2013-2014 F.Langot Assurancepriv´ee,Assurancesociale,retraiteetsant´e

Assurance priv´ ee, Assurance sociale, retraite et sant´ e

F. Langot

Univ. Le Mans (GAINS & IRA) Banque de France & PSE & Cepremap & IZA

2013-2014

Partie II : Retraite

Introduction & Plan

Partie 2 : Retraite

Retraite : en fin de vie, on ne peut plus travailler. Comment avoir des revenus ?

Profiter des revenus financiers

Partager les risques individuels de revenus, de mort

Plan

Chapitre 1 : Capitalisation vs R´epartition Le mod`ele `a 2 p´eriodes

La capitalisation La r´epartition Quel syst`eme choisir ?

Chapitre 2 : Le choix de d´epart en retraite D´eterminants du choix de d´epart en retraite Un syst`eme actuariellement neutre

Quels effets sur l’emploi des seniors ?

Partie II : Retraite

Chapitre I : Assurance priv´ ee (capitalisation) ou

assurance sociale (r´ epartition)

Retour sur les choix d’´ epargne dans le mod` ele ` a 2 p´ eriodes

Hypoth` eses 2 p´ eriodes

revenus :(w

_t

, p

_t+1

) pr´ ef´ erences : u(c

_t

, d

_t+1

)

r´ emun´ eration des march´ es financiers : r

_t

financement de p

t

par une taxe τ

t

Probl` eme du m´ enage

c

max

t,dt+1

u(c

_t

, d

_t+1

) s.c.

c

t

+ s

t

= w

t−

τ

t

d

_t+1

= (1 + r

_t+1

)s

_t

+ p

_t+1

Retour sur les choix d’´ epargne dans le mod` ele ` a 2 p´ eriodes

La CB intertemprelle est alors c

_t

+ d

_t+1

1 + r

t+1

= w

_t−

τ

_t

+ p

_t+1

1 + r

t+1

Propri´ et´ es de la capitalisation

Chaque individu verse τ

_t

pour recevoir p

_t+1

= (1 + r

_t+1

)τ

_t

donc

⇒

Si cap. c

_t

+ d

_t+1

1 + r

t+1

= w

_t

le comportement des agents est identique ` a celui qu’ils auraient lorsqu’il n’y a pas de syst` eme ”institutionnel” de retraites.

La solution de ce programme donne donc la valeur de

Retour sur les choix d’´ epargne dans le mod` ele ` a 2 p´ eriodes

Supposons u(c

t

, d

t+1

) = log(c

t

) + β log(d

t+1

), alors d

_t+1

c

_t

= β(1 + r

_t+1

)

⇒

c

_t

= w

_t

1 + β

⇒

s

_t

= β 1 + β w

_t

| {z }

=s_t^NoSS

−τ_t

Il y a de l’´ epargne s

t

> 0 ssi β

1 + β w

_t

> τ

_t ⇔

s

_t^NoSS

> τ

_t

Sinon, s

t

= 0.

Propri´ et´ es de la retraite par capitalisation

Supposons que s

_t^NoSS

> τ

_t

.

Il existe de l’´ epargne ”volontaire”.

Equilibre sur le march´ e des titres :

K

_t+1

= N

_t

s

_t

+ N

_t

τ

_t

= N

_t

s

_t^NoSS

car dans un syst` eme par capitalisation, les caisses de retraites

”transforment” les taxes en ”offre de fonds pr´ etables”.

⇒

Neutralit´ e du syst` eme de retraite par capitalisation.

Propri´ et´ es de la retraite par capitalisation

Si s

_t^NoSS

< τ

_t

, alors s

_t

= 0 car l’´ epargne ne peut pas ˆ etre n´ egative.

Situation ”d’´ epargne forc´ ee” o` u la SS impose un taux d’´ epargne via la taxe τ

_t

Equilibre su le march´e des titres, avecN_t+1= (1 +n)N_t,

Kt+1=Ntτt⇒ Nt+1

Nt

Kt+1

Nt+1

≡(1 +n)kt+1=τt

car dans un syst`eme par capitalisation, les caisses de retraites

”transforment” les taxes en ”offre de fonds pr´etables”.

⇒ Le syst`eme de retraite par capitalisation d´etermine l’accumulation de capital.

Dans quelle situation ”l’´ epargne forc´ ee” peut-elle ˆ etre optimale ?

Que ferait un planificateur ?

c

max

t,dt+1

∞

X

t=0

u (c

t

, d

t+1

)

s.c.







Y

_t

= I

_t

+ N

_t

c

_t

+ N

t−1

d

_t

K

_t+1

= (1

−

δ)K

_t

+ I

_t

Y

_t

= K

_t^α

N

_t^1−α

⇔

max

kt,dt+1

∞

X

t=0

u

k

_t^α

+ (1

−

δ)k

t−

(1 + n)k

t+1−

d

t

1 + n , d

t+1

Dans quelle situation ”l’´ epargne forc´ ee” peut-elle ˆ etre optimale ?

