Montréal
Série Scientifique
Scientific Series
2001s-36
Wealth Distribution,
Entrepreneurship and
Intertemporal Trade
Sanjay Banerji, Ngo Van Long
CIRANO
Le CIRANO est un organisme sans but lucratif constitué en vertu de la Loi des compagnies du Québec. Le
financement de son infrastructure et de ses activités de recherche provient des cotisations de ses
organisations-membres, d’une subvention d’infrastructure du ministère de la Recherche, de la Science et de la Technologie, de
même que des subventions et mandats obtenus par ses équipes de recherche.
CIRANO is a private non-profit organization incorporated under the Québec Companies Act. Its infrastructure and
research activities are funded through fees paid by member organizations, an infrastructure grant from the
Ministère de la Recherche, de la Science et de la Technologie, and grants and research mandates obtained by its
research teams.
Les organisations-partenaires / The Partner Organizations
•École des Hautes Études Commerciales
•École Polytechnique
•Université Concordia
•Université de Montréal
•Université du Québec à Montréal
•Université Laval
•Université McGill
•MEQ
•MRST
•Alcan inc.
•AXA Canada
•Banque du Canada
•Banque Laurentienne du Canada
•Banque Nationale du Canada
•Banque Royale du Canada
•Bell Québec
•Bombardier
•Bourse de Montréal
•Développement des ressources humaines Canada (DRHC)
•Fédération des caisses populaires Desjardins de Montréal et de l’Ouest-du-Québec
•Hydro-Québec
•Imasco
•Industrie Canada
•Pratt & Whitney Canada Inc.
•Raymond Chabot Grant Thornton
•Ville de Montréal
© 2001 Sanjay Banerji et Ngo Van Long. Tous droits réservés. All rights reserved.
Reproduction partielle permise avec citation du document source, incluant la notice ©.
Short sections may be quoted without explicit permission, if full credit, including © notice, is given to the source.
Ce document est publié dans l’intention de rendre accessibles les résultats préliminaires
de la recherche effectuée au CIRANO, afin de susciter des échanges et des suggestions.
Les idées et les opinions émises sont sous l’unique responsabilité des auteurs, et ne
représentent pas nécessairement les positions du CIRANO ou de ses partenaires.
This paper presents preliminary research carried out at CIRANO and aims at
encouraging discussion and comment. The observations and viewpoints expressed are the
sole responsibility of the authors. They do not necessarily represent positions of CIRANO
or its partners.
Wealth Distribution, Entrepreneurship
and Intertemporal Trade
*
Sanjay Banerji
†
, Ngo Van Long
‡
Résumé / Abstract
On étudie les échanges intertemporels entre deux pays ayant des
distributions de richesses différentes. On montre que, à cause du risque moral et
de l'imperfection du marché de capital, les gens très riches ou très pauvres
choississent de ne pas devenir entrepreneurs. À un taux d'intérêt donné, le pays
dont la distribution de richesse est moins égalitaire a une offre d'entrepreneurs
plus élevée, ce qui lui donne, en équilibre d'autarcie, un taux d'intérêt plus élevé.
Par conséquent, quand le commerce international est permis entre deux pays, celui
dont la distribution de richesse est moins égalitaire deviendra le pays créditeur.
Les politiques redistributives peuvent donc influencer le profil du commerce
intertemporel. Par exemple, si le gouvernement d'un pays adopte une politique
d'aide aux entreprises qui font faillite, l'offre d'entrepreneurs de ce pays
augmentera, ce qui, à son tour, haussera le taux d'intérê}t en équilibre d'autarcie.
Un pays créditeur deviendra un pays endetté quand une telle polique est
introduite.
This paper examines the pattern of intertemporal trade between countries
with different distribution of wealth. We also examine the consequences of
redistribution policies in this framework. The driving force of our model are risk
aversion, capital market imperfections, and costs associated with default. We
show that under capital market imperfections due to moral hazard, the very rich
and the very poor do not undertake any risk and become passive lenders. Only
individuals whose wealth lies within a medium range choose to become
entrepreneurs. At any given rate of interest, a country with with a wealth
distribution that is relatively less skewed to the left will have a greater supply of
entrepreneurs, leading to a higher equilibrium interest rate under autarky. Hence,
when the countries are opened to trade, those economies with highly skewed
distribution (to the left) will become net lenders. Redistributive policies therefore
will have impact on intertemporal trade. For example, if a government adopts a
bail-out policy (a redistribution from successful enterpreneurs to unsuccessful
ones) will increase the supply of enterpreneurs, driving up the autarkic interest
*
Corresponding Author: Ngo Van Long, CIRANO, 2020 University Street, 25
th
floor, Montréal, Qc, Canada
H3A 2A5
Tel.: (514) 985-4020
Fax: (514) 985-4039
email: longn@cirano.qc.ca
We are grateful to Abhijit Banerji, Kwan Choi, Christ Jacobs, Jérôme Detemple, Henry Wan, Jr., and participants at
the Seatle workshop and Queen's University seminar for many heplful comments.
rate. Consequently, a country which would be a lender if there were no bail-out
policy is a net borrower under the bail-out policy.
Mots Clés :
Échange intertemporel, distributions des richesses, risque moral, prudence,
marché de capital imparfait, entrepreneurship
Keywords:
Intertemporal trade, wealth distribution, moral hazard, prudence, imperfect capital
market, entrepreneurship
This paper examines the pattern of intertemporal trade between countries
with di erent distributions of wealth. We alsoexamine the consequences of
redistributionpoliciesinthis framework. The drivingforcesofourmodelare
riskaversion,capitalmarketimperfections,andcostsassociatedwithdefault.
We show that under capital market imperfections due to moral hazard, the
very rich and the very poor do not undertake any risk and become passive
lenders. Onlyindividualswhosewealthlieswithinamediumrangechooseto
become entrepreneurs. At any given rate of interest, a country with with a
wealthdistributionthatisrelativelylessskewed totheleftwillhaveagreater
supply of entrepreneurs, leading to a higher equilibrium interest rate under
autarky. Hence, when the countries are opened to trade, those economies
with highlyskewed distribution(to the left)willbecomenet lenders.
