Professor Scholz
Economics 441, Problem Set #4
Posted November 27, 2006
Due in class, Monday, December 11, 2006
All problems are worth 40 points unless noted differently.
1 (10 points). Suppose your preferences can be characterized by the simple utility function U = C.
You enjoy rock climbing, where you have a 10 percent chance of getting injured and losing $50,000.
Your income (and therefore consumption) in the uninjured state is $90,000.
a) What is the most you are willing to pay for an insurance policy? Will insurance companies be
willing to insure you?
Without insurance, expected utility is .9 90,000 + .1 40000 =290. The most someone would be
willing to pay is X, where X solves 90000 − X = 290 , so X=5900. Insurance companies would
be happy to offer insurance for prices between 5000, the expected payouts, and 5900, the most
people would be willing to pay.
b) Now, suppose there are a bunch of people who look just like you in their observable
characteristics (they talk the same, they look the same, they make the same amount of income).
But these people are passionate about their hobby of “base jumping.” Base jumpers make
parachute jumps off immobile objects, like tall buildings, bridges, cliffs, etc. Suppose base
jumpers have a 50 percent chance of getting injured and losing $50,000. What will this do to the
market for insurance?
The expected cost of insuring base jumpers, $25,000, is much higher than the expected cost of
insuring rock climbers ($5,000). The problem is that insurance companies would lose money by
offering policies attractive to rock climbers, since base jumpers would purchase them (since they
are observationally equivalent to the insurance company). But policies attractive to base jumpers
will be too expensive for rock climbers. The market would evolve in a way that rock climbers
will not find attractive complete policies (that would cost between 5,000 and 5,900).
c) This is an example of what phenomenon?
Adverse selection.
2 (10 points): In the schools of Beat Town, students are educated to play music, which they then perform
at home for their parents. When more money is spent on the schools, the students learn more songs, and
their parents are more entertained. The family utility function for each Beat Town family is U = C 0.5 S 0.5
where S is Beat town’s per-student expenditure on schooling and C is the amount of money the family
has left over for other consumption after paying the school tax. All students attend the same school in
Beat Town.
a) Although all the families in Beat Town have the same utility function, they have different
incomes. Suppose 50 families each earn $50,000; and 60 families each earn $80,000. If the town
votes on the level of a lump-sum tax to finance schools, what level will win?
Recall, since utility is invariant to monotonic transformations. So take the natural log of utility
to find (note: you don’t have to do this – doing so just makes the algebra a little easier):
U = (.5) ln(C ) + (.5) ln( S ), and C = Y − S , so U = .5ln(T − S ) + .5ln( S ),
dU
1
1
Y
=−
+
= 0. This implies S = .
dS
2[Y − S ] 2 S
2
A simple majority vote will result in the $80,000 families winning the vote (since there are 60 of
them and only 50 families with income of $50,000). From above, the $80,000 households prefer
a lump-sum tax of $40,000. Consequently, all families will be assessed a lump sum tax of
$40,000.
b) Now suppose lump-sum taxes are found to be unconstitutional. Along with this court ruling, and
the residents of Beat Town are ordered to substitute a flat percentage income tax levied at a rate
of 50 percent. Explain how the welfare (utility) of Beat Town residents changes (if it changes).
With a 50 percent tax, households with income of $80,000 will pay a tax of $40,000 and
households with income of $50,000 will pay a tax of $25,000. So the total average tax payments
are (50*(25,000)+60*(40,000))/110 = 33181.82. So the rich are worse off, since school
spending is less than their optimal value (of 40,000), and private consumption remains
unchanged. The poor are better off: consumption of 25,000 and school expenditures of
33,181.82 gives greater utility than consumption of 10,000 and school expenditures of 40,000.
3, 40 points total, 10 for each part) The superhero profession is a dangerous business. Every year, in fact,
there’s a probability p that a given superhero will be caught by a ruthless supervillain, who inflicts
damage that requires $50 in medical costs to heal. Being a superhero doesn’t pay well, but fortunately,
superheroes receive an annual income of $100 from their (covert) civilian job. They first spend money on
any medical costs, and use the rest for consumption.