Les CPO sont, avec f

_t⁰

= αk

_t^α−1

, u

_c,t⁰

[f

_t⁰

+ (1

−

δ)] = u

⁰_c,t−1

(1 + n)

u

_d,t⁰

= u

⁰_c,t+11+n¹

⇔







f_t⁰+(1−δ)

1+n

=

^u

0 c,t−1

u⁰_c,t

(1) 1 + n =

^u

0 c,t

u_d,t⁰ ₋₁

(2) Sur le march´ e, avec r

t

= f

_t⁰−

δ, on a

f

_t⁰

+ (1

−

δ)

≡

1 + r

t

= u

⁰_c,t−1

u

_d,t−1⁰

= (1)/(2)

⇒

La nouvelle restriction est donc (2).

Dans quelle situation ”l’´ epargne forc´ ee” peut-elle ˆ etre optimale ?

Le cˆ ot´ e droit de la relation (2) repr´ esente le taux marginal de substitution entre la consommation d’un jeune et celle d’un vieux vivant ` a la mˆ eme date t.

Planificateur : cette variable doit ˆ etre ´ egale au taux marginal de transformation (1 + n).

Cette condition n’a aucune raison d’ˆ etre v´ erifi´ ee dans

l’´ economie de march´ e, car l’horizon de chaque agent est limit´ e

`

a ses deux p´ eriodes de vie : il ne prend jamais en compte les possibilit´ es de substituer les consommations entre les g´ en´ erations.

Diff´ erences planificateur/march´ e vient de l’allocation des

ressources entre les g´ en´ erations et non de l’allocation des

Dans quelle situation ”l’´ epargne forc´ ee” peut-elle ˆ etre optimale ?

Supposons c

t

= c

t+1

, d

t

= d

t+1

et k

t

= k

t+1

,

∀t, alors

(1)

⇒

f

⁰

(k

^?

) + (1

−

δ) = 1 + n

⇔

f

⁰

(k

^?

) = n + δ

March´ e

⇔

Planificateur r = n Est-ce possible ?

(1 + n)k

_M

= β

1 + β [f

_M−

k

_M

f

_M⁰

]

⇒

k

_M

=

β(1

−

α) (1 + β)(1 + n)

_1−α¹

⇒

r

_M

= αk

_M^α−1−

δ = α(1 + β)(1 + n) β(1

−

α)

−

δ d’o` u r

_M

= n ssi

β(n + δ) α

Dans quelle situation ”l’´ epargne forc´ ee” peut-elle ˆ etre optimale ?

Si k

_M

> k

^?

, il y a sur-accumulation

⇒

neutralit´ e de la SS.

Si k

_M

= k

^?

, optimalit´ e

⇒

neutralit´ e de la SS.

Si k

M

< k

^?

, il y a sous-accumulation.

Une hausse de l’´epargne permettrait d’augmenter la

consommation de toutes les g´en´erations... sauf de la premi`ere.

L’effort d’´epargne est fait par la g´en´eration initiale qui r´eduit sa consommation et donc son bien ˆetre.

Un ´equilibre de sous-accumulation est donc efficace au sens de Pareto puisqu’il n’est pas possible de modifier l’allocation sans diminuer le bien-ˆetre d’au moins une g´en´eration.

τ^?= (1 +n)k^?. Dans ce cas, on akM=k^? grˆace `a SS.

La r´ epartition

Dans ce cas, on a N

t

τ

t

= N

t−1

p

t⇔

p

t

= (1 + n)τ

t

.

⇒

La r´ epartition r´ ealloue entre les g´ en´ erations La CB intertemprelle est alors

c

t

+ d

t+1

1 + r

_t+1

= H

t

avec H

t

= w

t−

τ

t

+ (1 + n)τ

t

1 + r

_t+1

c

_t

= H

t

1 + β

⇒

s

t

= w

t−

τ

t−_1+β^Ht

= H

t−^(1+n)τ_1+r ^t

t+1 −_1+β^Ht

s

_t

= β

1 + β H

_t−

(1 + n)τ

_t

1 + r

t+1

L’´ equilibre sur les march´ es financiers est K

_t+1

= N

_t

s

_t⇒

(1 + n)k

_t+1

= β

1 + β w

_t−

β(1 + r

_t+1

)

−

(1 + n)

(1 + β)(1 + r ) τ

_t

La r´ epartition

Supposons x

_t

= x

_t+1

,

∀x

et

∀t, et avec

f

⁰

= αf /k, alors (1 + n) = β(1

−

α)

α(1 + β) f

⁰−

β(1 + r )

−

(1 + n) (1 + β)(1 + r) τ

Comme l’allocation optimale doit v´ erifier f

⁰

= n + δ et r

^?

= n, on a τ

^?

t.q.

1

−

α α

β

1 + β (n + δ)

−

n = 1 + τ

^?

La r´ epartition est-elle optimale ?

Oui si r

M

< n : sous-accumulation.

