Time Inconsistent Preferences and Social

Security

By Imrohoroglu, Imrohoroglu and Joines

Presented by Carolina Silva

11/9/2004

1.Introduction

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:

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 Discrete time Stationary overlapping generations economy n: population growth rate J: maximum possible life span : time invariant conditional survival probability

from age j-1 to j. Closed economy

j

2.2 Preferences

Jjjj lc

1,

Jjjj lc

1,

*j

Individual of age preferences over the sequence are given by:

So, if we do not have that , then preferences are time inconsistent, because the valuation of depends on the age .

*j

1 and fb

regret 1 :2

:1

effect

effect bf

2.2 Preferences

Note from (2) that: the summation on the left do not affect

behavior, but it is important for welfare evaluation.

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’.

1

2.2 Preferences

Finally they assume:

1

)1(),(

11jj

jj

lclcu

Where is the coefficient of relative risk aversion and is consumption’s share in utility.

2.3 Measures of Utility

Individuals of different ages need not to agree on the valuation of …how to measure welfare effects?

Jjjj lc

1,

1. Compute welfare measures as viewed from each age j*: average of individual , average wrt the stat. distribution of agents of age j* across employment and asset state.

2. weighted average of , W, with the weight on being the uncon. probability of surviving from birth to age j*.

*jW

*jW

*jW

*jU

2.4 Budget Constraint Agents are subject to earnings uncertainty:

they receive shocks ,

can work1

unemployed0js

js

js follow a Markov process.

2.4 Budget ConstraintLet: =stock of assets held at the end of age j. =wage per efficiency unit of labor =efficiency index of an agent of age jTj = taxes paid by an agent of age jQj = retirement benefitsMj =unemployment insurance =lump sum, per capita government transfer

jjjjjjjjj MQTlwsarac 1)1(

jawj

constraint borrowing :0ja

2.4 Budget ConstraintWhere: unemployment benefits are such that:

At any 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.

0s

1 0

jjj lw

sM

1 Rjj

2.4 Budget Constraint

Taxes paid satisfy:

Where denote the tax rates for consumption, capital income, labor income, social security and unemployment insurances, respectively, and denotes accidental bequests.

jjusljajcj lwracT )(1

uslac and ,,,

2.5 Individual’s Dynamic Program

If then the agent’s dynamic program is a standard backward recursion.

If then we have to attribute a particular belief to the agent concerning how he thinks his future selves will behave:

Naïve Sophisticated

1

1

2.6 Aggregate Technology

Cobb-Douglas production function:

)1,0( with 1 LBKY

Where B>0 is assumed to grow at a constant rate of steady state per capita output grows at rate . Aggregate capital stock K depreciates at rate d.

Firm maximization requires:

1

and )1(L

KBwd

L

KBr

0

2.7 Government

Revenue from 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, ,

and unemployment, , insurances; each balances every period.

alc and ,

su

2.8 Stationary Equilibrium

Welfare comparisons will be made between stationary equilibria with different . A stationary equil. consists of:

s

rwsystemprice

xtypesagentofmeasure

LCArulesdecision

policygovernment

J

jj

jjj

usla

,:

)(:

,, :

,,,,,, G, :

1

c

3.Calibration Model period: 1 year

Set as to match empirical wealth/output ratio to 2.52:

and ,f

0.60 ,90.0 ,85.0

3 ,2 ,1

f

4.1 Results for the time consistent case

As : monotonic increase in K, I (=

saving rate), C and Y.

Welfare criterion: expected lifetime utility as viewed from 21 welfare is maximized at .

Compensating variation: welfare costs increases faster than linearly in .

1 fb

0% to10% from s

0s

s

%)91.5%10( CVs

4.2 Benchmark: Perfect commitment technology with

From 21 until death, these agents are committed to follow decision rules implied by behavior is the same as exponential.

1

Compared with an economy with no commitment technology, consumption increases at all ages, and this increase is larger during retirement years

So, steady sate welfare costs to quasi hyperbolic discounters of their time inconsistent behavior are substantial

)( wK

fb and '1 tedsophistica'

4.3 Effects of social security in a quasi hyperbolic economy

Here we consider different economies with %10s

(normalization: K, Y and C are 1 without uss)

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: time consistent behavior, but an old

individual may regret having consumed so much when young.

They define the degree of this type of regret, , implicitly by

1 and bf

0ˆ

)1(1 where)ˆ1(

1 fb

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, , to the extreme case , they get a slight increase in the preference for uss

1 b

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( fb

4.4 Welfare Analysis

Why is uss not effective in offsetting the utility losses due to ?

As we saw, uss depresses K in economies with , 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)

1

1

4.5 Sensitivity Analysis 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 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.

And with naïve and ? A tax of 10% raises welfare as viewed from all ages uss

does significantly raise welfare.

Problem with and sophisticated agents could not be solved. But the fact that naïve and sophisticated behave similar with higher suggest that welfare effects for both types might continue to be similar with lower .

60.0

60.0

s'

s'

5. Concluding Remarks

costs. welfarelsubstantia : 20%in 9.0 )1 s K

1

9.0

2) Social security is a poor substitute for perfect commitment technology for sophisticated agents with .

3) If preferences exhibit only effect 2 ( ) :

• uss generally does not raise welfare for agents with , either for naïve or sophisticated.

• uss does raise substantially the welfare of naïve agents with .

6.0

1

5. Concluding Remarks

4) If preferences exhibit only effect 1 : Ex ante annual discount rate must be at

least 8% greater then seems warranted ex post for a majority of population to prefer

Adding slightly increases the preferences for uss.

%10s

b to1

)( bf