Transcript Slide 1

Time Inconsistent Preferences
and Social Security
By Imrohoroglu, Imrohoroglu and
Joines
Presented by Carolina Silva
11/9/2004
1.Introduction
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Social security may:
provide additional utility for individuals
who regret their saving decisions.
Substitute for missing or costly private
markets in helping in the allocation of
consumption under uncertainty.
1.Introduction
But on the other hand, social security distorts
aggregate savings and labor supply
Whether social security is welfare enhancing is a
quantitative question
1.Introduction
In this paper, authors examine the welfare effects
of unfunded social security (uss) on individuals with
time inconsistent preferences:
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Quasi hyperbolic discounting: disc. factor between
adjacent periods close by is smaller.
Retrospective time inconsistency: put relatively less
weight on the past than on the future.
2.Model
2.1 ENVIRONMENT
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Discrete time
Stationary overlapping generations economy
n: population growth rate
J: maximum possible life span
 j : time invariant conditional survival probability from
age j-1 to j.
Closed economy
2.2 Preferences
Individual of age
are given by:
j
*
J
preferences over the sequence c j , l j j 1
So, if we do not have that b   f and   1, then preferences
are time inconsistent, because the valuation of c j , l j Jj 1 depends
on the age j * .
effect 1 :  f   b
effect 2 :   1
regret
2.2 Preferences
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Note from (2) that:
the summation on the left do not affect behavior,
but it is important for welfare evaluation.
  1 implies not only time inconsistent preferences,
but also time inconsistent behavior:
optimal policy functions derived at age j* for ages
j’>j* will no longer be optimal when the agent arrive
at age j’.
2.2 Preferences
Finally they assume:

c
u (c , l ) 
j

j

j

1 1
(1  l j )
1 
Where
is the coefficient of relative risk aversion and
consumption’s share in utility.

is
2.3 Measures of Utility
Individuals of different ages need not to agree on
J
the valuation of c j , l j j 1 …how to measure welfare
effects?
1. Compute welfare measures W j* as viewed from each age j*:
average of individual U j* , average wrt the stat. distribution
of agents of age j* across employment and asset state.
2. weighted average of W j*, W, with the weight on W j* being the
uncon. probability of surviving from birth to age j*.
2.4 Budget Constraint
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Agents are subject to earnings uncertainty: they
receive shocks s j ,
0  unemployed
sj  
1  can work
s j follow a Markov process.
2.4 Budget Constraint
Let:
a j=stock of assets held at the end of age j.
w =wage per efficiency unit of labor
 j =efficiency index of an agent of age j
Tj = taxes paid by an agent of age j
Qj = retirement benefits
Mj =unemployment insurance
 =lump sum, per capita government transfer
c j  a j  (1  r )a j 1  s j w j l j  Tj  Qj  M j  
a j  0 : borrowingconstraint
2.4 Budget Constraint
Where:
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unemployment benefits are such that:
0
Mj 
w j l j
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s 1
s0
At any j  jR  1 agents may take the irreversible decision of start
collecting uss benefits next period.
Mimic US’s social security system: piecewise-linear benefit formula, tax
applies only up to cutoff, elderly may continue to work without reduction in
benefits.
2.4 Budget Constraint
Taxes paid satisfy:
Tj   cc j  a ra j 1  ( l  s  u )w j l j  
Where  c , a , l , s and  u denote the tax rates for
consumption, capital income, labor income, social security
and unemployment insurances, respectively, and  denotes
accidental bequests.
2.5 Individual’s Dynamic Program
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If   1 then the agent’s dynamic program is a
standard backward recursion.
If   1 then we have to attribute a particular belief
to the agent concerning how he thinks his future
selves will behave:
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Naïve
Sophisticated
2.6 Aggregate Technology
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Cobb-Douglas
production function:
Y  BK1 L with   (0,1)
Where B>0 is assumed to grow at a constant rate of   0  steady
state per capita output grows at rate  . Aggregate capital stock K
depreciates at rate d.
Firm maximization requires:
K
r  (1   ) B 
L

