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Social Security & Insurance
Exploring Insurance for Social Security and Its Effect on Political Participation.
Mariah NapolitanoMarch 13, 2012
Prof Bryan Engelhardt
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Abstract: With the political climate consistently changing, the future of social security and other
government sponsored programs is always at risk. In order to reduce this risk, this paper
proposes introducing a new financial product to the market that would provide insurance for
social security benefits and suggests its affect on campaign contributions. This new market was
modeled and a solution was found where individual purchase full insurance and reduce their
political contributions aimed at continuing social security. To test the model a survey was given
to individuals who were asked about their interest in this new type of insurance and how it would
affect their political participation. The survey provided empirical evidence that a product like this
would be attractive to some individuals and gave important feedback on ways to improve the
product and the model.
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The ever-changing political climate in the United States creates uncertainty in the
existence and size of government-sponsored payouts, subsidies, and tax cuts. Whether it is social
security, Medicare, or the capital gains rate, the futures of these political risk items are constantly
in flux. Brought to the forefront of American society during the 2012 Presidential campaign due
to Republican Vice-President Nominee Paul Ryan’s plan to completely reform the system, the
future of social security is an issue Americans are genuinely and rightly concerned about.
According to the 2012 Social Security Trustees Report, “beneficiaries will face a painful 25
percent benefit cut in 2033 when the Trust Funds are exhausted – three years sooner than
projected just last year” (Paul Ryan website, 2012).
Many Americans who have paid into the social security system since they first started
working are at a risk of receiving a significantly smaller or nonexistent benefit than what they
had planned on for their retirement. Baring any drastic changes to the system this risk of
decreasing or eliminating social security seems like a real possibility, leaving many Americans
without sufficient funds for their retirement. But what if these individuals could guarantee their
social security benefit independent of governmental changes to the system? This paper proposes
creating a market for social security insurance, which would provide individuals the opportunity
to purchase an insurance policy now that would cover any decreases to the social security benefit
they receive when they retire. This type of insurance could also have an effect on the political
participation of the individuals who purchase it so it is important to investigate the changes, if
any, it might have on individuals’ voting or campaign contribution habits.
The paper is composed of two distinct parts; one in which a model is developed to depict
the social security market in a few variations and a second part where 30 individuals were
surveyed about their predictions for social security and their willingness to purchase social
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security insurance. Through this, I hope to show that insurance for social security (and other
types of political risk) is a feasible financial product that can benefit both American citizens and
insurance companies.
I. Background/Literature Review
Political risk insurance (PRI) is not a new concept, and is commonly used when
companies invest in developing nations. The Overseas Private Investment Corporation (OPIC) an
"independent" U.S. Government agency that sells investment services to assist U.S. companies
investing in developing countries, offers PRI to U.S. investors in over 150 developing nations.
The political risks typically insured are expropriation, political violence, and currency
inconvertibility (Garver, 2009). PRI policies cover “losses to tangible assets, investment value,
and earnings that result from political peril” (OPIC website, 2012). These policies allow U.S.
business to invest in developing nations while hedging the risks associated with unstable
governments.
Much has been written about how the PRI market is changing; books and papers detail
the effects of the September 11th attacks on emerging market investment and PRI (Riordan,
2004) and other discuss the movement away from government insurance policies to private
firms. (Salinger, 2008). But most relevant to the idea of insuring political risk items, like social
security, within the United States, is the discussion of what products are “insurable” and how
that definition changes slightly when considering items of political risk. Kathryn Gordon, of the
Organization for Economic Cooperation and Development (OECD), talks about these issues in
“Investment Guarantees and Political Risk Insurance: Institutions, Incentives, and
Development”. Insurable risk is defined as an item clients will want to buy which the insurance
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industry can profitability provided. Gordon lists the conditions necessary, according to the
OECD committee, to make a product insurable; “assessability (probability and severity of losses
should be quantifiable); randomness (the time at which the insured event occurs should be
unpredictable when the policy is underwritten, and the occurrence itself must be independent of
the will of the insured); mutuality (numerous persons exposed to a given hazard should be able to
join together to form a risk community within which the risk is shared and diversified)” (Gordan,
2008).
When discussing political risk items, the definition of insurable can diverge from these
conditions. Political risk items may not have the greatest degree of assessability; calculating the
probabilities of events happening can be challenging and there can be debate over whether an
event has actually occurred. One deviation from the standard conditions, which is not as
applicable to a product like social security insurance, is that many times there are no “risk
communities” because each policy depends on the specifics of the investor, the country of
investment and the product offered (Gordan, 2008). Within the market for social security
insurance the country and the product being offered will be the same for every investment so risk
communities can be formed. In reality social security insurance will create one large risk
community because it is likely that any change in social security policy will affect everyone who
buys the insurance (and more broadly anyone who receives social security). This does create the
issue that given a policy change, the insurance company will have to payout to all of its
customers. In this way, social security insurance is similar on a small scale to flood insurance;
when a flood occurs, it is not typically one building or house that gets flooded but rather an
entire area of buildings, resulting in insurance companies having to payout to large amounts of
customers at a time. One suggestion Erwann O. Michel-Kerjan details in his paper, Catastrophe
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Economics: The National Flood Insurance Program, for the National Flood Insurance Program
(NFIP) to stay financial sustainable in the event of catastrophe (like Hurricane Katrina) is to
“transfer part of its catastrophe exposure to reinsurers or to investors in the financial markets by
using alternative risk transfer instruments like a catastrophe bond (“cat bond”), a form of
contingent claim” (Michel-Kerjan, 182). A similar plan could be used by insurance companies to
relieve the pressure on the companies should social security get cut in the future. The key for
both the NFIP and any company that provides insurance for social security is to hedge their risk
whether be through reinsurers or finding an investment that is related to the item being insured.
This can be through shorting an investment directly related to the insurance item or investing in
an item inversely related to the item being insured.
One deviation on the definition of insurable that is important for the political risk
associated with social security is that it is not “independent of the will of the insured.”
Individuals have the opportunity to affect the outcome of social security through political
participation, such as voting and campaign contributions. Because of this reason, this paper will
focus on the effect campaign contributions might have on a product like social security
insurance.
Much of the literature on campaign contributes says approximately the same thing,
campaign contributions lead to policies that differ from those preferred by the median voter’s
favored policies (Grossman and Helpman 1996, Prat 2002, Coate 2004, Campante, 2011). In
particular, Grossman and Helpman (1996) showed in their model that a candidate is more likely
to win an election if they cater to special interest groups that are more likely to donate money to
their campaigns. These campaign contributions help candidates get elected because “risk-adverse
voters prefer candidates with a clearer policy position” (Pratt, 2002). Because of this a banning
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or limiting of campaign contributions moves policies closer to the preferred policies of the
median voters (Pratt, 2002).
Campante (2011) also showed that in combination with voting, contributions lead to a
“wealth bias” in political policies. When wealthy individuals donate money to political
campaigns, the politicians that receive the donations are encouraged to move their policies closer
to those of the wealthy. So the more financial inequality that exists, the more the political system
shifts in favor of the wealthy.
II. Model
Price Discriminating Monopolist with Constant Probabilities
Assumptions
The model assumes five different possible outcomes for social security in the future; it
doesn’t exist at all, it exist with benefits that are 25% of current benefits, with benefits that are
50% of current benefits, with benefits that are 75% of current benefits, and it exists with the
same amount of benefits it provides today.