Dans ce cas, les volume des cotisations futures d´ epassera le rendement (int´ erˆ et et principal) obtenu en pla¸ cant les cotisations pr´ esentes sur le march´ e financier.

Le syst` eme de retraite par r´ epartition ”domine” alors celui fond´ e sur la capitalisation.

En France, on observe r > n. Depuis 2000, 0.38% < n < 0.574% et 5.2% < r < 6%.

Changer de syst` eme : trop coˆ uteux car il y a un transition

⇔

g´ en´ erations sacrifi´ ees.

Partie II : Retraite

Chapitre II : Choix de d´ epart ` a la retraite

The extensive margin of the labor supply

The intensive margin : the number of hour worked by worker.

There is another margin : the extensive margin, which give the number of labor market participant in the population of the 15-65 years old.

Two important decisions drive this extensive margin :

1 The retirement decision which determines the employment rate of the older workers.

2 The education decision which determines the employment rate of the young workers.

The extensive margin of the labor supply

15−24 ans 25−54 ans 55−59 ans 60−64 ans

0 10 20 30 40 50 60 70 80 90 100

Rang des pays déterminé par le taux d’emploi des 60−64 ans

%

Taux d’emploi sur le cycle de vie − hommes − 2006

FraBel ItaPB

AllFin Esp

CanDanRUEU SueJap

Retirement Choice and SS

Taxes on labor incomes have also an impact on the retirement age decision

0 1 2 3 4 5 6 7 8 9 10

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

Taxe implicite sur la prolongation d’activité (indice)

Taux de non−emploi des 55−65 ans

Fra

Can

E−U

Jap Sué

R−U

Esp

P−B

All

Bel

Ita

Retirement Choice

Assumptions and notations :

Let T and R denote respectively the age of the death and of the retirement age,

Let w denotes the real wage

Let r and δ denote respectively the interest and subjective discount rate.

We assume that there is no uncertainty, and that agents can save and borrow in perfect financial markets.

At this stage, we assume that Social Security does not exist

⇔

There is no distortions.

Retirement Choice

Preferences : U =

U (c (t)) if t

∈

[0, R) U (c (t)) + v if t

∈

[R, T ]

where v denotes the leisure when the individual is a retiree.

The problem of the agent is :

c(t),R

max

Z T

0

e

^−δt

U (c (t))dt +

Z T

R

e

^−δt

vdt

s .t.

Z T 0

e

^−rt

c(t)dt =

Z R

0

e

^−rt

wdt

Retirement Choice

The FOC are :

U

⁰

(c(t)) = λe

^−(r−δ)t

ve

^−δR

= λwe

^−rR

⇒

v

|{z}

Opportunity Cost

= U

⁰

(c (R))w

| {z }

Marginal return

In order to determine R, it is necessary to determine c (R).

Assuming, for simplicity, that r = δ, implying c(t) = c,

∀t, the

budgetary constrain of the agent leads to :

c = w

RR

0

e

^−rt

dt

RT

0

e

^−rt

dt = w 1

−

e

^−rR

1

−

e

^−rT

The optimal age of retirement is then given by :

v = U

⁰

w 1

−

e

^−rR

w

Retirement Choice and SS

Assume now that there is a Social Security system. The agent budgetary constraint becomes

Z T 0

e

^−rt

c (t)dt =

Z R

0

e

^−rt

(1

−

τ )wdt +

Z T

R

e

^−rt

p (R)dt The FOC are :

U⁰(c(t)) = λe^−(r−δ)t ve^−δR = λ

(1−τ)we^−rR−p(R)e^−rR+ ZT

R

e^−rtp⁰(R)dt

⇒

v

|{z}

Opportunity Cost

= U

⁰

(c (R))

(1

−

τ )w

−

p(R) +

Z T−R

0

e

^−rt

p

⁰

(R)dt

| {z }

For a givenU⁰(c(R)), the

Retirement Choice and SS

We can then define this tax on the continued activity tax : TI = τ w

|{z}

Effective Tax

+ p(R)

| {z }

Implicit

Tax

− Z T−R

0

e

^−rt

p

⁰

(R)dt

| {z }

Bonus

The SS system is actuarially fair if and only if τ w + p(R) =

Z T−R 0

e

^−rt

p

⁰

(R)dt implying no distortion on the labor supply

v = U

⁰

(c (R

^SS?

))w

⇒R^SS?=R

Retirement Choice and SS

Because for all SS system, we have c = w 1

−

e

^−rR

1

−

e

^−rT

because τ w 1

−

e

^−rR

1

−

e

^−rT

= p(R) e

^−rR−

e

^−rT

1

−

e

^−rT

If the SS system is actuarially fair, the optimal age of retirement is the same than in a economy without SS At the opposite, if the SS system is not actuarially fair, but under the constrain that the SS system has no deficit, the optimal age of retirement is given by :

p + v

U

⁰

(c ) = (1−τ )w

⇔

v = U

⁰

w 1

−

e

^−rRSS

1

−

e

^−rT

!

