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Theoretical Economics Letters, 2021, 11, 889-909
https://www.scirp.org/journal/tel
ISSN Online: 2162-2086
ISSN Print: 2162-2078
DOI:
10.4236/tel.2021.115057 Sep. 30, 2021 889 Theoretical Economics Letters
The Demand for Crop Insurance Bundled with
Micro-Credit
Mame Mor Anta Syll1,2
1University Gaston Berger (UGB), Saint-Louis, Senegal
2Initiative Prospective Agricole et Rurale (IPAR, Think Tank), Dakar, Senegal
Abstract
Access to financial services is challenging for small farmers in developing
countries. This paper studies the demand for micro-
insurance when it is
bundled with micro-credit in the context of rain-
fed agriculture. It presents
the conditions under which linking micro-insurance with micro-
credit can be
beneficial for small producers who cannot acce
ss agricultural credit due to
lack of collateral. The results show that if crop insurance and agricultural
loans are bundled, the demand for crop insurance increases with the profit-
ability of the investment made through the agricultural credit, and it de-
creases with the level of collateral required during the application for credit.
Keywords
Micro-Insurance, Agricultural Micro-Credit, Finance, Demand
1. Introduction
Microfinance provides a large range of financial services to people who would
have difficulties accessing conventional banks. Those financial services include
microcredit, micro-savings, micro-insurance, micro money-transfers, etc. Mi-
cro-insurance enables populations excluded from the traditional financial sys-
tem to protect against the risks weighing on their person, their property, or their
activities. In the agricultural sector, weather index-based (micro)-insurance (WII)
is an innovative micro-insurance product allowing farmers without access to the
traditional financial systems to protect against agricultural risks. It is innovative
insofar as unlike traditional crop insurance, compensation is triggered by indices
that are correlated with the damage so that their simple monitoring makes it
possible to have legibility of claims and trigger the compensations.
How to cite this paper:
Syll, M. M. A.
(20
21).
The Demand for Crop Insurance
Bundled with Micro
-Credit.
Theoretical
Economics Letters
, 11,
889-909.
https://doi.org/10.4236/tel.2021.115057
Received:
August 16, 2021
Accepted:
September 27, 2021
Published:
September 30, 2021
Copyright © 20
21 by author(s) and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons
Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
M. M. A. Syll
DOI:
10.4236/tel.2021.115057 890 Theoretical Economics Letters
According to Roberts (2005), in 2001, the global premiums of agricultural in-
surance were evaluated at 6.5 billion USD and developing countries represented
only 13% of the total amount. To increase the uptake of agricultural insurance
products, practitioners and policymakers, along with the researchers committed
to the development of weather index-based insurance products in replacement
of traditional crop insurance that failed to scale-up. However, empirical results
related to the demand of WII revealed modest voluntary take-up rates after two
decades of experiments (Giné and Yang, 2008; Cole et al., 2013; Giné and Yang,
2009). Hill, Ruth, & Robles (2011), recall that only 10% in average of the poten-
tial clients of WII are buying the product. Matul et al. (2013) add the fact that
even for the contracts that are largely subsidized, the take-up rates are barely
above 30% with exceptionally modest renewal rates.
To understand the enigma of that low performance, an important empirical
literature grew up around the question of the determinants of the demand (see
Marr et al., 2016; Eling et al., 2014, for a complete literature review). That is how
basis risk (Clarke, 2016; Collier et al., 2009; Hazel et al., 2010; The World-Bank,
2009), contractual non-performance, (Doherty and Shlesinger, 1990; Jensen et
al., 2014), trust (Dercon et al., 2019), or understanding (Takahashi et al., 2016)
have been presented as potential constraints to the demand. Micro-credit has
been also presented as either a substitute to micro-insurance (Giné and Yang,
2009; Liu and Myers, 2012; Zimmerman et al., 2016) or a complement to it
(Carter et al., 2011; Jensen et al., 2014). Next to all of this abundant empirical
literature the theoretical literature continues to play its traditional role of path-
finder. Clarke (2016) explained in his model how basis risk makes the rational
demand for WII absent. Gollier (2003) and Liu and Myers (2012) put in relation
the demand and the liquidity constraints faced by the farmers with the first au-
thor supporting the idea that it increases the demand while the second authors
say that it does not. On the relationship with micro-credit, Carter et al. (2011)
show that wealthier farmers who can provide high amount of collateral purchase
insurance to protect their collateral while less wealthy farmers with little collat-
eral to loose decide to insure only if the price of the credit is low.
In the context of rain-fed agriculture which characterizes the developing
countries such as West African countries where the farmers have often only one
rainy season as regular production period and hence as main regular source of
income, what are the conditions under which agricultural micro-credit could
boost the purchase of crop insurance through bundling? We contributed to the
literature by providing an answer to that question through theoretical results
which showed a positive relationship between the demand for crop WII and
agricultural credit. We demonstrated that the demand for crop insurance at the
moment of application for an agricultural credit increases with the profitability
of the credit of the previous season and decreases with the amount of collateral
requested during the loan application. In the remaining of the paper, we present
first the theoretical literature, then we introduce the model before discussing its
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findings and concluding.
2. Review of the Theoretical Literature on the Demand for
Crop Micro-Insurance
The theoretical works on the relation between micro-insurance and micro-credit
in the agricultural sector can be organized into two groups of findings. In one
side, there are findings that highlight the obstacles that liquidity constraints
represents for the demand of crop micro-insurance and on the other side, the
capacity of micro-credit to unlock take-up of crop insurance when the farmers
have access to both of them has been demonstrated.
2.1. Credit Constraints as a Barrier to Index Insurance Demand
Among the theoretical results that have highlighted the effect of liquidity con-
straints and access to credit on demand, that of Clarke (2016) is among the most
famous. Clarke starts from the classical model of insurance demand of Mossin
(1968) to then take into account the risk of contractual default of the insurer in-
troduced for the first time by Doherty and Shlesinger (1990), but considering
that the default would come this time from basis risk. He finds that in this con-
text, mainly because the basis risks creates more uncertainty around the insur-
ance product by adding two more states of nature (the possibility of being
stricken without being reimbursed and the possibility of being compensated
without disaster), the optimal demand for producers who are risk averse is zero.
Clarke did not work in a context of coupling insurance with credit. But he finds
that even in a context of access to credit his results remain valid.
On the other hand, in the theoretical model of Gollier (2003), only households
that face liquidity constraints or too much risk of damage such as disasters agree
to insure themselves. His results stem from a dynamic two-regime model. In the
first regime, which is a self-insurance plan, the individual accumulates enough
resources to self-insure because he is in a hurry to get rid of the external insur-
ance. It does everything to accumulate more wealth over time and diversify the
risks it faces. In the second regime, which is the external insurance plan, indi-
viduals do not seek to accumulate wealth over time, this is the case when the rate
of impatience is sufficiently higher than the interest rate, when the insurance
premium is affordable, or when risk aversion is high.