Redistributivepoliciesthereforewillhaveimpactsonintertemporaltrade.
Forexample, if agovernmentadopts abail-outpolicy(a redistributionfrom
successfulenterpreneurstounsuccessfulones),thiswillincreasethesupplyof
enterpreneurs, drivingup theautarkicinterestrate. Consequently,acountry
which would be a lender if there were no bail-out policy is a net borrower
under the bail-out policy.
Whilethe literatureon riskaversion and entrepreneurship (see belowfor
abriefsurvey)emphasizesthe roleofthe coeÆcientofriskaversioninaloan
market with perfect information,weshow that itis the concept of prudence
thatplaysanimportantrolewhenrisk-averse individualsborrowinamarket
under imperfect information.
An interesting feature of the paper is the endogenous determination of
occupational choice. Individuals with identical utility functions but
di er-ent wealth endowments self-select to be (ornot to be) entrepreneur. In our
model, the outcome of any investment project depends onthe e ortlevel of
the entrepreneur incharge of the project. This e ortlevelis not observable
by lenders or nancial intermediaries. Contracts are designed to give
en-trepreneursincentivestoexert e ort. Whilethese contracts mitigateagainst
opportunismbyentrepreneurs, theycannotreplicatetheoutcomethatwould
beobtained under symmetric information. An important implicationof our
results isthat, inan extended version ofthe model,a well-designed
redistri-bution of wealth may stimulate risk-taking activities and result in a higher
growth rate.
capital market imperfections. It is useful to o erhere abrief comparison of
assumptions and results. Kihlstrom and La ont (1979) consider a society
with individuals having utility functions with di erent degrees of risk
aver-sion. They showthatmorerisk-averse individualsbecomeworkers while less
risk-averse onesbecomeentrepreneurs. Inour modelallindividualshavethe
same utility function, but di erent endowments of wealth. It is the
di er-enceininitialwealthsthatcausesindividualstochoosedi erentoccupations.
Kanbur (1979) alsostudies the relationshipbetween risk taking and income
distribution. Heassumesthatindividualshaveidenticalattitudetorisk,but
theirproductionfunctionsareex-postdi erentfromeachotherbecauseof
in-dependent productivity shocks whichare observed before individualsdecide
on becomingentrepreneur or worker.
Galor and Ziera (1993) consider a model with irreversible investment
involving xed costs. They assume that agents borrow to nance their
in-vestmentinskillacquisition,andbequeathsomeoftheirwealth. Thelending
rateishigherthantheborrowingrate,andthecostofborrowingishigherfor
borrowers with low initial wealth. The amounts these borrowers bequeath
are also smaller. The authors show that agents whose wealth lies below a
threshold level nd it optimal to choose unskilled jobs, with low wages. As
a result, their descendents will also choose to be unskilled workers due to
their low level of inherited wealth. A similar model is studied by Banerjee
and Newman (1993) who assume that borrowers need to o er collaterals to
lenders. Children ofpoorindividualsdonot inheritmuch,andtherefore
can-not o er suÆcient collaterals to potential lenders. They are thus forced to
choose not to be entrepreneurs. de Meza and Webb (1999) show that lack
of information on the part of banks may lead to an over-provision of loans
that encourages entry into entrepreneurship. If the associated incentive
ef-fects are strong, then there will be a positive relationship between wealth
and entrepreneurial activities.
Aghion and Bolton (1997) formulate a model similar to ours, but they
assumethatallagentsarerisk-neutral. Theprobabilityofsuccessofaproject
is dependent on the e ortlevelchosen by the entrepreneur. In their model,
individualsintheright-handtailofthewealthdistributionareentrepreneurs
who do not borrow, while individuals in the middle section of the wealth
distribution need to borrow to be entrepreneurs, because they do not have
suÆcientwealthtopayforthelumpyinvestment. Thisisinmarkedcontrast
aversion (b) partial collateral, and (b) bankruptcy cost, while Aghion and
Boltonassumeriskneutrality andstrictly limitedliability(zero paymentsto
banks inthe event of project failure).
Newman(1995)assumesriskaversion,and ndthat,undermoralhazard,
the poorer individuals tend to be risk-takers and the wealthier individuals
are risk-averters. This may beexplained asfollows. Optimalcontract under
moral hazard serve to resolve the tension between consumption-smoothing
across states of nature, and eÆcient deployment of e ort. We know that
wealthier individuals tend to need less insurance at the margin. Optimal
contractspromptthemtobearmoreriskatanygivene ortlevel. Individuals
with greater wealth nd it too costly (in terms of e ort) to bear risks, as
designed inthe contracts.
All the models cited above share a common assumption: there is no
transfer of resources from the entrepreneurs to the lenders in the event of
projectfailure. Thismay becalledthe\stronglimitedliability"assumption.
Our modelallowaweakerversionoflimitedliability: weassumethatpartof
an entrepreneur's private savings must be used topay debts in the event of
projectfailure. In addition,weassume theexistence of asmallreal resource
cost in the settling of a bankruptcy case. These twin assumptions play an
important role in our model. We nd that the very poor do not take risks,
because the marginal utilty of wealth in the event of bankruptcy is very
high. We also nd that the very rich do not become entrepreneurs, because
the e ort isvery costly.
The papers cited above deal with closed economies, and most authors
consider only a partial equilibriumsetting. Grossman (1984) addressed the
issue ofthe relationshipbetween internationaltradeand theformationof an
entrepreneurial class. He showed that in a free trade equilibrium, there is
a tendency to over-specialize in the non-risky sector and under-specializein
risk-taking activities. However, in his model, there was no risk sharing
ar-rangements. Thenon-existenceof nancialmarketwasexogenouslyimposed.
As Dixit(1987) pointed out, once the incentive compatibility conditions are
explicitly introduced, the market outcome is arguably a Pareto optimum.
