The Microfinance Disappointment: An
Explanation based on Risk Aversion∗
Alexey Khazanov

Omer Moav

Zvika Neeman

Boston College

University of Warwick
and IDC

Tel Aviv University

Hosny Zoabi
The New Economic
SchoolAbstract
Recent research indicates that microcredit has not contributed significantly topoverty reduction. Take up of affordable credit by the poor for investment in businesses, education and health, turned out to be very low. We argue that this can be explained by risk aversion, when investment affects the probability of success of a risky
project. Our model abstracts from fixed costs in the production technology, commonly
assumed in the existing literature. There are no imperfections in the loan market, andwe abstract from assumptions about false beliefs by the poor regarding the production
function or other behavioral assumptions. We conclude that to facilitate investment
and thereby reduce poverty, policy should be aimed at reducing the risk faced by the
poor.

∗

We are grateful to Gianni Marciante and Ilia Sorvachev for very helpful research assistance. We benefited from comments from: Arun Advani, Susanto Basu, Pablo Beker, Valery Charnavoki, James Choy, Diana
Egerton-Warburton, Moshe Hazan, Kirill Pogorelskiy, Ofer Setty, Selma Walther, David Weiss and participants in the CAGE-AMES seminar at the University of Warwick. Khazanov: Department of Economics,
Boston College, Chestnut Hill, MA, email: khazanov@bc.edu. Moav: Department of Economics, University of Warwick, Coventry CV4 7AL, United Kingdom, email: omer.moav100@gmail.com. Neeman: Berglas
School of Economics, Tel Aviv University, Tel Aviv, Israel, email: zvika@post.tau.ac.il. Zoabi: The New Economic School, 100 Novaya Street, Skolkovo Village, The Urals Business Center, Moscow, Russian Federation,email: hosny.zoabi@gmail.com. Alexey Khazanov wishes to acknowledge the support of the Ministry of
Education and Science of the Russian Federation, grant No. 14.U04.31.0002, administered through the NESCSDSI, at the early stage of research.

1

Introduction

A large body of research, as well as popular views regarding the persistence of poverty
in developing countries, suggest that credit constraints prevent the poor from accessing
profitable investment opportunities. This has led to the conclusion that a key policy intervention for alleviating poverty is the provision of affordable credit. This policy has been
implemented on a large scale in the last few decades. Thousands of microcredit NGOs
were established, offering the poor billions of dollars in loans at affordable interest rates,and offering hope for significant reduction in poverty. The UN declared 2005 the "Year of
Microcredit," and the 2006 Nobel Peace Prize was awarded to Muhammad Yunus and theGrameen Bank for their contribution to the reduction in world poverty.
But the years that followed revealed disappointing results.1 First, microcredit availability encouraged borrowing for consumption and thereby increased debt and poverty.
Second, microcredit failed to deliver the anticipated levels of productive investment because the poor tended to turn down the opportunity to borrow at a reasonably low interest
rate for the sake of investing in high return projects. Banerjee et al. (2015) find, in a randomized evaluation in India, low take up of microcredit, no increase in households whoare business owners, and despite some increase in the size of existing businesses, the average business remained small and not very profitable. Low take up and limited impact on
business creation is also found in other places ( Banerjee et al. (2013), Crépon et al. (2015),
Angelucci et al. (2015)). Tarozzi et al. (2015) find higher take up in Ethiopia, but, similarly
to other regions, no significant impact on business creation.
In this paper, we show that risk aversion can explain why the poor do not take up
affordable high return investment opportunities, and thus why poverty persists. In ourmodel, investment increases the probability that a risky project will succeed. This structure of the model is the key for our theoretical contribution and its application. The model
abstracts from any fixed costs (or more generally from any increasing marginal returns toinvestment), which are inconsistent with recent evidence, but nevertheless play a key rolein the existing literature.
Existing explanations for the persistence of poverty are typically based on credit constraints combined with increasing marginal returns in the production function, typicallyin the form of a fixed cost. (E.g., Dasgupta and Ray (1987), Galor and Zeira (1993), Banerjee and Newman (1993), Piketty (1997), Maoz and Moav (1999), Ghatak and Jiang (2002),
and Mookherjee and Ray (2003), among many others).2 The fixed cost prevents the poor
from investing a modest amount that they can afford, so that they can gradually escapepoverty. We address a related but different question: why do the poor leave high returninvestment opportunities unexploited when they do have access to credit?
When the fixed cost is coupled with risk, the risk averse poor might choose the safe
1

Banerjee et al. (2015). See also The Guardian https://www.theguardian.com/global-developmentprofessionals-network/2013/nov/19/microcredit-south-africa-loans-disaster
2
In Piketty (1997), the effort level, rather than capital investment, is indivisible. In Mookherjee and Ray(2003), the multiplicity of steady states requires indivisibilities in the return to education. Other scholars
focused on group-level influences as a source of non-convexities in the production function in generating
poverty traps (e.g., Benabou (1996), Durlauf (1996), Mookherjee et al. (2010)). For a summary of theories onpoverty traps see Bowles et al. (2006)2

alternative and avoid the investment, even if affordable credit is available. Indeed, Banerjee and Duflo (2011), argue that "[r]isk is a central fact of life for the poor, who often run
small businesses or farms [...] with no assurance of regular employment. In such lives
a bad break can have disastrous consequences." (p. 133). They further claim that investment is often equivalent to buying a lottery ticket (p. 87). For instance, the outcome ofschooling is employment by the government or a large firm, if successful, or subsistence
self-employment, if not. Thus, the downside outcome – if the investment fails to yield the
expected high return – could be too painful: It leaves the poor with a low income, less assets, and often a debt to pay. The fixed cost, according to this explanation, plays a crucialrole, as it prevents the poor from investing a modest amount that exposes them to a levelof risk they are willing to take.3
However, Banerjee and Duflo (2011) claim that many investment opportunities, suchas in education or health, offer a high expected return and do not have a significant fixed
cost. The marginal return to investment in education, they argue, is high at low levels
of investment: "... every little bit of education helps. People who are comfortable withreading are more likely to read newspapers and bulletin boards and to find out when there
is a government program available for them. People who go on to secondary education
are more likely to get a formal-sector job, but even those who don’t are able to run theirbusinesses better" (p. 82). Similarly, they argue that modest investments in health reducesignificantly the risk of illness, and thus provide a high expected return. The cost of illness– treatment and forgone earnings – could be devastating for the poor.
The absence of any fixed cost in the production function isn’t limited to education or
health. Kraay and McKenzie (2014) survey the empirical literature and conclude that theevidence is inconsistent with technology-based poverty traps. Not much capital is needed
to start a business in a developing country and returns to investment are very high: 5% to
20% per month, at investment levels as low as 100 dollars (McKenzie and Woodruff (2006),
de Mel et al. (2008, 2009, 2012), Fafchamps et al. (2014)). Similarly, despite high returns,
many farmers fail to invest in fertilizer that is availability in small quantities Duflo et al.
(2011)), and many shopkeepers fail to make small inventory investments (Kremer et al.
(2011)). Microfinance, in many cases, is available to finance these unexploited high-returnsmall investments.
To explain why the poor repeatedly avoid small affordable investments with high expected return, Banerjee and Duflo (2011) suggest that the poor typically believe that the
production function has an "S-shape" – the marginal return is low at low levels of investment and high at higher levels. Thus, the poor believe, despite the facts, that in orderto enjoy a high return, the investment should be large. The combination of false beliefs
and the risk associated with the investment push the poor to avoid it all together. Thatis, even if the poor could borrow the required funds at a low cost, they give up the opportunity because of the risk associated with a large investment. "In reality, there should
not be an education-based poverty trap: Education is valuable at every level. But the factthat parents believe that education is S-shaped [...] create[s] one." (p. 89). This claim is3

The argument that risk aversion leads to underinvestment isn’t new, of course. It is proposed by Stiglitz(1969) and is further developed, with an emphasis on the poor, by many others (See the literature review inBanerjee (2000)).