In one particular metropolis, there are two types of superheroes: clumsy and skillful. Clumsy
superheroes have a 90% probability of being caught by a supervillian and suffering $50 in medical costs.
Skillful superheroes, on the other hand, are only caught with probability 30%. Additionally, these two
types have different utility functions. The utility of consumption for clumsy superheroes is
U clumsy = (Cclumsy ).7 while the utility of consumption for skillful superheroes is U skillful = (Cskillful ).5 .
Fortunately for this metropolis, there are eleven times as many skillful superheroes as clumsy
superheroes.
ACME Insurance Company has moved into the city, and is thinking about offering health insurance to
superheroes.
a) If ACME can perfectly identify whether each superhero is skillful or clumsy, then it can charge a
different premium to each type. Suppose it charges an actuarially fair price for insurance. How
much will ACME charge each type for $1 in medical coverage? How much insurance will each
type buy, and how much will each type have for consumption if they get caught, and how much
will they have if they don’t get caught?
b) Now suppose that the type of the superhero is unobservable by the insurance company (although
the superheroes themselves know), so ACME can only offer a single price for insurance.
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a. What is the maximum amount of money each type will pay in order to be fully insured
against medical costs? Explain why this amount is the same or is different from the costs
of actuarially fair insurance.
b. Given this, what is the maximum amount that ACME can charge such that all types will
fully insure? Will ACME stay in business if it does this? Is there a market failure in the
insurance market – why or why not?
c) ACME instead considers offering two types of insurance coverage. The “minimal coverage” plan
provides $20 in insurance coverage for total costs of $7. The “extensive coverage” plan offers
$50 in insurance for total costs of $34. Which of these plans, if any, will each choose? Will
ACME stay in business? Now is there a failure in the insurance market? Why or why not?
d) Dr. Brain (a mad scientist) develops a new blood test that flawlessly identifies whether a
superhero is clumsy or skillful, so that everyone (including insurance companies) will then be
able to identify his or her type. However, this test is not free: the test costs $1.50 to administer to
a superhero. Suppose Dr. Brain offers to perform this procedure on any superhero who is willing
to pay $1.50.
a. Assuming that there is perfect competition in the insurance market, and that once a test is
undertaken all insurance companies know the superhero’s type, will any type be willing
to purchase this test? Why or why not? Note, in the absence of testing, the insurance
plans from part c) above will be what is offered. To answer this, think about what will
happen in the insurance market in response, i.e., will a single price be offered or will
different policies be offered depending on who takes the test – and what will be the
equilibrium prices?
b. Relative to c), which type is better off now that it is possible to reveal types – or are both
better off?
c. Now is there a failure in the insurance market?
Answer: Each superhero's utility exhibits diminishing marginal returns (you can see this by
simply plotting the level of utility against consumption, or by finding that the second derivative
is negative). This is equivalent to saying that the individual is risk averse. Since this is true, and
since insurance is fair, the price that each type pays for a unit of insurance will be equal to the
probability that they get caught - and further, we know from class that risk-adverse individuals
(i.e. utility with diminishing marginal returns) will fully insure if insurance is fair.
You could also get this result by solving for the optimal level of insurance that each type will
choose. For clumsy superheros, the problem would be:
Max pU accident + (1 − p )U NoAccident ⇒ Max.9(100 − 50 + b − .9b).7 + .1(100 − .9b).7
dU (.9)(.7)(.1) (.1)(.7)(.9)
=
−
=0
db (50 + .1b).3 (100 − .9b).3
⇒ b = 50
so
Where b is the amount of insurance the clumsy superhero buys (which is equal to the amount he
gets from the insurance company if he is caught). The procedure is similar for skilled superheros.
3
So, to answer the specific questions, ACME will charge the clumsy $0.90 for every dollar of
insurance and will charge the skillful $0.30. Since they are risk averse and the insurance is
priced fairly, they will both purchase $50 of insurance. The clumsy will consume $55 in both
the accident and no accident state. The skillful will consume $85 in both states.