That is, once the sources of marketimperfections are speci ed, risk-sharing
arrangementswill incorporate those features in incentive compatibility
con-ditions, and it would be impossible to improve upon the market by means
of public provisionof insurance etc. Grossman and Maggi(2000) consider a
diverse workforceexports the goodfor whichindividualsuccess ismore
im-portantthanteamproduction. Adverse selectiontendstocreateamismatch
of talents which tend to be exacerbated under free trade. While Grossman
and Maggiplaceemphasis ontheinteraction between distributionoftalents,
adverseselectioninthelabormarket,and trade,ourpaperisconcernedwith
interactions between wealth distributions, moral hazard, and imperfections
in the credit market, in the contextof a tradingworld.
The plan of the paper is as follows. In section 2, we develop a modelof
a closed economy, and state the assumptions and notation. In Section 3,we
characterize equilibrium loan contracts under moral hazard, and show how
the equilibrium riskless rate of interest and the occupational choice are
de-termined. Section4considers atwo-country worldand shows howacountry
with a more skewed distribution of wealth tends to be the net lender. We
also consider some extensions of the model, such as the implications of a
bail-out policy. Some concludingremarks are o ered insection 5.
2 The Autarkic Equilibrium
2.1 Assumptions and Notations
Webegin by consideringamodelof aclosedeconomy. There isacontinuum
of individualsinthiseconomy. Theydi erintheirinitialwealth,denoted by
w. The(cumulative)distributionofinitialwealthisF(w),andf(w)denotes
the corresponding density function. It is assumed that there is a closed
interval [w L
;w H
] over which f(w)is strictly positive,and that f(w)=0 for
all w outsidethis interval. We assume0<w L <w H . Then Z w H w L f(w)dw=1
and the percapita wealth is
w= Z w H w L wf(w)dw:
Thereare onlytwo periods. For simplicity,we assumethat consumption
takes place only in period two 1
. In period one, each individual may choose
1
ThisassumptionisalsomadebyGertlerandRogo (1990),butintheirmodel,unlike
ours, entrepreneurship is not endogenous. Aghion and Bolton (1997) also assume that
activity, using part of his own wealth, plus some additional borrowing, to
nance hisinvestment project), or he may choose not tobe anentrepreneur
(in which case he would keep all his wealth in the form of a deposit in a
nancialinstitution,whicho ers himthe risklessgross rateof returnr>1).
The endogenous determinationof r willbe explainedin due course.
We assume that wealth, in its physical form, cannot be stored 2
. It
fol-lows that all wealth must eventually be lent (via the nancial institutions)
to entrepreneurs, who use them as input in their investment projects. We
adopt the standard assumption that nancial institutions are risk neutral
and perfectlycompetitive,so that their expected pro tsare zero.
Each entrepreneur can carry out only one investment project. Each
projectrequires
k units of wealth. We assume that
k>w
H
(1)
so that all entrepreneurs need external nancing. A project can turn out to
bea success or a failure. In the case of success, the pay-o of the project is
a >
k. In the case of failure, the pay-o is zero. The probability of success
is denoted by (e), where e is the e ort level of the entrepreneur, which is
not observableby themarket. Forsimplicity,weassumethatecantakeonly
two possible values, 0 or 1. We write p = (1) > q = (0). This indicates
that the model exhibits the moral hazard property: the entrepreneur, who
is a net borrower, may have an incentive to work at an e ort level that is
lowerthanwhatwouldbeeÆcientinaworld ofperfectinformation. Inwhat
follows, unless otherwise stated, we assume that p q is suÆciently great,
sothat the equilibriumcontractshavethe followingproperty: entrepreneurs
are suÆciently rewarded forsuccess that they havean incentivetoset e =1
even though e isnot observable.
Since each entrepreneur needs
k units of capital as input, the
assump-tion that k > w
H
implies that in equilibrium, the endogenous number of
entrepreneurs, N, isless thanthenumberofindividualsinthiseconomy,M.
Eachindividual has the followingutility function
U =U(y;e)=v(y) e
2
An alternative assumption is that wealth can be stored but the rate of return on
is a positive parameter. Weset =1 by normalization 3
. The function v(y)
is increasing and strictly concave, with
v 0
(0)=1:
This property implies that each individual will invest some wealth in the
riskless asset (i.e., lending to a nancial institution), to avoid having zero
wealth in period 2.
Whiletheoutcomeofanygivenprojectisuncertain,weassumethe
proba-bilityofsuccessofanygivenprojectisindependentofthoseofotherprojects,
andthatthe numberofprojectsislargeenoughsothatthelawoflarge
num-ber applies. Thus, for the economy as a whole, if all entrepreneurs choose
e=1,aggregate outputis paN whereN isthe measure ofthe set of
individ-uals who,in equilibrium,choose tobe entrepreneur.
Since the expected return on a project is pa and entrepreneurs have to
incur e ort cost,itis thecase that the risklessrate of interestr must satisfy
the condition that
r
k <pa+(1 p)0 (2)
2.2 A Benchmark Case: Observable E ort and No
Bankruptcy Cost
If the e ort level of each entrepreneur is observable, and if there is no
bankruptcy cost, thenwe are inthe rst-best world. Since thereisa
contin-uum ofindividualsandacontinuumof investmentprojects, inthe aggregate
there is no risk, and a perfect insurance market implies that all individuals
are perfectly insured, given that their function v(y) is strictly concave. We
focus on the case where p=q is suÆciently great, so that optimal contracts
specify that entrepreneurs must exercise full e ort(e=1).
