Cremer, Helmuth

De Donder, Philippe

Maldonado, Dario

Pestieau, Pierre

English

Research Papers in Economics

Voting over type and generosity of a pension
system when some individuals are myopic:
Technical appendix
H Cremer, Ph De Donder, D Maldonado, P Pestieau
March 13, 2007
This appendix contains a presentation of the results obtained with the two
extreme possibilities in the family of CES utility functions: the case of perfect
substitution between consumptions in the two period of life (" = 1) and the case
of no substitution at all (" = ￿1).
1 Utility function with perfect substitution
across periods
In that case, we have that
u(c;d;‘) = w(1 ￿ ￿)‘ ￿ s ￿ ‘
2=2 + s + p:
It is easy to see that saving plays no role in that case. Without any role for saving,
we now show that the distinction between myopic and far-sighted agents vanishes,
in the sense that they have exactly the same preferences, for any given level of
productivity w.
In the case of a Beveridgean system, the equilibrium labor supply of both types
of agents is the same. Since no one needs to save to equalize marginal utility of
consumption across periods, the most preferred contribution rate is given by
￿
￿(￿;0) =
1 ￿ ￿
2 ￿ ￿
: (1)
In the Bismarckian case, far-sighted agents are indi⁄erent between any con-
tribution rates lower than 1/4. The main di⁄erence with the logarithmic utility
case comes from myopic individuals, who now dislike any positive tax rate: forced
saving has lost all appeal at the voting stage while keeping the disadvantage of
1inducing myopics to work less than would be optimal. The majority chosen level
of the Bismarckian contribution rate is thus zero.
When voting over the type of pension system, all individuals behave as far-
sighted individuals who save did in the logarithmic utility case. The individual
who is indi⁄erent between the two systems is given by
~ ￿R =
2 ￿ 2￿V(0;￿)
2 ￿ ￿V(0;￿)
, 0 < ￿
V(0;￿) ￿ 1=4; (2)
with all ￿ < ~ ￿R preferring Beveridge to Bismarck. We can then proceed as in
the logarithmic utility case to show that a majority always prefers Beveridge to
Bismarck.
Result 1 In a society composed of myopic and far-sighted agents with a utility
function showing perfect substitution across periods and a positively skewed distri-
bution of abilities, the majority voting equilibrium pension is Beveridgean with a
contribution rate equal to (1 ￿ ￿
med)=(2 ￿ ￿
med).
2 Utility function with no substitution across
periods
In that case, we have that
u(c;d;‘) = min[w(1 ￿ ￿)‘ ￿ s ￿ ‘
2=2;s + p]:
We ￿rst look at individuals￿most preferred contribution rate in both systems
before turning to majority voting.
2.1 Individuals￿ most preferred contribution rate under
the Beveridgean system
The myopic individual￿ s most preferred ￿ equalizes his consumption level across
the two periods given that he is not saving and that his labor supply is given by
‘￿
i = wi (1 ￿ ￿): We obtain that
w2
i (1 ￿ ￿￿(￿;0))
2
2
= ￿
￿(￿;0)(1 ￿ ￿
￿(￿;0))Ew
2;
which gives
￿
￿(￿;0) =
￿
2 + ￿
:
2The intuition for why the most preferred contribution rate increases with ability is
clear: a higher ability individual needs to pay a higher fraction of his ￿rst period
income in order for the net-of-tax and transfer income to be equalized across the
two periods.
Far-sighted individuals who do not save because of credit constraints are in
the same position as myopic agents: they use the pension system to equalize
consumption across periods. They thus share the same ￿￿(￿;0) as the myopic
individuals. Far-sighted individuals who save rather choose the contribution rate
that maximizes their total income over the two periods, using individual savings
to equalize consumption across income. They thus behave as in the logarithmic
utility case and their most preferred ￿ is given by equation (1). It is easy to see
that individual with ￿ < 2=3 do not save while individuals with ￿ > 2=3 do save
at their most preferred contribution rate.
2.2 Individuals￿ most preferred contribution rate under
the Bismarckian system
The myopic individual￿ s most preferred ￿ equalizes his consumption level across
the two periods given that he is not saving and that his labor supply is wi (1 ￿ ￿).
We then obtain
w2
i (1 ￿ ￿￿(￿;1))
2
2
= ￿
￿(￿;1)(1 ￿ ￿
￿(￿;1))w
2
i;
which gives
￿
￿(￿;1) =
1
3
:
The most preferred contribution rate is then the same for all myopic individuals.
As for far-sighted individuals, the analysis is the same as in the logarithmic