Liu and Myers (2012) took the same direction as Gollier (2003) by also devel-
oping a dynamic model that takes into account the liquidity constraints, but they
found an opposite result after enriching the model by taking into account the
risk of default of the insurer. They found that the existence of liquidity con-
straints reduces the demand for index insurance in the absence of the risk of de-
fault and even when the actuarial premium is fair. When the risk of default is in-
troduced, the absence of a liquidity constraint and the fair premium result in a
drop in demand because, whatever the amount of the premium, it may be lost in
the event of default. Liu and Myers (2012) went further in their work by pro-
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posing an alternative model for which farmers can delay the payment of the
premium until the end of the period covered by the insurance contract. They
found that postponing the payment of the premium, which means that it is
pre-financed with the credit, relaxes the liquidity constraints that prevent the
underwriting of insurance.
2.2. Linking Index Insurance with Agricultural Credit to Boost
Demand
Carter et al. (2011) have developed the theoretical model that is the closest to our
work by linking index insurance demand and access to credit according to the
level of collateral. They consider in their model three types of insurance:
self-insurance, implicit insurance through a credit and finally a formal insurance
contract. Their results show that farmers with a high level of collateral do not
accept taking credit in the absence of a formal insurance contract because they
avoid putting their collateral at risk. For this type of farmers, formal insurance
dominates implicit insurance through access to credit and they are willing to pay
the premium.
On the other hand, farmers with a low level of collateral agree to subscribe
only if the interest rate of the credit is low. Indeed, these small producers have a
low level of collateral and therefore the risk of losing it is not a major obstacle for
them. Buying an insurance contract gives them a small additional profit but is
very beneficial for their lender. The level of collateral held by the farmer thus
occupies an important place in this context because those with a high level of
collateral are rationed in terms of risks in the absence of formal insurance and
farmers with a low level of collateral are rationed in terms of prices if they can-
not afford to pay a high interest rate on the credit in addition to the insurance
premium. In conclusion, insurance increases producers’ demand for credit and
increases it much more when both contracts are linked or bundled.
The superiority of the bundle (index insurance + agricultural credit) on sepa-
rated contracts (credit only or insurance only) was also highlighted by De Nicola
(2015). The richest households purchase index insurance more than poor
households to the point at which they manage to insure their own assets. This
result matches that of Clarke (2016) according to which insurance demand is a
parabolic function of wealth. For De Nicola (2015) and De Nicola and Hill
(2013), access to credit gives the producer a certain level of wealth which makes
him to demand an insurance cover because the gain in terms of welfare provided
by an insured credit is greater than the gain from the single index insurance
contract (not linked to credit). These authors also found that this gain in terms
of well-being is more important for poor producers who decide to buy a
credit-related contract.
Elabed et al. (2013) also studied the demand for index insurance in relation to
access to credit. They found that index insurance could remove constraints on
access to agricultural finance by unlocking access to credit for producers.
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3. Modeling the Demand of Micro-Insurance Bundled with
Agricultural Credit
We propose here a model of demand for index-based insurance linked to agri-
cultural loan. Our objective is to theoretically analyse the relationship between
index-based insurance demand and agricultural loan to understand the condi-
tions under which the later could stimulate insurance demand. The details of the
model will be presented firstly before discussing its results secondly.
In order to do so, we start from two previous works on the demand for agri-
cultural insurance by small farmers in developing countries. It is primarily about
Sarris (2002) who proposed a theoretical model for determining demand for in-
surance against cocoa price fluctuations in Ghana and secondly from Carter et
al. (2014) work that analyzed the determinants of index-based insurance de-
mand. Indeed the work of Carter et al. (2014) is somehow an adaptation of the
work of Sarris (2002) to the demand of index-based insurance.
As a starting point, it is assumed that each year the agricultural activities are
carried out over a finite time horizon called the agricultural season which is di-
vided into two periods. The first period is that of production. The period of
production starts just before the first rainfall (one to two months before) and
goes till the end of the rainy period. The risk thus weighs on this period which
starts at the moment of the preparation of the season. During that moment the
producer register at the same time for the credit and the insurance products.
That first period ends at the time of the harvests. The second period is the period
of sale or marketing of the harvest. During this period, the farming activity
normally begins to generate incomes which will then be used to repay the credit
if there has been no loss and the index-based insurance has not triggered com-
pensation.
We start with the consumption model of Carter et al. (2014) and Sarris (2002):
( )
( )
**
t t tt
cy c y y
β
=+−
. In the model,
t
y
is the current income. It is exogenous
and random.
*
t
y
is the permanent income with
*
t
c
the corresponded perma-
nent consumption.
β
is the factor of consumption smoothing. In that model, the
consumption depends on the variation of the current income around the per-
manent income and the parameter of smoothing
β
. If
0
β
=
, the smoothing is
perfect and the current consumption is independent from the current income. If
0
β
=
there is no smoothing at all and the current consumption varies in the
same way than the current income through for example the sales of the assets
when a shock occurs.
Considering that model, we suppose the income that enable the consumption
comes from the use of agricultural credit,
a
. The agricultural credit (
a
) is pro-
vided to the farmer for its production and it is invested at the beginning of the
agricultural season during the first period. The credit invested will then provide
after its reimbursement a supplement income
( )
t tt
r ia−
with
( )
tt
ri−
repre-
senting the benefit of the agricultural credit,
t
r
the return rate of the agricul-
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tural credit and
t
i
its interest rate. That supplement income
( )
( )
t tt
r ia−
will
be used to cover the expenses of consumption during the period 2 of the current
agricultural season (period of marketing of the harvest), but also during the
production period of the next agricultural season. In another way, for each agri-
cultural season, the income that enables the consumption of the current period
depends on the profits from the investment made with credit of the previous pe-
riod
()
()
1
t
r ia
−
−
. The benefits of the credit enable then the consumption dur-
ing the period of marketing of the current production and during the next pe-
riod of production (that of the next agricultural season).
To introduce the credit, we replace in the consumption model given by
( )
( )
**
t t tt
cy c y y
β
=+−
, the expression
( )
*
tt
yy
β
−
by the expression
( )
( )
1 11t tt
r ia
− −−
−
and we call
( )
11tt
ri
−−
−
the benefit of the agricultural credit at
1t−
, 1t
r
− is the return rate of the agricultural credit and
1t
i−
is its interest
rate. We can then rewrite the consumption model as follow:
( ) ( )
*
1 1 11t t t tt
ca c r i a
− − −−
=+−
(1)
Figure 1 shows how the credit participates in the consumption during the two
periods that define the agricultural season: the period of production and the pe-
riod of marketing.
