a fully functioning
these simple institutions
market?
cr<^dit
m. The Model
We
is
consider an
economy populated by a continuum
continuous and consumers have finite lives of length T.
Time
of identical individuals.
These assumptions allow us to
treat the time individuals decide to purchase the indivisible good, the length of Roscas,
the fraction of the population
variables.
who have
the indivisible good at
his lifetime at
in the
economy
a constant rate of
receives an exogenously given flow of
y>0.
imagine that the durable good
Once purchased, we assume
it
economy.
We
lifetime.
Moreover, we assume that
assume,
it
for
and the services of a durable good.
and can be purchased
indivisible
market whose price
—one must own
c,
utility
at a cost of
B.^
does not depreciate, yielding a constant flow of services
remainder of an individual's
for a large (global)
agents
that
is
income over
At any moment, individuals can receive
from the flow of a consumption good, denoted by
for the
any moment, as continuous
This avoids some technically awkward, but inessential, integer problems.
Each individual
We
and
ti-e
is
We
assume that the durable good
is
produced
unaffected by the actions of agents in our
services of the durable are not fungible across
to enjoy its services.''
analytical
convenience,
that
separable, and quasi—linear in the flows of consumption
preferences
are
intertemporally
and of the durable's
services.
At
each instant an individual receives a flow of utility at the rate v(c)
durable good, and v(c)
4-
i^
count the future. Thus there
These
main
utility function v(
Assumption
0,
make our
assumptions
affecting the
v"(-) <
1:
v:
no motive
is
insights of the paper.
demand
for
saving except to acquire the indivisible good.
cumbersome without substantially
less
We make
the following assumption concerning the
):
•
K
-i IR
three times continuously differentiable, and satisfies v'(-)
is
and v"'(-) >
>
0.
we
strictly concave,
This assumption, which
positive third derivative.
satisfied
for
much
analysis;
As well as being increasing and
is
he does not have the
Moreover, we assume that individuals do not dis-
he does.
if
if
by most plausible
utility functions.
is
require the utility function to have a
not essential for
It is
much
of the analysis,
sufficient for individuals to
have a
precautionary savings (see Leland (1968)) and also guarantees proper risk
aversion in the sense of Pratt and Zeckbauser (1987).
A
consumption bundle
<s,c(-)>, where
K
s
G
^e
for
[0,T] U {u}
-t
mean
the individual
different
non—durable
never receives the durable good.
generality, that the population
is
a6[0,l].
bundles, one for each consumer type, and
s:[0,l]
We
-»
^and
An
may
allocation
is
By
c:[0,l]x[0,T] -
=
u"
we
without loss of
we index
then a set of consumption
The
[R
relabelling individuals as required,
generality that individuals with lower index
letting
suppose,
"s
be represented by a pair of functions
referred to as the assignment Junction, tells us the dates at
receive the durable.
By
at each date.
uniformly distributed over the unit interval and
consumers with numbers
<s(-),c(-,-)>, such that
described by a pair
denotes the date of receipt of the durable good, and
gives the rate of consumption of the
c:[0,T]
may be
an individual in this economy
function s(-), hereafter
which
different individuals
we may assume with no
numbers receive the durable
a denote the fraction of the population ever
to receive the durable,
earlier.
loss of
Thus,
we may suppose
that s(-)
s~
[u)
is
=
non-decreasing on
q
<
[0,q],
<
and everyone receives the good
analysis will focus.
The second component
and that s~
1,
—
=
(u)
(a,l].
If
a =
1,
then
being the case on which most of the
this
of an allocation gives us the consumption path
{c(a,r):re[0,T]} of each individual a.^
Under the allocation <s,c>, an individual
v(c(a,r))dr
u(a;<s,c>) =
(3.1)
+
of type
T
feasible, the allocation
time interval.
we
enjoys utility:
s(a)6 [0,T]
e(T-s(a)),
s(a)=i/.
v(c(a,r))dr,
To be
a
To make
must consume no more resources than are available over any
any assignment function s(-) and any date re[0,T],
this precise, for
define N(r;s) to be the fraction of the population
Then an
time r with assignment function s(-).9
which has received the durable by
allocation
<s,c>
is
feasible
if,
for all
r£[0,T]:
rT
(3.2)
The
rl
c(a,x)da)dx
(y
left
hand
>
N(r;s)B.
side of (3.2) denotes aggregate saving at r
Hence, an ailocation
aggregate investment.
exceeds aggregate saving. In the sequel
we
is
will let
and the
right
feasible if aggregate
F denote the
hand
side denotes
investment never
set of feasible allocations.
IV. Alternative Institutional Arrangements
The
object of this paper
allocating savings
and
it
is
is
to
compare
different
institutional arrangements for
the task of this section to describe four possible ways of allo-
cating resources in our model
—
Autarky, a
Random
Rosea, a Bidding Rosea, and a perfect
credit market.
Each
institution that
we
consider will result in a particular assignment
function and set of consumption paths, and thus a particular distribution of utility on
We can evaluate the
institutional alternatives
by comparing these
[0,1].
utility distributions.
IV. 1 Autarky
The simplest
institution
case saving occurs in isolation.
— i.e.,
Autarky
is
An
no financial intermediation
some period
of accumulation
program exceeds
durable.
T-v(y),
i.e.
all.
In this
individual desiring to acquire the durable, given strictly
concave utility and no discounting, will save the required sum
over
at
[0,t].
Acquisition
is
B
at
desirable
if
the constant rate
y—c,
the utility from such a
the life—time utility attainable without saving to
buy the
Consider, therefore, an individual's life—time utility maximization problem of
choosing c and
to:
t
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 this problem and
To develop an
Kfl
(4.2)
Assuming that
insightful expression for
t
.
< T
^
W
.
optimum
it is
i
>-
in (4.1),
lo
o^c?y r'^'y!i''''^l.
at the
,
let
W
'
W^ =
denote the maximal value.
convenient to define
"
then substitution and rearrangement
leads to the following:
(4.3)
.
T.[v(y) + e]-B-KO-
10
Expression (4.3) admits an appealing interpretation, anticipating a more general result
The value
obtained in section V.
the difference between the two terms on ihe right hand side of (4.3).
lifetime utility
would be
if
Autarky
of opiimal acquisition of the durable under
the durable v/ere a free good.
The
first
is
equals what
Hence, the second term
may
be
understood as the utility cost of acquiring the durable imposed by the need to save up.
From
(4.2), it
easily seen that this cost
is
is
directly proportional to the price of the
durable and to the loss of utility per unit time during the accumulation phase, and
The optimal autarkic consumption (and
inversely proportional to the length of that phase.
hence savings) rate, c.
It
chosen to minimize this cost.
is
should be noted that the functi on
—
function
,
it
non—negative,
is
/i(
•
)
It
desirable
is
i.e. t
.
=
W
.
t
.
,
may be
Moreover, from the
obtained by differentiating the
B-n'{(). These facts will be used repeatedly below.
apparent from (4.3) that
[i.e.
possesses all the usual properties of a cost
increasing and strictly concave ii.
Envelope Theorem the optimal acquisition date,
cost function,
is
>T-v(y)]
if
and only
a.^.quisition of
if
the durable under Autarky will be
B-/i(^) < ^-T.
We
formalize this requirement in
the assumption that follows:
Assumption
Let £(y,^) be defined by /i(£)/^
2: (Desirability)
Desirability requires that
some combination
of the lifetime over which these services
=
of the utility of durable services
may
enough relative to the cost of the durable (B) to make
Even
if
institutional arrangements.
If
the durable good
two phases.
We
is
is
(if),
the length
be enjoyed (T), and the income flow (y) be
large
durable good acquisition
Then (>i{y,'^)A^
J.
acquisition worthwhile.
its
not desirable under Autarky,
it
may
be so under other
shall return to this point below.
desirable, then lives
under Autarky
will
be characterized by
In the first of these individuals save at a constant rate until they have
sufficient funds to
buy the durable and,
since individuals are the same, this phase
and the
12
member
representative
of this Rosea perceives his receipt date for the pot to be a
Given Assumption
variable, f, with a uniform distribution on the set {t/n, 2t/n,...,t}.
member
each
saves at the constant rate equaJ to B/t over the
member's lifetime
utility is the
random
The continuum
it
is
meeting
may
at
+ (T-t).(v(y)+0 +
be understood as the limit of this
is
given by the expectation of (4.4), taking
W(t)
=
t.v(y-B/t)
In the above discussion
We
W(t).
budget constraint in
and, assuming
tT%
<
in
is
(4.1)).
it
t-p
and use
on
(T-t).(v(y)+e)
will be
of the
W^
joins the
[0,t].
+
r,
Thus the
[0,t].
Rosea of length
t
is
Hence,
t^/2.
Random
Rosea,
t,
as given.
It
chosen to maximize members' ex-ante
of choosing
to represent the
t
to
maximum
maximize
expected
almost identical to that in (4.1) (after solving for
We may
becomes a
therefore proceed
by analogy with
(4.1)
c
(4.5).
utility.
from the
through
(4.3)
deduce
that
T, write i^
Wj^=T.[v(y)+{]-B-M{/2).
(4.6)
As
+
Thus consider than the problem
denote the solution by
Notice that this problem
who
r to be uniform
we took the length
seems natural, however, to assume that
—
n
case as
finite
uniformly distributed on the interval
ex ante expected utility of a representative individual
expected utility
(t-r)^.
each instant of time, and the receipt date of the pot,
continuous random variable which
(4.5)
Thus each
of the Rosea.
As n grows, the Rosea meets more and more frequently. ^^ In the
approaches infinity, i^
limit
case
1,
variable:
W(t,T) = t.v(y-B/t:
(4.4)
life
random
the
case
of
Autarky,
we can use
the
Envelope
Theorem
to
.
13
tR=B-/i'(^/2).
Expression (4.6) reveals the elegant simplicity of our model of a
Comparing
(4.6)
upon Autarky.
with (4.3),
The
is
W-p
i.e.
—
W
.
=
B-[/i(^)
by the Rosea lowers the
—
/i(^)]
>
0,
sinee as
utility eost
we noted
above,
increasing.
So
far,
we have taken
This does, after
granted.
work and eaeh member
t-n/2 units of time
So even
if
of a
Random Rosea
is
savings in a Rosea are being put to
expects to enjoy the services of the durable
undesirable under Autarky,
This situation does indeed occur
it.
model of a Random Rosea developed here to be consistent,
if
This will be true only
has received his winnings
is
if
seems reasonable
it
Second, the finite formulation whose limit
of meeting dates of the
Rosea
.
First, for the
must be that each member
pot, as
is
assumed
in
the horizon remaining after the last
long enough to justify the investment, which will
depend upon the consumption value of the durable's
number
it
Wt^>T- v(y)>W
would actually choose to invest in the durable good upon winning the
member
rate under Autarky.
however, two more subtle issues related to desirability.
the foregoing derivations.
for
would prefer to form a Random Rosea and acquire the durable
good rather than go without
belies,
The
more than he would by choosing the same saving
acquisition of the durable good
This
Random Rosea
the question of desirability under a
seem reasonable.
all,
to suppose that the agents
this
Rosea.
Random Rosea improves
immediately apparent how a
finaneial intermedia ion provided
of aequiring the durable,
(jl{-) is
it
Random
is
is
services.
characterized in (4.6) assumes that the
equal to the
number
of
members.
Relaxation of
assumption allows for the possibility that members might elect to face an improper
probability distribution for the
random
receipt date
r.
Suppose, for example, that with an
an even number of members, meetings occurred on dates {2t/n,4t/n,6t/n,...,t}.
make
his contribution of
rate of B/2t.
B/n
at
Now,
to
each meeting, an individual would save at the constant
In the continuum case, conditional on winning the pot, each agent would face
a uniform distribution of receipt dates r£[0,t], but the unconditional probability of ever
14
receiving the pot would only be
would be preferred to the
one—half. Could
it
"fully funded'' version
be that such a "partially funded" Rosea
whose version whose welfare
is
given in
(4.5)?
To guarantee
all
member;
that Rosea
will
always want to design a Rosea so that they
eventually receive the pot and that bvying the durable
we
winning,
is
always an optimal response to
require a stronger desirabili-.y assumption than
turns out that
if
i^
is
we made under Autarky.