3

supported by some evidence (e.g., Nguyen (2008)) but other evidence presented by theauthors led them to conclude, as consistent with our approach, that the poor have a goodunderstanding of the relevant economic environment: "the poor are no less rational thananyone else – quite the contrary. Precisely because they have so little, we often find themputting much careful thought into their choices: They have to be sophisticated economists
just to survive." (p. ix).
Kremer et al. (2013) address the same facts but propose an alternative behavioral explanation. They argue that small businesses in developing countries reveal risk aversion
in small-stakes gambles that cannot be explained by any reasonable degree of risk aversion within expected utility theory, and propose that loss aversion within prospect theory
may play a role.
More recently, Banerjee et al. (2015) propose that the low take-up of loans for starting abusiness could be an outcome of the lack of complementary factors such as proper trainingor skills, and more generally, that there is less potential for high return businesses for thepoor than anticipated by microcredit enthusiasts. However, Bandiera et al. (2017) showthat when poor women receive a productive asset (a couple of cows) and some relevant
training, they have the skills to run a simple yet successful business that alleviates poverty
in the long run. These results seem consistent with our theory. Women who could borrowthe funds for the required investment and training, avoid it despite the high return. Risk
could be the main difference between borrowing for investment and receiving the asset
with no debt.
Our contribution is to show that the assumption that the poor have false beliefs or loss
aversion is not necessary in many important cases addressed by these theories. In particular, we show that investment projects with a binary outcome of success or failure, where
investment increases the probability of success at a constant or diminishing rate (no fixed
costs or any other s-shape in the production function), could lead the risk averse poor to
avoid any investment, despite the high expected return at low levels of investment – which
they are fully aware of. A simple illustrative calibration of our model, with parameter values estimated from Augsburg et al. (2015), demonstrates that reasonable values of the risk
aversion coefficient are sufficient to prevent the poor from investing (see Appendix B.2).
We show that risk averse agents choose a corner solution in such investment projects:
do not invest or invest a lot. For projects of this nature (investment increases the probability of success) the expected utility of a risk averse agent, as a function of investment,
is typically U-shaped. Moreover, if risk aversion is diminishing with wealth, an increase
in wealth would lead to a larger increase in expected utility at the high end of investment
than at the low end. The u-shape of the expected utility could lead to corner solutions
and the effect of wealth could lead to a shift from one corner to the other. The continuous
decline in risk aversion with wealth therefore leads to a discontinuous rise in investment,
as long as the risk cannot be fully alleviated within a reasonable cost. Thus, the observed
investment behavior looks as if there is a fixed cost or a belief that there is such a cost, evenwhen there is not.
To understand the reason for the U-shape of expected utility as a function of investment
size and the resulting corner solutions, consider a lottery with zero expected return. With
probability p the outcome of the lottery is a prize of one dollar, and with probability 1 − p
the outcome is zero dollars (no prize). The probability p is equal to investment. That is,4with no investment the probability of winning the prize is zero, and with investment of
one dollar the probability is one, and p is one half if the investment is half a dollar, and soon. In this case, any individual is clearly indifferent between no investment and investing
the maximum (p = 1), where with probability one the agent simply receives the dollar
invested back. The outcome is certain and identical in both cases. For any investment
strictly between the two corners, the expected return is the same as for the two corners,
but the realization is uncertain. It follows, therefore (by definition of risk aversion) that any
risk averse individual would strictly prefer the corners over any other investment strictly
between zero and one.
To understand why the poor might avoid a high expected return investment that thewealthy would take, consider two changes to the lottery described above. First, the prizein case of a successful outcome is higher than one, so that the expected return of the lottery
is positive. Second, there is a limit to investment that is strictly less than one. Risk averse
agents, as previously explained, would typically choose between one of the two corners,
now facing a tradeoff between avoiding risk (by not investing) and enjoying an expected
positive return (by investing the maximum possible). If the prize is not too high, since risk
aversion declines with wealth, the result would be a threshold wealth level above which
individuals invest in the project and below which they don’t.
This result doesn’t depend on any constraints on borrowing. Moreover, even if agents
could declare bankruptcy (in the case that they borrowed, were unsuccessful in their investment, and cannot repay the entire debt), results still hold as long as: (1) liability islimited and the wealth of individuals after bankruptcy is correlated with their wealthbefore bankruptcy (e.g., they can hide some wealth), and (2) bankruptcy, following an
unsuccessful investment, leaves individuals worse off in comparison to the option of notinvesting.
Our model, as mentioned above, assumes no fixed costs, or more generally, that the
production function is not S-shaped: the expected return on the first dollar invested is atleast as large as that of any other dollar. However, the realized income has to be sufficiently
large if the project is successful. Thus, we have an indivisibility in our model without any
fixed costs. We interpret this structure as investment that increases the expected returnfrom an existing asset. We propose that the large majority of investment opportunities thatare relevant for the poor are consistent with this interpretation. The indivisible asset is theindividual’s labor, which is the main productive asset of the poor, and the productivityof this asset can be augmented by investment. This is clearly the case if the individual isseeking employment with a firm or the state: Investment in general education, specificskills, or health, increase the probability of finding a well-paying stable job.
When the poor run their own small business, it is still true that the main asset of thebusiness is typically the owner’s labor. An individual whose business is providing services (such as plumbing or electricity services, or even jut simple manual tasks) could
invest in augmenting the relevant skills, health and physical abilities, or invest in marketing the services, purchase useful complementary tools, or anything else that increases theprobability the business is successful. Any other small business is not that different. The
owner has mainly her own time and could augment the expected income of the businesswith investment in complementary factors or intermediate goods, such as a larger stockand more shelves for storage in a shop, or more fertilizer, tools and irrigation equipment
5

in a field.
The view that risk plays a significant role, in particular among the poor, is supported
by evidence. Morduch (1990) empirical findings suggest that the poor avoid profitable
but risky technologies. Moreover, risk aversion declines with income and is significant
among the poor (see Andrisani (1978), Hill et al. (1985), Cicchetti and Dubin (1994), andShaw (1996). The World Bank (2001) report shows that "[the] poor are highly risk averse
and reluctant to engage in the high-risk, high-return activities that could lift them out ofpoverty" (p. 138), and "[a]s households move closer to extreme poverty and destitution,
they become very risk averse" (p. 145). Bryan et al. (2014) provide consistent evidence
concerning seasonal migration. The poor avoid low cost migration that is highly rewarding because of the fear of failure. Finally, insurance against adverse weather shocks can
induce farmers to invest more in high return risky production options (Karlan et al. (2014),
Cole et al. (2017)).
Banerjee and Duflo (2011) provide further support for the claim that risk plays a central role in the decisions of the poor who understand their economic environment. They
argue that the poor are constantly worrying about the future, particularly about imminent
disasters, and take a variety of ingenious and costly precautionary measures to limit therisks they are subject to, such as managing their businesses conservatively and diversifying their portfolio of activities, including by marriage and temporarily migration (pp. 141- 143).
We believe that our simple result – that risk aversion can lead to corner solutions –may have been overlooked by the existing literature because of the conventional strategy
for modeling risky investment. In the existing models of investment under uncertainty,
when fixed costs are absent, the probability of success is typically exogenous. The model
then limits the optimization of the agent to the scale of the project: the investor decides
how much to invest in an asset, but her investment has no effect on the rate of return (e.g.,
investment in a stock). In a project in which the rate of return is drawn from an exogenous
distribution, the typical result is that the expected utility of a risk averse agent is a concave
function of investment, leading to an interior solution. The optimal investment increases
with wealth (if risk aversion is declining with wealth), but the change is continuous. In
that framework, the S-shape of wealth dynamics is achieved by assuming a fixed cost (see,for instance, Acemoglu and Zilibotti (1997), and Banerjee (2000)).4
Our approach that investment affects the probability of success (rather than, or in addition to, the outcome conditional on success) is consistent with the evidence on educationand health. The outcome in the former, as mentioned above, is a good job if successful
and no such job otherwise. Of course, income could still be affected by education within
the two options, but as long as a significant part of the return to investment is in the formof higher probability of success our results hold. In the latter, investment in health could
increase productivity when not ill, but as argued by Banerjee and Duflo (2011), it has asignificant effect on preventing illness. In addition, our approach is consistent with thefindings of Banerjee et al. (2015), that businesses who borrowed and invested, have significantly more assets, and business profits are higher above the 85th percentile of profitabil4

Aghion and Bolton (1997) model the probability of success as an endogenous outcome of investment,
but they do not show our main result of the "u-shape" utility.