Each individual also has a utility function u(y) that measures the satisfaction received by
the individual if event y i occurs. This model assumes that individuals are risk-adverse requiring a
concave utility function. Both the firm and the individual believe with probability q i that event i
will happen. Additionally (q0+q25+q50+q75+q100) =1.
The first situation which the model has been dissected for is when the market for social
security insurance is controlled by a price-discriminating monopolist. In this scenario, the
existence of an insurance market allows the individuals to decide on the amount of insurance
they would like to purchase at the price set by an insurance company who has a monopoly on the
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market. As this market for social security insurance does not currently exist, it is reasonable to
assume that when the market is first created it will be dominated by one firm who will have the
power to price discriminate based on customers’ financials and beliefs about the future.
Model
This scenario calls for two models, one for the individual purchasing (or not purchasing)
the insurance and one for the firm who is providing the insurance. The price of that insurance is
represented by p. While the amounts of insurance in the individual’s model and the amounts in
the insurer’s model are different (one set is quantity demanded and the other set quantity
supplied) when the market is in equilibrium they will be equal so the model only uses one set of
variables to represent both quantities. The ‘quantity’ insurance purchased will be represented by
five different x values; x0 if social security does not exist at all when the individual retires, x25 if
social security exists at 25% of its current level when the individual retires, and so forth all the
way to x100 if social security exists at today’s levels when the individual retires.
Similarly the individual’s retirement portfolio will be represented by 5 variables; y0 if
social security does not exist at all when the individual retires, y25 if social security exists at 25%
of its current level when the individual retires, and so forth all the way to y100 if social security
exists at today’s levels when the individual retires. Additionally, five different probabilities are
necessary, the probability social security will not exist,q0, the probability social security will
exist at 25% of current levels,q25, and so forth until the probability social security will exist at
current levels, q100.
ln ( y ) is used as the utility function (which satisfies the conditions for the utility function
previously mentioned), we must assume that y i∧( y i+xi−p)>0 so the consumer’s decision
function will exist.
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The firm is in control of the market and will look to maximize its profits, π, by
optimizing all of the insurance amounts x iand the price of that insurance such that the
consumer’s decision function is satisfied.
π=p−(q0∗x0+q25∗x25 +q50∗x50+q75∗x75+q100∗x100) such t h at (1)
q0∗ln ( y0 )+q25∗ln ( y25 )+q50∗ln ( y50 )+q75∗ln ( y75 )+q100∗ln ( y100) ≤ q0∗ln ( y0+x0−p )+q25∗ln ( y25+x25−p )+q50∗ln ( y50+x50−p )+q75∗ln ( y75+x75−p )+q100∗ln ( y100+x100− p ) (2)
The firm’s profit function is equal to the price of the insurance, p, minus the expected cost of the
insurance. The consumer will purchase the insurance if their expected value of not purchasing
the insurance is less than or equal to their expected value of buying insurance. When the
expected values are equal the consumer will be indifferent between purchasing and not
purchasing the insurance.
In the firms profit equation the expected cost of the insurance is equal to
(q0∗x0+q25∗x25+q50∗x50+q75∗x75+q100∗x100 ). The consumer’s expected utility of not having
insurance is the probability of having a certain retirement package multiplied by the value of that
retirement package which can be described as
∑i=0,25,50,75,100
qi∗ln ( y i )
(3)
Similarly the consumer’s expected utility of having insurance is the probability of having a
certain retirement package multiplied by the value of that package plus the insurance payout the
individual would receive minus the price of that insurance. It can be described as
∑i=0,25,50,75,100
qi∗ln ( y i+x i−p)
(4)
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These conditions lead to a maximization problem with one constraint setting up a Lagrange
multiplier optimization problem,
L=p−(q0∗x0+q25∗x25+q50∗x50+q75∗x75+q100∗x100)−λ∗[q0∗ln ( y0 )+q25∗ln ( y25 )+q50∗ln ( y50 )+q75∗ln ( y75 )+q100∗ln ( y100)−q0∗ln ( y0+x0−p )−q25∗ln ( y25+x25−p )−q50∗ln ( y50+x50−p )−q75∗ln ( y75+x75−p )−q100∗ln ( y100+x100−p )] (5)
This problem has been evaluated and only one solution was found.
The first order conditions of this constrained optimization with respect to x i,
∂ L∂ x i
=−q i−λ ( −qi
y i+x i−p )=0
(6)
can be used to show that the maximizing solution to this problem is full insurance (proof in
Appendix A1.1), which is shown as x i= y100− yi, meaning the insurance payout if event i occurs
is equal to the difference of what the individual would have with today’s level of social security
and what they would have if event i occurred. This result can be used to calculate the price of the
insurance, resulting in
p= y100−( ( y0 )q 0) ( ( y25 )q25 ) ( ( y50 )q50 ) ( ( y75)q75 ) (( y100 )q100 ) (7)
(Proof in Appendix A1.2)
As the probability of the elimination of some or all of social security increases the price of the
insurance also increases which can be understood because it is more likely the firm is going to
have to pay out the insurance plan. The price of the insurance ranges from $0 (if the probability
of social security existing, q, equals 1) to y100− y0 (if the probability of social security not
existing equals,q0, equals 1).
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Actuarially Fair Market with Constant ‘q’s
Once this product has been on the market for a while it will no longer be sold by one
monopoly firm, instead a group of firms will join the market and drive the price down to its
market equilibrium. Using all the same assumptions as with the price discriminating monopolist
except now the firm is only maximizing profit with respect to x i, the solution is still full
insurance, except now the price is equal to the expected value of the insurance,
p=( q0∗x0+q25∗x25+q50∗x50+q75∗x75+q100∗x100) (8)
Price Discriminating Monopolist with q as a Function of Political Participation
In both of the prior cases, the probability that different levels of social security were
going to exist, q, was assumed to be constant but in reality those probabilities are changing based
on political participation. To incorporate that in the model, let q(c) be the probability that social
security will exists with the same benefit levels, where c is the amount of campaign
contributions, c>0 , q (c ) ϵ [0,1 ] , ∂q∂ c
>0 , ∂2 q∂ c2 <0 ,and q(c) has an inverse. By letting q(c) equal the
probability of social security existing in its current form, 1-q(c) has to cover the four possible
outcomes. Assuming people have a hard time predicting the difference between having social
security survive at 75%, 50%, 25%, and 0% of its current value, we’ll assume the probability of
each of those outcomes is 1−q (c)4
. Let c1be the contribution if the agent does not buy insurance
and c0be the contribution if they do purchase the insurance.
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Given the same assumptions as the previous price discriminating monopolist with the
except of the probabilities, the firm will look to maximize its profit in terms of x i , p, & c0 with
respect to the consumers decision function,
π=p−( 1−q(c0)4 ( x0+x25+x50+x75 )+q(c0)∗x100)suc h t hat (9)
( 1−q (c1)4 )[ ln ( y0−c1 )+ ln ( y25−c1 )+ ln ( y50−c1 )+ln ( y75−c1 ) ]+q(c1)∗ln ( y100−c1 ) ≤( 1−q ( c0 )
4 ) [ln ( y0+x0−p−c0 )+ln ( y25+x25−p−c0 )+ ln ( y50+x50−p−c0 )+¿ ln ( y75+x75−p−c0 ) ]+q(c0)∗ln ( y100+x100−p−c0 )
(10)
The firm’s profit equation is equal to the price of the insurance minus the expected cost
of the insurance, which is a function of c0, the amount of campaign contributions the individual
will give if they buy the insurance. The individual’s decision function is once again a comparison
of the expected utility of buying the insurance and the expected utility of not buying the
insurance. The expected utility of buying the insurance is equal to the probability of each
outcome occurring, which is a function of c1 the amount of campaign contributions the
individual donates if they don’t buy the insurance, multiplied by the utility of each outcome
occurring which equals the utility of the individual’s retirement package minus the amount of
campaign contributions. Similarly to the expected utility of buying the insurance, the expected
utility of not buying the insurance is equal to the probability of each outcome occurring as a
function of c0, the amount of campaign contributions if the individual buys the insurance,
multiplies by the utility of each outcome occurring which is equal to the retirement package of
the individual plus the insurance payout minus the price of the insurance and the campaign
contribution.