It
twice as big as the minimal value needed to induce acquisition of the
durable good under Autarky, then (4.6) does indeed characterize the performance of an
optimal
Random
Assumption
we
Rosea. Thus
replace
2': (Strong Desirability)
Assumption 2 with:
T
( > 2£{y,Tj).
Then we have:
Lemma
1:
Under Assumption
receiving the pot during the
Random Rosea
Proof:
2'
an optimal
of the Rosea.
life
will desire to use the
Random Rosea
involves every
member
Moreover, every participant of the optimal
proceeds to acquire the durable good.
See the Appendix.
In fact, specifying the
essential for
compares
much
Random
Random Rosea
is
Our
of our comparaiive welfare discussion in section VI.
is
not
analysis there
Roscas constrained to provide probability one of eventual receipt of the
pot to alternative arrangements.
institutions
so that everyone receives the pot once
The
fir
ding that a
Random Rosea
is
preferred to other
only strengthened by allowing the possibility that not everyone wins.
the other hand, partially funded Roscas do not appear to be
need Assumption 2', however,
for its implication that all
common
in practice.
We
On
do
winners will prefer to invest in
-
15
the durable good.
The
Otherwise the analysis culminating in (4.6)
tantamount
is
Assumption
2 assures that "all"
an Autarkic individual.
Assumption
"anything
"all" is preferred to
2'
less", i.e.
is
like being offered a fraction of
is
— the
preferred to "nothing"
must be stronger
since
it
only choices facing
has to guarantee that
it
intermediate levels of durable consumption must
be dominated.
Let <Srj,Crj> denote the allocation achieved with the optimal
Strong Desirability.
Recalling that an individual's type, a,
individuals have identical consumption paths which
fall
into
Random Rosea
As under Autarky,
two
distinct phases.
more
precise, CT^(Q:,r)=CTj, for Te[0,t-p]; and, CT^(a,r)=y, for re(tx^,T],
The
allocation
consumption
population
is
with
the
optimal
Random Rosea
is
under
identified with his order of
is
receipt of the durable, the assignment function will be s-n(a)=at-p.
lb).
comes from noting
to introducing divisibility of the durable good, since
being offered a probability of receiving the durable good
also
inconsistent.
intuitive reason for the stronger desirability requirement
that randomization
tx ante.
is
exhibited
all
To be
where CT%=y—B/t^.
Figure
in
1.
While
a piecewise horizontal function for each agent, the fraction of the
still
who have
received the durable at time
t is
now
increasing
and
linear (see Figure
Moreover, while exhibiting the same general pattern for consumption, we shall show
below that the accumulation phase
will not last the
same length of time
as under Autarky,
i"
IV.3 Bidding Rosea
We
turn next to the Bidding Rosea, where the order in which individuals receive the
accumulated savings
assumption to make
is
is
determined by bidding.
As discussed
that the bidding takes place
when
in section II, the simplest
the Rosea
is
formed
at
time zero,
and involves individuals committing themselves to various contribution rates over the
of the Rosea.
We
do not model the auction
explicitly,
what we believe would be the most plausible outcome in
life
but instead simply characterize
this context.
Specifieally
we
will
^
16
require that, for a Bidding Rosea of length
r6[0,t]
A
commits
an individual receiving the "pot" at date
t,
to contributing into the Rosea at the constant rate b(r) over this interval.
an equilibrium
set of bids {b(r):T6[0,t]} constitutes
by out bidding another
for his place in the queue;
(i)
if:
and
no individual could do better
contributions are sufficient to
(ii)
allow each participant to acquire the durable upon receiving the pot.
an obvious requirement of any bidding equilibrium.
of saving outside the Rosea.
Now
(i)
v(y-b(r))
while condition
(ii)
would
r6[0,t] is given by:
then implies that, for
(4.7)
ia>^ing
+
all
(1
u
condition
first
The second precludes the
and thus be
lie idle
Bidding Rosea of length
in the equilibrium of a
receiving the pot at time
Condition
Any such
The
=
t,
is
desirability
inefficient.
the utility of an individual
tv(y-b(r))+(T-t)(v(y)+(^)+(t-r)^\
r,r'G[0,t]:
-?)>'
=
v(y-b(r'))
+
(1
"f )e;
implies that
rT
b(r)dr=B.
(4.8)
Jo
These two equations uniquely determine the function b(-), given the Rosca's length
optimal Bidding
representative
To
member
is
will
length
its
maximized subject to
who
receives the pot at date
chosen
(4.7)
and
so
that
the
utility
level
of
a
(4.8).
r,
which we denote by c(r) = y
—
b(r) for r
describe the Bidding Rosea just as well with the function c(-) as
considering the bids directly.
variable
have
characterize the optimal Bidding Rosea, consider the consumption rate of an
individual
We may
Rosea
An
t.
a
=
r/t, yields
Substituting this into (4.8) and
6 [0,t].
we could by
making the change
of
17
t.[y-
(4.9)
=
c(a)da]
[
B.
Jn
Now
consider the consumption level, c
xt, (xg[0,1])
Hence
all
and note that
=
c(x), of
an individual
=
(4.7) implies that c(a)
who
receives the durable at
y~^{v{c)-{x-a)i), for
all
a6[0,l].
other individuals' consumption levels can be determined from the equal utilities
condition once c
known.
is
average consumption level and using (4.9),
utility level of the
(v(c)—(x— a)^)]da as the
Moreover, defining A(c,x)e/^[v
member who
we may
write
t
= B/(y—A(c,x)).
Hence the
receives the pot at date xt can be written as:
W(c) = T.[v(y)+^] -B.[v(y)-v(c)+xe]/[y-A(c,x)]
(4.10)
Since, all
particular
members have
member
identical utility levels,
by construction, maximizing the
the same as maximizing the
is
common
Thus we can view
utility level.
the problem solved by the Bidding Rosea as choosing c to maximize (4.10).
denote the optimal consumption level for type x individuals and
level achieved at the
WB
(4.11)
let
Wp
utility of a
Let Cd(x)
denote the welfare
optimum. Using by now familiar arguments we may
write:
= T.[v(y)+^]-B.MB(0,
where
(^(0
=
Min
vYvj^-v(c]+x^
0<c<y
y-A(c,x)
e>o.
This admits the same interpretation that
Bidding Rosea, as the difference between
saving up.
Note that ^-d
2-nd
utility
we noted
for
both Autarky and the
were the durable free and the cost of
mean consumption
at
the optimum, denoted by
Ag
(EA(Cp(x),x)), are both independent of x by construction.
length of the Bidding Rosea, which
Once
by tg=B/(y—Ag).
tg, will be given
again, the above formulation assumes that individuals will always
when they
durable
we denote by
Note also that the optimal
'B
Thus
receive the pot.
if
buy the
turns out to be the case that
it
some
individuals could actually do better by not purchasing the durable, the analysis will be
Intuitively,
inconsistent.
condition
given
in
seems reasonable to suppose that the (weak) desirability
it
Assumption
2
should
rule
intermediation afforded by the Bidding Rosea
mean
sub—section
establish the validity of this conjecture.
large" relative to y, B, and T,
provide appropriate bounds.
it is
is
While
it
the
financial
not random, none of the issues
Unfortunately, however,
arise.
all,
that the costs of saving up are less
than under Autarky and, since the allocation of the pot
raised in the previous
After
out.
this
is
clear that
we have been unable
i^
must be
to
"sufficiently
not cbvious that Assumption 2 or even Assumption 2'
Hence the exact condition necessary
to ensure desirability in
the optimal Bidding Rosea awaits determination.
Let <SxD,c-p> denote the allocation generated by the optimal Bidding Rosea.
with the
shall
the
Random
compare
Random
Rosea.
t-p
Rosea, the assignment function
with
tr>,
the length of the optimal
phase in which agents consume
IV.4
i.e.
Random
s-p(a)
Rosea.
=
crt^.
Below, we
Unlike Autarky and
is
all
is
similar with an accumulation phase followed by a
of their incomes.
Hence the allocation of non—durable
described by c-n{a,r)=c-r,{a), for rG[0,tr>] and ep(a,r)=y, for re(tT5,T].
The Market
The
final institution that
specify the behavior of such a
interest rate at time
time
linear,
Rosea, each individual receives a different consumption path under a Bidding
However the general pattern
consumption
is
As
r, i.e.
6{t) =
r.
It is
we consider
market
is
a credit market.
in the following
way.
In the present context,
we
Let r(r) denote the market
convenient to define 6{t) as the present value of a dollar at
exp(— /p.r(z)dz) and
to think of the
market
as
determining a sequence of
19
present value prices {(^(r):rG[0,T]} at which the supply of and
are equated.
who buys
Hence, an individual
demand
the durable good at time
for loanable funds
s
pays 6{s)B for
it.
Given the price path, an individual chooses a purchase time s(a) and a consumption path
{c(a,r):rG[0,T]} to maximize utility.
Hence, the optimization problem solved by each individual
is
given by
'^
Max
v(c(r))dr
{c(-),s}
+
^(T-s)
(4.12)
<5(rk(r)dr
subject to
Jo
+
5{s)B < y
(5(T)dr, s G [0,T].
Jo
'
This assumes that each individual decides to purchase the durable at some time.
again, this will require that ( be big enough, given T, y
is
true in what follows. 20
and B.
[y"'0
Since
all aG[0,l],
and second,
first,
<S|^(a),c^(a,r)> must
individuals
Cj^(a,r)da]dr=N(t;Sj^).B.
"^0
are
identical,
the
first
condition
implies
savings equals investment
at
all
points
in
time.
that,
can be written in a form analogous to
(4.3), (4.6)
and
in
equilibrium,
The second condition
We
use
equilibrium level of utility enjoyed by agents in a market setting.
this
this
at each date tG[0,T],
individuals are indifferent between durable purchase times.
that
assume that
rl
rt
(4.13)
shall
Hence, we define a market equilibnum to be an allocation
<Si^,C|^> and a price path 6{-) satisfying two conditions:
be a solution to (4.12) for
We
Once
W,,
to
all
just says
denote the
Below, we show that
(4.11).
Direct computation of competitive equilibrium prices and the associated allocation
is
difficult,
even in the simple case of logarithmic
utility explored in section VI.
However,
20
we
are able to infer the existence and
seme
of the properties of competitive equilibrium in
our model by using the fact that, with identical agents,
efficient allocation
which gives equal
it
utility to every agent.
must coincide with a Pareto
Hence, explaining properties of
the Market allocation must await considi ration of efficient allocations
more
generally.
This completes our description and preliminary analysis of the different institutional
frameworks which are dealt with
Our next
parisons of them.
in this paper.
task, however,
iency for the model described so
far.
is
In section
VI we
shall
make
com-
detailed
to discuss the implications of Pareto effic-
This will aid the study of competitive allocations,
while also providing insight into the baoic resource allocation problem posed here by the
existence of an indivisible durable good.
V. Efficient and
We
Ex Ante Optimal
AIloCc.tions
have now shown how
institutions
vai'ious
for
mobilizing
savings
produce
To
particular feasible allocations and associated distributions of utility in the population.
evaluate the allocative consequences of alternative institutions for financial intermediation
we
require
some welfare
criteria.
Two
such criteria are natural here.
Pareto efficiency, or more simply, efficiency.
efficient if there is
no alternative
individuals strictly better
well
off,
We
feasible allocation
while leaving
all
The
first is
consider an allocation to have been
which makes a non—negligible
<s,c>
but a negligible set of individuals at least as
is
criterion
is
defined in terms of ex ante expected utility.
better than <s',c'>, in this sense
if
/^ u(a;<s,c>)da
>
The
allocation
/^ u(Q;;<s',c'>)da.
The
a
indi-
preferred allocation gives a higher expected value of the lifetime utility of a type
vidual, with
if
set of
off.
The second
This
ex post
is
a regarded
a "representative
as a
man"
random
variable uniformly distributed on the unit interval.
criterion
which asks:
which allocation would be preferable
one did not know to which position in the queue one would be assigned?
We
are
21
particularly interested,
expected
utility.
We
call
same
individuals enjoy the
in
therefore,
the
this
that
allocation
which yields the highest level of
Notice that, since
ex ante optimal allocation.
level of utility
all
under Autarky, the Bidding Rosea, and the
Market, the two criteria coincide when applied to allocations emerging from any one of
This
these institutional forms.
allocation generated
preferred to
we
Indeed,
In
will present
Rosea.