[(1−τ )w

−p]

If there is no bonus when the agent delays her retirement age

Les surcotes : vers un syst` eme actuariellement neutre

Individu d’ˆ age z d´ ecidant de reproter son d´ epart ` a l’ˆ age z + 1.

⇒

beneficier d’une surcote, not´ ee λ(z + 1).

⇒

La CNAV doit alors financer sur une periode allant de l’ˆ age z + 1 ` a la fin de vie de l’agent (T (z + 1)) une pension

pens(z + 1) = [λ(z + 1) + 0, 5]w

^ref

(z + 1) Cependant, ` a l’ˆ age z, cet individu paie ` a la CNAV une cotisation ´ egale ` a τ w (z ).

En outre, la CNAV lui aurait vers´ e une pension de

pens(z) = [λ(z ) + 0, 5]w

^ref

(z ) si l’agent ´ etait parti ` a la

retraite ` a l’ˆ age z.

Les surcotes : vers un syst` eme actuariellement neutre

Financierement, la CNAV est indifferente entre ces deux options si la surcote verifie :

"

s

_t,t+1PT(z+1) i

[λ(z+1)+0,5]w^ref(z+1) (1+r)ⁱ

−τ

w (z)

#

=

T(z)

X

i

[λ(z ) + 0, 5]w

^ref

(z) (1 + r)

ⁱ

Table:Taux annuels de d´ecotes et de surcotes actuariellement neutres

r= 3% r= 4%

Age d´ecote surcote d´ecote surcote 60 ans 3,76% 6,28% 4,76% 7,36%

61 ans 3,88% 6,40% 4,92% 7,52%

62 ans 4,04% 6,56% 5,08% 7,68%

63 ans 4,20% 6,76% 5,24% 7,84%

64 ans 4,40% 6,96% 5,40% 8,04%

Pourquoi la surcote actuarielle g´ en` ere-t-elle des in´ egalit´ es ?

Les in´ egalit´ es face ` a la mort : entre un ouvrier et un cadre la diff´ erence est de 6 ann´ ees ` a 60 ans.

La SS doit verser une surcote sur 2 p´eriodes pour le cadre, 1 p´eriode pour l’ouvrier

Recettes :Ri = (λ+τ)wi, pouri=c,o. Cadre :Dc = 2δcwc ⇒δc = (λ+τ)/2.

Ouvrier :Do=δowo⇒δo= (λ+τ).

⇒

La prime incitative doit ˆ etre tr` es grande pour un ouvrier car

son ”risque” est de la toucher pendant peu de temps.

La s´ ecurit´ e sociale n’est pas une compagnie d’assurance

La SS ne peut pas connaˆıtre les risques de mort de tous les individus

⇔

Les individus sont mieux inform´ es sur leur sant´ e personnelle.

Elle propose donc une ”surcote moyenne”, calcul´ ee sur la dur´ ee de vie esp´ er´ ee moyenne :

δ

c

< δ

moy

= (λ + τ )/[(2 + 1)/2] < δ

o

Le cadre est gagnant car il vit plus longtemps / l’ouvrier perdant car il vit moins longtemps

Seul le cadre sera ”sensible” ` a ces incitations.

Ce syst` eme n’est donc pas efficient

⇒

il faut laisser aux

assureurs le soin de g´ erer l’assurance vie.

Bilan : Quelle type d’incitation choisir ?

La surcote actuarielle en rente

⇒

des in´ egalit´ es dues ` a l’inobservabilit´ e du risque individuel de mort.

⇒

Pas tr` es lisible pour les agents

Deux autres solutions plus efficaces et plus lisibles

⇒

La surcote avec ”sortie en capital” : ` a l’ˆ age de retraite, versemment d’un capital qui peut alors g´ er´ e par un assureur.

⇒

Le cumul emploi-retraite : plus favorable pour les agents

contraints financi` erement.

La sortie en capital

Table:Surcotes en capital mesur´ees en ann´ees de salaires nets

r = 3% r = 4%

Age de d´ epart 0,5 1 2 0,5 1 2

61 ans 0,85 0,71 0,64 0,86 0,72 0,65

64 ans 1,73 1,44 1,31 1,76 1,46 1,33

63 ans 2,64 2,20 1,99 2,69 2,24 2,03

62 ans 3,57 2,97 2,70 3,66 3,05 2,77

65 ans 4,53 3,78 3,43 4,67 3,89 3,53

Les sorties en capital sont calcul´ees pour trois proportions diff´erentes du salaire moyen (0,5, 1 et 2). Le ratio de remplacement pour ces diff´erents niveaux de salaire est respectivement de 0,64, 0,51 et 0,45 (OCDE 2006).

La r´ eforme de 2003 : Un mod` ele por la France

Assumptions

A large number of individuals with identical preferences.

Life cycle stages of working age and retirement.

The retirement age derives from an endogenous decision Agents age stochastically

Upon death, individuals are replaced by other individuals of the same dynasty and are imperfectly altruistic towards them.