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This sets up a similar optimization problem except it is now being optimized with respect
to x i , p, & c0. The solution to the optimization is once again full insurance, x i= y100− yi, (proof
in Appendix A2.1). The other variables that maximize the insurance are,
p= y100−c0−(( y 0−c0 ) ( y25−c0 ) ( y50−c0 ) ( y75−c0 ) )1−q(c0)
4 (( y100−c0 )q(c0))
(12)
(Proof in Appendix A2.2)
c0=( ∂ q∂ c )
−1
( 4x0+x25+ x50+x75 )
(13)
(Proof in Appendix A2.3)
The optimal campaign contribution is equal to the inverse derivative of the sum of the insurance
payouts (at the optimal solution,x100=0) divided 4. If we allow q ( c )=1+c2+c , which satisfies the
above conditions, the optimal campaign contributions equal
c0=12∗√ x0+ x25+x50+ x75−2
(14)
(Proof in Appendix A2.4)
To compare this optimal amount of campaign contribution to the pre-insurance amount of
campaign contributions, the optimal campaign contribution before the insurance is offer needs to
be determined. This amount can be determined by maximizing the expected value of social
security without insurance with respect to campaign contributions,c1 . This amount is equal to,
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c1=( ∂ q∂ c )
−1( 4∗[( 1−q (c1 )4 )∗( −1
y0−c1+ −1
y25−c1+ −1
y50−c1+ −1
y75−c1 )−q (c1 )
y100−c1 ]ln( ( ( y0−c1 ) ( y25−c1 ) ( y50−c1 ) ( y75−c1 ))
( y100−c1 )4 ) ) (15)
(Proof in Appendix A2.5)
In order to compare the optimal campaign contributions before and after insurance it is
sufficient to compare the first derivatives of the probabilities with respect to campaign
contributions, ∂ q∂ c . Since, by assumption, ∂
2 q∂ c2 is always less than 0,
∂ q∂ c is a decreasing function.
This attribute implies an inverse relationship between the how ∂ q∂ c0
and ∂ q∂ c1
are related and how
c0 and c1 are related. From this relationship it can be shown that conditional on ∂ q∂ c
<1, ∂ q∂ c0
> ∂ q∂ c1
therefore c0<c1 (proof in Appendix A2.6). So as long as ∂ q∂ c
<1, optimal campaign contribution
before the insurance is less than the optimal campaign contribution after the insurance.
If we consider the same probability function as earlier, q ( c )=1+c2+c ,
∂ q∂ c
= 1(2+c1)
2 <1
(16)
(Proof in Appendix A2.7)
This satisfies the condition stated above so with this specific probability function, the campaign
contribution before the insurance is more than the campaign contribution after the insurance.
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III. Survey
Introduction to the Survey
While the models predict that if the market for social security insurance existed there is a
price and amount of insurance that individuals and firms could agree upon, this market currently
does not exist. To attempt to determine if this product is applicable in a real world setting 30
individuals were surveyed about their thoughts on the future of social security, the idea of social
security insurance, and social security’s affect on their political decisions like voting and
contributing to political campaigns. The results could also help determine why this market does
not exist today.
Explaining the Survey
To allow for some variation in what social security might look like when the responders
retired, the model that allowed five possible social security outcomes was used. The responders
were asked a series of questions to establish their thoughts on the future of social security which
was used to determine each q value. They were asked about the likelihood when they retire that
social security would exist in its current form and with at least 75%, 50%, and 25% of its current
benefits. Each question provided a list of answers from which the responder chose; Very High
(80%-100%), High (60%-79%), Low (40%-59%), Very Low (20%-39%), or No Chance (0%-
19%). Once a responder answered (a) Very High (80%-100%) no more probability distribution
questions were asked because the resulting probabilities would have been 0%.
The survey started with basic demographic questions to determine age and gender, both
of which factored into the price of the insurance being offered to the responder. Two additional
questions were necessary to calculate the price of the insurance. One was a question about the
responder’s expected reliance upon social security when they retire was used to estimate the
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responder’s retirement portfolio and the other was about the responder’s expected age of
retirement.
After all the necessary information was inputted, the price of the insurance in a price-
discriminating monopoly market was calculated and the idea of social security insurance was
introduced to the responder with the following,
“As of today the current average Social Security benefit is about $1,200 per
month which accumulates to more than $250,000 over the course of retirement.
There is some concern that the benefits granted by social security will be cut
sometime in the future. So what I want to do is to offer you the opportunity to
guarantee your roughly $250,000 by purchasing insurance for your social
security. Similar to home or car insurance, this investment will cover your future
benefits should current social security laws change. This policy would payout the
difference between how much social security you would receive under today’s
law and your actual benefit at the time of your retirement. For example if your
monthly benefit was reduced by $600 your insurance payout would total over
$125,000. Keeping all that in mind would you be willing to spend $XXX per
month to ensure your full social security benefit?”
If the responder answered “No” the same deal was offer (full insurance) but this time at the
perfectly- competitive market price (a lower price). If the responder once again answered “No”
they were asked if there was any price at which they would be willing to purchase the insurance,
and if so, what price.
The next set of questions the responders were asked to answer were about reasons they
would not want to buy the insurance. They were asked, “Is there any reason why you wouldn’t
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want to buy this type of insurance?” Depending on their answer to that question, they were next
asked, if whatever reason they gave in the previous question (ie “I don’t have enough money
now” or “I don’t see social security as an important part of my retirement plan” etc.) did not
exist/happen/matter etc. would they purchase the insurance.
As one of the goals of the survey was to determine if social security insurance would
have an effect on people’s political decisions (both in voting and contributing to political
campaigns), before introducing the idea of social security insurance the responders were asked if
the regularly vote or contributed to political campaign and if they hoped their vote or
contribution would affect social security policy either positively or negatively. After the
insurance was offered, responders were then asked if buying the insurance would affect their
voting and/or contributing decisions because now decreases in social security benefits would not
affect their future benefits.
The full survey is in Appendix B.
Assumptions/Calculating Price
In order to calculate the price of the insurance a few assumptions in our model had to be
made. The first assumption was that males and females would live the national average life
spans, for women, 81 and for men, 76. An interest rate of 5% was assumed. When calculating the
size of the individual’s retirement package not including social security average savings of those
65 and older, average 401(k) balance of those 55 and older, and the average private sector
pension were considered. The size of the individual’s portfolio was dependant on their answer to
whether or not they thought they would be reliant on their social security. Our final assumption
was that everyone was going to receive the current average social security benefit, which is
$1,230 per month.
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The only slight alteration to the five outcome price-discriminating monopolist problem
set up in the Model section is that instead of have y i as the total size of the individual’s
retirement package in the event i we will have, y i=r+SS i, where r equals the individual’s
retirement portfolio without social security and SSi represents the amount of social security the
individual receives if event i occurs. This results in the price of the insurance being,
p=SS100+r – ( (r+SS0 )q0 ) (( r+SS25 )q25 ) (( r+SS50 )q50 ) (( r+SS75 )q75 ) (( r+SS100)q100)
The survey also asked individuals to consider buying the insurance at the actuarially fair
price.