It is
possible that an
efficient allocation in
itself
be
terms of the ex ante optimality criterion.
an example in which precisely this reversal occurs.
we
follows,
characterize efficient allocations for our
While
ante optimal allocation.
reader whose primary interest
economy and the
ex
this section is the analytic cornerstone of the paper, the
lies
in the results
focusing particularly on Corollary
allocations
Random
by a Random Rosea might be Pareto dominated, and yet
some Pareto
what
not the case for the
is
1
may
wish just to skim the next few pages,
which describes the essential properties of
and the discussion of the optimal allocation following Theorem
efficient
2.
V.l Efficient Allocations
To
Qe[0,l],
-t
characterize efficient allocations,
normalized so that lr.6{a)da
K
6 integrable
and jr.6{a)da
I
=
=
1}.
1.
we introduce weights
Define the set of
By analogy with
if
<s',c'>
is
/Q^(a)u(a;<s',c'>)da
efficient
>
then there must be a
/^ ^(a)u(a;<s,c>)da, for
we
set
sum
of weights
^'^^^
<s,c>£F
=
^(a)u(a;<s,c>)da,
-Jo
:[0,1]
in
B
such that
<s,c>. In order
therefore study, for fixed ^£0, the
rl
W(^;<s,c>)
each agent
of individuals' utilities.
problem:
max
for
such weights: Q={0
all feasible allocations,
to investigate the properties of efficient allocations
/r .\
>
a standard argument in welfare
economics, an efficient allocation should maximize a weighted
Hence,
all
^(a)
22
for
u(a;<s,c>) as deCned
W^ =
problem, and
Although
Let <s«,Ca> denote an allocation which solves this
in (3.1). 21
W(^;<s^c^>) denote
(5.1)
may appear
the maximized value of the objective function.
to be a
"non—standard" optimization problem,
solved explicitly under our simplifying assumptions on preferences.
Theorem
ited in
below.
1
length in what follows.
We
develop the
The proof
Theorem and
The
it
can be
is
exhib-
solution
discuss its implications at
some
of this result, presented in the Appendix, exposes the
nature of the resource allocation problem which any institution for financial intermediation
must confront
The
in our
first
economy.
point to note about the efficiency problem
stages, loosely corresponding to static
level of aggregate
weighte.l
sum
and dynamic
that
The
efficiency.
it
can be solved in two
first
consumption be optimally allocated across individuals,
of instantaneous utility
at
which everyone receives the durable
at
We
of the efficiency
distributed
among the
r, i.e. c
Jt)£J ^c p{a,T)da.
problem should convince the reader that
in period r.
maximizes the
the optimum.
different types of individuals so as to
from consumption
i.e.
once again, on the case in
shall focus,
Let c«(r) denote aggregate consuinption in period
inspection
requires that any
each date, while the second determines the
optimal acquisition path for the durabk good.
utilities
is
To
state this
more
this
A
brief
should be
maximize the weighted sum of
precisely, define the
problem
1
Max
0{a)y{x{ct))da
(5.2)
rl
subject to
for all
w >
0.
x{ck)da
—w
Let Xo(*)W) denote the solution and
objective function.
Then
a type
a
indi\'idual's
let
Va(w) denote the value of the
consumption
Cn{a,T)=Xp{ci,cJT)), and total weighted consumption utility
at
is
time Te[0,T]
is
given by
given by V^(Cfl(r)).
It
23
sJa) and cJr).
remains, therefore, to determine
Note
that <s,c>
that since preferences are strictly monotonic, the feasibility constraint
first
F
e
[see (3.2)]
may
rT
JQ
That
discounting,
jss of generality
be written
if
as:
c(a,r)da]dy
=
N(r;s)B, Vre[0,T].
Jo
savings must be put immediately to use.
all
is,
1
rl
[y-
(5.3)
without
Moreover, in the absence of
the flow of aggregate savings /^[y—c(a,r)]da equals zero at
then efficiency demands that
it
must
also be zero at
any
t'>t.
later date
some date
r,
Otherwise, by
simply moving the later savings forward in time one could assign some agents an earlier
date
receipt
consumption.
for
the
durable
'
without
The foregoing implies
reducing
anyone's
=
s~^{t), for rG[0,s(l)], and
N(r;s)
=
1, for
We
we have y-c(r)=B/s'(s
Therefore,
we may conclude
(r)),
and
for
all
can use these facts to
re(s(l),T].
expression for N(t;s) in (5.3) and differentiating with respect to
re[0,s(l))
must be
This in turn implies that such an assignment
function will be invertible and different ial)le almost everywhere.
N(r;s)
non—durable
that any assignment function solving (5.1)
continuous, increasing, and satisfy s(0)=0.
write
from
utility
r,
Te(s(l),T]
we
Substituting this
find that for
all
we have y-c(r)=0.
that the following condition must be satisfied by any solution
to (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,
can be no outright waste of resources.
i.e.
Hence, for part two of the solution,
interested in the problem of choosing functions
sJa) and cJr)
to
maximize
there
we
are
24
rT
rl
V^(c(r))dr+eT-^
(5.5)
subject to (5.4).
By
0{a)s{a)da
we
analogy, with our earlier analysis
define
rV^(y)-V^(a)+7.
min
fiJ-y) E
(5.6)
y-a
0<(T<y
and denote the solution by
Theorem
for
1:
cr
Ai).
We now
,
7>0,
have:
Let <s,c> be an efficient allocation with s~ (j^)=0, and
which <s,c> provides a solution in
(^.l).
Then
let
^£0 be the weights
the maximized value can be written in
the form
W^=T.[V^y) + {]-B.j
and the assignment function
/.^{/^^(z)dz)dx
satisfies:
a
s(a)
Moreover, for
all ae[0,l],
c(a,s(x))
Proof:
=
=
B.
[y-a m]0{z)^)] ^dx,
a6[0,l].
non—durable consumption obeys
x^(Q!,^^(ax^(z)dz)), for xe[0,l]; and c(a,r)
=
x^a,y),
for re(s(l),T].
See the Appendix.
As noted, Pareto
waste of resources.
allocated
efficiency requires
First,
efficiently
two conditions beyond the absence of physical
any given aggregate
among
individuals
and
level of
non—durable consumption should be
second,
it
must optimally manage the
intertemporal trade—off between aggregate consumption of the
diffusion of durable ownership.
We
discussed above
non—durable and
how V^-) summarized
the
faster
first
of
25
More needs
these stages.
to be said about the
dynamic
efficiency considerations
—
in
particular, the relevance of the minimization conducted in (5.6).
To
see this consider the expressi;;n for
the difference of two terms.
utility
sum
if
The
Crsl,
W« in
1.
This welfare measure
is
T-[V^y)+,^], would be the maximal weighted
The second term
the durable were a free ijood.
equivalent) cost of acquiring the durable
Theorem
It is this
which
cost
is
is,
therefore, the (utility
minimized in
(5.6).
It
has
two competing components: non—durable consumption foregone in the process of acquiring
the durable (since c(s(a))
<
y)
and durable services foregone in allowing some non—durable
consumption (since s'(a) < B/y).
sum
of these
two components
duration of the
therefore
Consider a small interval of time a6(a,a+da), then the
approximately [Y Jy)—V Jc{s{a))+^j
is
time interval
s'(a)da=Bda/[y—c(s(a))].
is
means minimizing the product of these terms
at
0{z)dz], while the
Efficient
each aG[0,l].
accumulation
This
is
precisely
the problem described by (5.6).
A
Figure
2).
geometric treatment of the minimization problem (5.6)
The
function
V J-)
is
falls
Therefore, choosing a to minimize the ratio
means finding that point {a,VJa)) on the graph of V^(-) such
that the straight line containing
graph of V^(-).
also be helpful (see
smooth, increasing and strictly concave because we have
assumed that v(-) has these properties.
[Vfl(y)—V«((7)+7]/[y— a]
may
it,
and containing the point (y,V^(y)+7),
Notice from the diagram that
o'fl'f)
is
must be decreasing,
tangent to the
rising to
y as 7
to 0.
This observation, together with Theorem
1,
permits us to deduce some useful
properties of efficient allocations:
Corollary
(i)
1:
Let <s,c> be an efficient allocation with s" (i/)=0.
the assignment function s(-)
1
im
o-tl
is
s'(a)
Then
increasing, strictly convex
=
+0D,
and
and
satisfies
26
for
(ii)
c(a, •)
ae[0,l],
all
is
increasing on the interval
and constant
[0,s(l)],
thereafter.
Proof:
(i)
In view of (5.4) and
=
s'{a)
satisfies
B/[y-CTi,^/
and approaches y as 7
(ii)
6{z)dz)], for
to 0.
1,
is
decreasing,
we know
that any efficient allocation <s,c>
some Oe&.
As noted above (^Al)
is
decreasing
Hence, the result.
This follows immediately from Theorem
that crJ'y)
The
falls
Theorem
1 after
noting that Xni<^r)
is
increasing and
d
properties of the assignment function imply that, in an efficient allocation, the fraction
who have
of the population
received the durable by time r
is
increasing and strictly
In addition, the rate of accumulation (the time derivative of N(r;s)) approaches
concave.
zero as r goes to s(l).
Finally,
tion in
Theorem
Rosea in
worth drawing attention to the relationship between the characteriza-
it is
1
and the welfare expressions found
and
(4.6)
for the
Bidding Rosea in (4.11).
for
Autarky
These expressions
general form, allowing the intuitive interpretation that welfare
achieved
when the durable
durable's services.
is
in (4.3), for the
is
all
Random
take the same
the hypothetical utility
a free good, net of the implicit utility cost of acquiring the
This observation
is
the basis for most of the results in section VI.
V.2 The Optimal Allocation
The optimal
weighted.
must
is
that efficient allocation in which individuals are equally
Since individuals are identical and are assigned types randomly, this solution
maximizes the
we can
allocation
ex ante expected utility of a representative agent.
Hence, noting that 1G0,
write ex ante expected utility as W(l;<s,c>), and the optimal allocation <s
satisfy:
W(l;<s
,c
>)
>
,c
>
W(l;<3,c>), V <s,c>eF.
Notice also that V,(w)ev(w), and Xi(a,w)Ew, for
all
(a,w).
This implies that
if all
27
agents have equal weights and utility
consumption should be allocated equally among
/i,(q)E//(q),
where
Theorem. These
Theorem
2:
/i(-) is
facts,
Let <s
>
all
defined in (4.2), and that
together with
,c
concave, then aggregate
strictly
is
Theorem
agents.
^i'
It is
non—durable
also useful to note that
{'y)=[y—(Tjj)]~
,
using the Envelope
1, yield:
be the optimal allocation.
Then, ex ante expected
utility
can be
written in the form
rl
W(l;<s^,c^>)
E
W^ -
and the optimal assignment function
s
Moreover,
for all aefO,!],
Co(^'SoW)
Proof:
(i')=0,
population
when
we
observations,
we show
who
s
-B.
/i(e(l-a))da,
satisfies
= 3-[
(a)
/i'(^(l-x))dx.
non—durable consumption obeys
y-l/M'(^(l-x))
Provided that
foregoing
s
=
T-[v(y)+e]
for xe[0,l],
(i^)=0, the
after
Theorem
noting
that
that welfare under the
ever get the durable,
a.
/
and c^(a,r)
for re(sQ(l),T].
y
follows at once
from Theorem
0{z)dz=l l(z)dz=l— a
optimum
Hence,
=
let
W
a fraction a of the population get the durable.
is
fa)
=
T.[v(y)+Q^]
-
B-
[
(a) denote
(a
that
utility
Arguing as in the proof of Theorem
//(e(^a))da.
_
/i'(^(«-a)){da-B/x(0),
Jq
which, after using the Fundamental Theorem of Calculus, simplifies to
W^(a) = Te-B./i({a).
To prove
maximal expected
Differentiating with respect to a, yields
W'(a) = T^ + B-
and the
increasing in the fraction of the
obtain:
W
1
1,
28
By Assumption
W'(a)
>
0^ '
In
we know
2,
that
T(
>
Bfj(^).
Since
fi(,-)
increasing,
is
it
follows that
0.
addition
to
the
properties
outlined
allocation also gives individuals identical
c (0) equals c.
,
Corollary
in
1,
therefore,
consumption paths. Theorem 2 also
the optimal
tells
us that
the consumption level in the accumulation phase under Autarky22.
