# Welfare Effects of Social Security Reforms Across Europe : the Case of France and Italy

**ABSTRACT** We introduce a new hybrid approach to joint estimation of Value at Risk (VaR) and Expected Shortfall (ES) for high quantiles of return distributions. We investigate the relative performance of VaR and ES models using daily returns for sixteen stock market indices (eight from developed and eight from emerging markets) prior to and during the 2008 financial crisis. In addition to widely used VaR and ES models, we also study the behavior of conditional and unconditional extreme value (EV) models to generate 99 percent confidence level estimates as well as developing a new loss function that relates tail losses to ES forecasts. Backtesting results show that only our proposed new hybrid and Extreme Value (EV)-based VaR models provide adequate protection in both developed and emerging markets, but that the hybrid approach does this at a significantly lower cost in capital reserves. In ES estimation the hybrid model yields the smallest error statistics surpassing even the EV models, especially in the developed markets.

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**ABSTRACT:**We introduce a new hybrid approach to joint estimation of Value at Risk (VaR) and Expected Shortfall (ES) for high quantiles of return distributions. We investigate the relative performance of VaR and ES models using daily returns for sixteen stock market indices (eight from developed and eight from emerging markets) prior to and during the 2008 financial crisis. In addition to widely used VaR and ES models, we also study the behavior of conditional and unconditional extreme value (EV) models to generate 99 percent confidence level estimates as well as developing a new loss function that relates tail losses to ES forecasts. Backtesting results show that only our proposed new hybrid and Extreme Value (EV)-based VaR models provide adequate protection in both developed and emerging markets, but that the hybrid approach does this at a significantly lower cost in capital reserves. In ES estimation the hybrid model yields the smallest error statistics surpassing even the EV models, especially in the developed markets.CESifo Group Munich, CESifo Working Paper Series. 01/2009; - [Show abstract] [Hide abstract]

**ABSTRACT:**I consider a theory based on households that differ in their education, receive an unin- surable, idiosyncratic endowment of efficiency labor units, and face health and survival risks. Households also understand the link between the payroll taxes they pay and the public pensions that they receive, and decide when to retire from the labor force. I cali- brate this theory to the Spanish economy so that it replicates its demographic features, its macroeconomic aggregates and ratios, the Lorenz curves of its income and earnings distri- butions, and many of its institutional features. I show that this theory accounts for the retirement behavior of Spanish households almost exactly. I then use the model economy to study the aggregate, distributional, retirement and welfare consequences of increasing the number of years of contributions that are used to compute the pensions, and I evaluate this policy reform in the context of both the Spanish demographic and educational tran- sitions. I find that the reform increases the stock of capital by 6.3 percent, and output by 1.6 percent, but it decreases consumption per capita by 2.6 percent. The average pension decreases by more than 19 percent, and the average retirement age from 61.9 to 60.9 years. The reform also reduces the Social Security deficit from 8.6 to 5.4 percent of GDP in the year 2050. Finally, I find that the welfare loss for the population alive at the moment of the reform is 2.5 percent of lifetime consumption.

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WORKING

P A P E R

Welfare Effects of Social

Security Reforms Across

Europe

The Case of France and Italy

RAQUEL FONSECA

THEPTHIDA SOPRASEUTH

WR-437

October 2006

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Welfare Effects of Social Security Reforms Across Europe

The Case of France and Italy

Raquel Fonseca∗

Thepthida Sopraseuth† ‡

Abstract

This paper uses a calibrated life cycle model to quantify the distributional effects of Social

Security reforms. We focus on two countries, Italy and France, because they adopted two different

strategies to cope with aging. While France marginally modified its defined benefit pension plan,

Italy switched from a defined benefit pension plan to a contributive system. We find both reforms

redistribute welfare unevenly: high skilled workers are the primary winners of the French reform

and self employed individuals, especially unskilled workers, are the losers under the new Italian

Social Security arrangement. Finally, we estimate that the French reform only finances 20% of

the expected deficit. This is in sharp contrast with the Italian reform which finances the expected

deficit by cutting drastically the generosity of Social Security benefits.

∗Email : fonseca@rand.org RAND Corporation.

†Email : thepthida.sopraseuth@univ-evry.fr. EPEE, University of Evry, CEPREMAP and PSE Jourdan.

