APPENDIX: A Model of the Demand for Odd Fellows Sickness Insurance
In our exposition, we assume that an individual had a maximum amount that he is willing
to pay, a reservation price, for each good: rE for the sick benefit B and rS for the social goods of
membership. We also assume that his reservation prices for the separate goods were independent
of each other--that complementarities in consumption did not exist. Thus his reservation price
for the bundled good, DR, was the sum of rE and rS. Thus he paid his membership dues as long as
his reservation price, DR, exceeded the price, D*, that the lodge asked for its bundled good.
For our basic results on the demand for the fraternal sick benefit over the life-cycle, we
assume for the moment that the social component of his reservation price for the bundled good,
rS, was stable over his life course. Thus any age-related change in DR expressed an age-related
change in rE, its insurance component. We relax this assumption after the basic results are
derived.
Basic Results
Was the IOOF's sick benefit old man's insurance? If so, the member would have an
increasing reservation price (willingness to pay) for membership as he aged and his risk of
disability increased (rE increases hence DR increases). The longer he remained a member, the
stronger his attachment to the lodge membership and the less likely that he would quit.
Implicitly, his demand for market insurance over his life course would be independent of
(insensitive to) alternatives such as self-insurance or family insurance.
To show this we follow Ehrlich=s and Becker=s (1972) model to illustrate the demand
for insurance over the life-cycle when both market insurance and self-insurance are available to a
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consumer. Ehrlich and Becker demonstrate that when the risk of suffering a loss increases, a
consumer will increase his expenditure on market or self-insurance when only market insurance
or self-insurance is available. They also demonstrate that when both market and self insurance
are available, an increase in the probability of suffering a loss leads to an increase in expenditure
on self-insurance and a decrease in the expenditure on market insurance.
In the context of our study, Ehrlich=s and Becker=s results suggest that as Odd Fellows
aged, and their risk of losing income due to illness increased, they may have developed a stronger
attachment to fraternal membership for insurance purposes if there was no available substitute for
the fraternal sick benefit; or they may have had a weakening attachment to membership for
insurance purposes as they increased their self-insurance expenditures which allowed them to
substitute resources away from fraternal insurance.
Ehrlich=s and Becker=s results are not directly applicable to our study since they allowed
consumers to choose continuous amounts of both self-insurance and market insurance. IOOF
sickness insurance, in contrast, was offered as a Atake it or leave it@ contract. A member could
pay membership dues D and in return, receive a weekly benefit B for each week the member was
sick and unable to work. In addition, Ehrlich and Becker modeled market insurance as having a
price that is actuarially related to the benefit secured under the insurance contract. With fraternal
insurance, there was no explicit actuarial link between membership dues and benefits. We want
to adapt Ehrlich=s and Becker=s framework to take account of this nature of the fraternal
insurance contract.
A consumer of age A is endowed with income I=wT, where w is his weekly wage and T
is the number of weeks he works for a full year. If he is sick and unable to work for t(A) weeks
2
with probability p(A) he suffers a loss of income Le=wt(A). Both the probability of sickness
p(A) and weeks of sickness t(A) are non-decreasing with age (p(A)/A0, t(A)/A0). In
the absence of insurance possibilities, this consumer has an expected utility of:
EU n = (1 - p(A))  u(I) + p(A)  u(I - w  t(A))
u(I) is a utility of income function with:
2
u
 u
= u  0,
= u  0.
I
II
An Odd Fellow paid annual dues of D* to be a lodge member. If this member was sick
and unable to work, then he received B* dollars sick benefit for each week of sickness. A
member=s expected utility is:
*
*
*
EU m = (1 - p(A))  u(I - D ) + p(A)  u(I - D - w  t(A)+ B  t(A))
To simplify notation that follows, define incomes out of (n), and in (m), lodge membership as:
I 1n = I, I 2n = I - w  t(A)
I 1m = I - D, I 2m = I - (w - B)  t(A) - D
u/ I ij = u ij' , i = 1,2 and j = m,n.
*
*
*
EU m = (1 - p(A))  u(I - D ) + p(A)  u(I - D - (w - B )  t(A)) 
(1 - p(A))  u(I) + p(A)  u(I - w  t(A)) = EU n
Given (D*,B*), the member chooses to pay D* and remain a member as long as:
Another way to characterize this condition would be to define DR as the maximum amount a
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member would be willing to pay to secure the weekly sick benefit B*. Thus, by definition, DR is
the cost of membership at which EUm=EUn. If D*<DR, then EUm>EUn and the individual
chooses to be a member.
Since the member cannot choose D (or B), only whether or not to pay D* to get B*, we
can use DR as a measure of the member=s demand, or attachment for the fraternal sick benefit.
In particular we can examine how the consumer=s willingness to pay, or reservation price,
changes as his risk of illness increases with his age. Totally differentiating the definition of DR
with respect to DR and age A yields:
p(A)
t(A)
 [u( I 1n ) - u( I 1m ) + u( I 2m ) - u( I 2n )] +
 p(A)  (w  u 2' n - (w - B* )  u 2' m )
dD