6

ity, without a significant effect on profits for other businesses. Finally, we show that thedata from Augsburg et al. (2015) is consistent with a positive correlation between investment and the probability of successes in a project (see Appendix B.3).
Our theory also sheds light on two additional related facts described by Banerjee andDuflo (2007). First, the poor often spend the little they have above their subsistence needson consumption of goods that do not alleviate poverty and could most likely be avoided.
Second, the modest savings they might have, are invested at low rates of returns. Thus, thepoor avoid high return investment, even if this investment is possible, either by borrowing
when affordable credit is available, or by a temporary cut in consumption (such as theconsumption of "temptation" or "conspicuous" goods) or by reallocating some of their low
return assets. We propose that as they are reluctant to take risky investments they prefer
the short-term utility of consumption and low-risk/low-return savings.
A potential caveat should be acknowledged. Our theory explains why some high return investments are not exploited by the poor. These are investments that are mainly
aimed at increasing the probability of success rather than the income if successful. These
projects require large investment by risk averse agents, who therefore decline the opportunity. But why don’t the poor invest in projects in which the probability of success isgiven and the investment increases the return conditional on the outcome? Perhaps, formost of the poor the available options are of the type we study here. Be that as it may, in
Section 5 below, we describe circumstances under which agents would prefer to invest in"probability" than in "return conditional on success."
We are not the first to propose an explanation for persistence of poverty, which isn’t
based on a fixed cost in the production technology. A growing literature shift focus fromthe technology to the behavior of the poor. Moav (2002) shows that non-homothetic preferences, such that the marginal propensity to save increases with income, can lead to apoverty trap. Chakraborty and Das (2005) and Moav (2005) obtain similar results, without assuming non-homothetic preference, based on the interaction between health andhuman capital in the former and the trade-off between fertility and education in the latter. Banerjee and Mullainathan (2010) assume non-homothetic preferences with respect to"temptation goods" and Bernheim et al. (2015) focus on self-control problems. Moav andNeeman (2010, 2012) show that conspicuous consumption signaling equilibria could leadto persistence of poverty despite homothetic preferences. These papers propose different
explanations for the low savings of the poor that prevent them from access to high returninvestment opportunities. Here we focus on risk aversion to explain why the poor choose
not to invest even if they could.
In the next section, we present a simple version of our model that illustrates our main
claim: in projects in which investment increases the probability of success, the expectedutility of the risk averse agent leads to corner solutions. We further show that the degreeof risk aversion has an effect on the optimal solution: the higher risk aversion is, the more
likely it is that the agent would choose not to invest. Thus, risk aversion declining withwealth leads to a wealth threshold above which individuals invest a lot and below which
they do not invest at all. We contrast this result with a model in which the probability ofsuccess is exogenous and the return for any given outcome increases proportionally withinvestment. This structure of investment leads to an internal solution.
In section 3 we show that results hold when a competitive financial market and possible
7

bankruptcy are included in the model. We assume that individuals’ risk aversion declines
with wealth and initial wealth varies between individuals. As in the simple version, individuals can invest in the probability of success of a risky project with an exogenous binary
outcome of high or low income. They can augment their investment with a loan and pay
the competitive risk-adjusted interest rate in a perfect loan market, which takes into account the probability of a low-income outcome and that of bankruptcy. We assume that
individuals’ wealth is unobserved by the financial intermediary, and the interest rate isdetermined by a zero-profit condition, under limited liability: only the income from theinvestment project can be used to pay back the debt. Individuals could also lend theirwealth to the financial market and enjoy the risk-free return.
We show that an individual’s choice of investment is discontinuous in risk aversion,and therefore in wealth. This implies that individuals face a tradeoff between a safe, low
return option, and a risky, high expected return option. If an individual chooses the safe
option, her initial wealth is augmented by low income. If she invests in the project, the end
outcome in case of failure is that she is left with her initial wealth and the low income fromthe unsuccessful project, net of the investment cost. If this initial wealth is low then risk
aversion is high: the disutility from losing the low income in the risk-free option is high.
This leads to our main result: despite the absence a fixed cost in the production technology,
optimization prevents the poor from investing in a high return project even if credit isavailable at a competitive rate. We remove any frictions from the credit market, but one
market friction is crucial: full insurance against risk (at an affordable cost) isn’t available
in our model. We believe that this is a reasonable assumption. For many reasons related
to moral hazard and adverse selection, such insurance doesn’t exist for many profitable
investments – in human capital in particular.5
In section 4 we address the question of insurance against a bad outcome and show thatthe poor are willing to pay more than the wealthy for insurance. More importantly, the option to insure increases the likelihood that the poor would exploit high return investment
projects. This leads to the policy conclusion that providing microinsurance alongside microcredit could help to reduce poverty. Section 5 offers some concluding remarks.

2

A Simple ModelIn this section, we illustrate our main claim: in projects with a binary outcome of either
a high or a low income, in which investment increases the probability of the realizationof high income ("success"), the maximization of the expected utility of a risk averse agent
yields a corner solution. We further show that the degree of risk aversion has an effecton the optimal solution: for a project with a positive expected return, there is a threshold5

Udry (1990), Townsend (1995), and Morduch (1995) provide evidence that the poor are often insured
against the risks they take, but as suggested by Morduch (1990) and Banerjee (2000), these studies only
observe the risky activities people have chosen to take. The poor may have foregone other investment opportunities to limit the risk they bear. Moreover, Townsend (1995) shows that full insurance is limited tosome risks. In addition, the World Bank (2001) reports that "poor people, even though they need insurance
most, are more likely to drop out of informal [insurance] arrangements." (p. 144). Finally, Banerjee and Duflo (2011) show that the poor avoid insurance, in particular health insurance, because "[c]redibility is always
a problem with insurance products." (p.153).

8

level of risk aversion above which agents choose zero investment and below which maximal investment. We contrast this result with the one obtained in the "standard" modelin which the probability of success is exogenous and the return for any given outcome
increases proportionally with investment. In such a model the optimal level of investment
is obtained at an interior solution.
An agent has a CARA utility function u(x) = −e−λx where the parameter λ > 0 describes the agent’s Arrow-Pratt coefficient of risk aversion. A higher value of λ indicatesthat the agent is more risk averse. As λ decreases to zero, the utility function converges tothe risk neutral identity function u(x) = x.6
The agent’s income is determined by the success or failure of an investment project: it
is high, H, with probability p and low, L, with probability 1 − p, where H > L ≥ 0. The
agent controls the probability of success of the project p ∈ [0, p], 0 < p < 1, at a linear costc(p) = αp for some α > 0. We focus on the case in which investment in the probability ofsuccess of the project p generates a positive expected return, or pH + (1 − p)L − αp > L
for any p > 0. This assumption is equivalent to the assumption that α < H − L.
The objective of the agent is to choose the probability p ∈ [0, p] that maximizes its
expected utility from the project, which is given by
U (p) ≡ pu(H − αp) + (1 − p)u(L − αp).
Proposition 1. The agent’s expected utility function U (p) is U-shaped in p. Namely, it is nonincreasing on an interval [0, p̂] and increasing on the interval [p̂, p] for some p̂ ∈ [0, p]. Notice that
U (p) may be either nonincreasing or nondecreasing on the entire range.
Proof. We show that if U is increasing at some p, then it is increasing for all p0 > p. The
agent’s expected utility is equal to
U (p) ≡ −pe−λ(H−αp) − (1 − p)e−λ(L−αp)and its derivative with respect to p is equal to
U 0 (p) = (1 + λαp) eλαp e−λL − e−λH − λαeλαp e−λL .
Suppose that U is increasing at p, or U 0 (p) > 0. It follows that
U 00 (p) = λα (2 + λαp) eλαp e−λL − e−λH
− λαeλαp e−λL= λα U 0 (p) + eλαp e−λL − e−λH
> 0,which implies that U 0 (p0 ) > U 0 (p) for all p0 > p or that U continues to increase throughout
the remainder of its range.
The fact that U (p) is U-shaped implies that the agent would choose either the smallest or the largest probability p ∈ [0, p]. The realized utility of an agent who chooses the6

To see this, recall that von Neumann-Morgenstern utility functions are only unique up to affine trans−λxformations. Hence, the utility function u(x) = −e−λx is equivalent to the utility function u(x) = 1−eλ ,which converges to the identity function as λ tends to zero by L’Hôpital’s Rule.