utility case: they are indi⁄erent between any two contribution rates lower than
1/4 (since for any such rates public saving perfectly crowds out private saving)
and dislike any rate greater than 1/4:
￿
￿(￿;1) 2 [0;
1
4
]:
We now turn to the study of majority voting equilibria. We ￿rst look at
the majority voting equilibrium contribution rate in a Beveridgean system, before
turning to the Bismarckian system. Finally, we look at the result of a vote between
Beveridge and Bismarck as a function of the proportion of myopics in society.
32.3 Majority voting equilibrium contribution rate in a
Beveridgean system
The most preferred Beveridgean contribution rate increases monotonically with
ability for myopic individuals, but it ￿rst increases then decreases for far-sighted
individuals (see Figure 1 panel B in the paper). If far-sighted are numerous
enough (i.e., if ￿ is small enough), we have a majority voting equilibrium of
the type ￿ends-against-the-middle￿ , where low ability myopics and far-sighted
together with high ability far-sighted would like a lower contribution rate (the
￿rst group because the contribution rate is high enough for their second period
consumption to be higher than their ￿rst period￿ s, the second group because
they would rather save privately), while middle abilities myopics and far-sighted
would like a higher contribution rate. Formally, the majority voting equilibrium
contribution rate is given by
(
￿V(0;￿) = ￿￿
2+￿￿ =
1￿￿+
R
2￿￿+
R
;
F(￿
￿) + (1 ￿ ￿)(1 ￿ F(￿
+
R)) = 1=2;
where ￿
￿ is the median voter among the myopic and far-sighted credit constrained
individuals while ￿
+
R is the median individual among the non credit constrained
far-sighted. The second equation guarantees that these individuals are median,
in the sense that half the population would like a higher contribution rate. It is
clear that we have 0 < ￿V(0;￿) < 1=4, and that the majority equilibrium tax rate
is weakly increasing in ￿.
On the other hand, if the myopics are numerous enough, they may be able
to push the Beveridgean rate above 25%. In that case, the majority voting equi-
librium is not of an ￿ends-against-the middle￿kind anymore, but rather we have
that all far-sighted together with the myopics with ￿ < ￿
￿ would like a lower
rate while myopics with ￿ > ￿
￿ would like a bigger contribution rate. ￿
￿ is then
obtained from
￿(1 ￿ F(￿
￿)) = 1=2:
It will prove handy to compute the utility levels reached by di⁄erent groups
of individuals when the Beveridgean system is adopted. First, the myopic and
far-sighted individuals with ￿ < ￿
￿ would rather prefer a lower tax rate, because
the majority chosen one lowers their ￿rst period consumption level below their
second period one. Hence,
U
Be(￿) = c =
￿
1 ￿ ￿V(0;￿)
￿2
2
w
2
i
for any individual (far-sighted or myopic) with ￿ < ￿
￿: Second, the far-sighted
with ￿ > ￿
￿ do save in order to equalize their consumption levels across periods.
4Their utility is then
U
Be(￿) = c = d =
￿
1 ￿ ￿V(0;￿)
￿2
4
w
2
i +
￿V(0;￿)(1 ￿ ￿V(0;￿))
2
Ew
2:
Finally, a myopic individual with ￿ > ￿
￿ does not save although his ￿rst period
utility is higher than his second period￿ s. His utility is then
U
Be(￿) = d = ￿
V(0;￿)(1 ￿ ￿
V(0;￿))Ew
2:
2.4 Majority voting equilibrium contribution rate in a Bis-
marckian system
The majority voting equilibrium contribution rate in a Bismarckian system is 1/4
if ￿ ￿ 1=2 and 1/3 otherwise. The utility level attained by all myopics is
U
Bi =
￿
d = 3
16w2
i if ￿V(1;￿) = 1=4;
c = d = 2
9w2
i if ￿V(1;￿) = 1=3:
As for far-sighted, their utility is
U
Bi = c
Bi = d =
w2
i
4
if ￿
V(1;￿) = 1=4:
On the other hand, if ￿ > 1=4, their ￿rst period utility becomes lower than
their second period￿ s. This has an impact on their labor supply, since they only
contemplate the impact of any labor supply decision on the ￿rst period only. In
other words, they behave like the myopic agents do and decrease their optimal
labor supply to li = wi(1 ￿ ￿): In that case,
U
Bi = c = d =
2
9
w
2
i if ￿
V(1;￿) = 1=3:
2.5 Voting over Beveridge vs Bismarck
The result of the vote will depend on whether myopics or far-sighted can impose
their view on the Bismarckian contribution rate. We start with the situation where
far-sighted are more numerous (￿ ￿ 1=2) before turning to the other possibility.
2.5.1 A majority of far-sighted: ￿ ￿ 1=2
We start with the preferences of far-sighted for the two systems before turning to
the myopics. We then analyze which among the two systems gets a majority of