Because the insurance (both at the monopoly price and the actuarially fair price) is being
paid for and distributed out over an extended period of time, it was necessary to take interest
rates and future/present values of the insurance into consideration. To do so we first solved for
the price of insurance per year in year of retirement dollars which was used to calculate the total
price of the insurance in year of retirement dollars. This amount was then put in today’s dollars
and distributed equally (like an annuity) through the years the individual has left until retirement.
From this price the monthly price of the insurance, which the individual will pay every month
until they retire, is calculated.
Results
The results of this survey proved to be promising for the possibility of the real word
application of social security insurance as 43% of those survey said they would purchase this
insurance if it were offered at the price-discriminating monopoly price. Another 65% of those
who said no to the monopoly price said they would buy the insurance if it were offered at a lower
price which brings the total percentage of those who said yes to some price to 80% of those
surveyed. The average monopoly price of the insurance was $91.91 but more descriptive is the
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difference in price between those who said that would buy the insurance, $46.29 and those who
said they would not at the monopoly price, $126.80. Of those who said that would not buy the
insurance at the price offer but gave a price at which they would buy the insurance, the average
price given was $51.36 per month. Below are the main descriptive statistics of some of the
important variables discussed in the survey,
Table 1: Descriptive Statistics of Important Variables
Mean Median Standard Deviation
Minimum Maximum
Age 47.1 47. 5. 36 54Retirement Age 67.17 67 2.44 62 70Monopoly Monthly Price $91.91 $61.62 $86.83 $14.93 $430.26Expected Value(Social Security) $10,774 $11,070 $2,516 $4,428 $14,022
In terms of the age statistics the one that appeared to have the most effect on whether or
not the individual chose to purchase the insurance was the difference between ‘years until
retirement’ and ‘years of retirement.’ The average difference was 9.73 years but the average
difference of those who purchased the insurance was 12 years while the average of those who
chose not to purchase at the monopoly price was only 8 years. Of those whose difference was
above the average only 26.67% said they would purchase the insurance while 60% of those
whose difference was below the average would buy the insurance at the monopoly price. This
result is not unexpected because the longer you are retired and the fewer years you have until
retirement the higher your price because you are paying for more years of insurance in less time.
In the survey women and men behaved a bit differently. The percent of women who said
they would purchase the insurance at the monopoly price was 33.33% while the percent of men
willing to purchase the insurance was 50%. However, the average price of a women’s insurance
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policy, $103.52, was higher than the average price of a man’s policy, $84.71. The obvious driver
of this difference in price is the average life spans used to calculate the years of retirement. Since
women live on average five years later than men, it was not unexpected that the average number
of years retired for women was 12.67 years while the men’s average number was only 8.78.
Another important factor in determining the price of the insurance was the individual’s
probability distribution of the likelihood social security will exist in its varied forms. Most
responders believed that most likely when they retired social security would be at its current
levels however there was a distribution amongst all five possible outcomes,
Table 2: Probability Distribution
Probability you get at least i% of Social Security Very High
80%-100%High
60%-79%Medium40-59%
Low20-39%
Very Low0-19%
100% 0% 20% 43.33% 30% 6.67%75% 16.67% 26.67% 40% 13.33% 3.33%50% 36.67% 33.33% 23.33% 6.67% 0%25% 63.33% 20% 13.33% 3.33% 0%
It can be noted the distribution of the probabilities resembles the probability distribution function
used when campaign contributions were considered where the probability of social security
existing at 100% was one number and the remaining probabilities were equally distributed
amongst the other four outcomes.
Those who believe that the probability social security would exist at its current level was
above the average (55%) had an average monopoly price of $72.45 while those who believed the
probability was below the average had an average price of $125.54. This difference in price can
be explained because as the probability of social security decreasing increases the likely hood
that the firm is going to have to payout the actual insurance benefit increases so they are going to
charge more because their expected cost increases. Additionally as the probability of social
Napolitano 21
security deceasing increases, the expected value of the utility of the insurance for the individual
will increase making the insurance more valuable to them and more likely to purchase the
insurance at a higher price. This is shown by the average cost of the insurance of those whose
q100was less than the average and who said they would purchase the insurance, $67.02, being
greater than the average cost of the insurance of those whose q100was greater than the average
and who said they would purchase the insurance, $37.08. Though for many people who were less
optimistic about the future of social security the increase in price resulting from their beliefs
about the future was too high resulting in only 36% of those whose q100was less than the average
agreeing to purchase the insurance, compared to the 47% of those whose q100was greater than the
average agreeing to purchase the insurance.
When asked about their projected reliance on social security, only 27% of those surveyed
thought that they would be reliant upon social security when they retired. Of those who projected
their reliance upon social security 62.5% agreed to purchase the insurance compared to those
who said they would not be reliant upon social security, where only 36% agreed to purchase the
insurance. While reliance upon social security clearly showed to be a factor in whether or not to
purchase insurance it really did not have much of an impact on the price of the insurance. For
example the monopoly price of insurance for a averaged aged (47) male with an average
probability distribution who plans on retiring at the average retirement age of 67 who does not
believe he will be reliant upon social security is $70.96 and a male with all the same
characteristics except who plans on being reliant upon social security has a price of $74.46, a
difference in price of less than $4 per month, which only accumulates to $600 in today’s dollars
of extra cost.
Napolitano 22
Table 3: Correlation coefficients related to buying the insurance at the monopoly price (where
buying the insurance is denoted with a 1)
Variable Correlation CoefficientGender (Male = 1) 0.1647705Age 0.0232346Probability SS exists at 100% 0.1620708Probability SS exists at 75% 0.1070184Probability SS exists at 50% 0.1647705Probability SS exists at 25% -0.3574661Probability SS exists at 0% -0.1089328Reliant on SS (Reliant = 1) 0.2332413Retirement Age 0.4445927Monopoly Price -0.4673082Expected Value of Insurance -0.4995743Expected Value of Social Security 0.2046543Years Until Retirement 0.1841197Years Retired -0.3291058Difference in yrs retired & years until retirement 0.3039435
Table 4: Average prices of those who agreed to the monopoly price, who said no to the
monopoly price, and the average independent of their monopoly price response,
Avg Price Of Yes – Mon. Avg Price Of No – Mon. Avg Price – Mon.Price $39.48 $130.00 $92.21Men $52.98 $115.36 $84.71Women $31.24 $139.67 $103.52Reliant $38.24 $185.03 $93.28Not Reliant $51.32 $114.32 $91.41
Table 5: Averages of those who agreed to the monopoly price, who said no to the monopoly
price, and the average independent of their monopoly price response,
Yes to Mon. Avg No to Mon. Avg AvgAge 47.23 47.00 47.10Retirement Age 68.38 66.20 67.17Age difference 12.00 8.00 9.25Prob of 100% SS 0.58 0.53 0.55
Napolitano 23
All survey participants were also asked what reasons they had to not purchase the
insurance. Their responses can be divided into four different categories; financial, government,
information/logistical, and moral objections. The most common responses were financial in
nature with four major issues were brought up multiple times. Many people thought the price
was simply too high and the insurance was not worth the price. Another common response was
that this insurance was not in their budget now so they could not afford it. Individual’s also
suggested that there were better investment opportunities with higher returns for the same price
as the insurance, so they felt their money could be used in a better way. Finally, while it was
asked in the survey, responders replied that they were not relying on their social security to be
there so they did not see the need to insure its presence.