The
pattern of consumption and the fraction of the population which has acquired the durable
good in the optimal allocation are illustrated, as a function of time, in Figure
individual has an identical consumption path beginning at c
The
the end of the accumulation phase.
increasing
It
allocation
earlier, if
and concave over the
is
and
1.
Each
smoothly to y
at
fraction of the population with the durable
is
.
rising
interval of accumulation.
helpful to note the relationship
between the problem solved by the optimal
and that of accumulating a perfectly
divisible
good under autarky.
As noted
the durable good were perfectly divisible, then there would be no gains from trade
and Autarky would be optimal. 23
creates the problem.
under perfect
It
Nonetheless, the
divisibility,
is
precisely the indivisibility of the durable which
economy may approximately
replicate the situation
even in the presence of indivisibilities, by randomly assigning
individuals to positions in the queue at the initial date.
This
is
tantamount to granting
each individual a "share" of the aggregate amount of the durable good available at any
subsequent date.
The exact
link between the
Here the function K(r)
holds at time
r.
is
two problems
is
spelled out in the following
lemma.
to be interpreted as the stock of the divisible asset the individual
29
Lemma
2:
The optimal aggregate consumption 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.
See the Appendix.
Proof:
Thus the consumption path
of
non—durable consumption
precisely that attained by an
is
individual accumulating a perfectly divisible durable good.
durable
services,
financial
intermediation
provides
the
In the absence of trade in
only
means
to
limitations imposed on the agents by the constraint of indivisibility.
completely overcome in
allocation
it
same
would be under
as
it
is
:,he
overcome the
In the optimal
sense that ex ante expected utility
is
just the
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) in any
equilibrium with indivisibilities.
of section VI.
It is
This
is
z.
general point, and
it is
why
it is
important therefore to understand
key to
many
true.
Assumptions about
of the results
preferences affect the extent to which randomization provides a perfect substitute for
divisibility,
but not whether
presence of indivisibility.
it
improves over deterministic, equilibrium outcomes in the
From an
ex ante viewpoint, the representative agent
the deterministic assignment function
bearing
effect,
risk.
beginning on or before
By
type
that
r,
sees
•
)
without knowing
a probability
and evaluates
his
N(r;s)
his types
of enjoying the
a
is
faces
already, in
durable's
services
expected utility accordingly.
contrast, adopting an ex post viewpoint on any equilibrium, an agent
a and
is,
He
s''
who
facing assignment function s(-),
must be
he must not want to be any other type. 24
This
knowing
his
willing to accept his assignment
is
—
what the substance of the equal
30
utility conditions of the
Thus
Bidding Rosea and the market equilibrium.
institutional settings, only those allocations
which
satisfy feasibility
in the latter
and equal
utility
two
can
This amounts to an additional constraint on the problem of maximizing ex
be considered.
ante expected utility.
This completes our characterization of efficient and optimal allocations.
Having
understood their properties, we are ready to examine how well the various institutions that
we have
VI.
discussed perform relative to these criteria.
The Performance
The purpose
detail.
We
of this section
is
to analyze the allocative performance of Roscas in
shall be interested in understanding
and expected
allocation
of Roscas
how they perform
utility criteria, as well as in establishing additional features of the resource
which they produce. In order to do
the results of the previous two sections.
consistent
we
optimum,
i.e. t
let t
this, it is helpful to set
This
denote the date a^ which
we do
all
in
Table
1.
out concisely some of
To keep the
wpif- p
,•
J
1
Date by which all have
acqui red the durable
W^=T[v(y)+^]-B/.(0
t^=B/.'(e)
Random Rosea Wj^=T[v(y)+^]-B/i(|)
tj^=B/^'(|)
Autarky
notation
have acquired the durable under the
=s (1).
TABLE
,.,
relative to the efficiency
Bidding Rosea
Wg=T[v(y)+e]-BMg(0
tg=B/(y-A3)
Optimum
W^=T[v(y)+^]-B/ J/x(QOda
t^^B/ J/.' (a^da
31
We begin
Proposition
Proof:
1:
By
by discussing the efficiency of resource allocation
The
allocations achieved by Bidding
Corollary
l(i) efficient
and Random Roscas are
have
allocations
in Roscas.
strictly
inefficient.
convex assignment functions,
while the analysis of sections IV. 2 and 1V.3 showed that Roscas, with uniformly spaced
meeting dates and constant contribution
The
rates, lead to linear
assignment functions, d
constraint that contributions occur at a constant rate during the
life
of a Rosea
intended to reflect an important feature of these institutions as observed in practice
A
simplicity.
result of this simple representation
convexity of efficient assignment functions
is
is
—
their
the inefficiency noted above.
a consequence of the fact
that,
is
as
The
the
remaining horizon becomes shorter, the value of the durable good to an agent who acquires
it
diminishes, so the
amount
of current
durable goods should also decline.
consumption foregone to finance diffusion of
Roscas, as
we have modeled them, cannot achieve
Notwithstanding, the best
subtle intertemporal shift in resource allocation.
does yield
maximal
ex ante expected utility to its
assignment function being linear.
common
level of utility for its
members
this
Random Rosea
subject to the constraint of the
Moreover, the best Bidding Rosea generates the highest
members, among
all
feasible allocations satisfying the linear
assignment function requirement.
Our next
Proposition
utility
Hence,
2:
result concerns the welfare comparisons.
(a)
The optimal
and Autarky the lowest,
a
representative
agent,
allocation yields the highest level of ex ante expected
(b)
The Random Rosea dominates the Bidding Rosea.
using
the
ex
ante
welfare
criterion,
would rank the
32
institutions in the following way:
Moreover,
that
[4,0-
Recall that
Proof:
it
is
v'">0.
y~c
/x'((^)
(v(c)— i^(x— a))da.
(•)
is
—^}
Wp
is
<
.
and concave,
Hence,
/ig(0 = minf ^^yj^T^^H^^ }, any x6[0,l],
Because v(')
is
/i((^)>/ip>((^)>/i(4).
In light of our earlier remarks,
and the ex ante expected
Random
increasing and strictly concave
utility
What
Roscas.
than by competitive bidding.
The
result then follows
is
it
we have
maximizing allocation must be the
Individuals, given our assumptions,
We
is
immediately
the ranking of the
would always prefer
shall return to the reason for this
for
individuals
information about their differences to other Rosea members.
differences
allows
best, of the alternatives
most noteworthy about Proposition 2
Bidding could serve as a means
found.
x=^
seems obvious that Autarky must be the worst,
should be noted that the assumption of homogeneous agents
it
we
Therefore A(c,l)<c, and (by Jensen's
use a savings association that allocates access to the accumulating resources by
present,
<^>0,
n
1.
being considered here.
i^>0.
decreasing and convex, under our assumption
Thus, taking x=] we see that ^((f)</z((^), while taking
us to see that /io(i^)>/i(4).
by reference to Table
W
positive, increasing
increasing and strictly convex.
is
Inequality) A(c,^)>c.
Bidding and
< Wy. <
Recall also the definitions
v~
that
.
Min{^^^'^^^^^r-^
easy to show that
where A(c,x)e/„v
know
W
would have to be unobservable
lot,
rather
At
momentarily.
crucial to
is
to
to
what
credibly convey
Notice, though, that those
to the other agents to
obviate our conclusion
Otherwise the self—selection would occur prior to the formation of the Roscas,
above.
resulting
in
homogeneous associations
which would perform
better
with
a
random
allocation of positions in the queue for funds than with bidding. 25
Formally,
than does a
utilities
it
can be seen that the Bidding Rosea solves a more constrained problem
Random
Rosea:
equilibrium of the bidding process requires that lifetime
be equal, which precludes the equality of marginal
utilities across
agents which
is
33
necessary for ex ante expected utility maximization. 26
related analysis of Bergstrom (1986). 2^
main
will
idea.
It
depicts a
only one individual in each time period.
since the assignment function
must be
receive the good in each period.
consumption
is
upon who
one wins the pot in period
i?iich
that
two
1
and
B
Random Rosea
in the
if
two
We
periods.
to allocate the durable to
same number
efficiency
of individuals
must
and which
utility possibility frontiers
is
With a Random Rosea, non—durable
Hence, the utility allocations
at
live for
economy Roscas can achieve
linear, i.e. the
individual 2 wins
it.
will
be at
A
if
individual
At either of these points,
Since both utility allocations have equal probability, each
individual's expected utility will be at point C.
that the
is efficient
gets the durable first.
utility is allocated unequally.
hence the allocation occurs
it
In such an
The::e are
equally allocated.
made
also
is
Figure 4 provides a diagrammatic exposition of the
two person ecoromy wherein individuals
assume that parameter values are
relevant depends
This point
Under a Bidding Rosea,
utility is equal
at the inters(5Ction of the utility possibility frontiers.
results in a higher level of expected utility
Indeed, in this simple economy, the
Random Rosea
and
It is clear
than the Bidding Rosea.
achieves the optimal allocation.
For exactly the same reason, the competitive equilibrium allocation, constrained by
definition to provide agents with equal utilities, generates lower ex ante expected utility
than the optimal allocation <s
person,
,c
>.28
This can also be seen from Figure 3 since in a two
two period world the Bidding Rosea and the Market
In general, however, the Market
is
result in identical allocations.
superior to a Bidding Rosea.
To understand
this, recall
that, in addition to being constrained to provide agents equal utilities, the Bidding
also constrained to
Proposition
3:
Bidding Rosea,
Proof:
have a linear assignment function.
While being
i.e.
W
inferior to the
We summarize
Rosea
is
these observations in
optimal allocation, the Market
is
preferred to a
> W,, > Wg.
Since each agents' utility
is
constant in both <S|yr,Cj,^> and <Sg,Cg>, and since
34
<s-p,c-r.>
Pareto inefficient while by the First Fundamental Theorem of Welfare Econ-
is
omics <S|^,c^>
Pareto
is
efficient,
we must have W^>Wt5- Moreover,
the constancy of
agents' utility in a competitive equilibrium implies:
W^ E
V<?60:
(6.1)
W(<.Us^,c^>)
>
W(^i<Sj^,Cj^>)
=
W(^j^;<Sj^,Cj^>) =
The
where 6y^ are the weights associated with the competitive allocation.
(6.1) reflects the fact that <s^,c,|>
the equality
is
due to the
fact
depend on the weights.
maximizes the weighted sum
.hat the
^^^
inequality in
of utilities with weights
The competitive
W(^;<s,c>)}!
^G0 <s ,c>6F
equilibrium solves an elegant mini—max problem.
Thus, not only
is
Wi>,<W
(equality
impossible since then, by the strict concavity of v(-) and the fact that Ci^^c
convex combination of <ST,^,c^,r> and <s
M
<s
,c
>), but W-i^
is less
W
«.
This
is
,c
> would
than any maximized weighted sum of
that
the
utilities,
competitive equilibrium uses weights which
utility,
shows
strictly
dominated,
in
by the optimal Random Rosea. Hence, our
final result
on
that
the
"equal
As already mentioned,
Wp
assignment function, while
that utilities are equal.
is
maximal
W.^ maximizes
the
can
be
more
of
an
"linear assignment function" constraint.
ante
The question naturally
on the relation of these values.
conditions
ex
is
constraint
utility"
impediment to generating ex ante welfare than the
result
d
key to our constructive demonstration, in Proposition 4 below, that
terms of ex ante expected
comparisons
a strict
be feasible and would dominate
there exist circumstances under which the competitive allocation
welfare
,
is
jvi
This proof demonstrates
minimize
0;
weighted average of a constant function does not
W^=Min{ Max
So
W M-
welfare
same
to
having a linear
criterion subject to the constraint
arises
One might have
subject
whether one can prove a general
suspected that under some plausible
the competitive allocation would dominate the inefficient
Random
Rosea.
35
However,
this
is
What
not the case.
follows
is
an illustration of the
institution of financial intermediation, allocating its funds
by
lot,
fact that
a simple
can actually out perform
a perfectly competitive credit market.
Proposition
In the case of logarithmic utility, there exists a
4:
Random Rosea dominates
Proof:
the Market;
i.e.
W^
of proof
from the proof of Proposition
sum, over
all
less
since explicit
indirect,
is
it late.
We then
a
when
^
is
may
be understood by
must
get
much
non—durable consumption than
lower
Therefore, the agents' marginal utilities from
those
non—durable consumption
be relatively low.