votes, as a function of the proportion of myopics.
5We ￿rst show that all far-sighted who do not save (i.e. with ￿ < ￿
￿) prefer
Beveridge to Bismarck, i.e. that
￿
1 ￿ ￿V(0;￿)
￿2
2
w
2
i ￿
w2
i
4
;
, ￿
V(0;￿) ￿ 1 ￿
p
1=2
which is always true given that ￿V(0;￿) ￿ 1=4:
For far-sighted with ￿ > ￿
￿, the analysis is exactly the same as in the case
of a logarithmic utility function since they equalize consumption across the two
periods. The threshold ￿ saver who is indi⁄erent between Beveridge and Bismarck,
denoted by ~ ￿R, is given by (2).
As for myopics, we ￿rst show that all the individuals with ￿ < ￿
￿ (i.e., those
who would like a lower Beveridgean tax rate) prefer Beveridge to Bismarck:
￿
1 ￿ ￿V(0;￿)
￿2
2
w
2
i ￿
3
16
w
2
i;
, ￿
V(0;￿) ￿ 1 ￿
p
3=8;
which is always true given that ￿V(0;￿) ￿ 1=4: Observe that myopics with ￿ < ￿
￿
di⁄er only from far-sighted of the same productivity by the utility level they get
in the Bismarckian system. This utility is smaller for the myopics since they don￿ t
work enough in their ￿rst period of life. Given that all far-sighted with low ability
already prefer Beveridge to Bismarck, it is no surprise that myopics share the
same preference.
Myopics for which ￿ > ￿
￿ prefer Beveridge to Bismarck if
￿
V(0;￿)(1 ￿ ￿
V(0;￿))Ew
2 ￿
3
16
w
2
i;
i.e., if their ability is lower than a threshold
~ ￿M =
16
3
￿
V(0;￿)
￿
1 ￿ ￿
V(0;￿)
￿
:
This threshold is always lower than 1 (given that ￿V(0;￿) < 1=4 with a ma-
jority of far-sighted) and increases with ￿V(0;￿) for the reasons mentioned in the
logarithmic utility case.
We can now look at the result of a majority vote over ￿, given ￿. If ￿ = 0,
there is no myopic individual and all far-sighted individuals with ￿ < ~ ￿R prefer
Beveridge to Bismarck, as in the logarithmic utility case. However, we cannot
simply reproduce the analysis contained in the logarithmic utility case to prove
6that they constitute a majority, because of the ends-against-the-middle property
of the majority voting equilibrium Beveridgean contribution rate: ￿V(0;￿) is not
the most-preferred ￿ of the median-endowed individual anymore, and one cannot
simply compare ~ ￿R with ￿
med. Rather, we have to compare ~ ￿R with ￿
+
R. Since,
at equilibrium, F(￿
￿) + (1 ￿ F(￿
+
R)) = 1=2, it follows that 1 ￿ F(￿
+
R) ￿ 1=2.
The analysis contained in the logarithmic utility case shows that the curve which
depicts ~ ￿R (labeled R in Figure 2 of the paper) is everywhere to the right of the
curve on which ￿
+
R is situated. Hence,1￿F(~ ￿R) < 1￿F(￿
+
R) ￿ 1=2, which means
that F(~ ￿R) > 1=2 and that Beveridge is preferred by a majority.
We now look at what happens as the proportion of myopic individuals in-
creases. As ￿ increases, ￿V(0;￿) weakly increases, which increases the political
support for Beveridge among the myopics but decreases it among the far-sighted.
As in the logarithmic utility case, a majority of the population may support
Bismarck if there are enough myopics in the population. Finally, observe that
￿V(0;1=2) < 1=4, because all far-sighted plus many low ability myopics prefer a
lower-than-25% Beveridgean contribution rate.
We now look at what happens when the proportion of myopics is pushed above
one half.
2.5.2 A majority of myopics: ￿ > 1=2
We start with the preferences of far-sighted for the two systems before turning to
the myopics. We then analyze which among the two systems gets a majority of
votes, as a function of the proportion of myopics.
We ￿rst show that all far-sighted who do not save with the Beveridgean system
(i.e. with ￿ < ￿
￿) prefer Beveridge to Bismarck, i.e. that
￿
1 ￿ ￿V(0;￿)
￿2
2
w
2
i ￿
2
9
w
2
i; (3)
where the right hand side gives the utility for far-sighted under the Bismarckian
system. This inequality is always true given that ￿V(0;￿) ￿ 1=3: The intuition for
this result is as follows: in both systems, these far-sighted are credit constrained.
In that case, they o⁄er the same amount of labor as a myopic individual of the
same ability. They prefer Beveridge to Bismarck because the former o⁄ers them a
lower contribution rate and thus increases their ￿rst-period consumption compared
to the latter.
For far-sighted with ￿ > ￿
￿, the analysis is the same as in the case of a loga-
rithmic utility function (since they equalize consumption across the two periods),
except that they get a lower utility with the Bismarckian system. The threshold
7￿ saver who is indi⁄erent between Beveridge and Bismarck is obtained from
￿
1 ￿ ￿V(0;￿)
￿2
4
w
2
i +
￿V(0;￿)(1 ￿ ￿V(0;￿))
2