To analyse the demand for index insurance when the producer has access to
agricultural credit as described in this context, we proceed in two steps:
First we consider a model in which 1) the producer is not required to provide
a collateral to access agricultural credit, the idea being that agricultural insurance
will be considered a substitute for the collateral and the producers who, at the
time of their loan application, decide to purchase the insurance will not need to
pledge their assets; 2) the insurance premium is not pre-financed by the agricul-
tural credit and is therefore not subject to the interest rate. For example, the
premium can be paid directly by the producer or pre-financed by another
non-financial institution to which the producer is affiliated such as a farmer or-
ganization (cooperative, savings group, etc.).
Then, we will complete the simple model by considering the case where 1) a
collateral is asked to the producer to access agricultural credit, whether he has
subscribed or not to index-based insurance. In fact, the presence of the basis risk
and the lack of coverage of idiosyncratic risks by index insurance justify the re-
quest for a collateral by the credit institution. 2) The insurance premium will be
pre-financed by agricultural credit to compensate for liquidity constraints that
may block the payment of the premium. As a result, the latter will be hit by the
interest rate that applies to the agricultural credit.
Figure 1. Description of the activities during the agricultural season. Source: Author.
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3.1. The Simple Model
There are different factors in the definition of the utility function based on
whether the producer has bought the insurance or not and whether we are on
the first or the second period of the agricultural season. Without the insurance
contract, the associated utility of the first period of the season (period of produc-
tion) is
( )
( )
0
Uca
since it depends on the return on investment of the previous
credit (
0
a
). For the utility of the second period, it is uncertain because it de-
pends on the variability of the rains which occur during the first period. It is
hence given by
( )
( )
1
EU c a
. With the insurance contract, the utility of the first
period becomes
( )
( )
0
Uca B−
and that of the second period
( )
( )
1
EU c a z+
with
B
the premium paid and
z
the indemnification. The index 0 and 1 refer to
the different periods of the agricultural season, namely the period of production
at
t
= 0 and the period of marketing at
t
= 1.
Table 1 provides the utility functions of the farmers for each of the agricul-
tural season depending on whether they are insured or not in a context of bun-
dling with micro-credit. If the revenue from the agricultural production are
above the amount of credit invested increased by its cost, the agricultural loan is
hence profitable to the producer
( )
( )
0
t tt
r ia−>
. We also suppose in addition
that the agricultural credit is used only for the production and not for consump-
tion. In fact, the producer gets the credit at the beginning of the agricultural sea-
son and reimburses it during the marketing period of the same season. It is the
profit he gets after repayment
( )
( )
t tt
r ia−
which interests us here because it is
this one which allows the smoothing of the consumption of the later periods.
3.1.1. Condition of Uptake of the Insurance
In order for the producer to agree to pay for the insurance product, it must allow
him to keep at least the same level of utility as if he did not subscribe to it. In
other words, the insured agricultural season should provide to the farmer the
same level of utility than the agricultural season for which he has not purchase
the insurance. For this, the premium (
B
) must be equal to the producer’s will-
ingness to pay. The latter is in turn regarded as the profit from the insurance
contract since it is the part that must be deducted from the income of the first
period of the agricultural season so that the utility of the entire season with the
insurance is equal to its utility without the insurance. Hence, with the same level
Table 1. Possible utility functions in the case of the simple model.
Without insurance With insurance
Period
1
( )
( )
*
1 00 0
Uc a r i+−
( )
( )
*
1 00 0
Uc a r i B+ −−
Period
2
( )
( )
*
2 11 1
EU c a r i+−
( )
( )
*
2 11 1
EU c a r i z+ −+
AS
( )
( )
( )
( )
**
1 0 0 0 2 11 1
Uc a r i EUc a r i
δ
+ −+ + −
( )
( )
( )
( )
**
1 0 0 0 2 11 1
Uc a r i B EUc a r i z
δ
+ − − + + −+
AS = Agricultural Season = Period 1 + Period 2. Source: Author.
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of utility, the willingness to pay allows the producer to transfer the risk that
weighs on the second period to his insurer.
In each case, with or without insurance of the agricultural season, we have a
discrete intertemporal utility function with uncertainty intervening in the sec-
ond period.
δ
represents the individual discount factor (or discount rate). The
producer will then choose the amount of
B
that will allow him to equalize the
utility derived from the crop year in both cases:
( )
( )
( )
( )
( )
( )
( )
( )
**
1 0 0 0 2 11 1
**
1 0 0 0 2 11 1
Uc a r i EUc a r i
Uc a r i B EUc a r i z
δ
δ
+ −+ + −
= + − − + + −+
(2)
At this point, we can rewrite this equality using Taylor’s expansions of each of
its terms.
For the left terms, we will have:
( )
( )
( ) ( )
2
2
1 000 000 000
1
2
Uc ari U ariU ari U
∗ ∗∗ ∗
′ ′′
+−=+− + −
( )
( )
() ( )
2
2
2 111 111 111
1
2
EU c a r i EU a r i EU a r i EU
∗ ∗∗ ∗
′ ′′
+−=+− + −
Similarly, we will have for the terms on the right:
( )
( )
( )
( )
( )
( )
2
*
1 00 0 00 0 00 0
1
2
Uc ar i B U ar i BU ar i BU
∗ ∗∗
′ ′′
+ −−= + −− + −−
( )
( )
( )
( )
( )
( )
2 11 1
2
11 1 11 1
1
2
EU c a r i z
EU a r i z EU a r i z EU
∗
∗∗ ∗
+ −+
′ ′′
= + −+ + −+
By replacing the expressions of the Taylor expansions of each term in Equa-
tion (2), we obtain after simplifications the following Equation (3):
( )
( ) ( )
( )
22
0 00 1 1 1
11
0
22
B aBr i B Ez aE zr i Ez
ρ ρ δ δρ δρ
− −+ − + −+ =
(3)
Equation (3) gives the value of
B
that the producer must give up during the
production period to ensure a level of utility equal to what he would have if he
were not to insure during the entire agricultural season. We can already notice a
result without novelty: if the coefficient of risk aversion is zero (
0
ρ
=
), then
B Ez
δ
=
. On the other hand, if the credit is not profitable and does not allow
the smoothing of consumption
( )
( )
( )
0 0 11
0& 0ri ri−= −=
then we have the
following expression:
22
11
22
B B Ez Ez
ρ δ δρ
= −+
.
B
will then increase with the
discount rate (
δ
) and the value of the expected pay-out (
Ez
).
Considering now the Equation (2), we are interested in the relationship be-
tween the demand of insurance (
B
) and the profitability of the agricultural credit
(
r
−
i
). Therefore, we perform the differentiation of Equation (3) with respect to
(
r
−
i
), assuming that
( )
( ) ( )
0 0 11
r i r i ri−=−=−
. We obtain the expression of
Equation (4):
( )
( )
10
BEza Ba
ri
ρδ
∂= −
∂−
(4)
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First we discuss the cases where the credit of the production period of the pre-
vious agricultural season is equal to that of the production period of the current
agricultural season
( )
01
aa=
, then the case where the credits of the two produc-
tion periods are different
( )
01
aa≠
.