Another group of reasons for not purchasing the insurance were attributed to the current
state of government affairs. It was suggested that the risk of social security failing/reducing were
not high enough to warrant purchasing the insurance (even though this was calculated into the
price). One individual stated that “Politicians are so slow moving that more than likely social
security is not going to change before I retire.” The second concern over the current state of
government affairs was that the current economic state of affairs is so precarious that “today my
answer maybe yes, but if we fall off the fiscal cliff tomorrow I might change my mind.”
A small but substantial number of responders rejected the insurance because they felt
they did not have enough information to ease their possible concerns about purchasing the
insurance. Their main concern was making sure the firm that would be providing this insurance
was reliable and sustainable over the long term. They did not want to take on the risk of
purchasing this insurance from a company they were not sure would be around when they
retired. The other main information objection was about the question of contingencies. For
Napolitano 24
example, what would happen if they died young? Would they be able to write the policy over to
a spouse or a dependant?
The last set of objections could be grouped together as moral objections to the insurance.
A few individuals thought that because of the price of the insurance not everyone would be able
to buy it so they felt it was unfair to people who could not afford it. They thought that if many
people bought the insurance the government would not have a large incentive to keep social
security at its current level so those who need social security the most would be severely
negatively impacted. The other moral objection of the insurance was that people believed they
were entitled to their social security and that the government has the responsibility to give them
back at least what they paid into the system.
These moral objections to the insurance can be used to help understand the unexpected
results of the survey with regards to how purchasing social security insurance might affect their
political decisions. Before the insurance was mention, 93% of those surveyed claimed to vote
regularly and 46% of them stated that keeping social security at its current levels was an issue
that considered when deciding who to vote for. No responders answered that they were more
likely to vote for someone who supports reducing social security. 47% of responders said that
they regularly contributed to political campaigns and 50% of that group said that keeping social
security at its current level was an issue they considered when deciding to contribute. Once again
no individuals said they were more likely to donate to a political campaign of someone who
supported reducing social security.
After “offering” the individuals the opportunity to buy social security insurance, they
were asked after buying the insurance about how social security would affect their political
decisions. Because these questions were contingent upon the individual’s response to the
Napolitano 25
previous questions about voting and if they would buy the insurance, when looking for the effect
the insurance has upon these political decisions it is important to only look at individuals who
were able provided answers to all the questions concerned. 77% of those surveyed were able to
answer both the before and after insurance question on how social security effected their voting
habits. Of that group, before the insurance was taken into account, 52% of them claimed to be
more likely to vote for a candidate who supports continuing social security at current levels and
the other 48% said social security did not affect their voting decisions. When the responders
were asked to think about their voting choices after buying the insurance, 48% said they would
be likely to support a candidate who is in favor of keeping social security at its current levels, 4%
said they would support a candidate who wants to reduce social security and the last 48% said it
would have no effect on their voting. Only 17% of responders changed their answer from the
before insurance question. 25% of those who changed went from social security not affecting
voting to supporting someone who wants to reduce social security, and 50% went from
supporting continuing it at current levels to not having it affect their votes. The last 25% of
voters who switched went from social security not affecting their vote before the insurance to
now supporting someone who wants to keep it at current levels.
A difference in proportions test was also used to determine if the difference in responses
before and after the insurance was offered was statistically significant. Before the insurance was
offered 46% of 28 individuals said keeping social security at its current levels is a significant
factor when deciding who to vote for and after the insurance was offered 48% of 23 individuals
said responded the same way. Using a two-tailed hypothesis test, where the null hypothesis is
that the results from both cases are the same and the alternate hypothesis is that they are
different, a z-score of -0.1428 can be calculated based upon this, resulting in a p-value of 0.88.
Napolitano 26
Using a significance level of 0.1, the null hypothesis can be accepted because 0.1 < 0.88,
meaning the results before and after the insurance is offered are statistically the same.
Only 43% of those surveyed were able to answer both the before and after insurance
questions on how social security affected their campaign contributions. Of those individuals,
before the insurance 54% supported keeping social security at current levels and the other half
said it did not affect their contributions. No individuals changed their answers when asked about
their contributing after buying the insurance.
IV. Conclusion
While the survey results did not completely confirm the results of the model, it did
provide some empirical evidence that social security insurance is a financial product some
individuals would be willing to purchase at the price they would be charged under a price-
discriminating monopolist or at some other price. Based on the results of the survey, where price
was the highest indicator of whether or not an individual said they would purchase the insurance,
the product should be targeted at those who would have a lower price per month, which can be
significantly affected by the amount of years they have until retirement and the amount of years
they plan on being retired as well as their thoughts on the future of social security. The insurance
policy would be the most attractive to individuals who fall in the middle ground of the expected
future of social security. Those who strongly believe social security will not exist have
increasing higher prices for the insurance resulting in many individuals responding that the price
was simply too high and those who strongly believe social security will exist in its current form
refuse to buy the insurance even when the price is significantly low because they do not see the
risk of a social security decrease as high enough to warrant an insurance policy.
Napolitano 27
In terms of political participation, the results of the survey indicated that this type of
insurance would not affect their voting and campaign contributing habits in the way the model
predicted, which was that they would donate less after they had purchased the insurance than
they had before. Most responders did not predict the insurance to affect their political
participation at all. Based on the responses to why individuals would not purchase the insurance
it is possible that even if they had the insurance, individuals would still want the social security
program to continue for the sake of those who are unable to or did not purchase the insurance, so
they would not change their voting or contributing habits.
While the results of this analysis of a social security market showed this insurance as a
feasible product, it did bring up some interesting objections and questions as well. Seeing as this
product was offered at an equilibrium price and individuals still rejected the proposal, what
changes can be made in the model to account for individuals’ concerns about the product,
particularly about what effect their buying of the product will have on the future of social
security as well as those who don’t buy it? This concern is legitimate because if more and more
individuals purchase the insurance it is not unreasonable to assume that the government will
become less concerned with the guarantee of social security and make cuts to the system. While
these cuts will not affect those who have the insurance, it will affect both those who did not
purchase the insurance and the insurance companies that have to payout to all their customers.
This issue proposes an extension of the model to include the number of individuals who buy the
insurance. The number (or percentage) of people purchasing the insurance will individually
affect both the probability of social security existing and the profit of the insurance company.
The results of both the analysis of the model and the survey indicate that marketing
insurance for social security would create the opportunity for a multi-million dollar market to
Napolitano 28
develop. It is a desirable financial product individuals would be willing to invest in to protect
against the uncertainty of a government backed subsidy. It would also open the door to
expanding the insurance market for other items associated with political risk which creates an
even bigger opportunity for millions of dollars of transactions.