B, part a) the maximum amount that each would be willing to pay for insurance is equal the
amount that makes him or her exactly as well off with insurance as without. Without insurance,
the expected utility of each type is
EU clumsy = .9(100 − 50).7 + .1(100).7 ≈ 16.428
EU skillful = .3(100 − 50).5 + .7(100).5 ≈ 9.121
Full insurance would mean that the superhero receives full compensation for injury if injured,
resulting in complete consumption smoothing across the two possible states (caught or not
caught). Hence, his income in each state is simply his full income less insurance costs. So to
solve for the maximum each type would be willing to pay for full insurance, solve the following:
EU clumsy , no insurance = EU clumsy , FI ⇒ 16.428 = (100 − X ).7 ⇒ X = 45.48
EU skillful , NI = EU skillful , FI ⇒ 9.121 = (100 − X ).5 ⇒ X = 16.80
So the clumsy type is willing to spend 45.48 for full insurance (for a per-unit price of .910) and
the skillful type is willing to spend 16.80 (for a per-unit price of .336). Note that each is willing
to pay more than the actuarially fair price - this is because their utility function exhibits
diminishing marginal returns to consumption (risk aversion):
B, part b) If ACME is looking for a single market price such that everyone will fully insure, it
will have to offer the maximum price that the skillful are willing to pay: 16.80. However, when it
does this it will collect 12*16.80=201.6 in revenue for every 11 * .3*50+.9*50=210 it expects to
payout - since expected revenue is less than expected payout, ACME won't stay in business if it
offers this plan. Hence, there's a market failure: we know that each type is willing to pay more
than the actuarially fair price for full insurance, and ACME is certainly willing to offer insurance
at a greater-than-fair price. The problem is that types are unobservable (that is, asymmetric
information exists in this insurance market), so this arrangement cannot occur. Because there are
transactions that would occur under full information that would make everyone better off, a
market failure exists.
c) Now, let's consider the expected utility that each type would receive under each plan:
EU clumsy , PlanA = .9(100 − 50 + 20 − 7).7 + .1(100 − 7).7 ≈ 18.747
EU skillful , PlanA = .3(100 − 50 + 20 − 7).5 + .7(100 − 7).5 ≈ 9.132
EU clumsy , PlanB = .9(100 − 50 + 50 − 34).7 + .1(100 − 34).7 ≈ 18.779
EU skillful , PlanB = .3(100 − 50 + 50 − 34).5 + .7(100 − 34).5 ≈ 8.124
4
Comparing these expected utilities to those without insurance, as calculated in (b, part a), we see
that clumsy types prefer plan B to plan A, and are better off by purchasing insurance than not
purchasing insurance. The skillful types prefer plan A to plan B, and are also better off by
purchasing insurance. So if these packages were offered, all the clumsy would purchase type B,
and all the skillful would purchase plan A. ACME revenues are 11 *7+1 *34=111. ACME
expected payout is 11 *.3*20+.9*50=111. ACME would make zero economic profit from this
package, but since zero economic profit allows the company to earn market rates of return on all
factors of production it is willing to offer the insurance. Now everyone has some insurance
However, there is still a market failure for the same reason as before - the skillful types still wish
to purchase full insurance at actuarially fair prices, and the insurance company would offer
insurance for those prices - but due to asymmetric information, this is impossible. The outcome
may be more favorable than before, since now everyone gets some insurance, but a market
failure still exists.