Individuals face two sources of capital market inefficiency :

market incompleteness that prevents them from insuring against idiosyncratic risks,

liquidity constraint : individuals are not allowed to run into debt.

Stochastic structure of the model : Labor ability

Labor ability : the labor ability process is a three-state, first-order Markov chain.

labor ability γ can be High, Medium or Low : γ

∈

Γ =

{H,

M, L}

Labor abilities are assumed to be correlated across generations as the result of the transmission of human capital from parent to child (Becker and Tomes [1979]).

once born with a labor ability, individuals keep the same ability during their working life.

the wage level and the wage profile over the working life as the return from seniority depend on the labor ability.

the unemployment risk at the end of working life will also

Stochastic structure of the model : Aging

Life cycle : before the early retirement ages (ERA)

All agents are ”born” as young workers (Y ) at a given age which corresponds to end of education.

Before the ERA, three classes of working age : the young (Y ), the experienced (E ) and the old workers (O)

The probability of remaining a young (experienced) worker in the next period is π

_YY

(π

_EE

) and, as aging occurs

sequentially, the probability of becoming an experienced (old)

worker is 1

−

π

_YY

(1

−

π

_EE

).

Stochastic structure of the model : Aging

Life cycle : after the early retirement ages (ERA)

With a probability 1

−

π

_OO

, older workers reach the ERA.

From the ERA onwards, individuals face a probability of dying, skill-specific.

Until the MRA, workers grow older of one year each period.

Conditional on being alive, they choose to retire (R

_ERA

) or not (W

_ERA

).

If they decide to postpone retirement, they remain in the labor force one additional year.

If they survive, they will face the same choice at the beginning of the next period.

Conditional on being alive and in activity, workers must retire

Stochastic structure of the model : earnings

Earnings dynamics

As a worker accumulates experience during his life cycle, his efficiency grows with his age.

when a young worker becomes an experienced worker, his efficiency is multiplied by 1 + x

_Y

An old worker’s efficiency is (1 + x

_E

) times that of a young agent.

Decreasing returns : x

_Y

< x

_E

. x are skill-specific :

{x_Y

(γ ); x

_E

(γ )}.

The stochastic age variable ξ follows a finite state Markov process :

ξ

∈

Ξ =

{Y

, E, O, ERA, ERA + 1, ..., MRA

−

1, MRA}

Stochastic structure of the model : unemployment

Unemployment risk

We introduce an unemployment risk only for older workers (ξ = O ), which is skill-specific.

The (un)employment shock φ

∈

Φ =

{e,

u} follows a two-state Markov process.

The individual labor input is set to l (φ) :

When unemployed (φ=u), the time endowment is devoted to leisure (l(u) = 0) and workers receive an unemployment benefit until the age of full pension rate.

When employed (φ=e), they inelastically supplyl units of labor input (l(e) =l) at a wage ratew.

Consistently with empirical evidence, we will consider that the

unemployment state is an absorbing state until retirement.

Social Security System in France

The General Regime : the first pillar of the SS system.

The pension is based on the following formula : ω

^GR

= min 1,

₁₅₀^d

×

w

^ref ×

ρ

the number of contributing quarters d : there is a normal number d

ⁿ

of contributing quarters. Before 1993, d

ⁿ

= 150, and after d

ⁿ

= 160.

the pension rate ρ, if retirement at age z :

ρ = 0.5

−

0.0125

×

max

{0,

min [(MRA

−

z )

×

4, d

ⁿ−

d )]}

the reference wage w

^ref

=

_N¹ PN

n=1

Min(w

n

, Cap

^SS

), where w

n

and Cap

^SS

are the wage of the best N years and the SS cap.

Behaviors : Firms

Technology.

Y = K

^α

(XL)

^1−α

Y : aggregate output.

L : labor the labor input obtained by aggregating the efficiency labor units.

X is a deterministic exogenous productivity trend growing at a rate of g .

K the aggregate capital which depreciates at a constant rate δ

α

∈

[0, 1]

Behaviors : Firms

Profit maximization leads to :

w (γ, ξ)(1 + Θ

_f

(w (γ, ξ)) = µ(γ, ξ)(1

−

α) Y L r + δ = α Y

K

with r the interest rate and Θ

_f

is the contribution rate paid by the

firm to finance the pay-as-you-go pension system.

Behaviors : Households

Preferences.

u(C , 1

−

l ) = (C

^1−ν

(1

−

l)

^ν

)

^1−˜σ

1

−

σ ˜ C

_t

consumption

l leisure. Time endowment is normalized to one.

˜

σ

∈

[0, 1[∪]1,

∞[.

ν

∈

[0, 1].