Given the rate of interest r;competitive nancial institutions o er
con-tracts that specify, for any given w, the amount
k k(w) to be lent to the
entrepreneur with wealth w, and the payments R s
(w) and R f
(w) that he
must make in the events of success and failure respectively. Here k(w)
de-notes the amount of equity that the entrepreneur puts in his project. The
3
An entrepreneurcan choosee tobe0or1. Non-entrepreneursneed notexpend any
to zero expected pro t for the nancialinstitutions: maxpv[ (w k)r+a R s ]+(1 p)v[ (w k)r R f ] 1 subjectto pR s +(1 p)R f =r( k k)
The solution yieldsconstant utility across statesof nature
(w k)r+a R s =(w k)r R f At the optimum,R s +rk and R f +rk are determinedby R s +rk =(1 p)a+r k R f +rk =R s +rk a
Thus, given r, the expected utility of an entrepreneur with initial wealth w
who accepts a contract that stipulates e =1, is
EV e (w;1;r)=v h wr+pa r k i 1 (3)
Similarly,azero-expected-pro tcontractthat maximizesthe expected
u-tilityofanentrepreneurwithinitialwealthw,giventhathise ortisspeci ed
to be e=0, yieldsthe net utility
EV e (w;0;r)=v h wr+qa r k i 0 (4)
De ne the function
(w;r)EV e (w;1;r) EV e (w;0;r)=v h wr+pa r k i 1 v h wr+qa r k i (5)
This function measures, for an individual with wealth w; the relative
at-tractivenes of being an entrepreneur with e = 1 (rather than being an
en-trepreneur with e = 0). It is decreasing in w for any given r: w
< 0.We
assume that, for all r < pa=
k, (w;r) > 0 for w = w L
and (w;r) < 0
for w suÆciently great. Then there exists a \criticalvalue" w c
(r) such that
individuals with wealth w < w c
(r) will prefer being an entrepreneur with
e=1 tobeing one with e=0. Since (w;r)is increasing inr forall w< k,
if w (r) < w H
(which is less than
k by assumption), an increase in r will
increase w c : dw c dr = r w = fv 0 h wr+pa r k i v 0 h wr+qa r k i g(w k) w >0
A pure lender,on the other hand, obtainsthe utilityv(wr):If r>qa= k,
[respectively, if r < qa=
k] the strategy of being a pure lender dominates
[respectively, is dominated by] that of being an entrepreneur who exerts no
e ort. We assume that q is so smallthat in equilibrium, r > qa= k. Let us de ne (w;r)=EV e (w;1;r) 1 v(wr)
This functionmeasures, for anindividualwithwealth w; therelative
attrac-tivenesofbeinganentrepreneurwithe=1(ratherthanbeingapurelender.)
Then,forallrintheinterval[qa= k;pa= k], (w;r)isdecreasinginw: w <0.
There exists a value w b
(r)such that (w b
(r);r)=0. Given the assumption
that r >qa=
k,we can deduce that w b (r)<w c (r), and dw b dr = r w = v 0 h wr+pa r k i (w k) v 0 (wr)w v 0 h wr+pa r k i r v 0 [wr]r <0
Thus, in this economy, under perfect information (i.e., e ort is
observ-able), the demand forloans is
B(r)= Z w b (r) w L ( k w)f(w)dw
and the supply of loans is
L(r)= Z W H w b (r) wf(w)dw Since B 0 (r) = ( k w b )f(w b ) dw b dr < 0 and L 0 (r) = w b f(w b ) dw b dr > 0, the
demand curve isdownward sloping,and the supply curve isupward sloping.
Theirintersectiondeterminestheequilibriuminterestrate,r . Atr ,demand equals supply, B(r )=L(r ),implying k Z w b (r ) w L f(w)dw= Z w H w L wf(w)dw=w
w w b (r )
and let F(w)denote the cumulativedistribution of wealth:
F(w) Z w w L f(z)dz then F(w )= w k
Tosummarize,underperfectinformation,thefraction ofpopulationwho
are entrepreneurs in equilibriumis w=
k, and only individualswhose wealth
is smallerthan w F 1 (w=
k)will choose to be entrepreneurs.
2.3 The Moral Hazard Case: Unobservable E ort
Now we turn to the case of unobservable e ort levels. We continue to
as-sumethateachentrepreneur'swealthisknowntothe nancialintermediaries.
This,and theassumptionthat individualshavethe sameutilityfunctionand
identicalabilitymeansthatthereisnoadverseselectionproblem: noonecan
lieabouthiswealthorhisutilityfunction. Theonlyproblemismoralhazard:
if how much an entrepreneur must pay back to the nancial intermediaries
is independent of his e ortlevel,then he may have an incentiveto exert no
e ort. Contracts must therefore be designed to provide suÆcient incentive
for enterpreneurstochoose e=1(Thisisof coursebased onthe assumption
that p q is suÆciently great tojustify the choice e=1:)
We now describe a contract for an entrepreneur with wealth level w.
Recall that we assumed w H
<
k. The contract says that \if your wealth is
w and youcontributeanamountk wasyour \equity"inyour investment
project(sothat your borrowing fromyour nancialinstitution is
k k 0)
andyoudeposittheremainingw kata nancialinstitution,thenyoumust
pay back toyour lender(the nancialinstitution, orFIforshort)anamount
which depends onthe outcome of your project. Ifthe outcome is \success",
your investment yields the gross return a, and you must pay an amount R
toyour FI;if youroutcome is\failure"(the investmentyieldsa grossreturn
of zero), then, with your periodtwo wealth (w k)r, you must pay back to
the FI anamount (
k k)>0". (In what follows, we assume <r.) The
assumption that > 0 is meant to capture the fact that most real world
to be very small.
It follows that if the entrepreneur exerts e ort (i.e., e = 1), then his
expectedutility is pv[(w k)r+a R ]+(1 p)v h (w k)r ( k k) i 1G(w;k;R ;1)
and if he does not exert e ort(i.e., e=0), then hisexpected utility is
qv[(w k)r+a R ]+(1 q)v h (w k)r ( k k) i G(w;k;R ;0)
We assume also that when a bankrupt entrepreneur pays the amount
(
k k)>0tohisFI,the latteronlygets afraction of it:Inother words,
the FI incurs a real cost (1 ) (
k k) in collecting (
k k) from the
failed entrepreneur. Bankruptcy costs are cost associated with the transfer
ofresources fromthe debtortothecreditorinthe badstateofnature. These
costs consist of legal fees, delay costs, etc. Several studies have shown that
suchcosts canhaveanimpactonthe valueof claimsownedbycreditors and
are re ected inthe pricing of loans.