Case 1:
( ) ( )
01 1
:B
a a a Ez B
ri
ρδ
∂
= = −
∂−
Therefore, if the insurance premium is not pre-financed, a collateral is not
requested as a condition of access to the credit and the producer has an amount
of credit for the current crop year (
a
1) equal to the amount received during the
previous agricultural season
a
0, then the benefit of
B
insurance increases with the
benefit of the credit
( )
ri−
if
Ez B
δ
≥
. In other words, the more the credit is
profitable, the more the farmers will be willing to pay for the insurance.
Case 2:
( )
( )
01 1 0
:B
a a Eza Ba
ri
ρδ
∂
≠=−
∂−
However, in the case where the insurance premium is not pre-financed, a col-
lateral is not requested as a condition of access to the credit but the credit
amounts of the two production periods are different (
01
aa≠
), if the benefit of
the credit
( )
ri−
increases the benefit of the insurance (
B
) increases only if
10
Eza Ba
δ
≥
. We conclude that, the benefit (willingness to pay) of the insurance
increases with the benefit of the credit only if the producer increases the amount
of the credit which he uses from one crop year to another.
3.1.2. The Optimal Demand in the Case of the Simple Model
In fact, the equalization of the utility functions of the insured agricultural season
and the uninsured agricultural season in Equation (2) gives just the amount of
the willingness to pay that allows the producer to start benefiting from the agri-
cultural insurance. The producer must seek to maximize this benefit and to do
so, he must maximize the difference between these two utility functions. How-
ever, since this maximization is performed by choosing the optimal value of
B
,
maximizing the difference between the two amounts returns to maximizing only
the utility of the insured agricultural season:
( )
( )
( )
( )
( )
**
1 0 0 0 2 11 1
:MaxB U Uc a r i B EUc a r i z
δ
= + −−+ + −+
The optimal value of the premium that will allow the producer to maximize
the utility he obtains from the agricultural season when he decides to purchase
the insurance is then:
( )
*
00 0
1
B ar i B
ρ
= −−
(5)
We conclude that in the absence of a request for a collateral to access the loan
and an absence of a pre-financing of the insurance premium by the credit, the
optimal amount that the producer is willing to pay (
*
B
) is an increasing func-
tion of the benefit of the credit (its profitability) he has received during the pre-
vious crop year
( )
0
ari
−
and an increasing function of the level of risk aver-
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sion
ρ
. The purchase of the insurance policy will occur if the optimal value of the
producer’s willingness to pay (
*
B
) is greater than the premium majored by the
the insurer’s charges (
m
).
( )
*
1, if premium cost 1
take-up 0, if not
Bm
≥ = ×+
=
3.2. The Complete Model
Now we include the collateral (
g
) as a condition of access to the credit and also
the possibility to pre-finance the insurance premium by the credit itself, which
increases it by (
iB
). The collateral can be in monetary or material form but in all
cases it must be provided at the time of the application for credit, namely at the
beginning of the production period. It is then fully returned to the producer if he
repays the credit during the marketing period. Without the insurance contract,
the utility associated with the consumption of the first period is
( )
( )
10
Uc a g−
and that of the second period
()
( )
21
EU c a g
+
. With the insurance contract, the
utility of the first period becomes
()
()
( )
( )
10
1Uc a g iB
− ++
and that of the
second period will be
( )
( )
( )
21
EU c a g z++
.
Table 2 presents the expansive forms of the utility functions of the complete
model. In that model, the producer must provide a collateral at the time he ap-
plies for credit and must also pay interests on the insurance premium which is
deducted directly from the amount of the credit he is granted. The utility func-
tions are given according to the periods and whether the producer purchases the
insurance or not. We also, assume that we are in a context of availability of the
credit, which is the only source of funding the farmers have access to make an
agricultural investment.
3.2.1. Condition of Uptake of the Insurance
As in the simple model, for the producer to subscribe to the agricultural insur-
ance it should allow him at least to reach the same level of utility as in the case in
which he is not insured. Hence
B
will always represent the amount that must be
withdrawn from the production period to equalize the utility obtained from the
insured agricultural season to that of the uninsured agricultural season.
B
is then
the willingness to pay which is also interpreted as the benefit from the insurance
contract. The condition of subscription to the insurance is then the following
one:
Table 2. Description on the utility function in case of the complete model.
Without insurance With insurance
Period 1
( )
( )
*
1 00 0
Uc a r i g+ −−
( )
( )
*
1 00 0
Uc a r i B+ −−
Period 2
( )
( )
*
2 11 1
EU c a r i g+ −+
( )
( )
*
2 11 1
EU c a r i z+ −+
AS
( )
( )
( )
( )
01
Uca g EUca g
δ
−+ +
( )
( )
( )
( )
( )
( )
00 1
1U c a g i B EU a g z
δ
− ++ + + +
AS = Agricultural Season = Period 1 (production) + Period 2 (Marketing).
M. M. A. Syll
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10.4236/tel.2021.115057 899 Theoretical Economics Letters
( )
( )
( )
( )
( )
( )
( )
( ) ( )
( )
**
1 0 0 0 2 11 1
**
1 0 0 0 2 11 1
Uc a r i g EUc a r i g
Uc a r i g iB EUc a r i g z
δ
δ
+ −−+ + −+
= + − − + + + −++
(6)
By introducing Taylor’s expansions, equality 6 can be presented in this way:
() ( )( ) ( ) ( )
( )
( )
()
( )
()
()
2
2
0 0 00 0 0 0
2
11 1
1
1 11 1
2
10
2
i B a B r i i i gB B i E z
Er i Eza Ez gEz
ρ ρ ρδ
δ ρ δ ρδ ρ
+ − − + ++ + + −
+− + + =
(7)
We can also note that in the absence of risk aversion (
0
ρ
=
),
( )
0
1i B Ez
δ
+=
and
( )
0
1
B i Ez
δ
= −
.
To get the relationship between the insurance benefit (
B
) and the profitability
of the agricultural credit (
ri−
), we differentiate Equation (7), assuming that
( )
( ) ( )
0 0 11
r i r i ri−=−=−
. We get the expression of Equation (8):
( )
( )
( )
100
1
BEza Ba i
ri
ρδ
∂= −+
∂−
(8)
We discuss now the case where the amount of credit that is necessary for the
consumption smoothing through its return on investment is the same from one
agricultural season to another firstly. The case where the credit of the current
agricultural season is different from that of the previous agricultural season is
discussed secondly.