9minimal probability p = 0 is certain and is equal to U (0) = u(L). The realized utility ofan agent who chooses the maximal probability p = p is uncertain; it is equal to u(H − αp)with probability p, and to u(L − αp) with probability 1 − p. We denote this lottery by(H − αp, L − αp; p, 1 − p). The assumption that investment in p generates a positive expected return (0 < α < H −L) implies that all sufficiently risk neutral agents would choosep = p. On the other hand, all sufficiently risk averse agents would choose p = 0. The factthat the expected utility U (p) is U-shaped implies that the agent’s choice is discontinuous
in its degree of risk aversion as described in the next proposition.
Proposition 2. The agent’s choice of p is discontinuous in its level of risk aversion λ. There exists
a threshold level of risk aversion λo > 0 such that more risk averse agents with λ > λo choose theminimal probability p = 0, and less risk averse agents with λ < λo choose the maximal probability
p = p.
Proof. As noted in footnote 6, as λ tends to zero, the agent’s utility function converges
to a linear function, which implies that the agent becomes risk neutral and so prefers thelottery (H − αp, L − αp; p, 1 − p) over the certain outcome L. By continuity, this is also thecase for all agents with small enough level of risk aversion λ.
An agent prefers the certain outcome L over the lottery (H − αp, L − αp; p, 1 − p) if andonly if−pe−λ(H−αp) − (1 − p)e−λ(L−αp) < −e−λL
if and only ifpeλ(αp+L−H) + (1 − p)eλαp > 1.
As λ increases to infinity, eλ(αp+L−H) tends to zero, but eλαp tends to infinity, which implies
that the last inequality is satisfied for all λ large enough.
Finally, the fact that an agent with a smaller λ is less risk averse than an agent witha larger λ implies that any lottery that is preferred over a certain outcome by the former
is also preferred by the latter. It therefore follows that there exists a threshold level ofrisk aversion λo > 0 such that more risk averse agents with λ > λo choose the minimal
probability p = 0, and less risk averse agents with λ < λo choose the maximal probability
p = p.
Thus, Proposition 1 shows that the agent’s expected utility is U-shaped, which induces
corner solutions; Proposition 2 shows how the optimally chosen corner solutions varies
depending on the agent’s level of risk aversion.
Remark. The U -shape of the agent’s utility as a function of investment stands in sharp
contrast with its shape if the probability of success p is exogenously given and is independent of the agent’s level of investment. Specifically, suppose that the payoff to the agent
upon success H(c) ≥ L is increasing and weakly concave in the agent’s cost of investmentc ≥ 0. In this case, the objective of the agent is to choose the cost of investment c so as tomaximize its expected utility
V (c) ≡ pu(B + H(c) − c) + (1 − p)u(B + L − c)where the utility function u is increasing and concave (not necessarily CARA).
10

Proposition 3. The agent’s expected utility function V (c) is concave in its cost of investment c.
It follows that it is either increasing throughout its range, decreasing throughout its range, or isinverse U -shaped in c.
Proof. The first and second derivatives of V (c) are given by
V 0 (c) = pu0 (B + H(c) − c) (H 0 (c) − 1) − (1 − p)u0 (B + L − c)and2

V 00 (c) = pu00 (B + H(c) − c) (H 0 (c) − 1) + pu0 (B + H(c) − c)H 00 (c) + (1 − p)u00 (B + L − c),respectively. The conclusion follows from the concavity of the functions u (·) and H (·). 
Notably, the agent’s choice of level of investment is still decreasing in its level of risk
aversion, but unlike in the case with an endogenous choice of probability, it is continuous
in the level of risk aversion.7
Finally, one wonders to what extent our result that when the agent controls the probability of success investment is discontinuous in the agent’s degree of risk aversion generalizes to other utility functions beyond CARA. In Appendix B.1 we provide a numericalexample that shows that our result extends also to the case of a CRRA function, albeit withan "approximate" rather than an exact U-shape result as is the case for CARA functions. It
should be emphasized that the agent’s investment is still discontinuous in its level of risk
aversion.
Moreover, as explained in the introduction, the intuition for our U-shape result doesn’t
depend on the specific utility function used. Consider the following specific case for ourlottery: L = 0, H = 1, and α = 1. In this case any investment p ∈ [0, 1] has an expected
return of zero. Clearly any individual is indifferent between the two corners: invest zeroand receive zero and invest one and receive one back. The outcome is certain and identicalin both cases. For any investment strictly between the two corners, the expected returnis the same as for the two corners, but the realization is uncertain. It therefore follows
that any risk averse individual would strictly prefer the corners over any other investment
strictly between zero and one.
One concern regarding our theory is, of course, its relevance to reality. In particular, can the theory explain the facts with reasonable parameter values? Following Rabin(2000), Kremer et al. (2013) argue that risk aversion cannot explain the fact that many poor
households neglect small, high expected return investment opportunities. They claim thatthe required coefficient of risk aversion is too high, and propose instead that loss aversion
provides a better explanation. In Appendix B we use data from Augsburg et al. (2015)8
and find, as consistent with our framework, that wealthier households invest more andthat there is a positive correlation between investment and the probability a project is successful. Using the same data, we present a simple illustrative calibration of our model,
7
Specifically, it is possible to show that if {un } is a sequence of utility functions that converges to a utility
function u and {cn } and c are the associated costs of investment, then if un exhibits more/less risk aversion
than un+1 then cn is smaller/larger than cn+1 and the sequence {cn } converges to c.
8
See Appendix A for data description.

11and show that under reasonable values of relative risk aversion, sufficiently poor agents
do not invest. Finally, we estimate the risk aversion parameter from the data and find thatfor the average project, under a relative risk aversion coefficient of 1.82, the model has agood fit with the data.

3

A Model with a Resource Constrained AgentIn this section, we show that the results of the simple model continue to hold when themodel is embedded in a competitive financial market with the possibility of bankruptcy(when the investment project fails and the agent pays only part of the debt). As in thesimple model, individuals can invest in the probability of success of a risky project withan exogenous binary outcome of high or low income. They can augment their investment
with a loan and pay the competitive risk-adjusted interest rate in a perfect loan market,
which takes into account the probability of a low-income outcome and bankruptcy. We
assume that individuals’ wealth is unobserved by the financial intermediary, and the interest rate is determined by a zero-profit condition, under limited liability: only the income
from the investment project can be used to pay back the debt.
We show that an individual’s choice of investment is discontinuous in its degree ofrisk aversion, and therefore in wealth, assuming that risk aversion declines with wealth.
This implies that individuals face a tradeoff between a safe, low return option, and a risky,high expected return option. If an individual chooses the safe option, her initial wealthis augmented by low income. If she invests in the project, the end outcome in case of
failure is that she is left with her initial wealth and the low income from the unsuccessfulproject, net of the investment cost. If this initial wealth is low then risk aversion is high: thedisutility from losing the low income in the risk free option is high. This leads to our main
result: despite the absence of non-convexities in the production technology, optimization
implies that the poor behave as if there is a fixed cost which prevents them from investingin a high return project despite the fact that credit is available at a competitive rate.
Suppose that the agent has an initial income of B ≥ 0 that it can use in order invest in arisky binary project as described in the previous section. An investment of size c(p) = αpgenerates an additional income H with probability p and an additional income L, 0 < L <H, with probability 1 − p, where 0 < α < H − L.
An investment that is larger than B requires the agent to borrow. Suppose that theagent has access to a competitive credit market in which the riskless interest rate is normalized to zero. A loan of size b ≥ 0 can be obtained at the interest rate r(b) that allows
lenders to break even. We assume that B is non-verifiable to lenders so that an individual
who borrows any amount return a maximum amount L if its additional income is realized
to be L, and a maximum amount H if it is realized to be H. We assume that the success ofthe project as well as the agent’s choice of probability p are verifiable (for example, because
the project requires investment in observable physical capital) so that lenders are able toasses the correct interest rate to charge the loan, and would refuse loans that are larger
than what is needed in order to finance the agent’s investment.9 Finally, we assume that
9Note that the agent may want to borrow a larger amount than the amount necessary to finance its in-12