Since the
non—optimal intertemporal
allocation
utilities to
characteristic of Roscas,
when
^
is
sufficiently big the
dominates.
Another
difference
accumulation periods.
In what follows
terminal time.
utility
very large, respecting the equal utility constraint means those
non—durable consumption
Random Rosea
use the fact,
maximized weighted
Intuitively the result
result.
does not effect the utility cost of the
i^
We
seems intractable.
construct a set of weights whose maximized utility
be very unequal, causing their ex ante expected
magnitude of
of
^,
representation of competitive
that the market gives the least
than Wrj, to infer the
receiving the durable early
acquiring
3,
possible weights.
recognizing that,
will
>
> W^.
allocations, even in the case of logarithmic utility,
is
for all ^
See the Appendix.
The technique
sum
^ such that
we
We
between,
Our next
the
task, therefore,
shall refer to the date
now
have:
institutions
is
concerns
to explore
by which
all
some
the
length
of
their
facts concerning these.
have acquired the durable as the
36
Proposition
5:
The terminal times under Autarky,
Proof:
<
t
Considering the formulae in Tab.e
1,
and the
tions a decreasing, strictly
c*(0
Let
li'{^)
given
c*(|)
Ado
t,->
can be ordered as follows: tTi/2
R'
=
give
V"' >
0,
Some
.
The moral of
all
savings
relatively short.
Now
:s,
ex ante
stands
..o
Rosea
its
is
fact that /i'(-)
two
is
>
0:
under our assump-
above follow at once.
inequalities
Table
=
^{()
t-^
1,
<
strict
v'(c*(0)
decreasing and,
is
2t.
and
if
and
only
if
convexity of c*(-). n
roughly speaking, that the more efficient the
the period of accumulation.
Since under
reason that the accumulation period would be
optimum, on the other hand,
basis (see the discussion in Section V.2).
must slow
last
from
more protracted
it
Optimum
.
which follows from the
services of the durable, as acquired, were
acquisition
Rosea, and the
straightforward algebra shows that c*(-)
the
lie idle,
Random
Then V^
(4 2).
proposition
this
The
<
[c*((^)+c*(0)]/2,
financial intermediation,
Autarky
.
in
convex.
strictly
< [c*(0+y]/2 =
t
convex function, the
minimum
the
[y—c*((^)]
<
the
allocates resources as if the
immediately available to
We
know from
also
all
agents on a prorated
Corollary
that the rate of
1
The Random
to zero as cumulative diffusion approaches unity.
somewhere between these two extremes, but has the property that on the average
members acquire the durable good
earlier
than would be the case
if
they saved in
isolation.
Note also that the
results
on the terminal dates can be used to
about the average rates of consumption
(c),
is
something
and therefore saving, during the accumulation
phase for the different institutions after remembering that
of Proposition 5
infer
that the average .savings rate
is
t
= B/y —
greatest under
c.
Hence a corollary
Autarky and lowest
under the optimal allocation.
The optimal length
would appear,
also
of the period of accumulation under the Bidding
under the market mechanism)
is
more
Rosea (and,
difficult to characterize,
it
because
37
it is
We
not related to the function Mr>(-) in any simple way.
have however been able to
obtain the following result.
Proposition
(i) If
6:
than under Autarky,
Random Rosea
is
l/v'(')
tyjH
i.e.
concave, then the terminal time in a Bidding Rosea
is
.
.
later than with a
l/v'{-)
convex, then the terminal time under a
is
Kidding Rosea,
later
i.e.
tr><tT>-
See the Appendix.
Proof:
Whether l/v'(c)
is
concave or convex does not follow directly from any well known
property of the utility function.
l/v'(c)
is
case, v(c)
satisfied,
VII.
If
(ii)
is
convex
=
if
t
.
p>l and concave
if
p<l.
implying that l/v'(c)
ln(c),
and
For iso—elastic
< t^ <
is
It is
utility functions,
v(y)
=
y ~^/{l—p),
interesting to note that in the borderline
linear in
c,
both parts of the Proposition are
tp-
Conclusions and Suggestions for Further Research
This paper has investigated the economic role and performance of Roscas.
We
have
sought their rationale in the fact that some goods are indivisible, a fact which makes
autarkic saving inefficient.
superior to Autarky.
utility
Moreover, a
Random Rosea may
than in a perfect credit market.
in conjunction
significant in
The
many
they are
latter result is
more
striking
still
—
when taken
a factor surely
contexts in which they are adopted.
of issues remain outstanding
The most important concerns
incentive for those
efficient,
yield a higher level of expected
with the fact that Roscas are such a simple institution
A number
work.
While Rosea allocations are not Pareto
who win
and we hope to consider them
the sustainability of Roscas.
There
is
in future
a strong
the pot early to stop contributing to the Rosea, although as
noted in the introduction, this does not appear to be a serious problem in practice.
we
The
38
kinds of primitive societies which spawn Roscas are comparatively rich in the ability to use
social sanctions to enforce contractual performance.
in the process of
economic development causing problems
issues better would,
likely to be the
This ability
we
may however
for Roscas.
Understanding these
conjecture, further enhance our understanding of
most "appropriate" savings institution
for
an economy.
deteriorate
when Roscas
are
39
References
S., 1964, The Comparative S.udy of Rotating Credit Associations, Journal of the
Royal Anthropological Society of Great Britain and Ireland ?, 201—229.
Ardener,
Arnott, R. and J. Riley, 1977, Asymmetrical Production Possibilities, the Social Gains
from Inequality and the Optimum Town, Scandinavian Journal of Economics 79,
301-311.
Begashaw, G., 1978, The Economic Role of Traditional Savings Institutions in Ethiopia,
Savings and Development 2, 249—262.
Bergstrom, T., 1986, Soldiers of Fortune? in Heller, W.P., R.M. Starr and D.A. Starrett,
(eds). Equilibrium Analysis: Essays in Honor of Kenneth J. Arrow, Cambridge:
Cambridge University Press.
Bouman,
F.J. A., 1977, Indigenous Sailings and Credit Societies in the Third World:
Message, Savings and Development 1, 181—218.
A
Fernando, E., 1986, Informal Credit and Savings Organizations in SriLanka: The Cheetu,
Savings and Development 10, 253—263.
C,
Geertz,
The Rotating
Credit Association: a 'Middle Rung' in Development,
Development and Cultural Change 1, 241—263.
1962,
Economic
Goffman, C. and G. Pedrick, 1965, First Course in Functional Analysis, Englewood
Cliffs,
N.J., Prentice Hall.
Grossman, R., 1989, Deposit Insurance, Regulation, and Moral Hazard
Industry: Evidence from the ISSOs, Mimeo.
in
the Thrift
Hillman, A. and P. Swan, 1983, Participation Rules for Pareto Optimal Clubs, Journal of
Public Economics 20, 55—76.
Leland,
Light,
H.E., 1968, Saving and Uncertainty: The Precautionary
Quarterly Journal of Economics 82, 465^73.
I.,
Mirrlees,
Demand
for
Saving,
1972, Ethnic Enterprise in America, Berkeley, University of California Press.
1972,
J. A.,
The Optimum Town, Scandinavian Journal of Economics
74,
115-135.
Osuntogun, A., and N. Adeyemo, 1981,
Mobilization of Rural Savings and Credit
Extension in Pre—Cooperative Organizations in South—Western Nigeria, Savings
and Development 5, 247—261.
Pratt, J.
and R. Zeckhauser, 1987, Proper Risk Aversion, Econometrica
Rhadhakrishnan,
S.
and
others, 1975, Chit Funds, Institute for Financial
Research, Madras.
Symons, E. and
J.
55,
White, 1984, Banking Law, 2nd Edition,
?
143—154.
Management and
40
Appendix
Proof of Lemma 1: To prove that the optimal Random Rosea will be fully funded, we must
extend our model of a Random Rosea to allow for partial funding. To do this all we
have to do is to let N6(0,l] be the limiting ratio of meetings to members, but otherwise
adopt the model of the optiiT.rJ Random Rosea with a continuum of agents as it was
described in the text. Then if the Rosea has length t, members save at the rate N-B/t,
and face probability N of ever winning the pot. Conditional on winning, the receipt date
a is uniform on [0.1]. Reasoning in like manner to the arguments for (4.3) and (4.6), and
denoting by Wrj(N) the maximal expected utility of the members of this Rosea, we have:
first
Wj^(N) = Max[tv(c)+(T-t)v(y)+N-(T-j)^]
As
before,
if
the optimal length
Wj^(N)
(A.l)
as the welfare associated
vides its
is
members
optimal
iff
=
t<T
then (4.2)
may
be employed to write:
Random Rosea which pro"Full funding" of the Rosea
combining (4.6) and the above, iff:
with that optimal "partially funded"
a probability
T^
t-(y^)=NB, 0<c<y-NB/T.
T.[v(y)+Ne] -NB./i(Ne/2)
N
of ever obtaining the pot.
Wj^hWj^(1)>Wj.(N), VNg(0,1]. That
(A.2)
s.t.
>
K-^/2)
- yNe/2)
^
is,
^^^(o^^j
this condition is stronger than the desirability requirement of Assumption 2 may be
seen as follows: For N » 1 the RHS of the equation above x /i({/2) + {(/2)fi'{^/2) > fj,U),
by concavity. So if the inequalioy holds for N G (1— e,l], e near zero, we may conclude that
T/B > /i(if)/i^, which is Assumption 2. On the other hand, if Assumption 2 fails then,
although a Random Rosea which guarantees eventual receipt of the durable may be
preferred to doing without it, one which provides something less than certainty of receipt
(N<1) would be even better. Moreover, since (ii{i) is not generally a convex function, the
above inequality could hold for N k 1, but fail for N < 1. Thus, without imposing some
additional assumption we cannot assure even the local optimality of a "fully funded"
Random Rosea.
now show that (A.2) holds, given Assumption 2'. Strong desirability implies
(T/B>2}j{(/2). Let x>0, and e*(x) give the minimum in (4.2) when ,^=x. Then:
That
We
W -- y^*(x) ^< v(y)-^(c*(x))+x
y-c*(x)
XI,' (y\
XM
Therefore, use (A.l), set
x=N(72
^^^^§^= Te-B[M(^)
for
^.
above, and recall that
+^./.'(N|)]
^
-
(
^
^^''''•
/zis strictly increasing, to get:
> B.[^-2/.(N|)]
0<N<1. So Wj^(l)>Wj^(N), VNg(0,1], and (A.2)
>
B.[^-24)]
> 0,
obtains.
To prove the remaimng part of the lemma, consider the choice problem faced by the
member of the optimal "fully funded" Random Rosea who vnns the pot at date refO,t]. He
must continue
to
pay into the Rosea
at rate y
— B/T
on the interval
(r,t].
Ii
he ever
41
acquires the durable he does best to buy
payoff is:
W*(r)
=
immediately, in which case his continuation
it
(t-r).v(y-^)
+
(T-t).v(y)
+
(T-r)e.
he never acquires it, he should consume for the rest of his life at the maximal constant
rate consistent with available resources, which generates the continuation payoff:
If
r(t-r)(y-f)
+
(T-t)y
+
B^
W(r) =(T-r).v
We
need to show that W*(r)>W(r), VrG[0,t].
true if and only if W*(t)>W(t). (That is, if any
durable, it will be the last one to receive the pot.)
can show that v(y)
+ ^l
^iy+rn—r), for
t
It
may
member
be easily verified that this
will
want not
Thus, the proof
will
the optimal length of this
is
to invest in the
be complete
Random
if
Rosea.
we
But
follows from (4.2), (4.4) and (4.5) thaf, t=B/[y-<:*(^/2)]=B/i'(^/2), while (4.2) implies
v'(y)=/x(0). So by the concavity of v(-), the properties of //(•) derived above, and in light
of Assumption 2':
it
j
v(y+T^)-v(y)
<
y'iYHrfy =
M(c)/[^/.'(^/2)]
<
^
g\W%^\[%f = I < 6
-
Indeed, the same argument can be used to show that, under Strong Desirability, the last
winner of the "partially funded" Random Rosea (N<1) will also choose to acquire the durable, as presumed. This completes the proof.