Case 1:
( ) ( )
( )
01 1 0
:1
B
a a Eza Ba i
ri
ρδ
∂
= = −+
∂−
With an interest rate that applies on the insurance premium and also with a
collateral as a condition of access to the credit, the willingness to pay of the pro-
ducer increases with
( )
ri−
only if the premium is actuarially favourable
( )
1Bi≥+
. But with the interest rate on the premium, the willingness to pay in-
creases less than proportionally as the benefit from credit increases.
Case 2:
( ) ( )
( )
01 1 0
:1
B
a a Eza Ba i
ri
ρδ
∂
≠ = −+
∂−
This result indicates that if the benefit from the agricultural investment made
with the credit
( )
ri−
increases, the benefit of the insurance,
B
, increases only
if
( )
10
1a Ez a B i
δ
≥+
. With an actuarially fair premium
( )
( )
1
Ez B i
δ
= +
for
example, this result shows that if the producers increase the level of their in-
vestment between crop years, then the willingness to pay increases with the
profitability of the agricultural credit.
3.2.2. The Optimal Demand in the Case of the Complete Model
The optimal amount that the producer will spend on the insurance premium is
obtained by maximizing the expected utility of the insured agricultural season.
( )
( )
( )
( )
( )
( ) ( )
( )
*
1 00 0
*
2 11 1
:1Max B U U c a r i g i B
EU c a r i g z
δ
= + − − ++
+ + −+ +
(9)
The expression of the optimal value of
B
will be given by:
M. M. A. Syll
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10.4236/tel.2021.115057 900 Theoretical Economics Letters
( )
( )
00 0
*
00
1
11
ar i g
Bii
ρ
−−
= −
++
(10)
The optimal amount that the producer will be willing to pay for the insurance
during the current agricultural season is an increasing function of the return on
investment of the previous agricultural season
( )
( )
00 0
ar i−
, a decreasing func-
tion of the amount of collateral and an increasing function of the level of risk
aversion (
ρ
). As in the simple model, here also the underwriting of the insurance
policy will take place if the producer’s optimal willingness to pay
*
B
is larger
than the premium majored by the insurer’s charges (
m
).
( )
*
1, if premium cost 1
take-up 0, if not
Bm
≥ = ×+
=
4. Discussion
Through the linkage between crop index-based insurance and agricultural credit,
producers in some developing countries such as Senegalese farmers are given the
option of deferring payment of premiums until repayment of the loan, which
improves access to the insurance product. Access to credit thus has a positive ef-
fect on the demand for insurance. Similarly, for credit applicants who decide to
insure, the amount that is required as a collateral can be lowered because insur-
ance is also considered a form of collateral that protects the lender against pay-
ment default. This is supposed to improve access to credit for the poorest farm-
ers who do not have enough collateral to propose to micro-finance institutions
and could then be considered as a positive effect of insurance on the demand for
credit and on agricultural investment. A complementary relationship is thus es-
tablished between crop index-based insurance and agricultural credit through
the fact that they mutually improve their demand.
The theoretical verification of this positive relationship between insurance and
credit when they are linked through the model developed in this chapter has
shown that demand for insurance increases with the increase of the profitability
of agricultural credit. For that profitability to increase, it is necessary for its cost
(the interest rate) to be lower than its return rate so that the producer can be in-
terested in the insurance after each agricultural season. With regard to collateral,
producers would expect their uptake of crop index-based insurance to allow
their lender to reduce the level of collateral they request on each application to
an agricultural credit. The model shows that the lower the amount of the collat-
eral on the credit requested, the higher the benefit of the insurance and the
chances of purchasing it.
Connected to the theoretical literature, our results are in line with those of
(Dercon et al., 2019), who emphasized the fact that linking insurance with credit
was more beneficial than offering insurance in a separate contract. Their results
also showed that the contracts that bundles insurance and agricultural credit at-
tract more poor farmers who are excluded from the financial system and who do
M. M. A. Syll
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10.4236/tel.2021.115057 901 Theoretical Economics Letters
not have a formal separated insurance contract at the time of their application to
agricultural credit. This is consistent with our finding that demand for agricul-
tural insurance is higher among producers with low level of collateral. Similarly,
with respect to the interest rate, (Dercon et al., 2019) show that, in the absence of
crop index-based insurance, interest rates on credit increase with the decreasing
of the level of collateral presented by the credit applicants. They continue their
analysis by showing that with index-based insurance linked to credit, lenders no
longer feel the need to raise interest rates even when the value of the collateral
are low. This result is also in line with our result which states that demand for
crop insurance is an increasing function of the profitability of the credit, hence a
decreasing function of its interest rate.
Several crop index-based micro-insurance pilots in developing countries
aimed to get farmers to abandon low-risk low-return investment strategies to
adopt farming techniques that require more resources but are more profitable
(Karlan et al., 2012). This objective could be achieved if farmers feel more confi-
dent and are hence more willing to invest more because they benefit from an
insurance contract that transfers part of the risk of default on loans to insurers.
In addition, index-based insurance would allow lenders to overcome the sys-
temic risks they face when they have a large number of credits exposed to agri-
cultural risks in their portfolios. Another advantage of linking micro-insurance
contracts with agricultural credit is the possibility it offers the lender (or any
other intermediary who has to manage the risks that his members or clients
transfer to him) to worry less about collateral because the insurance can be a
form of loan security.
5. Conclusion
A major problem of agricultural activities in developing countries is the reluc-
tance of poor rural households to adopt new production techniques. Liquidity
constraints due to low access to credit have been cited as one of the main reasons
why farmers adopt low-risk investment strategies for low returns (Duflo et al.,
2004; Feder and Umali, 1993). To deal with that issue the distribution patterns of
(index) micro-insurance are changing in most of the developing countries, par-
ticularly in West Africa. For example, in Senegal, the National Agricultural In-
surance Company (CNAAS) relies on farmers’ organizations, which serve as in-
termediaries to reach small farmers. The objective of our work was to analyse the
demand for micro-insurance when it is linked to agricultural credit as is the case
in Senegal, through a theoretical model. The exercise consisted of putting in re-
lationship the demand for crop index-based insurance with the characteristics of
agricultural credit, such as the interest rate and the level of collateral requested.
A first simple model in which a collateral is not requested and where the pro-
ducer pays the insurance premium before the credit is granted was first pre-
sented. It is then completed by 1) integrating the collateral as a necessary condi-
tion to access the loan and by 2) taking into account the possibility of including
M. M. A. Syll
DOI:
10.4236/tel.2021.115057 902 Theoretical Economics Letters
the amount of the insurance premium in the amount of the agricultural credit
demanded thus allowing to postpone its settlement until the repayment of the
agricultural loan. In Senegal, it is this model of offer that is developing since the
introduction of crop index-based insurance in 2012. Producers who are mem-
bers of farmers’ organizations or clients of micro-finance institution who pro-
vide agricultural loans, either in the form of cash or in the form of inputs
(equipment, fertilizers, seeds), have the possibility to purchase a crop in-
dex-based insurance which enables them to cover the credit contracted during
the agricultural season against climate risks such as rainfall variability (Syll and
Weingaertner, 2018).