Thus an FI that lends the amount
k k to an entrepreneur who does
exert e ortcan expect toget
pR+(1 p) ( k k)
Ontheotherhand,theFItakesrasgiven,andmustpaytheamountr( k k)
to depositors. In equilibrium, we have the following zero expected pro t
condition (if e=1): pR+(1 p) ( k k)=r( k k) (6)
Competition among the FIs imply that, for given r, the FIs will o er
to any entrepreneur with wealth w a contract that maximizes his expected
utility,subjecttothezeropro tcondition. Wealsoassumethatthecontract
provides enoughincentive for the entrepreneur to exert e ort(e=1). More
formally,FIj willo ertothe entrepreneurwith wealth wa pairof
number-s (R ;k) that maximizes G(w;k;R ;1) subject to the incentive compatibility
constraint
G(w;k;R ;1)G(w;k;R ;0) (7)
Wenowturntoafullercharacterizationofequilibriumcontracts. Weproceed
as follows. First, we take the interest rate r as given, and show how the
incentive compatibility constraint and the zero pro t condition determine
the contract for entrepreneurs at each wealth level. Then we show how the
theinterestraterisdeterminedendogenously. Inwhatfollows,wenormalize
by setting=1.
3.1 Equilibrium loan contracts for entrepreneurs,
giv-en the interest rate on the safe asset
In equilibrium,pro twillbezero, andthe incentivecompatibility
constrain-t binds for each entrepreneur. It follows that, for given r, conditions (6)
and (7) determine the equilibrium contract for entrepreneurs with wealth
w. The following lemma characterizes the equilibrium contract, under the
assumption that is small:
LEMMA 1: (Equilibrium contract) In equilibrium, for given r, there
exists a lower bound w>0such that entrepreneurs with wealth w w will
belentthe amount
k k(w)and willbe asked to pay the amountR (w)0
in the event of success, and h
k k(w) i
0 in the event of failure, where
the pair (R (w);k(w)) satisfy the following conditions:
(i)the expectedpro t of the FIis zero
R = ( k k) p [r (1 p) ] for k k (8)
and (ii) the incentive compatibility constraintholds with equality
v[(w k)r+a R ] v h (w k)r ( k k) i = 1 p q (9)
Proof: See the Appendix.
REMARK1: Equation(8)impliesthat,giventhe risklessrate ofinterest
r,the repayment tothe FI(in the event of success) perdollarborrowed isa
constant, independentof the amount borrowed:
r s R k k = 1 p [r (1 p) ]:
s
theeventoffailure,is perdollarborrowed.) Given , whichisexogenous
inour model, nancialinstitutions willbewillingtolend onlytoindividuals
whose wealth exceeds a certainlowerbound w>0:
REMARK2: Equation(8)isrepresentedinFigure1 by thedownward
s-lopingcurveKKinthe(k;R )space,withtheverticalintercept[ r (1 p) ]
k=p. Ifp q issuÆciently close toone, and is suÆciently small, then the
intercept of the secondcurve (de ned by (9) and denoted as the curve VV)
is below[r (1 p) ]
k=p and hence the two curveswillhave an
intersec-tion (k(w);R (w))with k(w)>0 and R (w)>0,if w exceeds a certainlower
bound w>0.
PLEASE PLACE FIGURE 1HERE
LEMMA2: Entrepreneurswithgreaterwwillborrowless(i.e.
k k(w)
decreases with wealth) and hence invest more (put more equity) in the risky
project, i.e. k 0
(w)>0, given that contracts must satisfy (8)and (9).
Proof: Note that the curve KK is independent of w. For any given k,
an increase in w will shift the curve VV down, resulting in an intersection
tothe rightof theformer intersection. Hencek
increases, andR
decreases.
This means the individual borrows less when hiswealth increases.
REMARK 3: One of the standard text-book results is that if the payo
per dollar invested in a risky asset in each state of nature is independent
of the amount invested in the risky asset, then the amount an individual
invests inthe risky assetisanincreasing functionof hiswealthif and onlyif
his absolute risk aversion is a decreasing function of wealth. Our result in
Lemma 2isdi erent,becausethe pay-o perdollarof equity inthe event of
success is Æ s = a R k = a k 1 pk [r (1 p) ]( k k)
which is dependent on k (decreasing in k). (And similarly, the pay-o per
dollar of equity in the event of failureincreases in k.) Lemma 1 states that,
with a concave utility function, but independently of whether absolute risk
aversion is a decreasing or increasing function, entrepreneurs with greater
wealth will contribute more equity inthe project.
LEMMA3: Giventhatcontractsmustsatisfy(8)and(9),entrepreneurs
withgreaterwwillinvest moreinthe risklessasset,(i.e., w k
(w)increases
v 0 (s)(r )>[v 0 (u) v 0 (s) ] p (10)
where u is the nal wealth inthe failurestate,
uu(w)(w k(w))r (
k k(w)) (11)
and s isthe nal wealthin the success state,
ss(w)(w k(w))r+a R (w) (12)
Proof: From (8) and (9), weget
dk dw = [v 0 (u) v 0 (s)]pr [v 0 (u) v 0 (s)]pr+v 0 (s)(r ) [ v 0 (u) v 0 (s) ] p (13)
which isless than 1if and only if (10) holds.
REMARK4: Condition(10) holds if issmall, which isassumed here.
3.2 An example: logarithmic utility
Let
v(y)=lny
then we obtain from(9)
ln " (w k)r+a R (w k)r ( k k) # = 1 p q hence wr kr+a ( k k)r s = h wr kr ( k k) i e 1=(p q) Thus h r s +(r )e 1=(p q) r i k= h r s e 1=(p q) i k a+rw h e 1=(p q) 1 i (14) i.e., k(w)= h r s e 1=(p q) i k a+rw h e 1=(p q) 1 i [r s +(r )e 1=(p q) r] (15)
dk dw = h e 1=(p q) 1 i r [e 1=(p q) 1]r+[r s e 1=(p q) ]
whichispositiveand less than1provided issmall. Thisis consistentwith
Lemmas 2 and 3.