The individual’s state variable is (a, φ, γ, ξ)

Behaviors : Households until the retirement age

Vw(a, φ, γ, ξ) = maxc≥0

u(c,1−l(φ)) + ˜βP

φ⁰

P

ξ⁰P(φ⁰|φ, γ, ξ)P(ξ⁰|γ, ξ)Vw(a⁰, φ⁰, γ, ξ⁰)

subject to

(1 +g)a⁰= (1 +r)a+y(φ, γ, ξ) [1−Θw(y(φ, γ, ξ))]−c−IeTu

a⁰≥0

V

w

denotes the value function of workers.

P (φ

⁰|φ, γ, ξ) is the probability that a worker of labor market

status φ, ability γ and age ξ becomes type φ

⁰

the next period P (ξ

⁰|γ, ξ) is the probability that a worker of ability

γ and age ξ becomes age ξ

⁰

the next period.

Θ

_w

(y(φ, γ, ξ)) is the contribution rate paid by the worker (employed or unemployed) to finance the SS system

Unemployment benefits are financed through a lump-sum tax

T by workers when employed.

Behaviors : Households from ERA to MRA

Vw(a, φ, γ, ξ)

= maxc≥0







u(c,1−l(φ)) + ˜β

(1−πM(γ)) max [Vw(a⁰, φ, γ, ξ+ 1),Vr(a⁰, φ, γ, ξ+ 1)]

+πM(γ)ηP

γ⁰P(γ⁰|γ)Vw(a⁰,e, γ⁰,Y)





 subject to

(1 +g)a⁰= (1 +r)a+y(φ, γ, ξ) [1−Θw(y(φ, γ, ξ))]−c−IeTu

a⁰≥0

Conditional on being alive (1

−

π

_M

(γ)), workers become one year older ξ

⁰

= ξ + 1.

V

_r

(a

⁰

, φ, γ, ξ + 1) is the expected utility to be retired.

V

_w

(a

⁰

, e, γ

⁰

, Y ) : value of a new-born worker (ξ = Y ), employed (φ = e), with ability γ

⁰

linked to γ (father) by P (γ

⁰|γ).

The young individual inherits the estate of his deceased father.

Behaviors : Retired Households

Vr(a, φ, γ, ξ) = maxc≥0

u(c,1) + ˜β

(1−πM(γ))Vr(a⁰, φ, γ, ξ) +πM(γ)ηP

γ⁰P(γ⁰|γ)Vw(a⁰,e, γ⁰,Y)

subject to

(1 +g)a⁰= (1 +r)a+ω(φ, γ, ξ)−c a⁰≥0











Retirees receive a pension ω(φ, γ, ξ) that is the sum of the public SS pension and benefits paid by mandatory

complementary schemes.

The retiree choices his consumption and the amount of financial assets he wants to give to his child.

Uncertainty : the stochastic intergenerational expected

changes in ability.

Equilibrium 1

The exogenous Markov processes are summarized by

s: Φ×Γ×Ξ → S (φ, γ, ξ) → s(φ, γ, ξ)

withP(s⁰|s) = Pr{st+1=s⁰|st =s}

The steady state equilibrium is characterized by Worker and retiree choices

{c_w

(a, s ), c

_r

(a, s), a

⁰_w

(a, s ), a

⁰_r

(a, s )} and for retirement age Ψ(a, s).

A vector of prices (r , w (s )), A SS policy (θ, ω

^GR

(s )),

A stationary distribution of individuals Λ(a, s),

A set of aggregate variables ( ˜ K , L)

Equilibrium 2

Households solve value functions.

Factor prices are competitive.

∃Λ(a,

s ), associated with (A(a, s), P (s

⁰|s)) :

Λ(a⁰,s⁰) =X

s

X

{a:a⁰=A(a,s)}

Λ(a,s)P(s⁰|s)

where A(a, s ) is such that :

A(a,s) =

Ψ(a,s)a⁰_w(a,s) +[1−Ψ(a,s)]a⁰r(a,s)

with Ψ(a,s) =

1 ifVw(a,s)≥Vr(a,s) 0 otherwise

Factor inputs are aggregated over individuals :

L=X

s

X

a

Ψ(a,s)lµ(s) K˜ =X

s

X

a

Λ(a,s)A(a,s)

The payroll tax rate θ adjusts to balance the SS budget :

XX

Ψ(a,s)θy(s) =XX

(1−Ψ(a,s))ω^GR(s))

Equilibrium : numerical solutions

A = [0 < a

2

< a

3

< ... < a

n

] and S = [s

1

, s

2

, s

3

, ..., s

m

] Define m vectors v

j

, with dim(v

j

) = n

×

1. The ith row are such that v

_j

(i) = v(k

_i

, s

_j

),

∀i

= 1, ..., n

Let m matrix R

_j

, with dim(R

_j

) = n

×

n, define by R

_j

(i , h) = u(c(a

_i

, s

_j

, a

_h

)) for i = 1, ..., n and h = 1, ..., n Define an operator T ([v

1

, ..., v

m

]) that maps a set of vectors [v

₁

, ..., v

_m

] into a set of vectors [tv

₁

, ..., tv

_m

] :





 tv1

.. . tvm





= max











 R1

.. . Rm





+β(Π⊗1)





 v₁⁰

.. . vm⁰













where

⊗

is the Kronecker product and Π is the transition

matrix that governs life-cycle, employment opportunities and

Numerical solutions : Howard improvement algorithm

To make a guess to an initial feasible policy function a

⁰

= g (a, s),

To compute the n

×

n matrix I

_h

, for h = 1, ..., m, defined by I

_h

(a, a

⁰

) = 1 if a

⁰

= g (a, s ) and I

_h

(a, a

⁰

) = 0 otherwise, Using the definition of R

_j

(i , h), to evaluate the vectors [v

1

, ..., v

m

] implied by using that policy forever :





 v1

.. . vm





=





 R1

.. . Rm





+β







Π11I1 . . . Π1mI1

..