in case of indifference, the agent prefers a larger to a smaller loan (this last assumption is
not necessary for our results, but it simplifies the analysis below).
Suppose that the individual has a CARA utility function and chooses the probability
p at cost c(p) = αp as described above.
Proposition 4. The agent finances its entire investment c(p) = αp through a loan.
Proof. If c(p) ≤ L then the individual is indifferent with respect to how it finances its
investment because regardless of whether it finances the investment from its own funds
or through a loan, its income is B + H − c(p) and B + L − c(p) following success and failureof the project, respectively. It therefore follows that if c(p) ≤ L then the individual will
finance its investment entirely through a loan.
If c(p) > L then borrowing allows the individual to insure itself against risk because
an individual who borrows the entire amount necessary for investment c(p) returns only
L if the project fails and so enjoys an income of B in that case, whereas an individual
who borrows a smaller amount and relies on its own funds (but still chooses the sameprobability p) still has to pay back L if the project fails so only enjoys a smaller income
than B in that case (note that a same probability p produces equal expected incomes inthe two cases and that lenders earn zero profits). A bigger loan implies that the induced
lottery second-order-stochastically-dominates the lottery induced by a smaller loan. So
every risk averse individual would prefer a bigger loan over a smaller loan. It therefore
follows that in this case the agent would borrow the largest amount possible, which isequal to c(p).10,11

We show that, as in the case described in the simple model, the agent’s expected utility
is U-shaped in p. The fact that individuals borrow the entire amount needed to financetheir investment implies that the individual’s induced expected utility function U (p) is
given by:
pu(B + H − αp) + (1 − p)u(B + L − αp)
if p < Lα(1)
U (p) =L
if Lα ≤ ppu(B + H − L − α + p ) + (1 − p)u(B)because an individual who borrows an amount c(p) = αp < L returns αp in both statesof the world, and an individual who borrows an amount c(p)
 = αp ≥ L returns L if theLproject fails, and L + α − p if the project succeeds, so that p αp−(1−p)L+ (1 − p)L = αppoverall.12
vestment because such a loan provides insurance to the agent: an agent who borrows such a larger amountenjoys a certain income that is paid back only upon success. Lenders may be reluctant to lend larger sums
because of moral hazard considerations, and in any case, the point of this paper is that poor agents cannot
reduce or eliminate their exposure to risk, which such larger loans would facilitate.
10
The reasoning above implies that the agent would like to borrow possibly even more than c(p), but weassume that lenders would refuse larger loans (see the discussion in footnote 6).
11
Lenders will obviously not lend more than min{c(p), (1 − p)L + pH}. However, the assumption that
(1 − p)L + pH − c(p) > L ensures that this minimum is obtained on c(p) = αp.
12Observe that αp−(1−p)L≤ pH+(1−p)L−(1−p)L= H so the individual can indeed return the loan if theppproject succeeds.

13Proposition 5. The individual’s induced expected utility function U (p) that is described in (1) is
U-shaped in p.
Proof. For values of p that are such that c(p) < L or p < Lα , it is possible to show thatthe function U (p) = pu(B + H − αp) + (1 − p)u(B + L − αp) is U-shaped using a similarargument to the one used in the proof of Proposition 1.
For values of p that are such that c(p) ≥ L or p ≥ Lα , we need to show that the function
U (p) = pu(B + H − L − α + Lp ) + (1 − p)u(B) is increasing in p. This implies that U (p) is Ushaped over its entire range because U (p) is continuous so the argument is valid regardlesswhether U (p) is decreasing, increasing, or U-shaped for p ∈ [0, Lα ].
The derivative of U (p) with respect to p ≥ Lα is equal to




L L
L
U (p) = −pu B + H − L − α +− u(B).
+u B+H −L−α+
p p2
p
0

0

For our CARA utility function, u(x) = −e−λx and u0 (x) = λe−λx . So, U 0 (p) > 0 if andonly if:




L
λL
λ H −L−α+> ln 1 +.
p
p
The conclusion follows from our assumption that H − L > α together with the fact thatx > ln(1 + x) for all x > 0.
It follows that like in the simple model, also in this model we have:
Proposition 6. The agent’s choice of p is discontinuous in its level of risk aversion λ. There exists a
threshold level of risk aversion λo such that more risk averse agents with λ > λo choose the minimal
probability p = p, and less risk averse agents with λ < λo choose the maximal probability p = p.
Proof. Follows from the same argument as in the proof of Proposition 2.



If agents with a smaller initial income (bequest) B are also more risk averse in the sensethat their CARA utility function has a larger risk coefficient parameter λ then we have:
Corollary. The agent’s choice of p is discontinuous in its initial income (bequest) B. There exists
a threshold level of income B o such that poorer agents who have a smaller initial income B < B ochoose the minimal probability p = p, and richer agents who have a larger initial income B > B ochoose the maximal probability p = p.

4

The Role of Insurance

We assumed that investment in the success of the project generates a positive expected net
return for any probability p > 0, or that H − L > α. This assumption implies that if the
maximum probability of success p̄ ≤ 1 is in fact equal to 1 then it is optimal for all agentsto set p = 1 regardless of their level of risk aversion or initial income.
14

In fact, the ability to set p = 1 is more valuable for risk averse or low income agentsthan for other agents. To see this, suppose that the probability p has the following convex
cost function(
αp, 0 ≤ p < p̃
c(p) =,(2)αp̃ + α0 (p − p̃), p̃ ≤ p ≤ 1
for some α0 > α. This cost function implies that elimination of the per-unit risk of failure
becomes more expensive beyond the point p = p̃.
We ask, are the poor willing to pay more than the rich for reducing risk ?
Figure 4 depicts the expected utility U (p) in the case where the cost function is given by
2. The left panel depicts the expected utility for a risk averse or low income agent (λ = 1),
while the right panel draws the expected utility for a high income or less risk averse agent(λ0 = .2).
Low income

0.44High income

0.54

0.42

0.52

0.4
0.5
0.38

Expected utility

0.48
0.36
0.46
0.34
0.44
0.32
0.42
0.3
0.4

0.28

0.26

0.38
0

0.2

0.4

0.6

0.8

1

0

Probability

0.2

0.4

0.6

0.8

1

Probability

Figure 1: Expected utility. H = 1.55; L = 0.4; p̃ = 0.6; α = 1; α0 = 1.25; λ = 1; λ0 = .2
As can be seen from the figure, for the low income agent U (1) > U (0) > U (p̃), while
for the high income agent U (p̃) > U (1) > U (0). Namely, the rich agent is less concerned
about the risk associated with failure of the project and chooses optimally p = p̃ < 1. On
the other hand, a poor agent chooses optimally to reduce the risk associated with failure
to zero by choosing p = 1 and is willing to pay more for it.

15

5

Investment in Probability vs. Investment in Return Upon
Success

In certain circumstances, an individual who has an opportunity to invest either in increasing the probability of success or in the reward upon success, with equal expected return,would prefer to invest in probability.
To see this, consider the case of an agent who faces a binary project. The project either
succeeds with probability p ≥ p0 > 0 and yields H(p), or fails and yields L, where H(p) >L ≥ 0 for all p ≥ p0 .
Suppose that the expected return of the project is constant so that pH(p) + (1 − p)L ≡C > L is fixed.13 Hence, the individual faces the dilemma of whether to invest in the prob, which is decreasing
ability p at the expense of the reward upon success H(p) = C−(1−p)Lpin p, or maximize the reward upon success but minimize the probability of success.
Suppose that p > p0 ≥ p0 . We compare the p-lottery in which the individual receives
H(p) with probability p and L with probability 1 − p with the p0 -lottery in which the individual receives H(p0 ) > H(p) with probability p0 and L with probability 1 − p0 . Our
assumption that pH(p) + (1 − p)L ≡ C > L is fixed implies that these two lotteries generate the same expected income to the agent.
Proposition 7. A p-lottery that pays H(p) and L with probabilities p and 1 − p, respectively,
generates a higher expected utility to a risk averse individual than a p0 -lottery that pays H(p0 ) >H(p) and L with probabilities p0 and 1 − p0 , respectively, if p > p0 ≥ p0 .
Proof. Follows immediately from the fact that the p-lottery second-order-stochasticallydominates the p0 -lottery.
Hence, under the circumstances described in this section, any risk averse individual
would prefer to invest in the probability of success over the reward upon success.

