RESIT
Name of subject
: The Economics and Finance of Pensions
Subject code
: 323068
Date of examination
: 20-01-2015
Length of examination : 3 hours
Lecturer
: Lans Bovenberg
ANR: 148199
Roel Mehlkopf
ANR: 694448
Telephone number of 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.
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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 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 PAYG system is more vulnerable to aging than a funded
pension system. Please describe under which conditions this statement is
true and under which conditions it is false.
2. [25 points] Somebody argues that an EET system taxes the return on
pension saving and therefore discourages pension saving, compared to the
situation without taxes and transfers. Explain under which circumstances
this statement is true or not, and why. [Hint: assume that tax revenues are
returned to households in the form of lump-sum transfers]
3. [25 points] Give three reasons for why the government may want to force
people to save part of their labor income for old-age pensions. Describe the
conditions, under which forcing people to save does not lead to more
private savings.
4. [ 25 points] Explain how annuities insure against longevity risk. Explain why
annuities are called inverse life insurance. Explain why people tend to
insure mortality risk (i.e. buy life insurance) in the beginning of their
working career when they have young children whereas people tend to
insure longevity risk (i.e. buy annuities) when they retire from the labor
force at older ages.
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PART 2: FERTILITY SHOCK (50 points)
Consider a small, open economy where the constant return rs in any period
s is
exogenously determined on world capital markets ( rs  r  0 ). In any period, two
overlapping generations are alive: a young and an old generation. Every generation lives
for two periods. The population grows at a constant net rate n  0 per generation (i.e.
N s  (n  1) N s 1 for every period s), where N s denotes the number of people entering the
labor force at the beginning of period
s . Assume that r  n .
The welfare state provides a public pension, which pays out an exogenous defined benefit
X to every surviving old member of society. These public pension benefits are financed
by taxes on the young generation in such a way, that the government budget constraint is
balanced in every period. The tax levied on each young person at time s is denoted by Ts
The government’s balanced budget constraint in any period
s is given by
Ts N s  XN s 1 .
The balanced budget constraint states that total tax income (left-hand side) should equal
total expenditures on benefits (right-hand side). Rearranging the balanced budget
constraint yields the tax level T that finances the defined benefit X:
T
X
.
(1  n)
The public pension scheme is introduced at time t . The generational account of the
generation born at time t 1 equals
GAt 1 
X
.
1 r
It can be derived that the generational account of the generation born at time s  t
equals
GAs 
( r  n )
X.
(1  r )(1  n)
a) [10 points] Provide the economic intuition for why the generational accounts
of the generations born at time s  t are negative.
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Assume that at time t  m where m  1 there is a fall of fertility in the population. That is:
generations born at time t  m or later are 1  nˆ times larger than their preceding
generation rather than 1  n times larger, where n̂  n . More formally, we have that
N s  (1  nˆ ) N s 1 if s  t  m
N s  (1  n) N s 1 if s  t  m
.
It can be derived that the generational account of the generations born at time t  m and
afterwards worsens by
n  nˆ
X due to the fall in the fertility level, if the old-age
(1  nˆ )(1  n)
pension system is defined benefit (i.e. the contribution level rather than the old-age pension
level balances the budget in the PAYG system).
b) [10 points] Provide the intuition for why the generational accounts of the
generation born at time t  m and afterwards worsen due to the fall of
fertility.
c) [10 points] Explain in words why in a defined-contribution system (that fixes
T independent of fertility (i.e. the benefit level rather than contribution level
balances the budget in the PAYG system)) a fall of the fertility level affects
also the generational account of the generation born at time t+m-1. Explain
the sign of the impact on this generational account.
d) [10 points] After the fall of the fertility level, is the PAYG element in the
defined-contribution scheme smaller or larger compared to the case in which
we have a defined-benefit scheme? What does this imply for the generational
account of future generations if we compare a defined-contribution scheme
to a defined-benefit scheme? Explain your answers.
e) [10 points] Explain how the intergenerational distribution that results from the
fall in the fertility level differs between the defined-contribution setting and
the defined-benefit setting. Also, provide two reasons why in times of falling
fertility it may be preferable to have a defined-contribution scheme rather
than a defined-benefit scheme in terms of intergenerational fairness and
efficient intergenerational risk sharing.
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PART 3: RISK SHARING (50 points)
Consider a model with two countries: Home and Foreign (variables for Foreign are
generally denoted by a *). There are N households in the Home country and N
*
households in the Foreign country. The model has one period. The preferences of Home
and Foreign households are given by the utility functions:
C  ,
C1v
*
*
and u  C  
u C  
1  v*
1 v
*
* 1 v
*
respectively, where 0  v, v  1, and C and C represent, respectively, consumption of
*
Home households and consumption of Foreign households.
Each household owns an exogenous capital stock that is normalized to unity. Capital
owned by Home households is sold on the capital market for a rate of return R . Capital
*
owned by Foreign households is sold on the capital market for a rate of return R .
Assume that there are open, international capital markets, so that the rates of return in
Home and Foreign countries are equal ( R  R ). There are two states of the world: a bad
*
state in which the returns are low ( R  R*  RL ) and a good state in which the returns are
high R  R*  RH  RL ).
The expected utility of a Home household is given by:
  u  CL   1     u  CH  ,
where
 is the probability of the bad state occurring and CL and CH denote consumption
of a Home household in the bad state and the good state, respectively. CL* and C H * are
defined analogously for a Foreign household.
The international resource constraints for the good state and bad state of the economy are
given by:
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NCL  N *CL*  NRL  N * RL .
NCH  N *CH*  NRH  N * RH .
Assume that Home households have the same degree of risk aversion as Foreign
households ( v  v ). In that case, optimal risk sharing implies:
*
CL CL*

CH CH *
a)
[10 points] Explain the intuition behind the optimal risk sharing equation by
considering the case when the equation above does not hold. Explain why
no transfers between the two countries are required to attain the optimal risk
sharing solution.
The optimal risk sharing condition can be rewritten (by substitution of the international
resource constraints into the optimal risk sharing condition) as:
CL RL

CH RH
b) [10 points] Explain the intuition behind this result.
For the following question, assume that Home households are more risk averse than
Foreign households ( v  v ). In that case it follows that the optimal risk sharing implies:
*
CL  CL* 


CH  CH * 
v* / v
.
c) [10 points] How does the optimal allocation of risk change as a result of Home
households being more risk averse than Foreign households (ceteris paribus
all other elements such as the rates of return in both states)? In particular,
who transfers resources to whom in which state?
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Suppose from now on again that Home and Foreign households are equally risk averse
( v  v ).
*
d) [10 points] Explain how and why the amount of risk that each Home
household should bear (as measured by relative consumption across the two
states, CL / CH ) changes as ceterus paribus (i) the population of Foreign
households increases and (ii) the return in the good state increases.
e)
[10 points] How do these results (on the impact of more Foreign households
and the impact of a higher return in the good state) change if Home
households are more risk averse than Foreign households?
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