Ew
2 =
2
9
w
2
i:
If ￿V(0;￿) is su¢ ciently low (lower than 0.057), all far-sighted prefer Beveridge to
Bismarck. Above that threshold, the value of ￿ of the far-sighted who is indi⁄erent
is decreasing in ￿V(0;￿), and tends to 1 as ￿V(0;￿) tends towards 1/3. In other
terms, there is always a majority of far-sighted who prefer Beveridge to Bismarck.
Observe that myopics with ￿ < ￿
￿ do not di⁄er from far-sighted of the same
productivity here, since they have the same labor supply and are both credit
constrained in the two systems. We then obtain that all such myopics prefer
Beveridge to Bismarck.
Myopics for which ￿ > ￿
￿ prefer Beveridge to Bismarck if
￿
V(0;￿)(1 ￿ ￿
V(0;￿))Ew
2 ￿
2
9
w
2
i;
where the right hand side, giving their utility under the Bismarckian system, is
greater than in the corresponding case when ￿ < 1=2. The threshold ability level
is
~ ￿M =
9
2
￿
V(0;￿)
￿
1 ￿ ￿
V(0;￿)
￿
;
which is then lower than its corresponding value when ￿ < 1=2, for a given value
of ￿V(0;￿). This threshold is always lower than 1 (given that ￿V(0;￿) < 1=3 with
a majority of myopics) and increases with ￿V(0;￿) for the reasons mentioned in
the logarithmic utility case.
We can now look at the result of a majority vote over ￿, given ￿ ￿ 1=2. If
￿ = 1=2, there is no discontinuity in the majority voting equilibrium Beveridgean
contribution rate, which is the one most preferred by far-sighted and myopics who
are just credit-constrained (￿ = ￿
￿) and far-sighted with ability ￿ > ￿
￿, who
are saving. On the other hand, there is a discontinuity in the majority voting
equilibrium Bismarckian rate, which jumps from 1/4 to 1/3 as myopics become a
majority. As myopics get their most preferred tax rate in the Bismarckian system,
their utility makes a discontinuous upward jump in that case, and there is a
discontinuous downward jump in the proportion of myopics preferring Beveridge to
Bismarck. The opposite reaction occurs for far-sighted: they are now all worse o⁄
in the Bismarckian system (because forced to save too much), and a strictly higher
proportion of them prefers Beveridge. The total proportion in the population
voting in favor of Beveridge changes discontinuously as one crosses the ￿ = 1=2
line, but whether the proportion increases or decreases, and whether the majority
result changes or not depends on the distribution of abilities.
8As ￿ increases further, ￿V(0;￿) increases which increases the support for Bev-
eridge among the myopics and decreases it for the far-sighted. The net impact on
the proportion of votes for Beveridge is indeterminate. As noted in section 2.3,
it is possible that, when the proportion of myopics reaches a certain threshold,
the majority voting game on the Beveridgean contribution rate loses its ￿ends-
against-the-middle￿property, with all far-sighted preferring a lower tax rate. This
does not generate any discontinuity in either equilibrium contribution rate or in
the proportion of votes in favor of one system. Finally, as ￿ gets close enough to
one, we can prove that the majority chosen system is the Beveridgean one. The
decisive voter is then a myopic agent who just equalizes his ￿rst and second period
consumption in the Beveridgean system. Equation (3) shows that he most prefers
Beveridge to Bismarck, and so does a majority of voters. The intuition for this is
straightforward: this myopic individual gets his most preferred contribution rate
in both systems, but in the Beveridgean system he also bene￿ts from the redis-
tribution embedded in this system. Equation (3) shows that for all myopics with
a lower ability, the redistributive bene￿t from Beveridge outweighs the fact that
the Beveridgean tax rate is higher than their most preferred one, and that they
end up being credit constrained.
We then obtain the following result.
Result 2 Assume that society is composed of a fraction ￿ (1￿￿) of myopic (far-
sighted) agents, that both kinds of agents have the same min[c;d] utility function
and the same positively skewed distribution of abilities. Then a majority of voters
prefer a Beveridgean social security system if ￿ 2 f0;1g. If both types of agents
coexist, it may be the case that a majority of voters prefer Bismarck to Beveridge.
Moreover, there is a discontinuity in the political support for either system as
myopics become the majority in society.
9

Voting over the Size and Type of Social Security when some Individuals are Myopic.

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