Our results show a positive relationship between the demand for crop in-
dex-based insurance and agricultural credit through the return on investment
(or profitability) of the latter during the previous agricultural season and the
amount of collateral requested from the farmer during the starting agricultural
season. In other words, a good agricultural season obtained today thanks to
agriculture credit would increase the demand for insurance during the next sea-
son. However, the amount of collateral requested when granting loans lead to a
decrease in demand for crop insurance when linked to agricultural credit.
However, as theoretical results do not systematically coincide with empirical
results, which can be much more complex, the following questions must be an-
swered empirically to confirm our conclusions. In practice, does bundling mi-
cro-insurance to agricultural credit lead to more agricultural investment by the
farmers? Do more farmers get insured in that case? What about a decreasing in
collateral and interest rate on the credit? Does it also lead to more insurance
take-up?
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this pa-
per.
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Appendix: Proof of Results
1) Simple model: proof of result (3)
We recall that the condition for the purchase of the insurance by a farmer in
this model is given by:
( )
( )
()
( )
( )
( )
( )
( )
**
1 0 21
**
1 0 21
Uc a r i EUc a r i
Uc a r i B EUc a r i z
δ
δ
+ −+ + −
= + −− + + −+
The expansion of Taylor on each of its terms provide:
For the left terms we will have:
( )
( )
( ) ( )
2
2
1 000 000 000
1
2
Uc ari U ariU ariU
∗ ∗∗ ∗
′ ′′
+−=+− + −
( )
( )
( ) ( )
2
2
2 111 111 111
1
2
EU c a r i EU a r i EU a r i EU
∗ ∗∗ ∗
′ ′′
+−=+− + −
Similarly, we will have for the terms on the right:
( )
( )
( )
( )
( )
()
1 00 0
2
*
00 0 00 0
1
2
Uc a r i B
U ar i BU ar i BU
∗
∗∗
+ −−
′ ′′
= + −− + −−
( )
( )
( )
( )
( )
( )
2 11 1
2
11 1 11 1
1
2
EU c a r i z
EU a r i z EU a r i z EU
∗
∗∗ ∗
+ −+
′ ′′
= + −+ + −+
After replacement of all the terms by their Taylor expansion, the relation (2)
becomes:
() ( )
( ) ( )
()
()
( )
( )
()
( )
( )
()
2
2
00
2
*2
11
2
00
2
11
1
2
1
2
1
2
1
2
U ariU a riU
EU a r i EU a r i EU
U ari BU ari BU
EU a r i z EU a r i z EU
δ
δ
∗∗ ∗
∗∗
∗∗ ∗
∗∗ ∗
′ ′′
+− + −
′ ′′
+ +− + −
′ ′′
= + −− + −−
′ ′′
+ + −+ + −+
( )
( )
2 2 2 22
00 00 0 0 00 0 0 0
2 2 2 22
11 11 1 1 11 1 1 1
* 2 2 2 22
00 00 0 0 00 0 0 0
00 00
12
2
12
2
12
2
22
U a rU a i U a r U ri a U a i U
EU Ea r U Ea i U Ea r U Er i a U Ea i U
U a r U a i U BU a r U r i a U a i U
r Ba U i Ba
δ
∗∗∗ ∗ ∗ ∗
∗∗∗ ∗ ∗ ∗
∗ ∗∗ ∗ ∗ ∗
∗
′ ′ ′ ′ ′′
⇒+ − + − +
′ ′ ′′ ′′ ′′
++ − + − +
′ ′ ′ ′′ ′′ ′′
=+ − −+ − +
′′ ′
−+
( ) ( )
(
( ) ( )
(
( ) ( )
( ) ( ) ( )
)
)
2*
1 1 11
2 2 2 22
11 111 11
2
1 1 11
12
2
22
U B U EU a E r U a i E U
E zU aE rU iaE rU aiEU
a E r zU i a E zU E z U
δ
∗∗ ∗ ∗
∗ ∗ ∗∗
∗ ∗∗
′ ′′ ′ ′
++ + −
′ ′′ ′′ ′′
++ − +
′′ ′′ ′′
+−+
M. M. A. Syll
DOI:
10.4236/tel.2021.115057 906 Theoretical Economics Letters
( ) (
( ) ( ) ( ) ( ) ( )
( )
(
2 2 2 22 *
00 00 0 0 00 0 0 0
2 2 2 22
11 11 11 111 11
2 2 2 22
00 00 0 0 00 0 00
00 00
12
2
12
2
12
2
22
U a r U a i U a r U r i a U a i U EU
aErU aiEU aErU iaErU aiEU
U a r U a i U BU a r U r i a U a i U
r Ba U i Ba U
δ
∗∗∗ ∗ ∗ ∗
∗∗ ∗ ∗ ∗
∗ ∗ ∗∗ ∗ ∗ ∗
∗∗
′ ′ ′ ′ ′′
⇒+ − + − + +
′ ′ ′′ ′′ ′′
+−+ − +
′ ′ ′ ′′ ′′ ′′
=+ − −+ − +
′′ ′′
−+
) ( ) ( )
(
( ) ( )
(
( ) ( )
( ) ( ) ( )
)
2
1 1 11
2 2 2 22
11 111 11
2
1 1 11
12
2
22
B U EU a E rU a i E U
E zU a Er U i a Er U a i E U
a E r zU i a E zU E z U
δ
∗∗ ∗ ∗
∗ ∗ ∗∗
∗ ∗∗
′′ ′ ′
+++ −
′ ′′ ′′ ′′
++ − +
′′ ′′ ′′
+−+
That yield the following reduced equation:
( )
( ) ( ) ( )
2
00 00
2
1 1 11
1
2
10
2
BU r Ba U i Ba U B U E zU
a E r zU i a EzU E z U
δ
δδ δ
∗ ∗ ∗∗ ∗
∗∗∗
′ ′′ ′′ ′′ ′
−− + + +
′′ ′′ ′′
+−+=
Multiplying the expression by
1
U∗
−
′
, gives the following relation:
2
00 00
2
1 1 11
1
2
10
2
U U UU U
B r Ba i Ba B E z
U U UU U
UUU
aErz aiEz Ez
UUU
δ
δδ δ
∗ ∗ ∗∗ ∗
∗ ∗ ∗∗ ∗
∗∗∗
∗∗∗
′ ′′ ′ ′′ ′
− − −− −
−− + + +
′ ′ ′′ ′
′′ ′′ ′′
−−−
+−+=
′′′
U
U
∗
∗
′′
−
′
is the Arrow Pratt’s risk aversion coefficient (
ρ
). The expression be-
comes hence:
( ) ( ) ( )
( )
( )
2
00 00 11
2
11
1
2
10
2
B r Ba i Ba B E z E z E r a
iEza Ez
ρ ρ ρδ δ ρ
δ ρδ ρ
− + +− +
−+=
Assuming that
( )
( )
( ) ( )
( )
11 1 1
Ezr i Ez Er i−= −
and multiplying by
( )
00
ri−
and
()
11
ri
−
yield the expression below:
()
( ) ( )
( )
22
0 00 1 1 1
11
0
22
B aBr i B Ez aE zr i Ez
ρ ρ δ δρ δρ
− −+ − + −+ =
2) Simple model: proof of result (4)
We start from the condition of optimality given by the results (3) to draw the
following relation:
( )
( )
( )
( )
( )
)
( )
( )
( )
( )
22
0 00
11
1
11
22
Er i f B
B aBr i B Ez Ez
Er i aE z
ρ ρ δ δρ
δρ
−=
−+ − − + −
−=
To find the optimal value of
B
that maximizes the benefit of the credit
()
( )
)
11
Er i−
, we perform a partial differentiation.