‡We are grateful to Richard Disney, Elsa Fornero, Tullio Japelli, Sergio Jiménez - Martin, Francois Langot, John

Rust and Justin Wolfers for helpful comments. We also thank the audiences for presentations of drafts of this paper at

ESEM (2005), EALE/SOLE (2005).We thank the audiences for presentations of drafts of this version at ESEM (2005),

EALE/SOLE (2005) and L&P Brown Bag seminar at RAND (2005).

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1Introduction

All industrialized countries have adopted reforms to support the “Pay as you go” (PAYG) system

jeopardized by population aging. The funding of pension systems is one of the major concern in view

of the changing demographic trends. However, beyond the solvency issue, it is crucial to quantify the

distributional effects of Social Security reforms allowing to identify winners and losers from reforms.

Some careers are naturally longer than others and some workers expect to live longer after retirement

than others. These facts underline that Social Security (hereafter SS) is not only an individual in-

surance but also a social insurance. By affecting insurance possibilities as well as savings and labor

supply, SS reforms redistribute welfare within and across generations. Redistributive considerations

also help understand political support for these reforms.

This paper investigates how changes in Social Security regimes affect workers’ welfare. We focus

on two countries and their associated reforms that occurred during the nineties: Italy and France,

because they adopted very different strategies to cope with aging. While France marginally modified

its defined benefit pension plan, a contributive system was introduced in Italy in place of a defined

benefit plan. This paper is in direct relation with a large literature, focused primarily on the United

States that uses a life cycle framework for the analysis of SS policy reforms (DeNardi, Imrohoroglu &

Sargent (1999), Conesa & Krueger (1999), Huggett & Ventura (1999) among others).1

We develop a unified theoretical framework in order to quantify the welfare effects of SS reforms.

Along the lines of Rust & Phelan (1997), Huggett & Ventura (1999) and Fuster, Imrohoroglu &

Imrohoroglu (2003), we use a fully dynamic life-cycle model, in which individuals are heterogeneous

with respect to age, wealth and labor status. Our model also allows individuals to differ in skill levels,

which permits a rich analysis of intragenerational welfare changes. Retirement and savings decisions

are endogenous decisions made under uncertainty, while agents are altruistic (with respect to their

offsprings) and face financial constraints. The model is first calibrated to match key features in the

data across heterogeneous groups within countries. We then evaluate redistributive effects of policy

reforms in steady-state after carefully mapping the "real world" Social Security reforms in France and

Italy into the model.

Our results suggest that high-skilled workers were the primary winners of the reform that occurred

in France as these workers took advantage of the greater flexibility in pension schemes. However, we

estimate that the reform only finances 20% of the expected deficit and as a consequence, no particular

group suffered a loss from that reform. Further reforms in the French benefit system are needed to

1In a recent literature, some studies use microsimulation models to investigate institutional differences across countries

as Gruber & Wise (2004) and Blondal & Scarpetta (1998), Blanchet & Pelé (1997) and Brugiavini (1997) for France

and Italy respectively. For want of a complete theoretical setting, such studies can not assess welfare implications of SS

reforms.

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finance the expected benefit. This is in sharp contrast with the Italian reform which finances the

expected deficit by cutting drastically the generosity of Social Security benefits. This strongly and

negatively affects the self-employed workers, approximately a third of the working population in Italy,

who contribute little to the pension system and have lower earnings than other workers.

We first present the model, then describe pension reforms in France and Italy. This allows us to

contrast the French system based on a defined pension system with the Italian contributive system.

Third, we solve the model in a benchmark calibration, then check its ability to reproduce the prevailing

features of SS and retirement profiles. Finally, we compare welfare effects of SS reforms in both

countries.

2 Social Security Reforms

Why is it interesting to compare Italy with France? Both retirement plans are pay-as-you-go systems.

In the case of France, the SS reform modified the defined benefit pension plan. In contrast, Italy chose

to switch from a defined benefit pension plan to a notional defined contribution system. Descriptions

of the French systems and Italian systems can be found in Blanchet & Pelé (1997), Brugiavini (1997)

and Brugiavini & Peracchi (2003). In this section, we describe key concepts and formulas of each

countries system that will be incorporated in the model.

2.1France

For French private workers, the pension benefit consists of two elements :

1. a “General Regime”. This regime is based on defined pension plans and managed by a State

agency (Caisse Nationale d’Assurance Vieillesse, hereafter CNAV). Its US counterpart is the Old

Age and Survivors Insurance (OASI).