3.3 To be or not to be an entrepreneur
So far, we have characterized contracts that the FIs o er to potential
en-trepreneurs with wealth w, which would make him choose e = 1, assuming
that he does want to be an entrepreneur. But depending on his wealth, an
individualmay nd that theutility of being alender;v(rw),may exceed the
expectedutilityofbeing anentrepreneur, i.e.,itispossiblethat, forsomew,
we have
v(rw)>EU pv(s)+(1 p)v(u) 1 (16)
Clearly, if w is close to zero, condition (16) will be satis ed, because
nancial institutions lend a positive amount only to individuals with w
w > 0: Thus the curve v(rw) lies above the curve EU when w is small: At
some b
w >0, the curve EU cuts the curve v(rw) from below. The question
is whether there exists some value e w >
b
w such that the curve v(rw) again
overtakesthe curveEU. The answerturnsout todepends onthemagnitude
ofthecoeÆcientofprudence,aconceptintroducedbyKimball(1990),which
is inturn related tothe convexity of the followingfunction
(y) 1
v 0
(y)
which, in our model, measures the marginal rate of substitution between
e ort and income:
MR S ey = U e U y = v 0 (y) = 1 v 0 (y)
The function (r) is convex if and only if the coeÆcient of prudence is
smaller than twice the coeÆcientof absolute riskaversion:
2v 00 v 0 v 000 v 00 (17)
1990). 4
LEMMA 4:
(i)If (y)is concave, then
p v 0 (s) + 1 p v 0 (u) 1 v 0 (ps+(1 p)u) (18)
and if (y)is convex, then
p v 0 (s) + 1 p v 0 (u) 1 v 0 (ps+(1 p)u) (19)
(ii)if is small, then
1 v 0 (ps+(1 p)u) 1 v 0 (rw) (20)
Proof: Part (i) follows from Jensen's inequality.The inequality in part
(ii)holds if and onlyif
rw<ps+(1 p)u (21)
Now, if is small, then ps+(1 p)u= rw rk(w)+pa (
k k(w))[r
(1 p) ( 1)]'rw+pa r
k >rw because of (2).
LEMMA 5: If (y) is convex, then the curve EU may cut the curve
v(rw)from belowat some value b
w>0,and then from above,at some value
e w>
b
w. (see Figure 2).
Proof: see the Appendix and Figure2.
PLEASE PLACE FIGURE 2HERE.
The following propositionfollows immediatelyfromthe above lemma:
Proposition 1: if the coeÆcient of prudence is smaller than twice the
coeÆcientofabsoluteriskaversion,thenitispossiblethat,givenr,thereare
4
Notethat decreasingabsoluteriskaversionholdsi
v 000 v 00 > v 00 v 0
which is not inconsistentwith (17). For further resultson the role of v 000
, see Hartwick
twovaluesw(r)<w(r)suchthatonlyindividualswhosewealth liesbetween
them will choose to be entrepreneurs. Individuals with wealth exceeding
e
w(r),andthosewithwealthbelow b
w(r)willchoosetoinvestinthesafeasset
(i.e., lend to nancialintermediaries).
REMARK: Proposition 1 may be explained as follows: given that the
marginalrate of substitutionbetween e ortand incomeis convex iny,
indi-viduals with very low wealth do not want to become entrepreneur because
they are not willing to take risks, and individuals who are very wealthy do
not wantto become entrepreneur because they do not want toexert e ort.
We now show how the two critical values b w(r) and e w(r) change when r increases. Proposition 2: b w 0 (r) > 0 and e w 0 (r) < 0 provided that (1 p) is
suÆciently small, where f(r )=(r )g 10
Proof: see the Appendix.
3.4 Endogenous determination of the interest rate on
the safe asset
So far, we have taken the interest rate r as given. Now we turn to its
determination. This is given by the condition that the the interest rate
must equate the aggregate lending by non-entrepreneurs (to the nancial
institutions)totheaggregateborrowingbyentrepreneurs(fromthe nancial
institutions). The former is given by
L(r)= Z b w (r) w L wf(w)dw+ Z w H e w(r) wf(w)dw
and the latteris
B(r)= Z e w (r) b w (r) ( k w)f(w)dw
The excess demand (function) forfund is
D(r)=B(r) L(r)= k Z e w(r) b w (r) f(w)dw Z w H w L wf(w)dw= k Z e w (r) b w (r) f(w)dw w
Clearly,ifrisveryhigh, thenD(r)<0,andififr =0,thenD(r)<0(recall
that
k > w). By continuity, there exists a value r
such that D(r
Furthermore, such r is unique, because D(r) < 0: Thus we obtain the
followingresult:
Proposition3: There exists a unique r
>0 such that the loan market
clears. Individuals with wealth exceeding e w(r
) lend all their wealth to the
nancial institutions,at the safeinterestrate r
, and sodo individualswith
wealth below b w(r
). Individualswhosewealth liesbetween thesetwocritical
valueswillbeentrepreneurs, and they invest afraction oftheirwealth inthe
safeasset,theremainingfractionbeingusedasequitycapital,whichisalways
less than the total amount of capital invested in each risky project. The
equilibrium fraction of the population who choose to become entrepreneurs
is N M = w k = Z e w(r ) b w (r ) f(w)dw. (22) 4 Open Economies
4.1 International Borrowing Under the Moral Hazard
Case
Nowconsider twocountries withthesamepopulationsizeand thesamelevel
of per capita wealth, but di erent distributions of wealth. We will look at
theirautarkicequilibria,andexaminethe incentiveforinternational
borrow-ing. Letusassumetherearetwocountries,AandB. Thedensityfunctionsof
the twowealth distributionsarerespectivelyf A (w; A )andf B (w; B ),where A and B
are shift parameters. Now consider the autarkic equilibrium of
country A, and let r A
be the autarkic interest rate. Consider the following
thought experiment: Suppose the distribution f A (:)undergoes achange: A now increases to A , which makes f A (w; A ) < f A (w; A
) for all w in the
interval [ b w(r A ); e w(r A
)]. Then clearly at the value r A
, the numberof willing
entrepreneurs willbeless than N. To restoreequilibrium,r A
must belower.