D part a) There might be an incentive here for the skillful type to purchase the test. Consider
what happens if the skillful pay for the test and reveal their type: given that there is perfect
competition in the insurance market, the skillful will be offered actuarially fair insurance, so they
would fully insure. The clumsy type would never purchase the test, because if they do their type
will be revealed, and they will receive actuarially fair insurance - which is more expensive than
the insurance plan they receive from c! To determine whether the skillful types will purchase the
test, consider their expected utility if they do:
EU skillful , NoIns = .3(100 − 50).5 + .7(100).5 ≈ 9.121
EU skillful , PlanA = .3(100 − 50 + 20 − 7).5 + .7(100 − 7).5 ≈ 9.132
EU skillful ,test = (100 − 1.5 − .3*50).5 ≈ 9.138
Since the skillful are slightly better off by paying $1.50 for the test, receiving fair insurance and
fully insuring, they will be willing to pay for Dr. Brain's test. Now, ACME will offer two
insurance packages. The first will be for those revealed to be skillful, who will be able to
purchase insurance for a per-unit of coverage cost of .3. The second will be available only to
those who haven't revealed themselves with a test (i.e. the clumsy), who will be able to purchase
insurance for a per-unit of coverage cost of .9. Both types will fully insure!
D, part b) The skillful are better off than in (c), because they are now fully insured. The clumsy
are worse off, because although they are still fully insured, they have to pay more in order to be
fully insured.
D, part c) Now there is no longer a market failure. The presence of the market for blood tests has
eliminated the asymmetric information problem, and everyone is fully insured at actuarially fair
prices (although the skillful are technically paying more than actuarially fair prices to receive full
insurance, since they first have to purchase the test).
5
4, 40 points total, 5 points for each part, with a 5 point bonus for completing the entire problem)
Consider a society of identical workers that each earns a wage W when they work. Each worker faces a
probability of sustaining an injury α. If they sustain an injury, they have no earnings (W=0). In any case,
1
however, they always have some outside income of 5. Workers have utility of the form: U = log(C ) ,
2
where C=consumption=total income in the period (they do no saving). Assume there is no moral hazard.
a) What is the expression for the expected utility of each worker?
Now, suppose that the government introduces a worker’s compensation program. Under this system,
individuals pay some fraction of their wage when they are employed (i.e., their wage is taxed at a certain
rate), and get a benefit when they are injured. The system must break even at a point in time; that is, the
benefits paid to injured workers must be equal to the taxes collected from employed workers.
b) What is the optimal workers compensation system? That is, what is the system that, subject to
the constraint of breaking even, maximizes worker utility? Present both the tax rate and the
benefit level for the system.
c) Are there welfare gains or losses from introducing the Workers’ Compensation system (you don’t
actually have to measure the gains/losses – just sign them)? Why?
d) Would your answer to part c) change if utility was of the form: U=(1/2)C?
Now suppose that when workers get injured their spouses go to work. Each worker injured gets an
amount kW from their spouse, where k is some constant and k < (1 − α ) .
e) What is the expected utility now if there is no Workers’ Compensation program?
f) Now, reintroduce Workers’ Compensation, which once again must break even. What is the
optimal Workers’ Compensation system now (both tax rate and benefit level)? How does this
compare to your answer to part (b)? Why?
g) Are there welfare gains or losses now from introducing the system (once again, no precise
measurement is necessary)? Intuitively (and not mathematically), are these gains or losses greater
than in part c)? Why?
Answer: a) E (U ) = (1 − α ) log( w + 5) + α log(5)
Remember, utility is invariant to monotonic transformations, so for ease of notation, I dropped
the ½.
b) The break-even condition dictates the benefit level, b, so τ w(1 − α ) − bα = 0 ⇒ b = τ w
so solve max E (U ) ⇒ max[(1 − α ) log( w(1 − τ ) + 5) + α log(5 +
τ
τ
1−α
α
1−α
α
.
τ w)]
The first-order conditions are
⎛
⎞
⎜
⎟1−α
(1 − α )
α
(α − 1) w
(1 − α ) w
(− w) + ⎜
+
=0
w=0⇒
⎟
w(1 − τ ) + 5
w(1 − τ ) + 5 5 + 1 − α τ w
⎜⎜ 5 + 1 − α τ w ⎟⎟ α
α
α
⎝
⎠
α −1
α −1
1−α
⇒
=
⇒ w(1 − τ ) + 5 = 5 +
τ w ⇒ τ = α ⇒ b = w(1 − α )
w(1 − τ ) + 5 5 + 1 − α τ w
α
α
c) Yes there are welfare gains. These occur because, due to the log utility, the workers are
risk averse and therefore value the insurance provided by the workers’ compensation system.