6

Conclusion

High hopes of significantly reducing poverty through microfinance were countered bydisappointing results. In this paper, we address the fact that take up of affordable creditby the poor for investment in businesses, education and health, turned out to be very low.
We propose that this can be explained by risk aversion. Our key assumption is that investment affects the probability of success of a risky project. With this structure, we canabstract from any fixed costs (or more generally an S-shape) in the production function,
which is consistent with the evidence. Our focus is on the short run: we offer an explanation for why risk averse poor leave high return investment projects unexploited, despite
affordable credit. An extension of our model, which in the spirit of existing models illustrates persistence of poverty over generations, is straightforward to develop. The key
13

We require that p ≥ p0 > 0 because we want to consider projects that all have the same expected returnthat is strictly higher than the expected return when p = 0, which is L.

16

element for the persistence of poverty is an “S-shape” relationship between current income and future income, which crucially depends on our model’s result of discontinuityof investment in income. This discontinuity, unlike in models in the existing literature,
isn’t a result of fixed costs in the production technology, non-homothetic preferences, “behavioral” elements, or credit constraints.
We conclude that to facilitate investment, policy should be aimed at reducing the risk
faced by the poor. Microfinance is probably important, perhaps necessary, for the reduction of poverty, but it is not sufficient. We propose, therefore, that future randomized
controlled trials would offer the poor access to credit coupled with some form of insurance. A possible direction, derived from our model, is to condition repayment of debt on
outcomes, with a higher interest rate when investment yields successful outcomes, andforgiveness of most of the debt in case of failure.

References
D. Acemoglu and F. Zilibotti. Was prometheus unbound by chance? risk, diversification,and growth. Journal of Political Economy, 105(4):709–51, 1997.
P. Aghion and P. Bolton. A theory of trickle-down growth and development. Review of
Economic Studies, 64(2):151–172, 1997.
P. J. Andrisani. Work attitudes and labor market experience. New York, NY: Praeger Publishers., 1978.
M. Angelucci, D. Karlan, and J. Zinman. Microcredit impacts: Evidence from a randomized microcredit program placement experiment by Compartamos Banco. American
Economic Journal: Applied Economics, 7(1):151–82, January 2015.
B. Augsburg, R. De Haas, H. Harmgart, and C. Meghir. The impacts of microcredit: Evidence from bosnia and herzegovina. American Economic Journal: Applied Economics, 7(1):183–203, January 2015.
O. Bandiera, R. Burgess, N. Das, S. Gulesci, I. Rasul, and M. Sulaiman. Labor markets andpoverty in village economies*. The Quarterly Journal of Economics, 132(2):811–870, 2017.
A. Banerjee, E. Duflo, R. Glennerster, and C. Kinnan. The miracle of microfinance? evidence from a randomized evaluation. American Economic Journal: Applied Economics, 7(1):22–53, 2015.
A. V. Banerjee. The two poverties. Nordic Journal of Political Economy, 26:129–141, 2000.
A. V. Banerjee and E. Duflo. The economic lives of the poor. Journal of Economic Perspectives,21(1):141–168, March 2007.
A. V. Banerjee and E. Duflo. Poor Economics: A Radical Rethinking of the Way to Fight Global
Poverty. New York, NY: Public Affairs., 2011.

17

A. V. Banerjee and S. Mullainathan. The shape of temptation: Implications for the economic lives of the poor. Working paper, 2010. Working paper.
A. V. Banerjee and A. Newman. Occupational choice and the process of development.
Journal of Political Economy, 101(2):274–98, 1993.
A. V. Banerjee, A. G. Chandrasekhar, E. Duflo, and M. O. Jackson. The diffusion of microfinance. Science, 341(6144), 2013.
R. Bansal and A. Yaron. Risks for the long run: A potential resolution of asset pricing
puzzles. The Journal of Finance, 59(4):1481–1509, 2004.
R. Benabou. Equity and efficiency in human capital investment: the local connection. The
Review of Economic Studies, 63(2):237–264, 1996.
B. D. Bernheim, D. Ray, and S. Yeltekin. Poverty and self-control. Econometrica, 83(5):1877–1911, 2015.
S. Bowles, S. N. Durlauf, and K. Hoff, editors. Poverty Traps. Princeton University Press,
2006.
G. Bryan, S. Chowdhury, and A. M. Mobarak. Underinvestment in a profitable technology:
The case of seasonal migration in Bangladesh. Econometrica, 82(5):1671–1748, 2014.
S. Chakraborty and M. Das. Mortality, human capital and persistent inequality. Journal of
Economic Growth, 10(2):159–192, Jun 2005.
C. J. Cicchetti and J. A. Dubin. A microeconometric analysis of risk aversion and the decision to self-insure. Journal of Political Economy, 102(1):169–86, 1994.
S. Cole, X. Giné, and J. Vickery. How does risk management influence production decisions? evidence from a field experiment. The Review of Financial Studies, 30(6):1935–1970,
2017.
B. Crépon, F. Devoto, E. Duflo, and W. Parienté. Estimating the impact of microcredit on
those who take it up: Evidence from a randomized experiment in Morocco. American
Economic Journal: Applied Economics, 7(1):123–50, January 2015.
P. Dasgupta and D. Ray. Inequality as a determinant of malnutrition and unemployment:
Policy. Economic Journal, 97(385):177–88, 1987.
S. de Mel, D. McKenzie, and C. Woodruff. Returns to capital in microenterprises: Evidence
from a field experiment. The Quarterly Journal of Economics, 123(4):1329–1372, 2008.
S. de Mel, D. McKenzie, S. Woodruff, Christopherde Mel, D. McKenzie, and C. Woodruff.
Innovative firms or innovative owners? determinants of innovation in micro, small, andmedium enterprises. World Bank Policy Research Working Paper No. 4934, 2009.
S. de Mel, D. McKenzie, and C. Woodruff. One-time transfers of cash or capital have longlasting effects on microenterprises in Sri Lanka. Science, 335(6071):962–966, 2012.
18

E. Duflo, M. Kremer, and J. Robinson. Nudging farmers to use fertilizer: Theory andexperimental evidence from Kenya. American Economic Review, 101(6):2350–90, October
2011.
S. N. Durlauf. A theory of persistent income inequality. Journal of Economic growth, 1(1):75–93, 1996.
M. Fafchamps, D. McKenzie, S. Quinn, and C. Woodruff. Microenterprise growth and theflypaper effect: Evidence from a randomized experiment in Ghana. Journal of Development Economics, 106:211 – 226, 2014.
O. Galor and J. Zeira. Income distribution and macroeconomics. The Review of Economic
Studies, 60(1):35–52, 1993.
M. Ghatak and N. N.-H. Jiang. A simple model of inequality, occupational choice, anddevelopment. Journal of development Economics, 69(1):205–226, 2002.
M. S. Hill, S. Augustyniak, G. Duncan, G. Gurin, P. Gurin, J. Liker, J. Morgan, and M. Ponza.
Motivation and economic mobility. Ann Arbor, MI : Survey Research Center, Institute forSocial Research, University of Michigan, 1985.
D. Karlan, R. Osei, I. Osei-Akoto, and C. Udry. Agricultural decisions after relaxing credit
and risk constraints *. The Quarterly Journal of Economics, 129(2):597–652, 2014.
A. Kraay and D. McKenzie. Do poverty traps exist? assessing the evidence. Journal of
Economic Perspectives, 28(3):127–48, September 2014.
M. Kremer, J. Lee, J. Robinson, and O. Rostapshov. The return to capital for small retailers
in Kenya: Evidence from inventories. Working paper, 2011.
M. Kremer, J. Lee, J. Robinson, and O. Rostapshova. Behavioral biases and firm behavior:
Evidence from kenyan retail shops. American Economic Review, 103(3):362–68, May 2013.
Y. D. Maoz and O. Moav. Intergenerational mobility and the process of development. The
Economic Journal, 109(458):677–697, 1999.
D. J. McKenzie and C. Woodruff. Do entry costs provide an empirical basis for poverty
traps? evidence from mexican microenterprises. Economic development and culturalchange, 55(1):3–42, 2006.
R. Mehra and E. C. Prescott. The equity premium: A puzzle. Journal of Monetary Economics,
15(2):145 – 161, 1985.
Moav, O. , and Z. Neeman. Saving rates and poverty: The role of conspicuous consumption
and human capital. The Economic Journal, 122(563):933–956, 2012.
O. Moav. Income distribution and macroeconomics: the persistence of inequality in aconvex technology framework. Economics Letters, 75(2):187–192, 2002.