Tomakethe aboveargumentmoreprecise,letuswritethe equationthat
determines the equilibrium autarkicrate of interestas
F A [ e w(r A ); A ] F A [ b w(r A ); A ]= w k Then dr A d A =
( @ @w F A [ e w(r A ); A ] ) d e w(r A ) dr ( @ @w F A [ b w(r A ); A ] ) d b w(r A ) dr <0 by Proposition 2, and ( @ @ A F A [ e w(r A ); A ] ) ( @ @ A F A [ b w(r A ); A ] ) = @ @ A Z e w(r A ) b w(r A ) f(w; A )dw = Z e w(r A ) b w (r A ) @ @ A f(w; A )dw<0 It follows that dr A d A <0.
Note that in the above argument, the condition @ @ A f(w; A ) < 0 over [ b w(r A ); e w(r A
)] issuÆcient, but not necessary, for tobe negative.
Itfollowsfromtheabove reasoningthatif thedensityfunctionf B (w; B ) di ers fromf A (w; A )in that f B (w; B )<f A (w; A ) for allw2[ b w(r A ); e w(r A )]
then country B will have a higher autarkic gross rate of return: r B
> r A
.
Under these conditions, the opening ofworld nancialmarkets willresultin
capital owfrom A toB. We obtainthe following proposition:
Proposition 4: Two countries with identical per capita wealth and
i-dentical individual utilityfunction may engage inintertemporaltradeif the
distributions of wealth are di erent.
Corollary: A marginal redistribution of wealth away from the
enter-preneurial class of a country can turn that country status from being a net
lender toa net borrower.
4.2 Bailouts
In this section, we set up a framework for studying the implications of
gov-ernment policies that seek to help bankrupt entrepreneurs. Suppose the
government taxes successful entrepreneurs, collecting from them amount T
each, and pay the unsuccessful ones an amount : Balanced budgetrequires
that pT Z e w (r) b w(r) f(w)dw=(1 p) Z e w (r) b w (r) f(w)dt (23)
where [w(r);w(r)] are solutionsof the equation v(rw)=pv[(w k)r+(a R ) T]+(1 p)v h (w k)r ( k k)+ i 1
Doesthis bailoutpolicyresult inalowerorhigherequilibriuminterest rate?
What is its implicationon the country's net borrowing?
First,note that (23) implies that
dT
d =
1 p
p
Thecontractbetweenthe nancialintermediary(FI)andtheentrepreneur
with wealth w must nowsatisfy
(p q)[v(s) v(u)]=1 (24) and pR= k k [r (1 p) ] (25) where now s(w k)r+(a R ) T and u(w k)r ( k k)+
From (24) and (25), we have
@k @T = pv 0 (s) pv 0 (u)(r )+(1 p)v 0 (s)(r ) and @k @ = pv 0 (u) pv 0 (u)(r )+(1 p)v 0 (s)(r ) Hence dk d = @k @T dT d + @k @ = (1 p)v 0 (s)+pv 0 (u) pv 0 (u)(r )+(1 p)v 0 (s)(r )
To nd the respose of theintersectionpoints e
w(r) and b
w(r)with respect
to , we use the equation of indi erence, whichis now
q p q +v h (w k )r ( k k )+ i =v(wr) (26)
where k =k (w;T;(T)). From (26) we get dw d = 1 (1 p)v 0 (u)v 0 (s) ( 1) (1 p)v 0 (s)(r )+pv 0 (u)(r ) (27) where =v 0 (s) " r (r ) @k @w # v 0 (NE)r (28) which is positive at b w(r) and negative at e
w(r). Hence, for a given r, b w(r)
increasesand e
w(r)decreaseswhenthereisanincreasein. Thisimpliesthat
thenumberof entrepreneursincreases, leadingtoanincreases inthe demand
for fund. Torestore equilibrium,the rate of interestr must increase.
Proposition5: Bail-outpoliciesofthetypedescribed abovewillleadto
an increase in the equilibriumrate of interest. In a two-country world, this
means that the country with such bail-out policies will become the debtor
country.
REMARK: The role of the bankuptcy cost factor < 1 is crucial here.
If =1, then bailoutshave noe ect onthe equilibriuminterestrate.
5 Concluding remarks
We have set up a model to show that a country's wealth distribution
in- uences the occupational choice of individuals: to be or not to be an
en-trepreneur. We have alsoshowed that, under certain assumptions, only the
middle-class individuals choose to be entrepreneurs. An implication is that
countries that have the same level of per capita wealth and the same
indi-vidualutilityfunction mayengage inmutualintertemporaltrade, aslongas
they havedi erent wealth distributions. The key elements inour modelare
(i) the tension between consumption smoothing across states of nature on
the one hand, and contract design to overcome moral hazard on the other
hand, (ii) the relationship between of the coeÆcients of prudence and risk
aversion, and (iii)bankruptcy cost and partial liability.
Thereareseveral directions ofgeneralization,whichwewish topursuein
ourfuturework. Anobviousextensionistheprocessofcapitalaccumulation.
One would thenbeable tosee howmoralhazardand initialwealth
distribu-tions in uencegrowthrates, andtodetermine conditionsunderwhichcycles
may occur. Taxation policies, including transfers, may be studied in the
PROOF OF LEMMA 1
Consider the problem
max R;k G(R ;k;w;1)=pv((w k)r+a R )+(1 p)v((w k)r ( k k)) 1 subjectto v((w k)r+a R ) v((w k)r ( k k)) 1 p q (29) and pR ( k k)[r (1 p) ]0 (30)
In the (k;R ) space, the feasible set isthe intersection of area abovethe line
R= ( k k) p [r (1 p) ] (31)
withtheareabelowthe(positivelysloped)curvede nedby(29),seeFigure2.
The absolutevalueof theslopeof theline(31),dR =dp, isapproximatelyr=p
if issmall. NowitiseasytoverifythattheobjectivefunctionG(R ;k;w;1)
isstrictlyconcaveanddecreasing in(k;R ). Thismeansthattheiso-expected
utility curves are concave inthe (k;R ) space, with negativeslope given by
@R @k = G k G R
If issmall,theabsolutevalueofthisslopeisapproximately(r=p)[v 0
(u)=v 0
(s)]>
(r=p). It follows that the maximum occurs at the point where both
con-straintshold with equality.