6
d) If U=.5C then the workers are no longer risk averse and therefore they do not value
insurance. There is no welfare gain to introducing workers’ compensation.
e) E (U ) = (1 − α ) log( w + 5) + α log(kw + 5)
f) max E (U ) ⇒ max[(1 − α ) log( w(1 − τ ) + 5) + α log(kw + 5 +
τ
τ
1−α
α
τ w)]
The first-order conditions are
⎛
⎞
⎜
⎟1−α
(1 − α )
α
(α − 1) w
(1 − α ) w
(− w) + ⎜
w=0⇒
+
=0
⎟
1
α
1−α
−
α
(1
τ
)
5
−
+
w(1 − τ ) + 5
w
⎜⎜ kw + 5 +
⎟
τw⎟
τw
kw + 5 +
α
α
⎝
⎠
α −1
α −1
1−α
τ w ⇒ τ = (1 − k )α ⇒ b = w(1 − k )(1 − α )
⇒
=
⇒ w(1 − τ ) + 5 = kw + 5 +
α
w(1 − τ ) + 5 kw + 5 + 1 − α τ w
α
g) Yes, there are gains, but they are smaller than in part c). They are smaller because spousal
employment is already providing some insurance against injury.
5: 40 points total, 10 for each part) Consider the following stylized model of an economy, which exists
for two periods (period 1 and period 2). There are two types of people in this economy: type As, who
typically have long life spans, and type Bs, who typically have short life spans. There are 20 type As, and
10 type Bs. Each type cares only about consumption while alive, and each person has $100 in income,
which he/she receives in period one (to make things simple, there are no work decision to be made in this
economy, so we won’t worry about labor supply and wages). Each person has an expected utility
function of the following form:
1
EU i = ln(C1 ) + p ln(C2 ), where p=.75 for type As, and p=.2 for the type Bs. In other words, no one
3
wants to starve if they live into the second period, but no one is certain whether they will reach the second
period – and type As are more likely to reach the second period than type Bs are.
Suppose that no annuity markets exist in this economy because potential annuity providers are
unable to observe individual types (resulting in adverse selection problems).
a) Although the annuity market doesn’t exist, people can still invest money in period 1. In period 2,
if the person is still alive, he or she receives his or her initial investment, plus a rate of return on
the investment of 10% (i.e. the interest rate is 10%). How much will each type choose to save in
the first period? What is first period consumption for each type, and how much will each type
consume if they reach the second period?
b) Even though people in this economy can smooth consumption by saving and earning a return
equal to the interest rate, would people be better off if an annuity market existed that provided
actuarially fair annuities? (In answering this part of the question, there is no need to solve for
anything. Just explain intuitively why you think an actuarially fair annuity market would or
would not improve welfare).
c) The federal government, recognizing a market failure in the annuity market, decides to correct the
problem by implementing a social security program. The way this social security program will
work is that the government sets a lump sum tax T that everyone must pay in the first period. The
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government takes this money, invests it at an interest rate of 10%, and uses the entire amount to
pay B in benefits to everyone who is alive in the second period.
a. Assume that the government’s social security budget must break even (in expectation).
Write out the government’s social security budget constraint. What multiple of taxes
must benefits be for the budget to exactly break-even? (i.e., find the X such that B=XT
for the government to break even).
b. Suppose the government is trying to choose the optimal lump sum tax T (and hence, the
optimal social security benefit amount B). If the government cares only about the sum of
utilities for everyone in the economy (i.e. it maximizes a utilitarian social welfare
function), and if the budget must exactly break even, what level of T and B will it
choose? (note: you can safely assume that neither type will want to save privately in
addition to social security). Is type A better off with or without social security? How
about type B? In what sense is this program redistributive?
c. The social security plan in part (b, just above) is never implemented because the
administration that proposed the policy is voted out of office by the type Bs.