2. mandatory complementary schemes (AGIRC and ARRCO). The second pillar is managed by

trade unions and representatives of employers.

Approximately 70 percent of the labor force falls under this so-called “General Regime”. Both re-

tirement plans are pay-as-you-go systems, but they are characterized by separate budgets. Mandatory

schemes cannot be discarded when analyzing retirement decisions and computing pension benefits.

However, policy debates focus on how the pattern of General Regime benefits could be modified in

order to encourage people to postpone retirement. Besides, the General Regime is directly managed

by the government, so it constitutes a policy tool, while complementary schemes are managed by non

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governmental organization. Hence, we focus on changes in the General Regime pensions2.

Table 1 summarizes the pension formulas that we consider. The difference between the General

Regime and Mandatory Complementary Schemes (MCS) lies in the way the contribution rate affects

the level of the pension. In the General Regime, the level of the contribution rate, and so the magnitude

of the contribution to the pension system, does not affect the level of the pension. The length of

contribution and the average annual wage are the only elements of the individual’s working life that

are included in the pension formula. In contrast, in mandatory complementary schemes, the level of

the contribution rate directly determines the level of the pension. The contribution rate enables people

to buy points. Each year, a fixed proportion of the wage is devoted to the purchase of these points.

The pension paid by mandatory schemes converts the number of points into euros. The General

Regime is thus a defined benefit plan while complementary schemes are notional defined contribution

plans.

2.1.1The pre-reform system

The French pension is the sum of the pension paid by the General Regime and the benefits paid by

mandatory complementary schemes.

The pension paid by the General regime (Pbasic(k) for an individual of age k) is the product of

three elements :

1. The proratization term min?1,d

years, the individual is entitled to full pension. In case of retirement before 40 contributive years,

the proratization term reduces the pension by

years, the individual does not gain additional pension.

40

?with d the number of contributive years. With 40 contributive

d

40. In case of retirement beyond 40 contributive

2. The average wage computed over the 25 best years of each individual’s carrier. SS capping

(capSS

t ) applies in the computation of the average annual wage for redistributive purposes

3. The pension rate. The full pension rate amounts to 50%. If the individual retires before con-

tributing 40 years, the pension rate is reduced by 5% per missing year. In contrast, the pension

rate does not reward additional working years beyond the required 40 years of contribution.

Figure 1 presents the value of the pension rate as a function of the retirement age for an individual

who started working at age 23. This agent will have the 40 required contributive years at age 63.

Should the individual retire earlier, the pension rate falls by 5% per missing year while continued

2To preserve the tractability of the model, we discard French self employed individuals who represent only 10% of

the population. Besides, in contrast to Italy, their pension regime greatly differs from the one that prevails in the private

sector.

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activity is not rewarded (which is not the case in the US pension system). The kinked profile of

the pension rate indicates that French agents have little choice in terms of retirement age : they are

enticed to retire as soon as they reach the full rate.

This General Regime pension is supplemented by mandatory schemes PARRCO(k) based on a

contributive plan. Pensions paid by mandatory scheme are also reduced (by 4% per missing year

captured by penalty(k)) in case of retirement before 40 contributive quarters (see table B3, Appendix

B). “Points” are purchased by each individual during his career. Each year, a fixed proportion of

the wage τw is devoted to the purchase of these points. One euro of earning yields 1/pARRCOpoints.

At the age of retirement, points are converted into euros of pension by multiplying the number of

points by a coefficient denoted vd, the value of each point at the date of retirement. Pension at age

k then amounts to PARRCO(k) where points(k) =?k

The complex set of French rules yields a strong implication in terms of retirement age. As underlined

by Blanchet & Pelé (1997), the heavy tax on continued activity leaves no freedom as far as retirement

age is concerned. An individual retires as soon as he contributes 40 years to the Social Security system.

Retiring earlier implies a dramatic fall in his pension, delaying retirement does not yield any significant

increase in benefits.