Napolitano 29
Appendix A
A1: Price Discriminating Monopolist with constant probabilities
L=p−(q0∗x0+q25∗x25+q50∗x50+q75∗x75+q100∗x100)−λ∗[q0∗ln ( y0 )+q25∗ln ( y25 )+q50∗ln ( y50 )+q75∗ln ( y75 )+q100∗ln ( y100)−q0∗ln ( y0+x0−p )−q25∗ln ( y25+x25−p )−q50∗ln ( y50+x50−p )−q75∗ln ( y75+x75−p )−q100∗ln ( y100+x100−p )]∂ L∂ x0
=−q0− λ( −q0
y0+x0−p )∂ L
∂ x25=−q25−λ( −q25
y25+x25−p )∂ L
∂ x50=−q50−λ( −q50
y50+x50−p )∂ L
∂ x75=−q75−λ( −q75
y75+x75−p )∂ L
∂ x100=−q100− λ( −q100
y100+x100−p )∂ L∂ p
=1−λ( q0
y0+x0−p+
q25
y25+x25−p+
q50
y50+x50−p+
q75
y75+x75−p+
q100
y100+x100−p )∂ L∂ λ
=−(q0∗ln ( y 0)+q25∗ln ( y25 )+q50∗ln ( y50 )+q75∗ln ( y75 )+q100∗ln ( y100)−q0∗ln ( y0+x0−p )−q25∗ln ( y25+x25−p )−q50∗ln ( y50+x50−p )−q75∗ln ( y75+x75−p )−q100∗ln ( y100+x100−p ))
A1.1Solving forx i
0=−q0−λ ( −q0
y 0+x0−p )1=( λ
y0+ x0−p )λ= y0+x0−p
By same process,λ= y25+ x25−pλ= y50+ x50−pλ= y75+ x75−p
λ= y100+x100−pSet two of lambdas equal to each other,
y0+x0−p= y100+x100−px100 equals 0 because if Social Security exists in its current form (i = 100) the insurance payout is 0.
y0+x0= y100
Napolitano 30
x0= y100− y0
Repeat same process for all x i,
x25= y100− y25
x50= y100− y50
x75= y100− y75
Solution is full insurance.
A1.2Solving for price∂ L∂ λ
=−(q0∗ln ( y 0)+q25∗ln ( y25 )+q50∗ln ( y50 )+q75∗ln ( y75 )+q100∗ln ( y100)−q0∗ln ( y0+x0−p )−q25∗ln ( y25+x25−p )−q50∗ln ( y50+x50−p )−q75∗ln ( y75+x75−p )−q100∗ln ( y100+x100−p ))=0
Plug in for x i q0∗ln ( y0 )+q25∗ln ( y25 )+q50∗ln ( y50 )+q75∗ln ( y75 )+q100∗ln ( y100)=q0∗ln ( y100−p )+q25∗ln ( y100−p )+q50∗ln ( y100−p )+q75∗ln ( y100−p )+q100∗ln ( y100−p )
ln ( y¿¿100−p)(q0+q25+q50+q75+q100)=ln ( ( y0 )q0 )+ln ( ( y25 )q25 )+ ln ( ( y50 )q50 )+ ln (( y75)q75)+ln ( ( y100 )q100) ¿Since probabilities sum to 1
ln ( y¿¿100−p)(1)=ln (( ( y0 )q0 ) ( ( y25 )q25 ) ( ( y50)q50 ) (( y75 )q 75) ( ( y100 )q100 ))¿( y¿¿100−p)= (( ( y0 )q0 ) (( y25 )q 25) ( ( y50 )q50 ) ( ( y75 )q75 ) (( y100)q100 ))¿
p= y100−( ( y0 )q 0) ( ( y25 )q25 ) ( ( y50 )q50 ) ( ( y75)q75 ) (( y100 )q100 )
A2: Price Discriminating Monopolist with Moving Probabilities
L=p−( 1−q(c0)4 ( x0+x25+x50+x75)+q(c0)∗x100)−λ[( 1−q(c1)
4 ) [ln ( y0−c1 )+ ln ( y25−c1 )+ln ( y50−c1 )+ ln ( y75−c1) ]+q (c1 )∗ln ( y100−c1 )−(1−q ( c0 )4 )[ ln ( y0+x0−p−c0 )+ ln ( y25+ x25−p−c0 )+ ln ( y50+x50−p−c0)+¿ ln ( y75+x75−p−c0 ) ]−q(c0)∗ln ( y100+x100−p−c0 ) ]
∂ L∂ x0
=−1−q(c0)
4−λ( −1−q(c0)
4y0+x0−p−c0
)∂ L
∂ x25=
−1−q (c0)4
− λ( −1−q (c0)4
y25+ x25−p−c0)
∂ L∂ x50
=−1−q (c0)
4− λ( −1−q (c0)
4y50+ x50−p−c0
)∂ L∂ x75
=−1−q (c0)
4− λ( −1−q (c0)
4y75+x75−p−c0
)
Napolitano 31
∂ L∂ x100
=−q (c0)−λ( −q (c0)y100+ x100− p−c0 )
∂ L∂ p
=1−λ( 1−q(c0)4
y0+x0−p−c0+
1−q (c0)4
y25+x25−p−c0+
1−q(c0)4
y50+x50−p−c0+
1−q(c0)4
y75+x75−p−c0+
q (c0)y100+x100−p−c0
)∂ L∂ c0
=q ' (c0)
4 ∗( x0+x25+x50+x75 )−q ' ( c0 )∗x100−λ[ q' (c0)4 ∗( ln ( y0+x0−p−c0 )+ ln ( y25+x25−p−c0 )+ln ( y50+x50−p−c0 )+¿ ln ( y75+x75−p−c0 ))−( 1−q ( c0 )
4 )∗( −1y0+x0−p−c0
+−1
y25+x25−p−c0+
−1y50+x50−p−c0
+−1
y75+ x75−p−c0 )−q' (c0 )∗ln ( y100+x100−p−c0 )−q( c0 )∗−1
y100+x100−p−c0 ]∂ L∂ λ
=−[( 1−q(c1)4 )[ ln ( y0−c1 )+ ln ( y25−c1 )+ ln ( y50−c1 )+ ln ( y75−c1 ) ]+q (c1 )∗ln ( y100−c1 )−(1−q (c0 )
4 )[ ln ( y0+ x0−p−c0 )+ ln ( y25+x25−p−c0 )+ln ( y50+x50−p−c0 )+¿ ln ( y75+x75−p−c0 ) ]−q(c0)∗ln ( y100+x100−p−c0 )]A2.1Solving forx i
0=−1−q (c0)
4−λ ( −1−q (c0)
4y0+x0−p−c0
)1=( λ
y0+ x0−p−c0 )λ= y0+x0−p−c0
By same process,λ= y25+ x25−p−c0
λ= y50+ x50−p−c0
λ= y75+ x75−p−c0
λ= y100+x100−p−c0
Set two of lambdas equal to each other,y0+x0−p−c0= y100+ x100−p−c0
x100 equals 0 because if Social Security exists in its current form (i = 100) the insurance payout is 0.
y0+x0= y100
x0= y100− y0
Repeat same process for all x i,
x25= y100− y25
x50= y100− y50
x75= y100− y75
Solution is full insurance.