D
Proof of Theorem
1:
In view of the discussion preceding the statement of the Theorem:
rS(l)
W.P E
Max{
rl
V.(c(r))dr
Jq
+ (T-s(l))-Viy) + {T-^
^(a)s(a)da}
JQ
«^
subject to
Now employ
j rS'
s'(a) = B/[y-^(s(a))], ae\0,l)
the change of variables: r=s(a), dr=s'(a)da, r6[0,s(l)]; note that s(l)
{ajda; and use (5.4) and the definition in (5.6) to get the following:
rS(l)
W^=
Max{
rl
V^c(r))dr + (T-s(l))-V^y) + {-(T-
= Max{T[V^y) + e] - j
^(a)s(a)da)}
s'(a).[V^(y)-V^c(s(a)) + e/^^(z)dz]da}
= T[V^y)+^] -Min{B[ {fV^y)-V^c(s(a)))+ai^(z)dz]/[y^(s(a))]}da}
T[V^y)+^]-BJM^^/l(?(z)dz)da.
=
42
This proves
each a6[0,l].
(5.4)
Note that the minimization above
(i).
So
pointwise, with respect to c(s(a)), at
implies that for ae[0,l], c{s{a))=aJ(j
it
we conclude
is
that s(a)=/?s'(x)dx satisfies
(ii).
Now
0{z)dz).
also
from
In view of this and
(5.4)
we know
that
c(r)=y, for r>s(l), and we noted earlier in the text that c{a,t)=X/(ct,c{i)), VaG[0,l],
Taken together, these prove
VtG[0,T].
Proof of
Lemma
From
2:
functions c (r) and
n
(iii).
and the
(5.4), (5.5),
V,(w)=v(w), we know that the
fact that
(a)
t) maximize:
s
rT
v(c(r))dr+
^T-
^[ s(a)da
JO
subject to:
=
s'(a)
Now,
for
B/[y-c(s(Q))], for
any feasible c(r) and
re[0,s(l))
and K(r)=l
K'(t)=0
for re[s(l),T].
all
ae[0,l); and, c(r)=y, for all r6(s(l),T].
for re[s(l),T].
It
K(-) as follows: K(r)=s~
s(a), define the function
follows that
(r) for
Notice that K'fr)=(y-c(r))/B for re[0,s(l)) and
rewrite the constraints as:
we may
B-K'(r)=y-<;(r); K(0)=0; K(T)=1; 0<c(r)<y.
Now
note
that
the
using
change
of
r=s(a),
variables
dr=s'(a)da we may write
/Js(a)da=/'JrK'(r)dr, and since K(0)=0 and K(l)=l, we have that T=T/QK'(r)dr.
Thus the objective function may be written
as:
T
[v(c(r))
Proof of Proposition
4:
When
+
^(T-r)K'(r)]dr.
n
v(c)=ln(c), welfare under the
Random Rosea
is:
Wj^=THy)+e]-|[l+x(e/2)]
where x(0~^^(l+X(0)-'f> ^^0) defines x(-)- Similarly,
Wm = T[ln(y)+e+/
for
some
^Tyr^©-
Wt^>W^
which
if
We
and only
this is so.
First
0^^{aMe^^ia))da]
know from the proof
if
3^66 such that
we
need:
-jf
for the
[1+xUf
of Proposition 3 that
Wt^>W«. The
Market we have
\^(z)dz)]dx,
W^
= an W^.
Hence,
proof constructs some weights for
43
Lemma
and
3:
Let f(-) be an increasing, strictly concave function satisfying ffO
and g(0
be a function on [0,1], strictly decreasing satisfying g(l) =
let g(-)
0,
1.
Then
/
f(g(x))dx
=
g(x),
and
E(y)
=
z
=
/
1, if
x
g(x)dx
g(x)dx.
random
Let x be a
Proof:
y
> f(l)/
<
variable which
/^g(x)dx, and
=
=
z
0, if
uniformly distributed on
is
[0,1].
Define
x > /^ g(x)dx. Then
E(z),
''o
where E(-) denotes the expectations operator. Moreover, z is riskier than y in the sense of
second order stochastic dominance. Therefore, since f( ) is strictly concave:
•
E(f(y))
=
/
f(g(x))dx
>
=
E(f(z))
f(l)
''o
/
g(x)dx.
-'o
This proves the lemma, n
This
lemma
implies that
Ox
/
xi^f
> xiOf if
n
'0
e{z)dz)dx
Oiz)dz)dx.
..
X
This in turn implies that
W^<T[ln(y) + e+/
^(x)ln(tf(x))dx]-|[l+
/
-1.1
where we have
also
used the fact that
Hence, a sufficient condition for
^ee such that
r
Now
define EC?) e
Wp
-fxiCm > f
I
I
*^0
^T
> W^,,
-1
^(z)dzdx
=
I
x^(x)dx.
-^0
is
that:
<?(>-)ln(<?(x))dx-|x(0/ x^(x)dx
A
^{^ f
x^(x)dx -xCOJ,
A
0{x)ln{0{x))dx.
s.t.
/
xd{x)dx='B
Then
Wj^>W^
if:
44
37)6(0,1) such that
B
Let
rTr-E7G(0,l),
^y
that
{^).[Dx{0-x{m] >
E(7?).
max
and consider the problem:
*
-
{'JxiO'^
ECS)}^!^
We
.
concaide
^e[0,l]
sufficient for
it is
Lemma
Wj^>Wj^j
(i) E(1J) is etiictly
4:
Q >7x(^/2)-
that
convex, and E{'d)>E{^),
V7?g[0,1];
and,
E'{'B)=X,
(ii) if
then E(^)=A^+ln(A(e'^-ir-^).
Proof:
L =
Define the Lagrangean
^x)ln(^(x))dx
/
+
A[?-
*^0
The
first
/
+
xO{x)dx]
f 0{x)dx\.
^[1- *^0
*^0
order condition with respect to ^(x)
and integrating
ln(^(x))+l— Ax—/i=0,
is:
-1 fe^-n
&
this condition, using the constraint, yields:
X
Inverting
xe[0,l].
=
Solving this
1.
first order condition, multiplying by 0\x) and integrating yields
observe that integrating the first order condition, after inverting and
multiplying through by x, and using the above derived expression for /i yields:
for
substituting into the
fi,
To prove
(ii).
(ij
-
/^x^x)dx
It
E
-d
= e^-^^xe^^dx = f Ae^-(e^-l)
straightforward
is
now
1/2 as A -t 0. Part
envelope condition, n
(f){X) -*
to see that
(i)
is
(f){X)
jsjQi^g
Thus x(0 >
as
-•
A
>
Q
od;
+
1^/2.
Moreover, (e^— l)/z
=
(f){X)
that E'{D)
-•
1
A
as
= (f
-•
(d),
= ^(A).
+od
and
from the
.7X(0_i
7x('^/2)
^^gQ ^j^^^^ fj.Qj^ ^^Q definition of x(-)> that
x('f/2)
e\(e^-l) - A-1
—
-•
now proved by noting
This result and simple calculation reveals:
e7X(C/2)
^e^-1
/^e^^dx
if
and only
x'iO = (l+x(0)/x(0) >
is
>
1-
a strictly increasing function.
So:
^>^(^)-l
im'
The
first of
for
a
^ e^>^[2'^f^2\-l ^
7X1^/2) + 7C/2
g7x(|)
r_e2l/^
LtxR/^T+tIT^J
-[7(x(^/2)
+ e/2)r^
the two terms on the right hand side grows unboundedly as ^ -> w. Moreover,
sufficiently large £, the second term vanishes.
Hence, for large enough {,
[eTXCO _ij/^^(^) > Qixiil'^) ^nd Wj^ > Wj^.
depends only on 7EB/Ty.)
.,
a
(Note that
^,
the critical value of
{,
»
45
Proof of Proposition
that
t-p > t
.
We
.
bidder
lowest
6:
will
(i)
Suppose that l/v'(c) is a concave function of c. We will show
use the following notation. Let Cp be the consumption rate of the
Rosea
the
in
CgHCg(l).
i.e.,
From
CE/ic(a)da=A(Co,l)); c'ec(1)=Ct3, and c°=c(0).
know
must equal the consumption
that c(a)
date atp.
c(a)=Cr.(Q;)),
(i.e.,
moment
over the
following
Lemma.
Lemma
the Fundamental
HX
^c'-c°^
^
da
^
(v(Cg)-(^(l-a));
we
level of individuals receiving the durable at
population average rate of consumption at each
We
Notice that c'(x)=if/v'(c(x)).
Theorem
],
will use the
while l/v'(c) convex implies
of Calculus
-1
v'(c(a))
1-1
1.
while by definition
c{a)=v
the discussion of section IV. 3
l/v'(c) concave implies v'(c) < (^/[c'—c
.1
1.
c is the
of the Rosea.
life
5:
By
Proof:
and
Define:
/-M-1
[v '(c)]
=
v'(
c(a)dQ;
Now
.
Jensen's inequality implies
*0
/^-4^<
•^0
v'(c(a))
when l/v'(c)
the
[v(y)
[v'(/^c(a)dal
^
= [v'(c)r\
J
*^0
^
concave (convex), n
is
We will
satisfies
(>)
use this
lemma
as follows.
Note that
first
order condition:
+
- v(c^)] - V (c^)(y^^) =
i
Moreover, using (4.10) (with x=l) one
determining c-p is:
1.
[v(y)+e-v(cB)]
[y-^
=
txD > t
.
if
and only
if
c > c
.
,
while c
0.
may
calculate that
the
first
order condition
0.
Lc^^°
Let A(c) = v(y)+<^-v(c) and
Therefore,
when
l/v'(c)
^c)
=
is
^/<c) =
concave,
A(c)-v'(c)(y-c).
we
A(c)-v'(c)(y-c)
>
Then c<Cg, A'(c)<0 and
have:
A(c3)-v'(c)(y-?)
?A'(c)>0.
46
--v'(r)
(y
-
=
c) >
V<Ca)-
'-c'-c"
This proves
(i).
To prove
the proof of
Rosea,
(i)
who
(ii)
note that trjHo
above,
may
and only
date tg/2
receives the durable at
The
thus be rewritterl
first
With
if Crj>c.
Cp denote the consumption
let
requires: v(Cti)=v(ct3)+(^/2.
Rosea
if
(i.e.,
of a
c',
c
,
c(-) as defined in
median bidder
Ctj=c(1/2)).
in the optimal
Bidding equilibrium
order condition for the optimality of the Bidding
as:
e
[v(y)+ ^/2-v(Cg)]
(y-^,
c'^°J
and the function c(a) now
c(a)
satisfies:
= v-\v(c3)-(^a)0,
a6[0.1].
Notice that Jensen's inequality and the convexity of v
c =
J
c(Q:)da=j
V
(v(cg)-(2-a)0da>
moreover, the first—order condition determining
[v(y)+e/2-v(Cj^)]
With A(c) and ^c)
=
-1/
(
)
•
imply:
Cg,
Crj is:
v'(Cj^)[y^j^].
defined as before
we may
use the
Lemma
to conclude that if l/v'(c)
convex, then
V<5) =
A(^-v'(c)(y-^
<
A(3g)-v'(E)(y-^ = [-i-^v'(c)] -[y^
^c'-c
This cornpletes the proof,
d
<
=
ip{c^).
is
47
End Notes
iBouman (1977) provides a comprehens^. i^e listing of all the countries in which Roscas have
He reports, for
been observed as well as some sense cf their quantitative importance.
example, that an estimated 60% of the population in Addis Ababa belong to a Rosea of
some form. Light fl972) discusses their use by ethnic minorities in the US. Additional
specific country studies are provided by Begashaw (1978), Fernando (1986), and Osuntogun
and Adeyemo (1981). The nineteenth century savings and loan associations in the US also
appeared to have worked as Roscas — see Grossman (1989) and Symons and White (1984),
page 65. Roscas have long been popular in India where they travel under the name of Chit
Funds. Radhakrishnan et al. (1975) describes them in detail. In Kerala, for example, they
report that Roscas are used in preference to banks by industrialists, educational
institutions and churches. In 1967, there were 12491 registered Chit Funds in this state of
India alone.
2The
The
classic anthropological studies of
latter
paper
is
particularly
Roscas are by Geertz (1962) and Ardener (1964).
as an introduction to the literature.
recommrnended
3While bidding and drawing lots seem to be the two most common ways of allocating the
it is also sometimes allocated according to need or known criteria such as age or
kinship seniority. The reader is referred to Ardener (1964) for a more detailed discussion.
pot,
4This was clearly recognized by Ardeiier (1964).