As the number of contributive years is not a state variable in our model, we use a proxy for this

variable by considering in both countries the current age k minus the age of end of education for

employed individuals (see table A2 in Appendix A ). Buffeteau & Godefroy (2005) who develop a

microsimulation model for France adopt a similar approach. For unemployed agents, the number of

contributive years is given by the difference between the current age and the sum of age of end of

education and average number of years of unemployment at this age and skill level, as documented by

Labor Force Surveys.

i=1

τw(i)

pARRCOdenotes the total number of points

accumulated throughout the working life.3

3For non executives, contributions are collected by ARRCO. Different contribution rates are applied to the part of

the wage below and above the SS cap. For executives, ARRCO (respectively AGIRC) collects the contribution for the

part of the wage below (respectively above) the SS cap. Finally, ARRCO and AGIRC introduce a wedge between the

contribution rate paid by workers (δ?) and the contribution rate that grants them points (δ). In the data, δ?> δ :

by imposing this discrepancy between the tax paid and tax yielding points, complementary schemes anticipate the

expected deficit associated with the future demographic change. ARRCO and AGIRC apply different wedges. Let

δ?

AGIRC

δAGIRC) be the wedge used by ARRCO (respectively AGIRC) for the part of the wage below and above the

Social Security cap. Both wedges are endogenously determined at the stationary equilibrium in the pre-reform scenario

to balance the budget on complementary schemes. The calibration of parameters of complementary schemes is given

by table A1 Appendix A. 60% of contributions to the complementary schemes are paid by employers and 40% by the

employee. This sharing rule is left unchanged throughout the paper.

δ?

δARRCO

ARRCO

(respectively

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2.1.2 The post reform system (Raffarin Reform, 2003)

The required contributive years to draw full pension is extended from 40 to 42 years. However, in order

to make old age pension more actuarially fair, the reduction in pension in reduced in case of early

retirement (proratization and decrease in the full rate yield a 2.5% fall in pension per missing year

instead of 5% in the pre-regime) and the pension is increased in case of delayed retirement (3% per

additional year beyond 42 years). The last elements aim at introducing more freedom in the choice of

retirement age.

On figure 1, the agent who started working at age 23 accumulated the required 42 years of contri-

bution at age 65. The kink then appears at this age. Notice that the reform introduces some flexibility

in the choice of the retirement age. Indeed, in case of retirement before the age of the full rate (65),

the reduction in pension is less dramatic. Besides, additional working years are rewarded with a 3%

increase in the whole pension (with a full rate unchanged at 50% beyond 42 years of contribution).

2.2Italy

Italy pension systems has had different reforms from 1992 until 1997. Before 1992, Italy had a Defined

Benefit (DB) system and has now a mandatory notional defined contribution (NDC) system. This

system is not exactly a defined contribution scheme. Indeed, it is a pay-as-you-go system where

contributions are used to pay current pensions rather than accumulated in a fund. Pension benefits

formula depends on economic growth and wages. Both benefits and contributions are individualized

and determined by the NDC accounting mechanism (see OECD (2005)). Table 2 summarizes the

computation formulas that are considered in our model.

2.2.1 The pre reform system (before 1992)

During the period leading to the reforms, the early retirement age was 60. The pensionable earnings

were the average of last 5 years real earnings for private workers (converted to real values through

price index) and the last 10 years for self employed individuals. The pension benefit was τI= 2% of

the pensionable earnings by years of tax payments (at most 40 years). Pension benefits were indexed

to wages.

2.2.2The post reform regime (after 1997)

The post reform regime is the result of a series of reforms : in 1992 (Amato), 1995 (Dini) and 1997

(Prodi). In the post reform system, each individual has an account in which contributions equal

33 per cent of earnings for employees and 20 per cent for the self-employed. Contributions for each

year are indexed to a five-year moving average of GDP. A conversion coefficient is then applied to

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total contributions. Pensions are indexed to prices. The conversion coefficient cIis chosen by Italian

authorities and displayed in table A3 of Appendix A. Contributions are capitalized at rate γI= 1.5%.

The age of end of education differs across skill categories (table A2 in Appendix A). These values

are important to compute the workers’ years of contribution. In the French and Italian regimes, 2/3

of the SS contribution rate is paid by the employer, leaving 1/3 of the contribution to the employee.

In contrast, Italian self - employed workers pay 100% of their contribution rate.

Both reforms aim at increasing labor supply by introducing strong incentives in case to postponed

retirement. Our model with endogenous retirement and wealth will be able to capture the possible

changes in retirement age that French and Italian governments try to affect through SS reforms.