A

A
=
dA
[(1 - p)  u1' m + p  u 2' m ]
R
R
dD
> 0 if I 2m > I 2n and I 1n > I 1m
dA
As a member ages and his risk of illness increases, his willingness to pay for the fraternal sick
benefit increases when that is his only source of insurance so long as membership results in a
transfer of income from the good state to the bad state. Since fraternal insurance is the only way
to transfer income, his willingness to pay increases as the desire to transfer income increases.
If we incorporate a social benefit >0 of membership from which the member derives
utility whether they are sick or healthy (=1 for non-members), the expected utility function for a
member becomes:
4
EU m =   [(1 - p(A))  u(I - D) + p(A)  u(I - (w - B)  t(A) - D)], > 0
If 0<<1, then the social side of membership diminishes the utility a member derives from
having the sick benefit. If =1, as we assume it does for non-members, then members are
indifferent to the social side of membership, hence the decision to remain in the membership is
solely an insurance decision. Finally, if >1, then the social side of membership is an additional
source of utility for a member. It is straightforward to show that dDR/d>0. Members who derive
more of a social benefit are willing to pay more for membership. For some members with high
, the sick benefit may be unimportant for the decision to pay dues and remain a member.
What we can conclude so far, is that if fraternal sick benefits are the only source of
insurance coverage, then as men age, and the probability of income loss due to illness increases,
the demand for the fraternal sick benefit increases. As members age they should become more
attached to the lodge membership. In this sense, the fraternal sick benefit would be an old man=s
benefit.
If, on the other hand, the IOOF's sick benefit was young man's insurance, the member=s
reservation price could fall as he aged and alternatives to market insurance opened up to him (rE
decreases hence DR decreases). At some point the reservation price (DR) would fall below the
purchase price (D*), and the individual would stop paying dues and exit from membership.
Implicitly, the Odd Fellows who do not exit have high social reservation prices (rS) due to high .
Alternatively, they are unable to accumulate enough savings for self-insurance (or older children
for family insurance) and their reservation price for insurance (rE) remains high, hence DR
remains high.
Again, following Ehrlich and Becker, an individual can choose an expenditure on self5
insurance, c, which reduces the magnitude of the loss of income if the member is sick and unable
to work. Thus, for an expenditure of c dollars on self-insurance, the realized loss of income for
the individual becomes L(Le,c)=wt(A)-s(c), where:
2
s(c)
 s(c) 
= s(c)  0, and
= s (c)  0
c
cc
The latter property of the loss function reflects that self-insurance may have a diminishing
marginal productivity.
With both fraternal sickness insurance and self-insurance available to a consumer, he
chooses c and (D,B) = (D*,B*) or (0,0) to maximize expected utility:
EU = (1 - p(A))  u(I - D - c) + p(A)  u(I - L( Le ,c) - c - D + B  t(A))
To simplify notation, define incomes out of (n), and in (m), fraternal membership as:
I 1n = I - c, I 2n = I - w  t(A) + s(c) - c
I 1m = I - c - D, I 2m = I - (w - B)  t(A) + s(c) - c - D
u/ I ij = u ij' , i = 1,2 and j = m,n.
The first order conditions for this consumer=s problem are:
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EU
= -(1 - p(A))  u1' - p(A)  [1 - s (c)]  u 2' = 0
c
and a participation condition where (D,B)=(D*,B*) if:
EU m = (1 - p(A))  u( I 1m ) + p(A)  u( I 2m ) 
(1 - p(A))  u( I 1n ) + p(A)  u( I 2n ) = EU n
Since the consumer takes (D*,B*) as exogenous, that is they choose to be members given the set
dues and benefits, we can characterize an optimal level of self insurance expenditure as a
function of exogenous parameters. c*=c(I,A,D,B) is the solution to the first order condition (11).
Totally differentiating the same first order condition allows us to demonstrate that the desired
level of self insurance expenditure increases when the probability of suffering a loss increases
with age as long as [1-s(c)]<0 and w>B:
'
'
''
*
dc - [( p(A)/A)  (u1 - u 2  [1 - s (c)]) + (1 - s (c))  (w - B)  ( t(A)/A)  u 2 ]
=
>0
dA
((1 - p(A))  u1'' + p(A)  u 2''  [1 - s (c) ] 2 - p(A)  u 2'  s (c))
The former condition requires that self insurance must increase net income in the loss state is
also the condition that must be satisfied for an individual to do any self insurance. The latter
condition requires that fraternal sick benefit not fully replace the lost wage income. This result
holds for any combination of (D,B) so long w>B. 1
1
This condition also holds, as Ehrlich and Becker demonstrate, when any amounts of self
insurance and market insurance can be chosen simultaneously.
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As the risk of suffering a loss increases, the use of self-insurance is unambiguously
increasing. What happens to the demand for fraternal sickness insurance? Again, define DR to
be the maximum that a consumer would be willing to pay for the fraternal sick benefit B*:
(1 - p(A))  u(I - D R - c) + p(A)  u(I - (w - B* )  t(A) + s(c) - c - D R )
= (1 - p(A))  u(I - c) + p(A)  u(I - w  t(A) + s(c) - c)
From before, we characterized the optimal level of self insurance expenditure as a function of an
individual=s income, probability of a loss, the fraternal dues and benefit, and the endowed
income loss; c*=c(I,A,D,B) is the solution to the first order condition (11). Since we will focus
on A and D and hold all else constant, we will express c*=c(A,D). Substituting c* into the
expected utility function given in (9) and evaluating c* at c(A,DR) and c(A,0) yields an
V(p, D R )  (1 - p(A))  u(I - D R - c(A, D R )) + p(A)  u(I - (w - B* )  t(A) + s(c(A, D R )) - c(A, D R ) - D R )
= (1 - p(A))  u(I - c(A,0)) + p(A)  u(I - w  t(A) + s(c(A,0)) - c(A,0))  V(p,0)
expression in terms of indirect expected utility functions:
Totally differentiating (15) with respect to DR and A yields:
V(A, D R )
V(A, D R )
V(A,0)
dA +
 dD R =
dA
R
A
D
A
The change in the consumer=s willingness to pay for the fraternal benefit as the risk of suffering
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a loss of income increases is:
V(A,0) V(A, D R )
dD

A
A
=
R
V(A, D )
dA
 DR
p(A)
t(A)
[u( I 1n ) - u( I 1m ) + u( I 2m ) - u( I 2n ))] +
 p(A)  (w  u 2' n - (w - B* )  u 2' m )