Coincidentally (or perhaps not coincidentally), the new administration cares only about
the type Bs. What level of T and B will the new administration choose, if the social
security budget must still be balanced? (Although the administration cares only about B,
each type must get charged the same lump-sum tax, and each type must receive the same
amount in benefits if they live to period 2). Assume that everyone can supplement social
security benefits with additional saving if they wish (i.e., they can save some of their
post-tax income in period 1, as in part a). Is type A better off with or without social
security? What about type B? Is either type better off than with the system in b (just
above)? Is this new program redistributive, and if so, how does its redistributive nature
differ from part b (just above).
d) Now, think about how this stylized economy and social security system relate to the American
economy and current social security system. In what sense is the social security system in the
U.S. redistributive in ways similar to those illustrated above? Who are relevant type As and type
Bs?
Answer: Don't get confused because we're thinking about consumption in different periods. Just
treat the problem as deciding how much to spend on two goods: consumption today and.
consumption tomorrow. The easiest way to solve intertemporal utility maximization is to
consider how much the person can consume in the second period, given how much he chose to
consume in the first period. So in this example, if the person lives into the second period, his
consumption is: C2 = (l + r) * (l00 – C1) = 1.1 * (100 - C1). Now just plug this in for C2, and
maximize the expected utility function with respect to C1.
1
1
p ln(C2 ) ⇒ max U = ln C1 + p ln(1.1(100 − C1 ))
3
3
⎞
⎛ 100 p ⎞
1 p⎛
1
300
⇒
− ⎜
, C2 = 1.1⎜
⎟ = 0 ⇒ 300 − 3C1 = pC1 ⇒ C1 =
⎟
C1 3 ⎝ 100 − C1 ⎠
3+ p
⎝ 3+ p ⎠
max U = ln C1 +
So p=.75 for type As, so C1A = 80 , saving = 20, C2A = 22 . p=.2 for type Bs, so C1B = 93.75 , saving
= 6.25, C2B = 6.875. Type As are saving more, and hence consuming less in period 1 than Bs do
8
but more in period 2, because the probability that they will live to see the second period is
greater.
b) Yes, people would be better off, because annuities provide a higher return on investment.
Remember, the price of fair insurance is the probability of payout (i.e. the probability of having
an accident). Here, the probability of payout is the probability of living into the second period.
Type Bs only have a 20% chance of reaching the second period, so the price of one unit of
annuity payout is .2. Provided the type B reaches the second period, the return on his investment
is huge- for every dollar he spends on insurance, he gets five dollars if he lives into the second
period, for a net return of four dollars. This certainly is greater than his returns from saving. Type
As must pay $.75 for $1 of coverage, so As receive $1.33 for every $1 invested if they live into
the second period. The return on their annuity investment is 33%, which is again higher than
returns from saving.
The reason that fair annuities can provide higher returns is because risks are pooled: everyone
pays in, but only a fraction receive payment. Annuities are valuable because they assist in
consumption smoothing throughout old age. In this simplified two period model, consumption
smoothing can be accomplished through private savings, but doing so is inefficient, since each
person needs to save on their own, and many people will die before they are able to eat their
saving. An actuarially fair annuity market makes it much more efficient to smooth consumption
(and hence increase utility).
C, part a) The government receives 30T in revenue (since it taxes all 30 people equally), and
invests it at an interest rate of 10%, so that it has 30T* 1.1 =33T in funds to payout in period
two. It pays out .75*B in expectation for every type A, and pays out .2*B in expectation for
every type B - so in sum, it expects to pay out 20*.75 *B+ 10*.2 *B= 17B. Since the budget
must break even at the end of period two, this implies that 33T=17B, or the benefits the
government pays out to anyone who lives into period two is 33T/17.