"The most obvious function of these
associations is that they assist in small—scale capital—formation, or more simply, they
create savings.
Members could sa^'C their contributions themselves at home and
accumulate their own "funds", but this would withdraw money from circulation: in a
rotating credit association capital need never be idle." p. 217.
example, the discusion in Ardener (1964)
sSee, for
^Note that the quantity
B
is
p. 216.
a stock, while y and c are rates of flow.
consonant with the anthropological stories. Individuals do not, typically, share the
from Rosea winnings. In cases such as the purchase a new roof for one's home, this is
not typically feasible anyway.
''This is
fruit
^The function
c:[0,l]'<[0,T]
integrable for
all {oc,t).
sThat
is,
N(r;s)
is
->
K
is
requirsd to be such that c(-,r) and c(a,-) are (Lebesgue)
the measure of the set s~\[0,r])E{ae[0,l]|s(a)<r
t.=T
lOThe problem makes no economic sense otherwise, since
provide a condition which
is
both necessary and sufficient
for
&
s{a)fu}.
implies
W.
W.<T-v(y).
>T-v(y)
in
We
Assumption
2 below.
lift is
trivial to
let
^
(-.
<^„,
show that
define (
h
increasing and strictly positive. To prove
{l—a)(^ and note that
fi{-) is
aL +
v(y)—vfc )-^^
v(y)-v(c )-f.
uit )
r\SQ,/
where
c
is
=
a
V
-
h
c
(1—
a)
^
^
a
the optimal consumption level at ^
y -
•'a
.
c
,
'
Note also ;hat
strict concavity,
48
v(y)-v(c,)-e'
K^iX
Hence, K^f
J
>
ior(.
y-c^
<^m(^i)
+
=
U^,^^}.
(l-ci)/X'^2^' ^" claimed.
i2From the properties of //() stated in the text, we know that fji()/(
Moreover, f^O>y'{y); so the Envelope Theorem implies
exists a
minimal £>0, which
is
desirability fails, then
the symbol "c»
"
s
.
h
i/
=
and
c
Autarky
.
if
and only
(a,T) h y, for
to denote both the function
all
if
^
(>
agents.
decreasing for ^>0.
wrM
y and of
a decreasing function of both
desirable to acquire the durable under
i^If
-n^
is
<
g.
T
tt,
Thus there
such that
it is
£.
Note
also that
which designates consumption
we
are using
levels for the
agents at various dates, and the value of that function during the period of accumulation.
No confusion should result.
for the moment that the vnnner will choose to buy the durable.
discussed further below.
i^We assume
tion
is
This assump-
i^Notice that the particular limit obtained depends upon the assumption that the spacing of
meeting dates is uniform and the contribution at each meeting is constant. By having the
meetings occur with different frequencies at different times during the life of the Rosea, and
by varying the rate of contribution across meeting dates, it may be shown that one can, in
the limit, generate every feasible allocation <s,c> in which the consumption paths c(a,r)
(Moreover, one can in
are constant in a, as the ex post outcome of a Random Rosea.
similar fashion generate every feasible allocation <s,c> in which the utility u(a;<s,c>) is
constant in a, as the limiting outcome of a Bidding Rosea.) However, we do not attempt
To do so would run contrary to the spirit of our
to exploit these fact in this paper.
The point of this exercise is to see how simple institutions of financial
analysis.
intermediation compare with each other and with a more complicated market mechanism.
We believe, upon our reading of the anthropological literature, that the simplicity and
minimal informational requirements of Roscas, as represented here and as observed in practice, are important factors explaining their popularity around the world.
We therefore
restrict ourselves to the study of Roscas which can in effect be fully and simply characterized in terms of two parameters — the frequency of meetings, and the contribution
required per meeting. It is of some interest that, despite this restriction, we nevertheless
find that Roscas can compare favorably with more complicated mechanisms of resource
allocation.
i^It is
worth noting that the larger the number of individuals
in a Rosea, the earlier is the
average receipt date of the durable. (E(r] = t-[l + l/n]/2.) Thus our formulation implies
that individual participants are better oft when more members are added to the Rosea. In
practice, at least two factors are likely to work against this.
First, a larger membership
requires more meeting dates and there are likely to be transactions costs associated with
these meetings. Second, the more members the more impersonal the relationship between
participants, and hence the greater the likelihood of incentive problems. The general issue
of the optimal number of members is an interesting subject for further research.
^''Unless
the optimal accumulation period
is
strictly shorter
than the plarming horizon, our
model of the Random Rosea breaks down. Those winning the pot near the end of the
Rosea would not rationally choose to buy a durable good if there were no time left to enjoy
its services.
Assumption 2' introduced below is sufficient to insure that acquisition of the
49
durable by the winner
i^From
and
(4.2), (4.4)
(4.6)
Since
fj{') is
=
it
+
T[v(y)
concave, /i(^/2)
in the optimal
should also be noted that since
a individual
the utility level of a type
u(a;<Sj^,Cj^>)
Lemma
always rational, and so trj<T at the optimum. See
is
Random Rosea
+
C]
tj^
=
1.
B-^'{(/2), we can write
at
"
^Mm -
(^/2)^'(,^/2)
>
(l-2a)(e/2)/z'(^/2)].
/i(^),
< W^. Thus
so that u(l;<Sj^,Cj^>)
the last recipient of the pot
is
worse—off than he would have
been under Autarky.
i^Both of these can be verified straightforwardly by noting that the
Cn(x) can be written as
=
1
first
order condition for
/ig(^)/^c'(a;)da.
the case of the Bidding Rosea, wc; have been unable to establish whether Assumption
to guarantee that all individuals will purchase the durable in a market
know that the market
equilibrium. Again, intuitively, it seems that it should be.
These two facts
equilibrium is efficient and that all individuals have equal utilities.
together with Assumption 2 should, one would imagine, imply that all individuals will
receive the durable. For, if a group of individuals never received the durable, then, by the
equal utilities requirement, they would have to be compensated by having higher
consumi tion levels. Given desirability., however, it should be cheaper, in terms of the loss
of the rest of the population's utility, to provide them with the durable.
2''As in
2
is
sufficient
We
2iAs justification for the characterization in (5.1) consider the "separating hyperplane"
as follows. Each feasible allocation <s,c>eF rives rise to an allocation
= { u(-;<s,c>)
<s,c>6F }. Given the convention
of utifity u(q;;<s,c>), a6[0,l]. Let
of labelling agents according to their order of receipt of the durable, U constitutes the
argument sketched
U
|
Under our assumptions
our model.
utifity possibifity set for
U
a convex,
is
bounded
subset of the space of Lebesgue integrable functions on the unit interval, L'([0,1]), with
non-empty interior, closed in the norm, topology. Convexity is assured by the fact [see
equation (5.4) below] that on the efficient frontier of F s'(a) is inversely proportional to
Thus the convex combination of two
the aggregate savings rate
[y— /c(x,s(Q:))dx].
consumption allocations allows receipt dates for every agent which are less than the same
convex combination of the corresponding assignment functions.
Now an efficient allocation <s',c'>eF generates an allocation of utifity u'eU
satisfying (suppress dependence of u on <s,c> hereafter): u(a) > u'(a) a.e. if and only if
u^U.
exists,
The Hahn— Banach Theorem
such that
p(u')
> p(u),
identified with the set of
=
VueU.
It is
well
known
1, p.
147).
[0,1].
(See, e.g.,
Therefore, there exists such a function,
/Ju'(a)^(a)da
>
p:L^([0,l])
that the dual of L'([0,ll]
bounded, measurable functions on
Pendrick (1965), Theorem
p(u')
implies a continuous finear functional
/Ju(a)0(a)da
=
p(u),
R
->
may
be
Gofiman and
(p,
satisfying:
V uGU.
Obviously (f){-) must be non—negative, a.e. Moreover, if u' corresponds to an allocation in
which all agents enjoy positive utifity from the flow consumption good then, given any
A of agents of measure strictly less than one, there is an alternative feasible
subset
allocation making all agents in A strictly better off, and all agents in [0,1]\A strictly
worse off. Therefore (p must be strictly positive, a.e. The weights 6{ ) correspond to the
function (f){-) normafized to integrate to one.
•
22To see
this,
note that CQ(a,SQ(0))
=
y
-
l/fi'iO
=
c^-
50
23Even with indivisibility the allocation problem reduces in this way if the durable's services
were fungible across agents — if there W::re, e.g., a perfect rental market for its services.
There are, of course, good (adverse select'on/moral hazard) reasons why such trade in durable services might not obtain, especially in a LDC setting. Moreover, some reports on the
use of Roscas stress their role in financing personal expenditures (daughter's wedding, feast
for fellow villagers, tin roof for house) which, though not producing a fungible asset, generate private consumption benefits lasting for some time that are not transferable to others.
24The similarity of this discussion to classical incentive compatibility considerations may be
misleading, in that there is really only o)ie "type" of agent in our model. Our results are
based on incentive considerations only in the most primitive way — the equal utility
requirement of equilibrium forces an allocation which is in the "non—convex part" of the
So the gains from randomization are
overall utility possibility frontier for the economy.
induced by the indivisibility in the model. The non-convexity is due to the possibility of
reassigning agents to different orders of receipt of the durable good. As discussed in the
note justifying (5.1), given an ordering of the agents, as repesented by the convention that
s(a) must be non—decreasing, utility possibilities are convex. By relabelling people at the
same physical allocation ji;e obtains a different distribution of utility among them. The
union of utility possibilities over all possible relabellings is not convex. (See Figure 3.)
The equilibrium incentive constraint of equal utility forces a utility distribution which is in
the intersection of all utility possibility frontiers generated by a relabelling of individuals.
As the Figure shows, the overall utility possibility set must be locally non—convex near
such a point.
25This point can be
made
formally precise.
Suppose that an individual who has the durable
obtains instantaneous utility v(c)+a;^ where a;G[w,w]
taste parameter.
is
an unobservable, individual
Let F(a;) denote the fraction of individuals
who have
specific,
a valuation less
than or equal to uj and let lo denote the mean value of w. Then welfare under the optimal
Bidding Rosea, when bids (b) and receipt dates (s) are functions <b(a;),s(a;)> of announced
"type" constrained to be incentive compatible, can be written:
Wg
= T[v(y)+^]-B/^(ei<l-G)),
where G is the Gini coefficient associated with the distribution F(a;). Thus welfare under a
Bidding Rosea depends positively upon the extent of the dispersion in agents' valuations of
the durable as measured by the Gini coefficient. By constrast, welfare under the Random
Rosea which treats
all
agents alike remains:
Wrj=T[v(y)4-(^]—B/i(i^w/2).
The Bidding
Rosea can dominate with sufficient inequality in the distribution of valuations.
this result is available from the authors on request.
A
proof of
26This does, of course, depend on the form of the utility function.
However, it may be
checked that the only case in which equality across agents of the level of lifetime utilities,
and equality of the rate of marginal utilities of both goods at all dates, does not conflict is
when utility may be written: u(q:;<s,c> )=V[/c(a,t)dt + ,^(T—s(a))]. That is, the two
goods must be perfect substitutes for one B.nother between any two dates.
2''The failure of the market to achieve majiimal expected utility parallels results obtained in
other literatures where indivisibilities are important, such as location models (Mirrlees
(1972) and Arnott and Riley (1977)), club membership (Hillman and Swan (1986)) and
conscription (Bergstrom 1986).
28An income lottery, with competitive equilibrium in a market for consumption loans occuring after the realization of random incomes becomes known, would allow a decentralized
realization of the ex ante expected utility maximum,
W
.
51
End Notes
iBouman (1977) provides a comprehensive listing of all the countries in which Roscas have
been observed as well as some sense of their quantitative importance. He reports, for
example, that an estimated 60% of the population in Addis Ababa belong to a Rosea of
some form. Light (1972) discusses their use by ethnic minorities in the US. Additional
specific country studies are provided by Begashaw (1978), Fernando (1986), and Osuntogun
and Adeyemo (1981). The nineteenth century savings and loan associations in the US also
appeared to have worked as Roscas — see Grossman (1989) and Symons and White (1984),
page 65. Roscas have long been popular in India where they travel under the name of Chit
Funds. Radhakrishnan et al. (1975) describes them in detail. In Kerala, for example, they
report that Roscas are used in preference to banks by industrialists, educational
institutions and churches. In 1967, there were 12491 registered Chit Funds in this state of
India alone.