3 Description of the economy

We assume that labor market status consists of employment and unemployment. Agents cannot com-

pletely insure against the idiosyncratic risk of being unemployed.4Accordingly, agents in our economy

differ with respect to their employment status and their employment history. In both countries, the

economy is modelled using an overlapping generation model with stochastic transitions on the labor

market. As a result, the working life of each individual affects their saving behavior. Agents also differ

in terms of their wealth.

Beyond the heterogeneity arising from uninsurable shock to individual employment opportunities,

life cycle features are also considered. Each period, some individuals are born and some individuals die.

We take into account different age groups and consider stochastic aging along the lines of Castaneda,

Diaz-Gimenez & Rios-Rull (2003). In addition, individuals face borrowing constraints and cannot hold

negative assets at any time. We build on Hairault, Langot & Sopraseuth (2005) that extends Rust &

Phelan (1997)’s model by introducing wealth accumulation. We analyze a small open economy which

implies that interest rate is exogenous. Moreover, we assume that agents are altruistic with respect

to future generations, i.e. they have a bequest motive. Our model does not allow for no aggregate

uncertainty in order to preserve the tractability of the model.

Finally, retirement behavior is endogenous. In order to analyze retirement decisions we take into

account several key features of each country’s retirement process

(i) Pensions depend on life-time earnings which is mechanically linked to unemployment spells

during the working life. Transitions on the labor market from employment to unemployment is

a salient feature of our model. These shocks cannot capture all income risks faced by individuals.

4Unemployment insurance leads to incomplete insurance against unemployment risk. There is also a trend toward

less unemployment insurance in most of OECD countries. In fact, Italy has not unemployment benefits longer than one

year.

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Employment shocks are introduced in order to capture differences in the employment rate during

the life cycle, with in particular a drastic fall in the employment rate of old workers. We want to

take into account the exit from the labor force at the end of working life as labor market status

prior to retirement is likely to affect retirement choices. Moreover, taking into account the low

labor demand for old workers is necessary to measure the implications of any policy aiming at

delaying retirement age (see Hurd (1996)).

(ii) Pensions are assumed indexed on prices. This was used in both countries as a way of reducing

SS generosity. The exact pre and post reform pension formulas are used in the model allowing

to measure the effect of changes in pension rules on retirement and savings.

(iii) A last important feature of the model is the skill structure. In each country, we distinguish

several ability groups defined by different skills. Mortality risk, unemployment transitions, life-

time earnings, age of end of education is specific to each skill category. The model accounts for

the differential mortality by skill groups. Hence, the model will be able to predict savings and

retirement decisions across and within each skill group, allowing each of these group to differ

substantially in the incentives they face.

3.1 Population dynamics and endowments

3.1.1Mortality

Some individuals are born and some individuals die every period. When an individual dies, he gives

birth to a single child.5

Individuals belonging to the same dynasty do not overlap. For sake of

simplicity, individuals face a mortality risk from the age of early retirement age on. In order to take

into account a typical life-cycle wage profile, we assume that the population can be divided into three

age groups: the young, the adult and mature respectively denoted Y , A and M. The mature age

group encompasses the ages of early retirement age (ERA) to 70 and more, (M = {ERA,60,...,70

and more})6. From the age of ERA on, each agent chooses to remain in his current state (employed

or unemployed) or retire the following period.7Each individual is born as a young worker. Following

Castaneda, Diaz-Gimenez & Rios-Rull (2003), the stochastic aging process is independent, identically

distributed across individuals of the same skill group and follows a finite state Markov chain with

conditional transition probabilities given by πkk= π(k?| k) = Pr(kt+1= k?| kt= k) where k and

5In all scenarios, the population is constant (n = 0). This assumption is consistent with French data (Aglietta,

Blanchet, & Héran (2002)). We choose to retain the same assumption in the Italian case in order to make our results

comparable across countries.

6Individuals do not die at age 70. The last age group encompasses individuals of 70 years old and more.

7The early retirement age is specific to each country. It is 60 years old for France, 60 years old for Italy in the

pre-reform scenario and 57 years old in the post - reform period.

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k?∈ K = {Y,A,M}. The probability of remaining a young (experienced) worker the next period is

πY Y (πAA). From the age of ERA on, the individual of age k ≥ ERA faces a rising probability to die

(1−πk). The mortality matrix takes into account the inequality in surviving probabilities across skill

groups. For example, the Matrix P that governs the age Markov-process in France for a particular

skill group is given by:

t + 1

t

Y

(20 - 34 years old)

πY Y

0

1 − π59

1 − π60

1 − π61

...