PROOF OF LEMMA 5:
Let (w) v(rw) EU(w) be the di erence between the utility of a
pure lender with wealth w and the expected utility of an entrepreneur
hav-ing the same wealth. We wish to nd conditions which imply that (w) is
a U shaped curve, taking positive values for small w, negative values for
intermediatevalues of w,and positivevalues again forlarge values of w.
Byde nition
(w)=v(rw) [v(u)+q=(p q)] (33)
We want toshow that 0
(w) <0 for small w and 0 (w)> 0 for large w. Now 0 =rv 0 (rw) v 0 (u)[r(1 k 0 (w))+ k 0 (w)] where @k @w = pr[ v 0 (u) v 0 (s)] pv 0 (u)(r )+(1 p)v 0 (s)(r ) (34) 0 (w) isnegative i H v 0 (u) " r (r ) @k @w # rv 0 (rw)>0 (35) which isequivalent to r (r ) @k @w > rv 0 (rw) v 0 (u) i.e., " 1 v 0 (rw) v 0 (u) # r>(r ) @k @w i.e., " v 0 (u) v 0 (rw) v 0 (u) # >(r ) p[v 0 (u) v 0 (s)] pv 0 (u)(r )+(1 p)v 0 (s)(r ) i.e., " v 0 (u) v 0 (rw) v 0 (u) # > p[v 0 (u) v 0 (s)] pv 0 (u)+(1 p)v 0 (s)f(r )=(r )g (36)
Let 1+ =f(r )=(r )g1. Inequality (36) holds i
" v 0 (u) v 0 (rw) v 0 (u) # > p[v 0 (u) v 0 (s)] pv 0 (u)+(1 p)v 0 (s)(1+ ) (37)
" pv 0 (u)+(1 p)v 0 (s)(1+ ) v 0 (u)v 0 (s) # [v 0 (u) v 0 (rw)]> p[v 0 (u) v 0 (s)] v 0 (s) (38) Let Z = pv 0 (u)+(1 p)v 0 (s) v 0 (u)v 0 (s) (39) Then (38) holds i 1+ (1 p) " 1 v 0 (rw) v 0 (u) # >Zv 0 (rw) (40) i [Zv 0 (rw) 1] (1 p) " 1 v 0 (rw) v 0 (u(w)) # <0 (41)
Note that v(rw) = v(u)+q=(p q) implies that rw > u(w) and hence
1 v 0 (rw) v 0 (u(w))
>0: We want the inequality (41) to hold for small w and to be
reversed for large w. Now, from Lemma 4,if is small and (w) isconvex,
then p v 0 (s) + 1 p v 0 (u) > 1 v 0 (rw) and hence Zv 0
(rw)>1, and (w) may be U-shaped if (1 p) h 1 v 0 (rw) v 0 (u(w)) i
is larger [respectively, smaller] than [Zv 0
(rw) 1] for small w; and smaller
[respectively, larger] than itfor large w.
It follows that if (w) is convex, then the EU curve may cut the v(rw)
curve from belowat some b
w,and then fromabove, atsome e w>
b w.
PROOF OF PROPOSITION 2
From (8)and (9), we have
@k @w = 1 [v 0 (u) v 0 (s)]pr>0 and @k @r = 1 h fv 0 (u) v 0 (s)g(w k)p+v 0 (s)( k k) i where =pv 0 (u)(r )+(1 p)v 0 (s)(r )>0
r,we get dw dr = 1 H " wv 0 (NE) v 0 (u)(w k)+(r )v 0 (u) @k @r # (42)
where H, de ned in (35) above, is positive at b
w(r) and negative at e w(r).
The numerator in(42)must now besigned. It can be written as
wv 0 (NE)+v 0 (u) " p(w k)fv 0 (u) v 00 (s)g+v 0 (s)( k k) pv 0 (u)+(1 p)v 0 (s)(1+ ) (w k) #
The term inside the square brackets can be writtenas
v 0 (s)(w k) " k k w k 1 (1 p) # or v 0 (s)(w k) " k w w k (1 p) #
Now, since w k(w)is increasing in w,we have
k w w k(w) > k w w H k(w H ) > k w H w H k(w H ) which ispositiveif k >w H
and is suÆciently small.
References
[1] Aghion, P. and P.Bolton,1997, ATheory of Trickle-Down Growth and
Development, Review of Economic Studies 64, pp151-172.
[2] Banerjee, A. and A. Newman, 1993, OccupationalChoice and the
Pro-cess of Economic Development, Journal of Political Economy 101, pp.
274-98.
[3] de Meza, D. and D. Webb, 1997, Wealth, Enterprise and Credit Policy,
Economic Journal 109, pp 133-65.
[4] Dixit,Avinash,(1987),TradeandInsurancewithMoralHazard,Journal
Review of Economic Studies 60, pp. 35-52.,
[6] Gertler, Mark and Kenneth Rogo , 1990, North-South Lending and
Endogenous DomesticCapitalMarketIneÆciency,Journal of Monetary
Economics 26,pp 245-266.
[7] Grossman, Gene, (1984), International Trade, Foreign Investment, and
theFormationoftheEntrepreneurialClass,AmericanEconomicReview
74(4), 605-14.
[8] Grossman, Gene, and Giovanni Maggi, (2000), Diversity and Trade,
American EconomicReview 90(5),1255-75.
[9] Hartwick, J.M. , 1999, Insuring and u 000
(y), Economics Letters 65,
205-212.
[10] Kimball,M.S.,1990,Precautionarysavinginthesmallandinthelarge,
Econometrica 58,53-73.
[11] Kanbur, S. M. , 1979, Of Risk Taking and the PersonalDistribution of
Income, Journal of Political Economy 87(4), 769-797.
[12] Kihlstrom R. E. and J.-J. La ont, 1979, A General Equilibrium
En-trepreneurial Theoryof FirmFormationBased onRiskAversion,
Jour-nal of PoliticalEconomy87(4), 719-748.
[13] Newman,A.,1995,RiskBearingandKnightianEntrepreneurship,
type-script.
[14] Rogerson, W.,1985,RepeatedMoralHazard,Econometrica 62,pp
459-465.
[15] Thiele, Henrik and Achim Wambach, 1999, Wealth E ects in the