A2.2Solving for price
∂ L∂ λ
=−[( 1−q(c1)4 )[ ln ( y0−c1 )+ ln ( y25−c1 )+ ln ( y50−c1 )+ ln ( y75−c1 ) ]+q (c1 )∗ln ( y100−c1 )−(1−q (c0 )
4 )[ ln ( y0+ x0−p−c0 )+ ln ( y25+x25−p−c0 )+ln ( y50+x50−p−c0 )+¿ ln ( y75+x75−p−c0 ) ]−q(c0)∗ln ( y100+x100−p−c0 )]=0
Napolitano 32
Plug in for x i
( 1−q ( c0 )4 ) [ln ( y100− p−c0 )+ ln ( y100− p−c0 )+ ln ( y100−p−c0 )+ ln ( y100−p−c0 ) ]+q (c0 )∗ln ( y100−p−c0 )=( 1−q (c1)
4 )[ ln ( y0−c1 )+ ln ( y25−c1)+ ln ( y50−c1 )+ln ( y75−c1 ) ]+q (c1 )∗ln ( y100−c1)
ln ( y¿¿100−p−c0)(4∗1−q ( c0 )
4−q (c0 ))= ln ( y0
1−q ( c0 )4 )+ ln ( y25
1−q ( c0 )4 )+ ln ( y50
1−q ( c0)4 )+ ln ( y75
1−q ( c0)4 )+ ln ( y100
q ( c0) )¿
ln ( y¿¿100−p−c0)(1)=ln(( y0
1−q (c0 )4 )( y25
1−q ( c0 )4 )( y50
1−q ( c0)4 )( y75
1−q (c0 )4 )( y100
q ( c0) ))¿( y¿¿100−p−c0)=( y0
1−q (c0)4 )( y25
1−q ( c0 )4 )( y50
1−q ( c0)4 )( y75
1−q (c0)4 )( y100
q ( c0 ) )¿
p= y100−c0−( y 0
1−q ( c0)4 )( y25
1−q (c0 )4 )( y50
1−q ( c0 )4 )( y75
1−q ( c0)4 )( y100
q ( c0 ) )
A2.3Solving for optimal campaign contributions, c0
∂ L∂ c0
=q ' (c0)
4 ∗( x0+x25+x50+x75 )−q ' ( c0 )∗x100−λ[ q' (c0)4 ∗( ln ( y0+x0−p−c0 )+ ln ( y25+x25−p−c0 )+ln ( y50+x50−p−c0 )+¿ ln ( y75+x75−p−c0 ))−( 1−q ( c0 )
4 )∗( −1y0+x0−p−c0
+ −1y25+x25−p−c0
+ −1y50+x50−p−c0
+ −1y75+ x75−p−c0 )−q' (c0 )∗ln ( y100+x100−p−c0 )−q
( c0 )∗−1y100+x100−p−c0 ]=0
Set x100 equal to zero and substitute in for optimal x and lambda.
q' (c0)4 ∗( x0+x25+x50+ x75 )−q' (c0 )∗(0 )=( y100−p−c0 )[ q' (c0)
4 ∗( ln ( y100−p−c0 )+ln ( y100−p−c0 )+ln ( y100−p−c0 )+¿ ln ( y100−p−c0 ))−( 1−q (c0 )4 )∗( −1
y100−p−c0+
−1y100−p−c0
+−1
y100−p−c0+
−1y100−p−c0 )−q ' (c0 )∗ln ( y100−p−c0)−q
( c0 )∗−1y100−p−c0 ]
q' (c0)4 ∗( x0+x25+x50+ x75 )=( y100−p−c0 )∗[ q' (c0)
4 ∗4∗ln ( y100−p−c0 )−q' (c0 )∗ln ( y100−p−c0 )−( 1−q (c0 )4 )∗( −4
y100−p−c0 )−q( c0 )∗−1
y100−p−c0 ]q' (c0)
4∗( x0+x25+x50+ x75 )=( y100−p−c0 )∗[ 1−q ( c0 )+q (c0 )
y100−p−c0 ]q ' (c0 )∗( x0+x25+x50+x75 )
4=
( y100−p−c0 )∗1y100−p−c0
q ' (c0 )∗( x0+x25+x50+x75 )4
=1
q ' (c0 )= 4( x0+x25+x50+x75 )
c0=q'−1( 4( x0+ x25+x50+x75 ) )
A2.4Optimal campaign contributions when q=1+c
2+c :
∂∂ c0
( 1+c0
2+c0)= 4
( x0+x25+ x50+x75 )
Napolitano 33
( 1(2+c0 )2 )= 4
( x0+x25+ x50+x75 )
(2+c0 )2=( x0+x25+ x50+x75 )
4
(2+c0 )=√ ( x0+x25+x50+x75)4
c0=12 √( x0+x25+x50+x75)−2
A2.5 Solving for optimal pre-insurance campaign contributions, c1
Maximize the expected value of social security with respect to pre-insurance campaign contributions, c1
EV (c1)=( 1−q(c1)4 )[ ln ( y0−c1 )+ ln ( y25−c1 )+ ln ( y50−c1 )+ ln ( y75−c1 ) ]+q (c1 )∗ln ( y100−c1 )−( 1−q (c0 )
4 )[ ln ( y0+ x0−p−c0 )+ ln ( y25+x25−p−c0 )+ln ( y50+x50−p−c0 )+¿ ln ( y75+x75− p−c0 ) ]−q(c0)∗ln ( y100+x100−p−c0 )
∂ EV∂ c0
=−q '(c1)
4∗( ln ( y0−c1 )+ ln ( y25−c1)+ ln ( y50−c1 )+¿ ln ( y75−c1 ))+( 1−q ( c1)
4 )∗( −1y0−c1
+ −1y25−c1
+ −1y50−c1
+ −1y75−c1 )+q' ( c1 )∗ln ( y100−c1 )+q
(c1 )∗−1y100−c1
=0
q' (c1)4
∗( ln ( ( y0−c1 ) ( y25−c1 ) ( y50−c1 ) ( y75−c1 )))−q' ( c1)∗ln ( y100−c1 )=( 1−q (c1 )4 )∗( −1
y0−c1+ −1
y25−c1+ −1
y50−c1+ −1
y75−c1 )+q( c1)∗−1y100−c1
q ' (c1 )∗(ln ( ( y0−c1 ) ( y25−c1 ) ( y50−c1 ) ( y75−c1 )))−4∗q ' (c1 )∗ln ( y100−c1 )=4∗[( 1−q (c1 )4 )∗( −1
y0−c1+ −1
y25−c1+ −1
y50−c1+ −1
y75−c1 )+q(c1 )∗−1y100−c1 ]
q '(c1)∗( ln( ( ( y0−c1 ) ( y25−c1 ) ( y50−c1 ) ( y75−c1 ))( y100−c1 )4 ))=4∗[( 1−q ( c1)
4 )∗( −1y0−c1
+ −1y25−c1
+ −1y50−c1
+ −1y75−c1 )−
q ( c1 )y100−c1 ]
q '(c1)=
4∗[( 1−q (c1 )4 )∗( −1
y0−c1+
−1y25−c1
+−1
y50−c1+
−1y75−c1 )−
q ( c1)y100−c1 ]
ln( (( y0−c1 ) ( y25−c1 ) ( y50−c1 ) ( y75−c1 ))( y100−c1 )4 )
A2.6 Campaign Contributions Before and After Insurance
Want to show c0<c1.Assumptions about q ( c ),
c>0q ( c ) ϵ [ 0,1 ]
Napolitano 34
∂ q∂ c
>0
∂2 q∂ c2 <0
q(c) has an inverse
Post-Insurance Campaign ContributionsFrom Appendix A2.3
c0=q'−1( 4( x0+ x25+x50+x75 ) )
Pre-Insurance Campaign ContributionsFrom Appendix A2.5
q '(c1)=
4∗[( 1−q (c1 )4 )∗( −1
y0−c1+ −1
y25−c1+ −1
y50−c1+ −1
y75−c1 )−q ( c1)
y100−c1 ]ln( (( y0−c1 ) ( y25−c1 ) ( y50−c1 ) ( y75−c1 ))
( y100−c1 )4 )Proving c0<c1
Because, by assumption, q ' ' (c )<0then it is adequate to show that q ' (c0 )>q' (c1) to prove c0<c1
q ' (c0 )>q' (c1)
4( x0+x25+x50+x75 )
>
4∗[( 1−q (c1 )4 )∗( −1
y0−c1+ −1
y25−c1+ −1
y50−c1+ −1
y75−c1 )−q (c1 )
y100−c1 ]ln( ( ( y0−c1 ) ( y25−c1 ) ( y50−c1 ) ( y75−c1 ))
( y100−c1 )4 )We know ln ( ( ( y0−c1 ) ( y25−c1 ) ( y50−c1 ) ( y75−c1 ))
( y100−c1 )4 )is negative because
( y100−c1 )>( y75−c1 )>( y50−c1 )>( y25−c1 )>( y0−c1 ) coupled with the original assumption that ( y i−c1)>0 then( y100−c1 )4> ( y0−c1 ) ( y25−c1 ) ( y50−c1 ) ( y75−c1 ) indicates that
0<( (( y0−c1 ) ( y25−c1) ( y50−c1) ( y75−c1 ))( y100−c1)4 )<1. The natural log of a number between 0 and 1 is a
negative number.