1 − π69

1 − πRR

A

(35 - 58 years old)

1 − πY Y

πAA

0

0

0

...

0

0

596061

···

···

···

···

···

···

...

···

···

69 70 and more

Y

A

59

60

61

...

69

R

00

0

0

0

0

0

0

0

0

0

...

0

0

0

0

0

0

0

...

1 − πAA

0

0

0

...

0

0

π59

0

0

...

0

0

π60

0

...

0

0

π69

πRR

3.1.2Wages

There are two components in the real wage : a deterministic exogenous productivity trend growing

at rate γ and the experience component allowing for an upward sloping wage profile with age. As a

worker accumulates experience during his life-cycle, we assume that the efficiency of the labor input

grows with the age of the agents.

3.1.3Unemployment Risk

At all ages, people face the risk of being unemployed. Transitions to employment and unemployment

are determined by exogenous probabilities that are specific to each skill level i and age group k. Every

period the individual receives the realization of a random variable z ∈ Z = {e,u} that determines

the probability to be employed or unemployed. z is a two-state, first-order Markov process with the

transition probability matrix πiY = Pr{z?= j|z = i} where πuu,kis the probability of an unemployed

of age k of remaining unemployed at age k +1 and πee,kis the probability of an employed of age k of

remaining employed at age k + 1. Transition probabilities at age k and skill level i are given by

?

Πk,i=

πee,k,i

πue,k,i

πeu,ki= 1 − πee,k,i

πuu,ki= 1 − πue,k,i

?

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3.1.4Social Mobility

Finally, we assume that a newly-born individual’s skill status is determined by his father’s. This social

mobility Φ is modelled as an exogenous Markov matrix

Φ = π(i?|i) = Pr{it+1= i?|it= i}

where i?is the skill status of the new born in the dynasty, and i denotes the random variable embodying

the skill status of the father. It is assumed that, once born with a random skill status (linked to that

of his father), an individual’s skill status is not modified in his life-time.

3.2The individual’s decisions: retirement choice, consumption and wealth

Individuals derive utility from leisure, consumption as well as from the consumption of their offsprings.

The intertemporal utility function of an individual is given by

??

where the period utility function u is strictly concave, the time-discount factor is β ∈]0,1[, consumption

Ctand leisure ltare positive. Labor is inelastically supplied by individuals as in Fuster, Imrohoroglu

& Imrohoroglu (2003). However, labor participation is endogenous in old age as individuals choose to

retire or not. Indeed, we focus on policy reforms aiming at delaying retirement rather than developing

part time jobs for old workers. Retirement age is the core issue in the policy reforms we investigate.

st∈ S is a compact notation to denote age, employment status and skill level. This vector follows a

finite state Markov chain with conditional transition probability given by

∞

?

t=0

βt

st∈V

π(st|st−1)u(Ct,lt) + ?Φβ

?

st∈S1

π(st+1|st)V (At+1,st+1)

?

(1)

π(s?|s) = Pr{st+1= s?|st= s}

Finally, the last term of the intertemporal utility function describes the utility derived by a parent

from his bequest. The parameter ? > 0 captures in a simple way the individual’s concern for the

welfare of his off-spring. Thus, V (At+1,st+1) denotes the expected utility of a new-born child who

begins his career according to a stochastic productivity ladder linked to that of his father and inherits

a stock of wealth At+1. We adopt a simple altruism specification for computational reasons. We

compute endogenous retirement and savings along the life cycle with heterogenous labor status and

ability levels for three steady states in each country (before reform with the current life expectancy,

after the increase in life expectancy with and without changes in the pension formula). Considering a

model with more complex transfers would substantially increase the computational burden.

As in Huggett & Ventura (1999), we assume that the instantaneous utility function u is a CRRA:

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u(c) =(c1−η(1 − l)η)1−˜ σ

1 − ˜ σ

with ˜ σ the risk aversion and η the weight of leisure (1−l) in the instantaneous utility. Time endowment

is normalized to 1. The utility function is Cobb Douglas in consumption and leisure. The reasons for

this choice are that this function is compatible with a balanced growth path and the parameters needed

for the calibration have been extensively studied in the literature relying on calibration (Auerbach &

Kotlikoff (1987), Prescott (1986), Cooley & Prescott (1995), Hansen & Imrogoroglu (1992), Rios~Rull

(1996), Huggett & Ventura (1999)). In addition, the individual faces two sources of capital market

inefficiency. The first stems from market incompleteness that prevents them from insuring completely

against idiosyncratic risks. The second relies on a borrowing constraint : net asset holding cannot be

negative (At+1≥ 0).