2The
The
classic anthropological studies of
latter
paper
is
particularly
Roscas are by Geertz (1962) and Ardener (1964).
as an introduction to the literature.
recommmended
3While bidding and drawing lots seem to be the two most common ways of allocating the
it is also sometimes allocated according to need or known criteria such as age or
kinship seniority. The reader is referred to Ardener (1964) for a more detailed discussion.
pot,
"The most obvious function of these
^This was clearly recognized by Ardener (1964).
associations is that they assist in small-scale capital—formation, or more simply, they
Members could save their contributions themselves at home and
create savings.
accumulate their own "funds", but this would withdraw money from circulation: in a
rotating credit association capital need never be idle." p. 217.
example, the discusion in Ardener (1964)
5See, for
^Note that the quantity
B
p. 216.
a stock, while y and c are rates of flow.
is
is consonant with the anthropological stories. Individuals do not, typically, share the
from Rosea winnings. In cases such as the purchase a new roof for one's home, this is
not typically feasible anyway.
^This
fruit
^The function
c:[0,l]x[0,T]
integrable for
all (a,r).
SThat
is,
N(r;s)
is
-• [R
is
required to be such that c(-,r) and c(a,
the measure of the set s~ ([0,r])={ae[0,l]|s(a)<r
lOThe problem makes no economic sense otherwise, since
provide a condition which
is
t.^T
both necessary and sufficient for
&
.
are (Lebesgue)
s(q:)^i^}.
implies
W
)
Wi<T-v(y).
We
>T-v(y) in Assumption
2 below.
lilt is
let
L
trivial to
i
(^2,
show that
define ^
v(y)-v(c
K^ J = ""—Y^c
where
c
is
increasing and strictly positive.
{l—aj'^c. ^^^^ ^o^^ that
fi{-) is
b a^.
+
/(y)-v(cj-e2
J-Ci
+
(^-"^
the optimal consumption level at (
Y^r^—'
.
Note
also that
To prove
strict concavity,
52
Hence,
fi{(
)
>
aij(L)
+
i^From the properties of
Moreover,
exists a
(l-a)/i((f2)>
/i(-)
^Jl^)'>v'{y)] so the
minimal £>0, which
stated in
desirability fails, then
the symbol "Ca
"
is
tlie text,
we know
that ij{OI(
-jP
=
a decreasing function of both
SkSu and
to denote
claimed.
Envelope Theorem implies
desirable to acquire the durable under
i3If
^''
c
Autarky
.
if
and only
(Q;,r) = y, for all
i^
^ w)~H'
y and of
decreasing for ^>0.
<
T
Thus there
q.
-rr,
such that
it is
^ > i-
if
agents. Note also that
we
are using
both the function which designates consumption levels
for the
agents at various dates, and the value of that function during the period of accumulation.
No confusion should result.
for the moment that the winner will choose to buy the durable.
discussed further below.
^^We assume
tion
is
This assump-
i^Notice that the particular limit obtained depends upon the assumption that the spacing of
meeting dates is uniform and the contribution at each meeting is constant. By having the
meetings occur with different frequencies at different times during the life of the Rosea, and
by varying the rate of contribution across meeting dates, it may be shown that one can, in
the limit, generate every feasible allocation <s,c> in which the consumption paths c(a,r)
(Moreover, one can in
are constant in a, as the ex post outcome of a Random Rosea.
similar fashion generate every feasible allocation <s,c> in which the utility u(q:;<s,c>) is
constant in a, as the limiting outcome of a Bidding Rosea.) However, we do not attempt
to exploit these fact in this paper.
To do so would run contrary to the spirit of our
analysis.
The point of this exercise is to see how simple institutions of financial
intermediation compare with each other and with a more complicated market mechanism.
believe, upon our reading of the anthropological literature, that the simplicity and
minimal informational requirements of Roscas, as represented here and as observed in practherefore
tice, are important factors explaining their popularity around the world.
restrict ourselves to the study of Roscas which can in effect be fully and simply charact-
We
We
erized in terms of two parameters — the frequency of meetings, and the contribution
required per meeting. It is of some interest that, despite this restriction, we nevertheless
find that Roscas can compare favorably with more complicated mechanisms of resource
allocation.
i^It is
worth noting that the larger the number of individuals
in a Rosea, the earlier is the
(E(r) = t-[l -H l/n]/2.) Thus our formulation implies
that individual participants are better oft when more members are added to the Rosea. In
practice, at least two factors are likely io work against this. First, a larger membership
requires more meeting dates and there are likely to be transactions costs associated with
these meetings. Second, the more meml;ers the more impersonal the relationship between
participants, and hence the greater the likelihood of incentive problems. The general issue
of the optimal number of members is an interesting subject for further research.
average receipt date of the durable.
i^Unless the optimal accumulation period
is
strictly shorter
than the planning horizon, our
model of the Random Rosea breaks down. Those winning the pot near the end of the
Rosea would not rationally choose to buy a durable good if there were no time left to enjoy
its services. Assumption 2' introduced below is sufficient to insure that acquisition of the
.
53
durable by the winner
i^From
(4.2), (4.4)
is
and
always rational, and so tr><T at the optimum.
(4.6)
it
=
should also be noted that since tj^
See
Lemma
B-//'(i^/2),
we can
1.
write
the utility level of a type a individual as
u(ai<Sj^,Cj^>)
Since p{-)\s concave,
in the optimal
=
T[v(y)
K^/2) +
Random Rosea
+
^]
-
B- [/i(^/2)
{^/2)pL'{(/2)
>
-
(l-2a)(e/2)/z'(^/2)].
/i(^),
so that u(l;<Sp^,Cj^>)
the last recipient of the pot
is
< W^. Thus
worse—off than he would have
been under Autarky.
i^Both of these can be verified straightforwardly by noting that the
c-p(x)
can be written as
1
=
first
order condition for
fiT,{^)jl^c'{a)da.
20As in the case of the Bidding Rosea, we have been unable to establish whether Assumption
is sufficient to guarantee that all individuals will purchase the durable in a market
know that the market
equilibrium.
Again, intuitively, it seems that it should be.
These two facts
equilibrium is efficient and that all individuals have equal utilities.
together with Assumption 2 should, one would imagine, imply that all individuals will
receive the durable. For, if a group of individuals never received the durable, then, by the
equal utilities requirement, they would have to be compensated by having higher
consumption levels. Given desirability, however, it should be cheaper, in terms of the loss
of the rest of the popiilation's utility, to provide them with the durable.
2
We
2iAs justification for the characterization in (5.1) consider the "separating hyperplane"
argument sketched as follows. Each feasible allocation <s,c>eF gives rise to an allocation
of utility u(a;<s,c>), q;£[0,1]. Let U = { u(-;<s,c>)
<s,c>6F }. Given the convention
of labelling agents accoroing to their order of receipt of the durable, U constitutes the
utility possibility set for our model.
Under our assumptions U is a convex, bounded
|
subset of the space of Lebesgue integrable functions on the unit interval, L^([0,lj), with
non-empty interior, closed in the norm topology. Convexity is assured by the f^ct [see
equation (5.4) below] that on the efficient frontier of F s'(a) is inversely proportional to
Thus the convex combination of two
the aggregate savings rate
[y— /c(x,s(a:))dx].
consumption allocations allows receipt dates for every agent which are less than the same
convex combination of the corresponding assignment functions.
Now an efficient allocation <s',c'>eF generates an allocation of utility u'gU
satisfying (suppress dependence of u on <s,c> hereafter): u(a) > u'(a) a.e. if and only if
u^U.
The Hahn— Banach Theorem
exists,
such that
implies a continuous linear functional
p(u') > p(u), VueU.
I*,
is
well
known
p:L^([0,l])
that the dual of L^([0,1]]
-•
may
R
be
identified with the set of bounded, measurable functions on [0,1]. (See, e.g., Goifman and
Pendrick (1965), Theorem 1, p. 147). Therefore, there exists such a function, (p, satisfying:
p(u')
=
/Ju'(a)(/i(a)da > /Ju(a)(;^(a)da
=
p(u),
V ugU.
Obviously ({){•) must be non— negative, a.e. Moreover, if u' corresponds to an allocation in
which all agents enjoy positive utility from the flow consumption good then, given any
subset
A of agents of measure strictly less than one, there is an alternative feasible
allocation making all agents in A strictly better off, and all agents in [0,1]\A strictly
worse off. Therefore (p must be strictly positive, a.e. The weights 0{ ) correspond to the
function 0(-) normalized to integrate to one.
22To see
this,
note that
c
(q:,s
(0))
=
y
— 1/m'(0 =
c
.
54
23Even with indivisibility the allocation problem reduces in this way if the durable's services
were fungible across agents — if there were, e.g., a perfect rental market for its services.
There are, of course, good (adverse selection/moral hazard) reasons why such trade in durable services might not obtain, especially in a LDC setting. Moreover, some reports on the
use of Roscas stress their role in financirg personal expenditures (daughter's wedding, feast
for fellow villagers, tin roof for house) which, though not producing a fungible asset, generate private consumption benefits lasting for some time that are not transferable to others.
24The similarity of this discussion to classical incentive compatibility considerations may be
misleading, in that there is really only cne "type" of agent in our model. Our results are
based on incentive considerations only in the most primitive way — the equal utility
requirement of equilibrium forces an allocation which is in the "non—convex part" of the
overall utility possibility frontier for the economy.
So the gains from randomization are
induced by the indivisibility in the model. The non-convexity is due to the possibility of
reassigning agents to different orders of receipt of the durable good. As discussed in the
note justifying (5.1), given an ordering cf the agents, as repesented by the convention that
s(a) must be non—decreasing, utility po.ssibilities are convex. By relabelling people at the
same physical allocation one obtains a different distribution of utility among them. The
union of utility possibilities over all possible relabellings is not convex. (See Figure 3.)
The equilibrium incentive constraint of equal utility forces a utility distribution which is in
the intersection of all utility possibility frontiers generated by a relabelling of individuals.
As the Figure shows, the overall utility possibility set must be locally non—convex near
such a point.
25This point can be
made
formally precise.
obtains instantaneous utility v(c)+w^ where
taste parameter.
Suppose that an individual who has the durable
a;6[w,aJ| is
an unobservable, individual
Let F(w) denote the fraction of individuals
who have
specific,
a valuation less
than or equal to u and let u denote the mean value of uj. Then welfare under the optimal
Bidding Rosea, when bids (b) and receipt dates (s) are functions <b(w),s(a;)> of announced
"type" constrained to be incentive compatible, can be written:
W3
= T[v(y)+^]-BM3(e^l-G)),
where G is the Gini coefficient associated with the distribution F(w). Thus welfare under a
Bidding Rosea depends positively upon the extent of the dispersion in agents' valuations of
the durable as measured by the Gini coefficient. By constrast, welfare under the Random
Rosea which treats
all
agents alike remains:
Wrj=T[v(y)+(^]— B/i((fw/2).
The Bidding
Rosea can dominate with sufficient inequality in the distribution of valuations.
is available from the authors on request.
A
proof of
this result
26This does, of course, depend on the fi^rm of the utility function.
However, it may be
checked that the only case in which equality across agents of the level of lifetime utilities,
and equality of the rate of marginal utilities of both goods at all dates, does not conflict is
when utility may be written: u(a;<s,c>)=V[/c(a,tjdt -I- ^(T-s(a))]. That is, the two
goods must be perfect substitutes for one another between any two dates.
27The failure of the market to achieve maximal expected utility parallels results obtained in
other literatures where indivisibilities are important, such as location models (Mirrlees
(1972) and Arnott and Riley (1977)), club membership (Hillman and Swan (1986)) and
conscription (Bergstrom 1986).
28An income lottery, with competitive equilibrium in a market for consumption loans occuring after the realization of random incomes becomes known, would allow a decentralized
realization of the ex ante expected utility maximum,
W
.
N(t
)
c(t)
to
time
nq**P€
^^
56
V9
e/;0(z)dz
'
P^M-e
i
V^(y)-V^(a-g (a))
N
t)
1181
022
Date Due
DEC
1
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PEB.2
2199I
j^i.
m^^
Lib-26-67
MIT LIBRARIES
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