19

O. Moav. Cheap children and the persistence of poverty. The Economic Journal, 115(500):
88–110, 2005.
O. Moav and Z. Neeman. Status and poverty. Journal of the European Economic Association,8(2-3):413–420, 2010.
D. Mookherjee and D. Ray. Persistent inequality. The Review of Economic Studies, 70(2):369–393, 2003.
D. Mookherjee, D. Ray, and S. Napel. Aspirations, segregation, and occupational choice.
Journal of the European Economic Association, 8(1):139–168, 2010.
J. Morduch. Risk, production and saving: Theory and evidence from Indian households.
Mimeo, Harvard University, 1990.
J. Morduch. Income smoothing and consumption smoothing. Journal of Economic Perspectives, 9(3):103–114, September 1995.
T. Nguyen. Information, role models and perceived returns to education: Experimental
evidence from Madagascar. MIT Working Paper, 2008.
T. Piketty. The dynamics of the wealth distribution and the interest rate with credit rationing. The Review of Economic Studies, 64(2):173–189, 1997.
M. Rabin. Risk aversion and expected-utility theory: A calibration theorem. Econometrica,
68(5):1281–1292, 2000.
K. L. Shaw. An empirical analysis of risk aversion and income growth. Journal of Labor
Economics, 14(4):626–653, 1996.
J. Stiglitz. The effects of income, wealth, and capital gains taxation on risk-taking. The
Quarterly Journal of Economics, 83(2):263–283, 1969.
A. Tarozzi, J. Desai, and K. Johnson. The impacts of microcredit: Evidence from Ethiopia.
American Economic Journal: Applied Economics, 7(1):54–89, January 2015.
R. M. Townsend. Financial systems in northern Thai villages. The Quarterly Journal of
Economics, 110(4):1011–1046, 1995.
C. Udry. Credit markets in Northern Nigeria: Credit as insurance in a rural economy.
World Bank Economic Review, 4(3):251–69, 1990.
P. Weil. The equity premium puzzle and the risk-free rate puzzle. Journal of Monetary
Economics, 24(3):401 – 421, 1989.
World Bank. World Development Report 2000/2001 : Attacking Poverty. New York, NY: Oxford
University Press., 2001.

20

AppendicesA

Data description

We use data from Augsburg et al. (2015) who studied the effect of microcredits in Bosnia.
They conducted an experiment by providing randomly loans to those who were rejectedby MFI.14 To facilitate their study, the authors collected data on various socioeconomic
variables, ranging from household consumption and assets to income and savings choices.
The data we use includes:• assetvalue which corresponds to Endowment B in the model
Source: Baseline & Follow-up surveys, Section 4• y_max which corresponds to the return H in the good state of the world
Source: Baseline survey, Section 5
Question asked: “Imagine that you do receive the loan from EKI and have a very goodmonth/year, economic conditions are flourishing and stable and there is great demand foryour product/service. . . What would be the maximum amount of profit this business of
yours receives in such a situation over the next month/year?”• bm_expenses which corresponds to investment µ
Source: Baseline & Follow-up surveys, Section 6
Question: “average yearly expenses of main business”• past_success which will be used to compute probability of success P
Source: Baseline & Follow-up surveys, Section 6
Question: “Please respond to the following statements on a scale of 1 (Disagree) 2 (Neutral)
3 (Strongly Agree)
Previous year was successful financially”
The summary statistics of these variables are given in Table 1.
14

One may argue that this is a selected sample. Indeed the authors write "[w]e can also compare theaverage marginal client to the population of Bosnia and Herzegovina as a whole and to regular first-timeclients of our MFI. We [...] find the average marginal client is younger and more likely to be male andmarried.[...] We also use data from the MFI’s management information system to compare the marginal
clients to regular first-time borrowers. This shows that marginal clients are younger, less likely to be married,and have less education. They are also less likely to be employed full-time." Since wealth is a sufficient
statistic in our model this may be a selected sample as younger, unmarried and less educated are typicallypoorer individuals. However since our focus in our quantitative exercise is on those who have actually madeinvestment decisions this selection is of less importance.

21

Table 1: Summary Statistics
.
count
mean
H: max expected income, yearly
1105 61992.77
B: value of all assets owned
988 109817.00µ: avg. yearly expenses of main business 295
6647.80
p: last year was successful financially
540
0.63

s.d.
125618.14
133953.59
15840.73
0.48

min
max
100 1320000
100 1785700
120 180000
01

To calculate the probability of success, p, we use the question previous year was successful
financially. We assume that those who strongly agreed with the statement could be considered successful. Then, we create a dummy variable that takes value 1 if a person was
successful and 0 – otherwise. Next, we sort the sample based on the asset value B, fromsmallest to the largest. Then, we divide this sorted sample into 30 income groups by thevalue of total assets that they own: the first group contains 18 poorest respondents, thesecond – the next 18 poorest poorest, etc. Finally, for each group we compute the share ofpeople within a group who were successful. Such share is an estimate of the probability ofsuccess p for a given income group. We notice that the minimal value of p is 0.(22), whichcorresponds to the first income group and the maximal value is 0.88, which corresponds
to the 28th income group.
Similarly, we aggregate µ and B, by taking the mean of individual µ and B, respectively,
for a given income group.

B

Quantitative Analysis

In this section we show that the discontinuity result may also hold for a CRRA utility
function. However, the optimal investment is not always a corner solution and may changewith the initial wealth. Specifically, richer people may make higher investments. Then we
provide a simple calibration, based on Augsburg et al. (2015), which suggests that, underreasonable values of relative risk aversion, poor agents do not invest. Then we use thesame data to show some observations that are consistent with our framework. Finally, we
estimate the risk aversion parameter and show that under reasonable values our model
has a good fit with the data.

B.1

CRRA Example
1−σAssume a CRRA utility function of the form u(x) = x 1−σ−1 . In this case, our expectedutility function that corresponds to the one in section 2 becomes15

U (p) = p

(B + L − µ)1−σ − 1(B + H − µ)1−σ − 1+ (1 − p)
1−σ
1−σ

15

(B.1)

We denote by µ the cost of investment and keep, in this subsection, the assumption made in our model,µ = c(p) = αp.