. . .. ... Πm1Im . . . ΠmmIm











 v1

.. . vm







This first computation of the vectors [v

₁

, ..., v

_m

] is used as a

terminal value vector in the Bellman equation to find a new

policy function. This function is used to update the preceding.

Calibration : demographic

Age of end of school education

Ability H M L

Age of end of education 22.2 19.5 17.4 Mortality risk at 60

H M L

πM 0.0410 a 0.0483 0.0538

a : Each year, an H-ability agent faces a 4.10 percent probability of dying Lifetime wages

Young Experienced Old

H 2.14 b 3.25 3.91

M 1.40 1.86 2.25

L 1 a 1.24 1.26

a : The wage of low-skilled young workers is normalized to one

Calibration : intergenerational ability transition matrix

Table:Intergenerational change in ability Son’s Ability (

t + 1)

Father’s Ability (

t

)

H M L

H 0.4077 a 0.3187 0.2736 M 0.2191 0.3507 0.4302 L 0.0929 0.1952 0.7119 a : A H-ability worker faces a 40.77 percent probability of giving birth to a H-ability son

Calibration : unemployment risk

Table:Unemployment replacement rate (ρu) and employment risk (πu)

ρ

_u

π

_u

H 0.60 a 0.063 b

M 0.61 0.083

L 0.65 0.086

a : An unemployed H-ability agent receives 0.60 of his last annual net wage b : An H-ability worker faces a 6.3 % probability of becoming unemployed

The fit of the wealth distribution

Data France Model All 60-64 All 60-64

(1) (2) (3) (4)

Gini 0.73 0.86 0.74 0.89

Top 1 percent 0.30 0.28 0.10 0.08 Top 5 percent 0.51 0.49 0.34 0.27 Top 20 percent 0.78 0.75 0.77 0.67 Top 40 percent 0.92 0.90 0.96 0.90 Top 60 percent 0.96 0.97 0.98 0.99

Liq. Constr. 0.22 0.23

The actuarially fair scheme

The actuarially-fair adjustment λ

^∗

(γ, ξ) is such :

1 πM(γ)

X

i=0

(λ

^∗

(γ, ξ) + 0.5)w

^ref

(γ, ξ) (1 + r )

ⁱ

| {z }

Pension paid by Social Security if the individual retires at ageξ

= (1

−

π

_M

(γ))

1 πM(γ)

X

i=1

(λ

^∗

(γ, ξ + 1) + 0.5)w

^ref

(γ, ξ + 1) (1 + r)

ⁱ

| {z }

Pension paid by Social Security if the individual retires at ageξ+1

−

θ w (γ, ξ)

| {z }

taxes collected on wages during ageξ

The actuarially fair scheme

Table:Actuarially-fair scheme (in %)

λ

^∗

(γ, ξ + 1)

−

λ

^∗

(γ, ξ)

ξ+ 1 61 62 63 64 65 66 67 68 69 70

H 0 0 0 5.99 6.53 7.11 7.76 8.46 9.22 10.05

M 5.87 6.45 7.08 7.78 8.55 9.39 10.32 11.34 12.46 13.69 L 5.90 6.52 7.2 7.95 8.83 9.76 10.78 11.92 13.17 14.55

See the other PDF file on retirement decisions.

Partie II : Retraite

Chapitre III : Retraite et emploi des seniors

Extension : Job Search and Retirement Decision

Let V

_i^e

(w ) be the value of the optimization problem for a worker of age C

_i

and paid w ,

V

_i^u

the value of the optimization problem for an unemployed worker of age C

_i

,

V

^r

the value of a retiree.

Let V

_i

(w ) be the value of the optimization problem for a worker of age C

i

who was employed in the previous period and has today the option to work at wage w

V

i

(w ) = max

{V_i^e

(w ), V

_i^u}

for i = 1, .., 5

Extension : Job Search and Retirement Decision

Bellman equations can be written as : for i = 1, 2, 3 and if the worker is employed

V

_i^e

(w ) = u((1

−

τ

p−

τ

b

)w , T

−

h) + β

{π_i

[(1

−

λ

i

)V

i

(w ) + λ

i

V

_i^u

] +(1

−

π

i

)[(1

−

λ

i+1

)V

i+1

(w ) + λ

i+1

V

_i+1^u

]

(1

−

τ

_p−

τ

_b

)w : net wage, τ

_p

tax rate for pensions and τ

_b

tax rate for unemployment insurance.