M. M. A. Syll
DOI:
10.4236/tel.2021.115057 907 Theoretical Economics Letters
( )
( )
( ) ( )
( )
11 00 0
00 0
*
00 0
1
01 0
1
Er i ar i
ar i B B
B
B ar i
ρ
ρρ ρ
ρ
∂− −−
= →− + − − = → =
∂
→ = −−
3) Complete model: proof of result (7)
The condition of purchase of the insurance in the model complete is given by
the following relation:
( )
( )
( )
( )
( )
( )
( )
( ) ( )
( )
(
(
**
1 00 0 2 11 1
**
1 00 0 2 11 1
1
Uc a r i g EUc a r i g
Uc a r i g B i EUc a r i g z
δ
δ
+ − −+ + −+
= + − − + + + + −++
Taylor’s expansion on each of the above terms gives:
For the first two terms of the relation:
( )
( )
( )
( )
( )
( )
1 00 0
2
*2
000 000
1
2
Uc a r i g
U U ari g U ari g
∗
∗∗
+ −−
′ ′′
= + −−+ −−
( )
( )
( )
( )
( )
( )
2 11 1
2
2
111 111
1
2
EU c a r i g
EU EU a r i g EU a r i g
∗
∗∗ ∗
+ −+
′ ′′
= + −++ −+
For the last two terms of the relation, we will have:
( ) ( )
( )
( )
( ) ( )
( )
( )
( ) ( )
( )
( )
10
2
*
00
1
1
11
2
Uc a r i g B i
U U ari gB i U ari g B i
∗
∗∗
+ −− + +
′ ′′
= + −− + + + −− + +
( ) ( )
( )
(
( ) ( )
( )
( ) ( )
( )
2 11 1
2
11 1 11 1
1
2
EU c r i a g z
EU U a r i g z EU a r i g z
∗
∗∗ ∗
+− ++
′ ′′
= + −+ + + −++
After replacing each term by its Taylor development, the relation becomes:
( )
( )
( )
( )
( )
( )
( )
( )
( ) ( )
( )
( )
( ) ( )
( )
( )
( ) ( )
( )
( ) ( )
( )
2
2
000 000
2
*2
111 111
00 0 0
2*
00 0 0
2
11 1 11 1
1
2
1
2
1
11
2
1
2
U Uari g U ari g
EU EU a r i g EU a r i g
U U ar i g B i
U a r i g B i EU
EU a r i g z EU a r i g z
δ
∗∗ ∗
∗∗
∗∗
∗
∗∗
′ ′′
+ −−+ −−
′ ′′
+ + −++ −+
′
= + −−+ +
′′
+ −−+ + +
′ ′′
+ −++ + −++
(
) ( )
(
()
( ) ( )
(
( ) ( )
( ) ( )
)
2 2 2 22
00 00 0 0 00 0 0 0
2*
00 00 1 1 11
2 2 2 22
1 1 1 1 11 0 1
2
10
12
2
22
122
2
2
U a r U a i U gU a r U r i a U a i U
r ga U i ga U g U EU a E rU a i E U
gE U a E r U i a U a i E U ga E r U
igaEU gEU
δ
∗ ∗ ∗∗ ∗ ∗ ∗
∗ ∗∗ ∗ ∗
∗ ∗∗ ∗ ∗
∗∗
′ ′ ′ ′ ′ ′′
⇒+ − − + − +
′′ ′′ ′′ ′ ′
− + +++ −
′ ′′ ′′ ′′ ′′
++ −+ +
′′ ′′
−+
M. M. A. Syll
DOI:
10.4236/tel.2021.115057 908 Theoretical Economics Letters
( )
( )
( )(
( )
( )
)
( ) ( ) ( ) ( )
( )
( ) ( )
* 2 2 2 22
00 00 0 0 0 00 0 0 0
22
00 00 00 00 000 00 0
2
2
0 1 1 11
2 2 2 22
1 1 1 1 1 11 1
1
12
2
2 2 2 2 2 2 21
1
122
2
U U ar a i g i B U a r ri a a i
r ga i ga r Ba i Ba r i Ba Bi a g gB i
B i EU aEU r EU ai EU g EU z
aErU ErU ia aiU Erz
δ
∗∗
∗∗ ∗ ∗ ∗
∗∗ ∗
′ ′′
= + − −−+ + − +
− + − + − + ++ +
′′ ′ ′ ′
+++ + − + +
′′ ′′ ′′ ′′
+ − ++
( ) ( )
(
( ) ( ) ( )
)
11 1
22
0 0 00
2
22 2
Ua iEzUa
E z U E r U ga i ga U g gE zU
∗∗
∗∗ ∗ ∗
′′
−
′′ ′′ ′′ ′′
+ + − ++
(
) ( ) ( )
(
( ) ( ) ( ) ( )
(
( )
( ) ( )
)
2 2 2 22
00 00 0 0 00 0 0 0
2
00 00 1 1 11
2 2 2 22
11 111 11 00
2
00
12
2
22
122
2
2
U a r U a i U gU a r U r i a U a i U
r ga U i ga U g U EU a E rU a i E U
gEU aErU iaErU aiEU gaErU
igaEU gEU
δ
∗ ∗ ∗∗ ∗ ∗ ∗
∗ ∗∗ ∗ ∗ ∗
∗ ∗ ∗∗ ∗
∗∗
′ ′ ′ ′ ′ ′′
⇒+ − − + − +
′′ ′′ ′′ ′ ′
− + +++ −
′ ′′ ′′ ′′ ′′
++ − + +
′′ ′′
−+
( )
(
( )
( )
)
( ) ( )
(
( )
( ) ()
2 2 2 22
00 00 0 0 0 00 0 00
22
00 00 00 00 000 00 0
2
2 2 2 2 22
0 1 1 1 1 1 11 1 1
1 1 11
1
12
2
2 2 2 2 2 2 21
1
1 22
2
U a r U a i U gU i BU a r r i a a i
r ga i ga r Ba i Ba r i Ba Bi a g gB i
B i U aE rU iaEU r aiEU EU rza
EU aErU aiEU
δ
∗ ∗ ∗∗ ∗
∗ ∗ ∗ ∗∗
∗∗
′ ′′ ′
= + − − −+ + − +
− + − + − + ++ +
′′ ′′ ′′ ′′ ′′
