EXAM
Name of subject
: The Economics and Finance of Pensions
Subject code
: 323068
Date of examination
: December, 9, 2014
Length of examination : 3 hours
Lecturer
: Lans Bovenberg
ANR: 148199
Roel Mehlkopf
ANR: 694448
Telephone number of departmental secretariat: 0134662703
Students are expected to conduct themselves properly during examinations and to obey any
instructions given to them by examiners and invigilators.
Firm action will be taken in the event that academic fraud is discovered.
Enter ANR!
Each question should be answered on TU exampaper, each furnished with the candidate’s name
and ANR number. If candidates are unable or unwilling to answer a question, they must
nevertheless submit a sheet of paper containing details of their name and ANR, together with the
number of the question concerned.
The 6 digit ANR number is printed on the TU card.
INSTRUCTIONS – PLEASE READ CAREFULLY
The wording and notations in the questions are meant to be as consistent and concise as
possible. With that in mind, please note the following conventions:
- “Explain X” or “Why X”, means that you should provide an intuitive explanation in words for the
statement X. In addition, you may also use mathematical symbols and expressions in your
answer, but they are not necessary.
There are three sections in the exam. The first section is worth 100 points while the second and
third sections are worth 50 points each. The first section contains four separate essay questions.
Your answer to the essay questions should not exceed 300 words per answer. The second and
third sections contain short questions. The questions within each section are related, but each
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question can be answered independently of the others, so if you get stuck on a question, don’t
worry about leaving it and moving on.
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PART 1: ESSAY QUESTIONS (100 POINTS)
1. [25 points] A government wants to help the poor without affecting the
intergenerational distribution of resources. Please explain how this government
can accomplish this through a combination of the following two reforms: (i) the
introduction of a PAYG pension system and (ii) a shift in the tax mix between
proportional consumption taxes and proportional labor taxes. Please explain how
the benefits of and the contributions to the PAYG system should vary with income.
2. [25 points] The SMarT (Save More Tomorrow) saving plan involves newly hired
workers signing a contract with their employer that states that if their salary will
increase in the future, then half of their salary is put in a pension saving account
unless the worker indicates at any time that he or she prefers to have the increase
in salary immediately paid out so that it can be used for consumption. Please
explain why some economists thought this plan would not affect saving for
retirement while others argued it would increase pension saving. Please provide
one reason why this scheme may be more effective with high-income workers (who
have invested a lot in human capital) than with low-income workers (who invested
less in human capital), and one argument why it may be less effective with highincome workers than with low-income workers.
3. [25 points] The Dutch government provides a flat retirement benefit to all citizens
older than 65 in the public pay-as-you-go retirement system (AOW). Some people
are arguing that the Dutch government should provide the public retirement benefit
only to poor citizens by means-testing the AOW benefit. Provide two arguments
against this proposal and two arguments in favor.
4. [25 points] Explain the difference between systematic (i.e. macro) and idiosyncratic
(i.e. micro) risk. Explain the reasons why people are typically not fully shielded
against these risks. In other words, why are people exposed to both systematic and
idiosyncratic risks in a market economy? Explain how longevity risk typically has
both a systematic and an idiosyncratic component and why longevity risk is not
fully insured in practice.
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PART 2: RISK SHARING WITH FUTURE GENERATIONS (50 points)
Consider a model with two non-overlapping generations: a currently-living generation and a future
generation. Both generations are equal in size, and for simplicity this size is normalized to unity.
The economy can be in two possible states: a state with an economic crisis and a state without
one. The currently-living agent has accumulated financial assets as a result of savings in the
past. The value of these financial assets depends on the realization of the state of the economy.
The value of financial assets of the currently-living generation and the future generation is
c
denoted by W and W
f
respectively. The realizations of the value of financial assets of the
currently-living generations are denoted by:
10 withcrisis
Wc  
30 without crisis
The future generations do not own any financial assets yet ( W
f
 0 ) and are thus not affected by
the impact of a current economic crisis on financial assets.
In addition to financial wealth, the generations also have human wealth. The value of the human
c
wealth of the currently-living generation is equal to H =10, regardless of the state of the
economy that materializes. The value of the human wealth of the future generation is equal to
H f = 20, regardless of the state of the economy that materializes.
The consumption of both the currently-living and the future generation is equal to their total
c
f
wealth (the sum of financial wealth and human wealth) and is denoted by C and C
respectively. Assume that the preferences of both agents are given by expected utility over
consumption and that the utility function is characterized by constant relative risk aversion (where
the superscripts c and h denote current and future generations, respectively):
  C c 1
U  E
 1 

c

 W c  H c 1
  E


1 


4




  C f 1
U  E
 1 

f

 W f  H f 1
 E


1 



,


in which  represents the coefficient of relative risk aversion, which is the same for both agents.
Assume that agents are risk averse:   0 (and   1 ).
c , crisis
c , no crisis
Furthermore, let C
and C
denote the consumption level of the currently-living
generation if there is a crisis and if there is no crisis, respectively. Similarly, let C f ,crisis and
C f , no crisis denote the consumption level of the future generation if there is a crisis and if there is
no crisis, respectively..
In the following questions, ex-ante refers to the situation in which it is not yet known whether or
not there is a financial crisis.
a) [10 points] Explain why the optimal ex-ante risk-sharing solution satisfies:
C f ,crisis
C c ,crisis
40