T

−

h leisure, with h the number of worked hours λ

_i

is the age-specific probability to be fired

π

_i

is the probability to be in the same age-cohort in the next

period.

Extension : Job Search and Retirement Decision

Bellman equations can be written as :

for i = 1, 2, 3 and if the worker is unemployed V

_i^u

= max

si

{u

((1

−

τ

_p

)b

_i

, T

−

s

_i

) +β

π

_i

φ(s

_i

)

Z

V

_i

(w )dF

_i

(w ) + (1

−

φ(s

_i

))V

_i^u

+(1

−

π

i

)

φ(s

i+1

)

Z

V

i+1

(w )dF

i+1

(w ) +(1

−

φ(s

_i+1

))V

_i+1^u

b are the unemployment benefits

φ(s

_i

) is the probability to find a job offer, with s

_i

the time

devoted to the search activities.

Extension : Job Search and Retirement Decision

for i = 4

V

₄^e

(w ) = u((1

−

τ

_p−

τ

_b

)w , T

−

h) + β

{π₄

[(1

−

λ

₄

)V

₄

(w ) + λ

₄

V

₄^u

] +(1

−

π

4

)[(1

−

λ

5

) max{V

₅

(w ), V

₅^r}

+ λ

5

max{V

₅^u

, V

₅^r}]}

V

₄^u

= max

s4

{u((1−

τ

p

)b

4

, T

−

s

4

) +β

π

4

φ(s

4

)

Z

V

4

(w )dF

4

(w ) + (1

−

φ(s

4

))V

₄^u

+(1

−

π

₄

)

φ(s

₅

)

R

max{V

₅

(w ), V

₅^r}dF₅

(w ) +(1

−

φ(s

5

)) max{V

₅^u

, V

₅^r}

Extension : Job Search and Retirement Decision

for i = 5

V

₅^e

(w ) = u((1

−

τ

p−

τ

b

)w , T

−

h)

+β

{π₅

[(1

−

λ

₅

) max{V

₅

(w ), V

₅^r}

+ λ

₅

max{V

₅^u

, V

₅^r}]

+(1

−

π

₅

)V

₆^r6

V

₅^u

= max

s5

{u((1−

τ

_p

)b

₅

, T

−

s

₅

) +β

π

5

φ(s

5

)

Z

max{V

₅

(w ), V

₅^r}dF₅

(w ) +(1

−

φ(s

₅

)) max{V

₅^u

, V

₅^r}] + (1−

π

₅

)V

₆^r6

V

₅^r

= u(p

₅

, T ) + β

π

₅

V

₅^r

+ (1

−

π

₅

)V

₆^r5

where p denotes the retiree’s pension at age 5.

Extension : Job Search and Retirement Decision

for i = 6

V

₆^r5

= u(p

₅

, T ) + β π

₆

V

₆^r5

V

₆^r6

= u(p

₆

, T ) + β

π

₆

V

₆^r6

Benchmark : the pension is not increased by additional years of working beyond the full pension rate :p6=p5.

⇔V^e does not increase if the agent decides to postpone retirement (huge tax on continued activity).

Policy : an actuarially fair increase in pension (p6>p5) can make the early retirement option undesirable for employed workers.

⇔Employment is more valuable than any other options : there is now an employment surplus at the early retirement age, which conversely boosts the search intensity before this age.

Extension : Job Search and Retirement Decision

The optimal decision for search intensity is given by : for i = 1, 2, 3, 4

u

⁰₂

((1

−

τ

_p

)b

_i

, T

−

s

_i

) = φ

⁰

(s

_i

)βπ

_i Z

V

_i

(w )dF

_i

(w )

−

V

_i^u

The marginal disutility of job search activity equals its expected

return, which is captured by the increase in the probability of

getting a wage offer times the expected surplus of employment.

Extension : Job Search and Retirement Decision

for i = 5

u

₂⁰

((1−τ

_p

)b

5

, T

−s₅

) = φ

⁰

(s

5

)βπ

5

R

max[V

5

(w ), V

₅^r

]dF

5

(w )

−

max[V

₅^u

, V

₅^r

]

If the continued activity opportunity is sufficiently attractive after the early retirement age, the employment and the unemployment values converge later, only when the mandatory retirement (C

₆

) is imminent.

The horizon of older workers just before the early retirement age is

then broadened.

Job Search and Retirement Decision : The Double Dividend of Actuarially-Fair Pension Adjustments

Table:Incentive schemes and employment rates

Age groups C

₁

C

₂

C

₃

C

₄

C

₅

Age in years 20-29 30-49 50-54 55-59 60-64

1. Benchmark 0.828 0.867 0.874 0.549 0

2. Retirement Policy 0.830 0.867 0.874 0.714 0.201

See the other PDF file on retirement decisions.