A

A
=
((1 - p)  u1' m + p  u 2' m )
R
From the above expression we determine that:
R
dD
 0 if I 2n  I 2m and I 1m  I 1n
dA
R
dD
> 0 if I 2n < I 2m and I 1m < I 1n
dA
These conditions indicate that whether the willingness to pay increases or decreases as the
member ages depends upon whether the fraternal sickness insurance or self insurance is more
effective for transferring income across states of the world. If fraternal insurance was the only
mechanism for transferring income across states of the world, then the willingness to pay for the
sick benefit increases as the consumer=s risk and duration of illness increases with age. If selfinsurance is possible, the willingness to pay for the sick benefit decreases. Given that the
fraternal benefit was fixed in its amount, while any amount of self-insurance could be selected, it
is likely that total insurance without fraternal coverage purchased could be higher than the
income secured with the combination of fraternal insurance and self-insurance.
When self-insurance is available as a substitute for fraternal insurance, the demand for
fraternal sick benefit falls as a man ages, his risk of illness increases and his demand for total
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insurance coverage increases. When the members are young, the fraternal sick benefit would
have been generous enough to meet total desired insurance. As they age and total desired
insurance increases, they increasingly turn to self-insurance since they cannot increase coverage
through the fraternal membership beyond the stipulated/fixed amount. As the share of the
fraternal benefit in total insurance benefits falls, so too does the willingness to pay for the
fraternal benefit, since aging members increasingly rely on self-insurance, which we have seen,
unambiguously increases as the members age and their risk increases.
Incorporating Age Related Changes in rS
The objective of our model of insurance is to motivate reasons why the fraternal sickness
insurance contract may have resulted in an Odd Fellow having an increasing or decreasing
attachment to lodge membership. These insights are then applied to determining if the
attachment to lodge membership weakened or strengthened with the duration of lodge
membership. Our basic results which we have labeled the old man=s model (the reservation
price for membership rises with age) and the young man=s model (the reservation price for
membership falls with age), rely on an assumption that the social benefits for membership ()
were fixed over the life-cycle for a given member. Thus we assume that the rS the reservation
price for the social benefits of membership was fixed over a given member=s lifetime. As a
consequence, we infer any age-related changes in DR to be driven by age-related changes in rE.
We now want to briefly consider relaxing this assumption and allowing the social benefits of
membership to have their own age-related changes, hence = (A).
As we show below, the social benefit dynamics do not change the fact that members
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value the sick benefit more (or less) as they age in the absence (or presence) of self-insurance.
The social benefit dynamics could be, however, a confounding influence on the identification of
movements in rE as reflected in DR. In light of this, our empirical strategy is to identify whether a
man's attachment to IOOF lodge membership increased or decreased with an individual's time in
the lodge. We then reconcile the increasing or decreasing attachment to membership with one of
our models of insurance demand, taking into account the potential influence of changes in rS.
To illustrate our point, consider the old man=s model of the sick benefit (no selfinsurance possible). We defined DR to be the level of dues at which a member was indifferent to
having the fraternal sick benefit, and not having the fraternal sick benefit. If we incorporate (A)
into the participation constraint of the Old Man's model we get the following:
EU m  EU n   ( A)  (1  p( A))  u( I1m )  p( A)  u( I 2m )  (1  p( A))  u( I1n )  p( A)  u( I 2n )  0
Total differentiation with respect to DR and A and yields:
p(A)
t(A)
 [u( I 1n ) -  ( A)  u( I 1m ) +  ( A)  u( I 2m ) - u( I 2n )] +
 p(A)  (w  u 2' n -  ( A)  (w - B* )  u 2' m )
dD

A

A
=
dA
[  ( A)  (1 - p)  u1' m + p  u 2' m ]
R



(1  p( A))  u ( I 1m )  p(a)  u ( I 2 m ) 

A

[  ( A)  (1 - p)  u1' m + p  u 2' m ]


Before when (A) was fixed for all A, the reservation price for membership was
unambiguously increasing with a member's age. The first part of dDR/dA show that, as before,
the change in the demand for membership due to the increased probability and duration of
sickness as a member ages is positive. The latter effect is the change in demand for membership
11
due to a change in the member's utility from the social functions of the IOOF lodge. The sign of
the last term depends on whether (A) increases or decreases with A. Thus, the direction of
change in the reservation price for membership DR as a member ages depends on whether (A) is
increasing or decreasing in A. If it is decreasing in A, it matters whether the increase in DR due
to a rising rE or the decrease in DR due to a falling rS is the dominant effect. In other words, the
insight of the old man's model of demand for the sick benefit is unchanged. Once we weaken the
assumption that changes in the demand for the sick benefit are equivalent are equivalent to
changes in the demand for membership, the sick benefit related changes will be more difficult to
identify.
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