C, part b) The government is now trying to decide what T to choose to maximize the sum of
utilities. In other words, its maximization problem is:
⎛
⎛
1
1
⎛ 33T ⎞ ⎞
⎛ 33T
max social welfare = 20 * ⎜ ln(100 − T ) + *.75* ln ⎜
⎟ ⎟ + 10 * ⎜ ln(100 − T ) + *.2 * ln ⎜
3
3
⎝ 17 ⎠ ⎠
⎝ 17
⎝
⎝
5
10
2
1700
−20
+ −
+
=0⇒T =
≈ 15.89, B = 30.84
100 − T T 100 − T 3T
107
⎞⎞
⎟⎟ ⇒
⎠⎠
So each person pays 15.89 in taxes, and receives a benefit in period two of 30.84 (if they survive
that long).
9
1
A
EU noSS
= ln(80) + ln(22) ≈ 5.15
4
1
B
EU noSS
= ln(93.75) + ln(6.875) ≈ 4.67
15
1
A
ln(30.84) ≈ 5.29
EU SS
1 = ln(84.11) +
4
1
B
ln(30.84) ≈ 4.66
EU SS
1 = ln(84.11) +
15
A is better off with social security than without - this is because it is highly probable that a type
A reaches the second period, and because the return from "saving" with social security is greater
than the returns from private investment. Since B=33T/17 ≈ 1.94T, one dollar "saved" (taxed)
through social security returns 1.94 dollars if the person reaches the second period, for a return of
94% (which is certainly greater than the 10% return from the private market). From A's
perspective, this isn't the optimal social security system -- the optimal system would in fact force
more saving through an even higher tax - but because the high return results in high period two
consumption, As find this system better than a world with no social security.
Bs, however, are just slightly worse off than before. This is because the tax rate is much higher
than what they would've chosen (see answers to the next part). So even though the rate of return
is much higher than what private savings yields in the absence of social security, Bs are worse
off because the tax rate is too high.
One could view this system as redistributive since it taxes Bs more than they wish, and these tax
revenues are partially redistributed to As since As are more likely to receive social security
benefits.
C, part c) Now the government chooses T to maximize B's utility:
1
⎛ 33T
max B ' s utility = ln(100 − T ) + *.2 * ln ⎜
3
⎝ 17
−1
1
100
⎞
⎟ ⇒ 100 − T + 15T = 0 ⇒ T = 16 = 6.25, B = 12.13
⎠
which implies
1
A
EU SS
ln(12.13) ≈ 5.16
2 = ln(93.75) +
4
1
B
EU SS
ln(12.13) ≈ 4.71
2 = ln(93.75) +
15
Now B is better off with this system than without, and is better than the first proposed system.
A is still better off than without social security (and if you assume that A can supplement
social security income with additional private savings, A's utility will be higher than 5.16), but
A is worse off than with the first social security system. This is because the forced savings in
this system are lower than the other system - since this system doesn't account for A's
preferences whatsoever, but the prior system partially weighted A' s preferences, A must be
better off with the first social security system. However, assuming A can save in addition to
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social security, A is not worse off than without social security - this is because A can always
save on top of social security at a 10% interest rate, but the returns on some investment (the
6.25 "invested" with taxes) is greater than the market rate of 10%.
Again, this system redistributes from Bs to As - this is because the ratio of expected payout to
payin is much higher for As than Bs. However, Bs are better off with this sort of redistribution
than they are if the social security system didn't exist.
D) There are many ways in which the American social security system is redistributive. For
example, since the PIA is calculated as a progressive function of the AIME, the social security
system redistributes from the poor to the rich (the replacement rate for social security benefits
is higher for the poor than the rich). What is most relevant from this example, however, is that
the system also redistributes from the short lived to the long lived - because if two people are
identical except for life expectancy, they'll pay the same amount in taxes, but the short lived
person will receive less in benefits than the long lived person will. In America, women tend to
live longer than men, and whites tend to live longer than blacks, and richer people tend to live
longer than poorer people - so due to differences in life expectancy, the system is
redistributive from the first group to the second (of course, the progressivity of benefit
calculations may make the system actually less redistributive from the first to second groups).
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