Napolitano 35
Since q ' (c )>0 and ln ( ( ( y0−c1 ) ( y25−c1 ) ( y50−c1 ) ( y75−c1 ))( y100−c1 )4 )<0 then
4∗[( 1−q (c1 )4 )∗( −1
y0−c1+ −1
y25−c1+ −1
y50−c1+ −1
y75−c1 )−q (c1 )
y100−c1 ]<0.
Since we are multiplying by a negative the sign flips,
ln( (( y0−c1 ) ( y25−c1 ) ( y50−c1 ) ( y75−c1 ))( y100−c1 )4 )
( x0+x25+ x50+x75 )<( 1−q (c1 )
4 )∗( −1y0−c1
+ −1y25−c1
+ −1y50−c1
+ −1y75−c1 )−
q (c1 )y100−c1
If a < c and b > d then ab< c
d so it will be sufficient to show
( x0+x25+x50+x75)>1
And
ln ( ( ( y0−c1 ) ( y25−c1 ) ( y50−c1 ) ( y75−c1 ))( y100−c1 )4 )<( 1−q (c1 )
4 )∗( −1y0−c1
+ −1y25−c1
+ −1y50−c1
+ −1y75−c1 )−
q ( c1)y100−c1
We can show that( x0+x25+x50+x75)>1
( x0+x25+x50+x75)= y100+( y100− y25 )+( y100− y50 )+( y100− y75 )= y100+( y100−.25∗y100)+ ( y100−.50∗y100 )+( y100−.75∗y100)= y100+(.75∗y100 )+ (.50∗ y100)+(.25∗y100)=52∗y100>1→ y100>
25
Since the social security administration will not payout a benefit of less than $1 per month we
know y100>25 so
( x0+x25+x50+x75)>1 .
We can show that
ln ( ( ( y0−c1 ) ( y25−c1 ) ( y50−c1 ) ( y75−c1 ))( y100−c1 )4 )<( 1−q (c1 )
4 )∗( −1y0−c1
+ −1y25−c1
+ −1y50−c1
+ −1y75−c1 )−
q ( c1)y100−c1
We are multiplying by a negative number, so the sign of the inequality switches,
Napolitano 36
1>( 1−q ( c1)
4 )∗( −1y0−c1
+ −1y25−c1
+ −1y50−c1
+ −1y75−c1 )−
q ( c1)y100−c1
ln( (( y0−c1 ) ( y25−c1) ( y50−c1) ( y75−c1 ))( y100−c1)4 )
Which is the same as1>q '(c1)
So if we allow 1>q '(c1) and combine it with( x0+x25+x50+x75)>1
It implies thatq ' (c0 )>q' ( c1)
And since q ' ' (c )<0by assumption,c0<c1
, given1>q '(c1)
A2.7 Campaign Contributions Before and After Insurance when q=1+c
2+c
As shown in A2.5 campaign contributions before the insurance are less than they would be after the insurance (c0<c1 ¿ ,conditional on q '(c1) < 1.
So if we allow q ( c )=1+c2+c (which satisfies the assumptions and is used earlier in the paper), then
q ' (c1 )= 1(2+c1)
2
0<q' (c1)<1
The condition is satisfied, implying c0<c1
Napolitano 37
Appendix B
Social Security Survey
1. What is your sex?2. When were you born?3. Do you regularly vote?
a. Yesb. No
4. If so, is social security policy an issue you consider when deciding who to vote for?a. Yes, likely to vote for candidate who supports continuing social security at
current levelsb. Yes, likely to vote for candidate who supports reducing social securityc. No
5. Do you regularly contribute to political campaigns?a. Yesb. No
6. If so, are you hoping your contribution will affect social security policy either directly or indirectly through which politician is elected?
a. Yes, in support of continuing social security at current levelsb. Yes, in support of reducing social securityc. No
7. In your opinion, what is the likelihood that social security will exist in its current form when you plan on retiring?
a. Very High (80% - 100%)b. High (60% - 79%)c. Medium (40% - 59%)d. Low (20% - 39%)e. Very Low (0%-19%)
8. In your opinion, what is the likelihood that social security will exist with at least 75% of current benefits when you plan on retiring?
a. Very High (80% - 100%)b. High (60% - 79%)c. Medium (40% - 59%)d. Low (20% - 39%)e. Very Low (0%-19%)
9. In your opinion, what is the likelihood that social security will exist with at least 50% of current benefits when you plan on retiring?
a. Very High (80% - 100%)b. High (60% - 79%)
Napolitano 38
c. Medium (40% - 59%)d. Low (20% - 39%)e. Very Low (0%-19%)
10. In your opinion, what is the likelihood that social security will exist with at least 25% of current benefits when you plan on retiring?
a. Very High (80% - 100%)b. High (60% - 79%)c. Medium (40% - 59%)d. Low (20% - 39%)e. Very Low (0%-19%)
11. How reliant do you see yourself being on your Social Security benefits when you retire?a. Reliantb. Not reliant
12. At what age do you plan to start receiving social security benefits?13. As of today the current average Social Security benefit is about $1,200 per month which
accumulates to more than $250,000 over the course of retirement. There is some concern that the benefits granted by social security will be cut sometime in the future. So what I want to do is to offer you the opportunity to guarantee your roughly $250,000 by purchasing insurance for your social security. Similar to home or car insurance, this investment will cover your future benefits should current social security laws change. This policy would payout the difference between how much social security you would receive under today’s law and your actual benefit at the time of your retirement. For example if your monthly benefit was reduced by $600 your insurance payout would total over $125,000. Keeping all that in mind would you be willing to spend $XXX (monopoly price – calculated based on answers to previous questions) per year to ensure your full social security benefit?
a. Yesb. No
14. (If answer to 13 is ‘No’) If you had the opportunity to purchase insurance for your social security benefits for $XXX (actuarially fair price - calculated based upon answers to previous questions) would you?
a. Yesb. No
15. (If answer to 14 is “No”) Is there any price you would be willing to buy the insurance? If so, what is it?
16. Is there any reason why you wouldn’t want to buy this type of insurance?17. (Based on answer to 16) If [whatever reason given in 16] did/didn’t exist/happen/etc.
would you purchase this insurance?18. If you buy this insurance then any changes to decrease social security would not affect
your future benefits. Keeping that in mind, after purchasing this insurance would social security policy be an issue you consider when deciding who to vote for?
Napolitano 39
a. Yes, likely to vote for candidate who supports continuing social security at current levels
b. Yes, likely to vote for candidate who supports reducing social securityc. No
19. If you buy this insurance, will you contribute to political campaigns in the hopes that your contribution will affect social security policy either directly or indirectly through which politician is elected?
a. Yes, in support of continuing social security at current levelsb. Yes, in support of reducing social securityc. No
Napolitano 40
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