1
K
 d and w  B 
L
2.7 Government
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Revenue from  c , l and  a
Makes purchases of goods of G each period
Any excess revenue over purchases: lump sum
transfer to agents.
Maintains a pay as you go social security,  s , and
unemployment,  u , insurances; each balances
every period.
2.8 Stationary Equilibrium
Welfare comparisons will be made between stationary
equilibria with different  s . A stationary equil. consists of:
governm ent policy: G,  c , a , l , s , u ,  ,  
decision rules : Aj , C j , L j 
m easureof agenttypes:  j ( x)j 1
J
price system : w, r
3.Calibration
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Model period: 1 year
Set  f ,  and  as to match empirical wealth/output
ratio to 2.52:
  1, 2, 3
  0.85, 0.90, 0.60
f
4.1 Results for the time
consistent case   
b
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f
   1
As  s  from10% to0% : monotonic increase in K, I (=
saving rate), C and Y.
Welfare criterion: expected lifetime utility as viewed
from 21  welfare is maximized at  s  0 .
Compensating variation: welfare costs increases faster
than linearly in  s ( s  10%  CV  5.91%).
4.2 Benchmark: Perfect commitment
technology with' sophisticated   1' and b   f
From 21 until death, these agents are committed to follow
decision rules implied by   1  behavior is the same as
exponential.
Compared with an economy with no commitment technology,
consumption increases at all ages, and this increase is larger
during retirement years ( K  w)
So, steady sate welfare costs to quasi hyperbolic discounters of
their time inconsistent behavior are substantial
4.3 Effects of social security in a
quasi hyperbolic economy
Here we consider different economies with  s
(normalization: K, Y and C are 1 without uss)
 10%
4.3 Effects of social security in a
quasi hyperbolic economy
So, while perfect commitment increases the
steady state values of K,Y,C, social security
lowers them.
Any welfare gain from uss must come from a
reallocation of consumption over the life cycle.
Age-consumption profiles
4.4 Welfare analysis
I. Only Effect 1:  f   b and   1
 time consistent behavior, but an old individual
may regret having consumed so much when
young.
They define the degree of this type of regret,ˆ  0 ,
implicitly by
 b  1(1    ˆ ) where  f  1(1   )
4.4 Welfare Analysis
If an agent of age j prefers a positive tax rate, then all older agents will prefer
that positive rate too.
4.4 Welfare Analysis
When adding effect 2,   1 , to the extreme case b  ,
they get a slight increase in the preference for uss
Is effect 2 trivial??
Not necessarily, even though effect 1 can generate much
greater disagreement between young and old selves than
effect 2 can, effect 2 in addition to influencing the valuation of
given sequences, it alter these sequences.
Scope for uss improving welfare
4.4 Welfare Analysis
II. Only effect 2 (  1  b   f )
4.4 Welfare Analysis
Why is uss not effective in offsetting the utility losses
due to   1 ?
As we saw, uss depresses K in economies with   1,
thus exacerbating any undersaving due to a low  .
As we saw in the consumption profile, consumption
rises during old age, but its gains are too small to
offset the losses due to lower consumption earlier in
life (this is as viewed from most points in life cycle)
4.5 Sensitivity Analysis
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Under small open economies, the reduction in K is
almost 3 times larger than what we obtained before:
in this case there are not changes in r to damp the
decrease in K.
Lifetime utility as viewed from all ages is
higher without uss
4.5 Sensitivity Analysis
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What happens if agents are naïve?
Using the economies of table VII, it is shown that naïve’s and sophisticated's
behavior is almost identical
welfare consequences of uss are
qualitatively the same.
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And with naïve and   0.60 ?
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A tax of 10% raises welfare as viewed from all ages  uss does
significantly raise welfare.
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Problem with   0.60 and sophisticated agents could not be solved.
But the fact that naïve and sophisticated behave similar with higher  ' s
suggest that welfare effects for both types might continue to be similar
with lower  ' s .
5. Concluding Remarks
1)   0.9  K in 20%  s : substantial welfarecosts.
2) Social security is a poor substitute for perfect commitment
technology for sophisticated agents with   1 .
3) If preferences exhibit only effect 2 ( 
 1) :
• uss generally does not raise welfare for agents with   0.9, either for
naïve or sophisticated.
• uss does raise substantially the welfare of naïve agents with   0.6 .
5. Concluding Remarks
4) If preferences exhibit only effect 1 ( f  b ) :
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Ex ante annual discount rate must be at least
8% greater then seems warranted ex post for
a majority of population to prefer  s  10%
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Adding   1 to  b   slightly increases
the preferences for uss.