In order to define a stationary equilibrium, we divide all variables by the growth rate of techno-

logical progress (1 + γ). We denote stationary consumption and wealth by

ct= Ct/(1 + γ)tand at= At/(1 + γ)t

whereas the real wage and the pension are denoted in stationary terms by

wt= Wt/(1 + γ)t, ωt(s) = Ωt(s)/(1 + γ)t,

3.3Computation of the Stationary Equilibrium in a Heterogeneous Agent Econ-

omy

The economic problem of an individual is to choose a sequence of consumption, asset holdings and

retirement age given a set of policies for social insurance and expected life expectancy. The individuals’

problem can be formulated recursively with the help of the value function v(.):

v(a,s;w,r) = max

c,a?u(c,l) + ?Φ˜β

??

s?

π(s?|s)v(a?,s?;w,r)

?

(2)

subject to the budget constraint

(1 + γ)a?

= (1 + r)a + y(s) − c

≥ 0

a?

where r, y(s) denote the interest rate, the income and the payroll tax. The individual of age k

has 2 state variables : the pair (a,s) with s the realization of the individual-specific process (age,

employment status and skill level) and a the beginning-of-period financial wealth. We consider a

stationary equilibrium where prices w and r are constant. Indeed, in a small open economy framework,

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the world interest rate is considered as given which translates into an exogenous wage. As c is the

consumption per effective units of labor, the discount rate becomes˜β = β(1−η)(1−˜ σ)∈]0,1[.

Figure 2 summarizes the decision tree faced by each individual.8(e.g. 62RU59 is an individual

aged 62 who, at age 59, while he was unemployed, decided to retire at age 60). Note that once retired,

individuals remain retired which is consistent with French and Italian data. There does not appear

to be important un-retirement flows as in the U.S. (Gruber & Wise (1999)). As income depends on

labor status and age, we present value functions for each age group and employment status.

Individuals in Young (Y), Adult (A) and Mature (M) age groups make their decisions taking

into account future probabilities of being unemployed (or employed) as well as future payoffs from

their current actions given their skill level. To illustrate the retirement choice we consider a mature

individual. Other value functions are developed in Appendix B.

When mature (s ∈ M), individuals choose their employment status x = E (employed) or U

(unemployed). From the age of 59 on,9individuals have the legal right to claim a pension the following

year. Each individual compares the future value of being retired to the future value of remaining on

the labor market, given the probability of death (1 − πkk?,i).

1. For ages k = {59,...,68}

Employed individual with skill i : From the age of ERA on, individuals of age k ≥ ERA choose

whether to retire or not the following year by comparing the expected future value function at

age k+1 when retired to the future value function at age k+1 if he remains on the labor market,

taking into account the probabilities of transitions to employment or unemployment

v(a,kE

i) =max

c≥0, a?u(c,l)

+˜β

?(1 − πkk?,i)Φ??πY Y,Uv(a?,YU

= (1 + r)a + wk

i) + (1 − πY Y,U)v(a?,YE

i) + (1 − πeu,k,i?)v(a?,k?U

i)?

+πkk?,imax[πee,k,i?v(a?,k?E

i− Θ(wk

i),v(a?,k?RE

i

)]

?

(1 + γ)a?

i) − c

a?

≥ 0

kE

currently employed. The future value is composed of two terms, the future value if alive and the

bequest value of dead. Each individual can die with probability 1 − πkk?,i, in which case a new

young individual is born with skill group is determined by the mobility matrix Φ. (This matrix

imeans that we look at the value function of an individual of age k and skill level i who is

8Based on Labor force data and literature surveys, when individuals are old and unemployed, they have little chance

to find a job. Our calibration based on micro-data find this evidence. See section 4.3.

9We adjust the first age where retirement is possible depending on the country and the period (pre or post reform).

For example, the earliest age an Italian considers retirement in the post-reform period is 56 while it is 59 prior to the

reform.

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is calibrated from French and Italian data (see Appendix A)). The share of young agents born

unemployed πY Y,U. Similarly the following value is derived from being an unemployed individual

and a retired individual.