++ + − + +
′′
++ −
( ) ( ) ( )
(
( )
( ) ( ) ( ) ( ) ( )
)
)
11
22
0 0 00
2
22 2
gE U E zU i E zU a
EU z EU r ga igaEU gEU gEU z
∗∗∗ ∗
∗ ∗ ∗∗∗
′ ′ ′′
++−
′′ ′′ ′′ ′′ ′′
++ − + +
(
)( ) ( )
(
( ) ( ) ( ) ( )
(
( )
( ) ( )
)
2 2 2 22
00 00 0 0 00 0 0 0
2
00 00 1 1 11
2 2 2 22
11 111 11 00
2
00
12
2
22
122
2
2
U a r U a i U gU a r U r i a U a i U
r ga U i ga U g U EU a E rU a i E U
gEU aErU iaErU aiEU gaErU
igaEU gEU
δ
∗ ∗ ∗∗ ∗ ∗ ∗
∗ ∗∗ ∗ ∗ ∗
∗ ∗ ∗∗ ∗
∗∗
′ ′ ′ ′ ′ ′′
⇒+ − − + − +
′′ ′′ ′′ ′ ′
− + +++ −
′ ′′ ′′ ′′ ′′
++ − + +
′′ ′′
−+
( )
(
( )
( )
)
( ) ( ) ( ) ( ) ( )
(
( ) ( )
2 2 2 22
00 00 0 0 0 00 0 00
22
00 00 00 00 000 00 0
2
2*
0 1 1 11
2 2 2 22
1 1 1 1 1 11
1
12
2
2 2 2 2 2 2 21
1
12
2
U a r U a i U gU i BU a r r i a a i
r ga i ga r Ba i Ba r i Ba Bi a g gB i
B i U EU aErU aiEU gEU E zU
aE rU iaEU r aiEU
δ
∗ ∗ ∗∗ ∗
∗ ∗ ∗∗∗
∗∗
′ ′′ ′
= + − − −+ + − +
− + − + − + ++ +
′′ ′ ′ ′ ′
++ + + − + +
′′ ′′ ′′
+ −+
( )
(
( )
( ) ( ) ( ) ( ) ( )
)
)
11 1 1
22
0 0 00
22
22 2
E U r z a i E zU a
EU z EU r ga igaEU gEU gEU z
∗∗ ∗
∗ ∗ ∗∗∗
′ ′′
+−
′′ ′′ ′′ ′′ ′′
++ − + +
We thus obtain the following reduced relation after simplification:
( ) ( ) ( ) ( )
( )
( )
( )
( )
( )
( ) ( )
0 00 0 0 0000
2
2
0 11 1
2
11
11
2
10
2
i BU r i Ba U i gBU r i i Ba U
B i U E zU E r i a E zU
EzU gEzU
δδ
δδ
∗ ∗∗ ∗
∗∗ ∗
∗∗
′ ′′ ′′ ′′
−+ −− ++ −−
′′ ′ ′′
++ + +−
′′ ′′
+ +=
M. M. A. Syll
DOI:
10.4236/tel.2021.115057 909 Theoretical Economics Letters
By multiplying this expression with
1
U∗
−
′
, the equality becomes:
( ) ( ) ( ) ( )
( )
( )
( )
0 00 0 0 0000
2
2
0 11 1
2
11
11
2
10
2
U UU U
i B r i Ba i gB r i i Ba
U UU U
UU U
B i Ez Er i aEz
UU U
UU
Ez gEz
UU
δδ
δδ
∗ ∗∗ ∗
∗ ∗∗ ∗
∗∗ ∗
∗∗ ∗
∗∗
∗∗
′ ′′ ′′ ′′
− −− −
+ −− ++ −−
′ ′′ ′
′′ ′ ′′
−− −
++ + +−
′′ ′
′′ ′′
−−
+ +=
′′
U
U
∗
∗
′′
−
′
is the Arrow Pratt’s risk aversion coefficient (
ρ
). The expression be-
comes hence:
()( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
()
2
2
0 00 0 0 00 0 0
2
11 1
1
11 1
2
10
2
i B r i Ba i gB r i iBa B i
Ez Er i Eza Ez gEz
ρρ ρ ρ
δ δ ρ δ ρδ ρ
+ −− ++ −− + +
−+− + + =
Assuming
( )( )
( )
( ) ( )
( )
11 11
Ez r i Ez Er i−= −
and multiplying by
( )
00
ri−
and
( )
11
ri−
, yield the expression below
( ) ( ) ( ) ( ) ( )
( )
( )
( )
( )
( )
( )
2
2
0 00 0 0 0 0
2
11 1
1
1 11 1
2
10
2
iB r iBa i igB B i Ez
Er i Eza Ez gEz
ρ ρ ρδ
δ ρ δ ρδ ρ
+ − − + ++ + + −
+− + + =
4) Complete model: proof of result (10)
As in the case of the simple model, we start from the insurance purchase con-
dition to derive the following relation
( )
( )
( )
11
Er i f B−=
()
( )
() ( ) ( ) ( ) ( )
( )
( )
()
( )
2
22
0 0 00 0 0 0
11
1
11
1 11 1
22
B i a Br i i gB i B i Ez Ez gEz
Er i aE z
ρ ρ ρ δ δρ δρ
δρ
− ++ − +− +− + + − +
−=
To find the optimal value of
B
that maximizes the benefit of the credit
( )
( )
11
Er i−
, we perform a partial differentiation.
( )
( )
( ) ( )( ) ( ) ( )
11
2
0 0 00 0 0 0
0
1 1 1 10
Er i
B
i a ri i g i B i
ρ ρρ
∂−
=
∂
⇒−++ − +− +− + =
We obtain,
( )( ) ( ) ( )
( )
0 00 0 0 0
2
0
1 11
1
a ri i g i i
B
i
ρρ
ρ
− +− +−+
⇒= +
( )
( )
0 00
0
1
1
a ri g
Bi
ρρ
ρ
−− −
⇒= +
The optimal value of the benefit
B
of the insurance that maximizes the ex-
pected benefit of the credit
( )
( )
11
Er i−
is given by:
( )
( )
00 0
*
00
1
11
ar i g
Bii
ρ
−−
⇒= −
++