f , no crisis
c , no crisis
C
C
60
b) [10 points] Suppose that the risk-sharing solution satisfies the ex-ante optimality
condition in equation a) and also assume that both agents are not worse off (in
terms of ex-ante utility) in comparison to the situation without risk sharing. Explain
why wealth is transferred from the future generation to the currently-living
generation if there is an economic crisis, while wealth is transferred from the
currently-living generation to the future generation in the absence of an economic
crisis.
c)
[10 points] Suppose that a risk-sharing contract satisfies the condition in equation
a). In addition, suppose that the risk-sharing contract is designed such that the
expected consumption level of the future generation remains unaffected. Is the
future generation willing to participate in this risk-sharing contract voluntarily from
an ex-ante perspective? Please explain.
d)
[10 points] Suppose that a risk-sharing contract satisfies the condition in equation
a) and also satisfies the condition that both generations are better off in terms of
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ex-ante utility. Please explain why this risk-sharing contract may not be feasible in
the absence of government intervention.
e) [10 points] It has been assumed that the value of the human capital of both
generations is not dependent on the outcome of the economy (i.e. whether or not
there is a crisis). Suppose now that the human capital of the currently-living
generation is worth only 5 in the event of an economic crisis, while its human
capital is worth 15 if there is no crisis (instead of 10, regardless of whether or not
there is a crisis). Similarly, suppose that the human capital of the future generation
is worth only 5 in the event of an economic crisis, while its human capital is worth
35 if there is no crisis (instead of 20 regardless of whether or not there is a crisis).
Please explain what the optimal ex-ante risk-sharing solution (in terms of the ratio
between optimal consumption levels in the two states) looks like in this situation.
Explain which generation transfers resources to which generation in which state of
the economy.
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PART 3: CONSUMPTION RESPONSE TO SHOCK IN INTEREST RATES (50 points)
Consider a model in which an individual saves over the life cycle on an individual account.
Assume that there are no intergenerational or intragenerational transfers. For simplicity, our
model assumes that the age of retirement and the age of death are both predictable. The
individual works during a period of M = 40 years and is subsequently retired during a period of
N = 20 years. Bequest motives are absent.
The after-tax wage rate W during the working career is constant and riskless. Human capital is
thus paid out in the form of constant wage income until it is fully depleted at the age of retirement.
Moreover, labor-market risks are absent. Labor supply L is fixed.
We assume that the individual has access to the capital markets only
through her or his individual retirement account. Consumption Ct during the active period (i.e.
when working) equals wage income minus pension savings. Consumption Ct during
retirement is given by the pension payoffs that are generated from savings plus accumulated
interest income. The pension savings of the individual are fully invested in the risk-free return r
that is fixed and determined exogenously on global capital markets.
a) [10 points] Explain (in words) why the budget constraint of an individual who starts
his or her working career is given by:
M
M N
0
0
WL  e  rs ds 

e  rs Cs ds
Individuals aim to maximize lifetime utility, which is the weighted sum over time of expected utility
at each point in time of the life cycle:
M N
U

e   s u (Cs )ds .
0
Utility at a point in time depends only on consumption at that time. The weights of future expected
utilities decline exponentially at the so-called rate of time preference   0 . Hence, people are
impatient: at equal levels of consumption a marginal unit of future consumption adds less to utility
than current consumption does. Preferences feature positive and constant relative risk aversion
  0 (with   1 ):
u (Ct ) 
1
(Ct )1
1 
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Positive risk aversion implies that individuals have a taste for moderation across time and across
contingencies. They prefer a smooth consumption level (with a low variance) rather than a highly
volatile consumption stream (with a high variance) over time or across contingencies. The taste
for moderation across time is inversely related to the intertemporal elasticity of substitution, which
equals ( 1/  ) in this utility specification. An individual exhibiting a low intertemporal elasticity of
substitution ( 1/  ) prefers a stable level of consumption over time.
It can be shown that if the interest rate is constant over the life cycle, the optimal consumption
path is characterized by:
dCt / dt r  

.
Ct

b) [10 points] Provide the economic intuition behind the equation above.
Consider the following scenario. The individual is saving on the basis of an optimal savings
strategy under the assumption that the interest rate r is exogenous and constant. Then,
unexpectedly, there is a decline in the interest rate from r to r  r during the working period of
the individual. After the unexpected shock, the interest rate will remain permanently fixed at the
new level r .
c) [10 points] Explain (in words) that the substitution effect associated with the
decline in interest rates causes the optimal savings level to decrease. Also explain
that a permanent decline in interest rates of 1 percentage point leads to an increase
in the optimal consumption level of approximately 1 T as a result of the
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substitution effect in the model for an individual with a remaining lifespan equal to
T and an intertemporal substitution elasticity equal to 1
2
.
[Hint: use the optimal consumption rule of the previous question at a fixed value
for the present value of future consumption].
d) [10 points] Explain (in words) that the income effect associated with the decline in
interest rates causes the optimal savings level to increase. Also explain that during
the working life a decline in interest rates of 1 percentage point leads to a reduction
in the optimal consumption level by approximately 10% as a result of the income
effect in the model. [Hint: compare the duration of consumption with the duration
of labor earnings].
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e) [10 points] The answers to the previous two questions imply that the total effect
(substitution effect and income effect together) on savings in response to the
decline in the interest rate is ambiguous and depends on age during the working
period. Explain why it becomes more likely that the substitution effect dominates
the income effect if the shock occurs early during the working life. Explain why the
total effect is (approximately) constant (but ambiguous) during the retirement
period.
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