22

Figure B.1 draws the expected utility for different levels of wealth , which corresponds
to the values σ = 1 , H = 5.5, L = 0.6875, α = 3, p̄ = 0.8 .16
Expected utility for different B,

1.6

=1

1.5

0.7

1.4

0.6

1.3

0.5

Probability

Expected utility

0.8
1.70
1.75
1.80
2.00
2.15
2.30

1.2

0.4

1.1

0.3

1

0.2

0.9

0.1

0.8

0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

Probability

1.5

2

2.5

3

3.5

4

4.5

5

Total assets

Figure B.1: Left panel: Expected utility as a function of probability (levels adjusted byconstant for convenience of presentation on one figure). Right panel: Optimal probability.
The left panel of Figure B.1 shows that while for a relatively low initial wealth agents(black curves) the optimum is achieved at p = 0, for a relatively high initial wealth agents(blue curves) the optimum is achieved at strictly positive and relatively high level of P .
More importantly, the U-shaped pattern still exists but it does not have to occur allover
the range [0, 1] (blue curves). The right panel of Figure B.1 shows that the discontinuity
in optimal choice of probability still holds. Specifically, for a relatively low initial wealth,B < 1.93 agents choose not to invest. Finally, throughout the range 1.93 < B < 2.40, theoptimal value of p increases with wealth.17

B.2

A Simple Calibration

We assume that the probability is a linear function of investment p = γµ, L = 0 and takep̄ = 0.88, H = 62 × 103 from the data.18 Then, we assume that the maximum µ from thedata corresponds to p̄ and use it to calculate γ.
We plot the probability of success as a function of total assets for different levels of σ inthe range considered acceptable in the literature. The figure illustrates our main results:
for reasonably low levels risk aversion poor individuals choose to avoid investment, anda continuous rise in wealth leads to a discontinuous jump in investment.
16This corresponds to u(x) = ln(x).
Remember that in the CARA case the U-shaped pattern holds for the whole range between 0 and 1. Thisimplies that if it is optimal for the agent to invest, it occurs at p̄. This made our analytical problem tractable.
18
We take H to be the average in the sample. Thus, for simplicity, we assume that all individuals face thesame project.
1723

Figure B.2: H = 62000, L = 0, p = 0, p̄ = 0.88

B.3

Some Observations from the Data

As described in Appendix A, we take the investments in projects, µ, probability of success,p, wealth, B and the high return, H from the data.19 We divide the wealth distribution
into 30 equal groups. For each group, i we calculate the average Bi , pi and µi and treat thisobservation as the one describing a representative agent of the corresponding group.
Observation 1. µ increases in B.
Our first observation is that the data is consistent with the result that richer agents
make larger investments. Figure B.3 shows a positive correlation between wealth B andinvestment µ. We run a regression of a quadratic from to allow for flexibility: µi = β0 +β1 Bi + β2 Bi2 + i . The estimation yields β̂1 = 0.039 (significant at the 5% level), and β2 =−0.00 (insignificant). This implies positive relationship between µ and B.
19

We will keep the assumption that L = 0.

24

20000
15000
10000
5000
0
0

200000400000Asset value

Investment

600000800000Fitted values

Figure B.3: Investment (vertical axis) vs. wealth (horizontal axis).

Although this result is not unique to our model but also true in the exogenous probability model among other ones, it is reassuring to find that richer agents make larger
investments.
Observation 2. p increases in µ.
One clear distinction between our model and the standard one is that we assume thatagents invest, at least partially, in order to increase their probability of success. We check
whether there exists any correlation between p and µ in the data. Here we deviate fromour model’s assumption of linear relationship between the probability and the investment,
and assume a general relationship of the form p = p + γµβ , which allows for flexibility as
β and γ can take any values.20 We, thus, estimate the empirical specification:ln(pi − p) = β0 + β1 ln(µi ) + i .21
20

(B.2)

Notice that this functional form implies that ln(p − p) = ln(γ) + β ln(µ). We take p = 0.22 from the data
as explained in Appendix A.
21
As can be seen from Figure B.3 the data on µ is very noisy. We, thus, use the fitted value of µi from
Observation 1. above.

25

-.5
-1
-1.5

8

8.5

9
ln(Investment)

9.5

10

Figure B.4: ln(pi − p) (vertical axis) vs. investment (horizontal axis)

Figure B.4 shows the data and the fitted line. Interestingly, we get that β̂1 = 0.26 (which
is significant at the 5% level) and β̂0 = −3.18 (significant at the 1% level), which implies γ =0.04. Thus, the relationship between p and µ is increasing and concave. This result standin contrast to the exogenous probability model and support our endogenous probability
framework.
Observation 3. p as a function of B.
From the two observations above it is clear that p increases with wealth. The dots in
Figure B.5 illustrate the actual probability for each level of wealth given in log scale acrossthe 30 wealth groups.

26

1
.8
.6
.4
.2
6

8

10
ln(Asset value)
p

12

14

Fitted values

Figure B.5: pi (vertical axis) vs. ln(Bi ) (horizontal axis)

Figure B.5 verifies the positive correlation between p and B. We estimate
pi = β0 + β1 ln(Bi ) + i

(B.3)

and get β̂1 = 0.067, which is significant at the 1% level.

B.4

Estimating the relative risk aversion parameterIn this subsection we discuss the estimation procedure of the relative risk aversion parameter, σ, and the fit of the model. In our model agents choose the size of investment µ. Given
the structural relationship between p and µ, this pins down the probability of success. We,
thus, match the probabilities chosen by agents that possess different levels of total assets
in the model, with the probabilities observed in the data.
Our goal is to find out whether the model is capable of fitting the data on probabilities
given a reasonable value of the relative risk aversion parameter.
It is important to notice that we match only individuals who made positive investments
as we observe only them. Thus, in this quantitative exercise we won’t capture the discontinuity in investment, but rather find that positive investment choices, according to ourmodel, are consistent with reasonable values of the relative risk aversion.22
One necessary assumption that we make is that agents are homogeneous in all characteristics except for the level of the total assets. We feed in exogenously the value of the22

Thus, in this estimation procedure, we are targeting the range, which is qualitatively consistent with
1.93 < B in our numerical example described in section B.1.

27

good outcome to be the average in the data, H = 61922.77, and assume that the bad outcome is L = 0. We also use the relationship p(µ) estimated in equation B.2 and set p̄ = 0.8
from the data.23 We then calculate the optimal probability by maximizing the expectedutility given in equation B.1.24
Formally, we pick parameters to minimize the following loss function:θ̂ = argmin
θ

2
30 
X
p(θ, θ̄, Bi )
1−
p̂i
i=1

where p̂i is the probability from the data,25 p(θ, θ̄, Bi ) is the model-predicted probability; θ̄is the vector of calibrated parameters (θ̄ = [H, L, γ, β, p, p̄]), and θ̂ is the vector of estimatedparameters (consists of σ only in this case). p(θ, θ̄, Bi ) is obtained from the maximizationof the expected utility given Bi and the values of the parameters.
Note that the loss function we are using is an inverse probability-weighted distance,
where a larger weight is assigned to lower probability:
30
X
1
i=1

p̂2i

2
p̂i − p(θ, θ̄, Bi )This is a natural way to assign larger weights to the low-income groups of population,
given our previous observation of the positive relation between the probability of the goodoutcome and total assets. Also note that in the dataset that we are using all the agents
invested µ > 0. Thus, we are matching the increasing segment of p(B) within the range
1.92 < B (see Figure B.1).
23

The maximum value of p in the data is 0.88, which corresponds to the 28th wealth group. the value of p
for the 10th wealth group is 0.7.
24
As discussed above, the estimates of γ and β are inferred from the regression ln(pi −p) = β0 +β1 ln(µi )+i
are β̂ = 0.26 and γ̂ = 0.04.
25
Since the data on p is noisy, we smooth the data and target the fitted value from regression B.3, p̂i .

28

0.8
0.75
0.7
0.65

Probability

0.6
0.55
0.5
0.45
0.4

Model
Data

0.35
0.3
0

5

10

15

20

25

30

Total assets (group)

Figure B.6: Probabilities from the data and from the model for estimated σ̂.

Figure B.6 illustrates how the model can match the data from Augsburg et al. (2015)
described above. As can be seen from the figure, the fit is very good although we are
missing the upper tail of the wealth distribution.
We obtain an estimate σ̂ = 1.82. The macroeconomic literature usually considers thevalues of the RRA parameter below or equal to 10 following the discussion provided in
Mehra and Prescott (1985) (e.g. the baseline results of Bansal and Yaron (2004) are obtainedfor RRA = 10). Weil (1989) points out that the parameter is likely to be closer to 1, according
to numerous studies. Thus, our estimate is consistent with the literature.26

26

To make sure that the estimate of σ we obtain corresponds to the global maximum, we used different
initial points for the minimization algorithm, and we chose the result with the lowest value of the loss
function. For robustness we check whether the estimates change significantly if we use the regular Euclidean
distance as a loss function, and a weighted sum of these two loss functions. The obtained estimates do notdiffer significantly from the baseline 1.82 and do not exceed 2.

29