2. Unemployed individual with skill i at age k :

v(a,kU

i)= max

c≥0, a?u(c,l)

+˜β

?(1 − πkk?,i)Φ??πY Y,Uv(a?,YU

= (1 + r)a + θu

i− c

≥ 0

i) + (1 − πY Y,U)v(a?,YE

i) + (1 − πue,k,i?)v(a?,k?E

i)?

+πkk?,imax[πuu,k,i?v(a?,k?U

iwk

i),v(a?,k?RU

i

)]

?

(1 + γ)a?

a?

3. Retired individual with skill i at age k :

v(a,kRx

i )=max

c≥0, a?u(c,l) +˜β

= (1 + r)a + ωkRx

?

− c

(1 − πkk?,i)Φ??πY Y,Uv(a?,YU

i) + (1 − πY Y,U)v(a?,YE

+πkk?,iv(a?,kRx

i)?

i)

?

(1 + γ)a?

i

a?

≥ 0

where kRx

i after being previously E (employed) or U (unemployed). R denotes retirement. The pension

denoted ωkRx

i

depends on age k, skill level i and labor market status before retirement x.

i

(with x = E or U) denotes that the individual is a retiree of age k and skill level

4. Those still employed or unemployed by age 69 are forced to retire in the next year.

?(1 − πkk?,i)Φ??πMM,Uv(a?,MU

(1 + γ)a?

= (1 + r)a + y(69x

i) − c

a?

≥ 0

v(a,69x

i) =max

c≥0, a?u(c,l) +˜β

i) + (1 − πMM,U)v(a?,ME

+πkk?,iv(a?,Rx

i)?

i)

?

where y(69x

individuals delay past age 70, which is consistent with the data. In the model, pension benefits

will be computed with French and Italian Social Security formulas, respectively.

i) = w69

i

− Θ(w69

i) if employed (x = E) or θuw69

i

if unemployed (x = U). No

Stationary Equilibrium

individuals’ choices for consumption, savings and retirement {c(a,s),a?(a,s),Ψ(a,s)}, value functions

v(a,s), a stationary distribution of individuals λ(a,s) and a set of aggregate variables (P,D) where

P (respectively D) refers to expenditures (respectively the revenues) of the retirement scheme. The

stationary equilibrium satisfies:

Given the vector of prices (r,w), the stationary equilibrium consists of

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Page 15

(i) c(a,s) and a?(a,s) are optimal decision rules given the vector of prices (r,w).

(ii) The real interest rate is given, which is consistent with a small open economy hypothesis. This

leads to a constant capital to labor ratio, thus making real wages w exogenous.

(iii) Saving decisions a?= g(a,s) and retirement behavior ψ = Ψ(a,s) are solutions to the lifetime

maximization program where

?

for s = e(employed),u (unemployed). Finally, let a?≡ A(a,s) be the individuals’ policy that

depends on his state on the labor market (employed or unemployed). A(a,s) is such that:

Ψ(a,s) =

1

0

si

sinon

v(a,{s,k}) ≥ v(a,{R,k})

A(a,s) = Ψ(a,s)g(a,? s) + [1 − Ψ(a,s)]g(a,R)

where an agent’s policy in state ? s = e(employed),u (unemployed) is denoted by a?≡ g(a,? s).

(iv) The endogenous probability distribution λ(a,s) is the stationary distribution associated with

(A(a,s),π(s?|s)) such that:

?

λ(a?,s?) =

s

?

{a:a?=A(a,s)}

λ(a,s)π(s?|s)?

(v) In the pre-reform scenario, SS budget is balanced:

P = D

given pension arrangements specific to each country. P denotes the sum of pensions paid by

the Social Security system while D refers to the sum of collected payroll taxes. Each variable

is computed over the stationary distribution that yields the proportion of employed and retired

people.

P=

?

?

s

?

?

i

?

?

a

λ(a,s)ωs

i

D=

s

i

a

λ(a,s)Θ(ws

i)

We implement numerical techniques based on a discretization of state variables (Ljungqvist &

Sargent (2000)). This study compares steady states that prevail before and after SS reforms.

Our life cycle model allows for endogenous retirement and savings with heterogeneity in wealth,

labor status and skills, which yields results on distributional effects of SS reforms. As in Huggett

14

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