1/66 Assurance, retraite et santé - F. Langot Assurance privée, Assurance sociale, retraite et santé F. Langot Univ. Le Mans (GAINS & IRA) Banque de France & PSE & Cepremap & IZA 2013-2014
2/66 Assurance, retraite et santé - F. Langot Partie II : Retraite
3/66 Assurance, retraite et santé - F. Langot 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 Répartition Le modèle à 2 périodes La capitalisation La répartition Quel système choisir? Chapitre 2 : Le choix de départ en retraite Déterminants du choix de départ en retraite Un système actuariellement neutre Quels effets sur l emploi des seniors? Chapitre 3 : Les faits
4/66 Assurance, retraite et santé - F. Langot Partie II : Retraite Chapitre I : Assurance privée (capitalisation) ou assurance sociale (répartition)
/66 Retour sur les choix d épargne dans le modèle à 2 périodes Hypothèses 2 périodes revenus :(w t, p t+1 ) préférences : u(c t, d t+1 ) rémunération des marchés financiers : r t financement de p t par une taxe τ t Problème du ménage max u(c t, d t+1 ) c t,d t+1 s.c. { ct + s t = w t τ t d t+1 = (1 + r t+1 )s t + p t+1
6/66 Assurance, retraite et santé - F. Langot Retour sur les choix d épargne dans le modèle à 2 périodes 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 Propriétés 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 à celui qu ils auraient lorsqu il n y a pas de système institutionnel de retraites. La solution de ce programme donne donc la valeur de l épargne libre d un agent.
7/66 Assurance, retraite et santé - F. Langot Retour sur les choix d épargne dans le modèle à 2 périodes Supposons u(c t, d t+1 ) = log(c t ) + β log(d t+1 ), alors d t+1 = β(1 + r t+1 ) c t = w t c t 1 + β s t = β 1 + β w t τ t } {{ } =st NoSS Il y a de l épargne s t > 0 ssi Sinon, s t = 0. β 1 + β w t > τ t st NoSS > τ t
/66 Propriétés de la retraite par capitalisation Supposons que s NoSS t > τ t. Il existe de l épargne volontaire. Equilibre sur le marché des titres : K t+1 = N t s t + N t τ t = N t s NoSS t car dans un système par capitalisation, les caisses de retraites transforment les taxes en offre de fonds prétables. Neutralité du système de retraite par capitalisation.
9/66 Assurance, retraite et santé - F. Langot Propriétés de la retraite par capitalisation Si s NoSS t < τ t, alors s t = 0 car l épargne ne peut pas être négative. Situation d épargne forcée où la SS impose un taux d épargne via la taxe τ t Equilibre su le marché des titres, avec N t+1 = (1 + n)n t, K t+1 = N t τ t N t+1 N t K t+1 N t+1 (1 + n)k t+1 = τ t car dans un système par capitalisation, les caisses de retraites transforment les taxes en offre de fonds prétables. Le système de retraite par capitalisation détermine l accumulation de capital.
10/66 Dans quelle situation l épargne forcée peut-elle être optimale? Que ferait un planificateur? s.c. max c t,d t+1 u(c t, d t+1 ) t=0 Y t = I t + N t c t + N t 1 d t K t+1 = (1 δ)k t + I t Y t = Kt α N 1 α max k t,d t+1 u t=0 t ( k α t + (1 δ)k t (1 + n)k t+1 d t 1 + n, d t+1 )
11/66 Dans quelle situation l épargne forcée peut-elle être optimale? Les CPO sont, avec f t = αk α 1 t, u c,t[f t + (1 δ)] = u c,t 1 (1 + n) u d,t = u c,t+1 1 1+n Sur le marché, avec r t = f t δ, on a } f t +(1 δ) 1+n = u c,t 1 u c,t 1 + n = u c,t u d,t 1 (1) (2) f t + (1 δ) 1 + r t = u c,t 1 u d,t 1 La nouvelle restriction est donc (2). = (1)/(2)
12/66 Dans quelle situation l épargne forcée peut-elle être optimale? Le côté droit de la relation (2) représente le taux marginal de substitution entre la consommation d un jeune et celle d un vieux vivant à la même date t. Planificateur : cette variable doit être égale au taux marginal de transformation (1 + n). Cette condition n a aucune raison d être vérifiée dans l économie de marché, car l horizon de chaque agent est limité à ses deux périodes de vie : il ne prend jamais en compte les possibilités de substituer les consommations entre les générations. Différences planificateur/marché vient de l allocation des ressources entre les générations et non de l allocation des ressources au sein d une même génération.
Dans quelle situation l épargne forcée peut-elle être optimale? Supposons c t = c t+1, d t = d t+1 et k t = k t+1, t, alors (1) f (k ) + (1 δ) = 1 + n f (k ) = n + δ 3/66 Est-ce possible? (1 + n)k M = r M = αk α 1 M d où r M = n ssi Marché Planificateur β 1 + β [f M k M f M ] k M = r = n [ δ = α(1 + β)(1 + n) β(1 α) β(1 α) (1 + β)(1 + n) δ β(n + δ) (1 + β)(1 + n) = α Peu probable! 1 α ] 1 1 α
14/66 Dans quelle situation l épargne forcée peut-elle être optimale? Si k M > k, il y a sur-accumulation neutralité de la SS. Si k M = k, optimalité neutralité de la SS. Si k M < k, il y a sous-accumulation. Une hausse de l épargne permettrait d augmenter la consommation de toutes les générations... sauf de la première. L effort d épargne est fait par la génération initiale qui réduit sa consommation et donc son bien être. Un équilibre de sous-accumulation est donc efficace au sens de Pareto puisqu il n est pas possible de modifier l allocation sans diminuer le bien-être d au moins une génération. τ = (1 + n)k. Dans ce cas, on a k M = k grâce à SS.
15/66 La répartition Dans ce cas, on a N t τ t = N t 1 p t p t = (1 + n)τ t. La répartition réalloue entre les générations 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 H t 1+β c t = 1 + β s t = w t τ t Ht = H t (1+n)τt s t = β 1 + β H t (1 + n)τ t 1 + r t+1 L équilibre sur les marchés financiers est 1+r t+1 Ht 1+β 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 t+1 ) τ t
16/66 La répartition Supposons x t = x t+1, x et t, et avec f = αf /k, alors (1 + n) = β(1 α) α(1 + β) f β(1 + r) (1 + n) τ (1 + β)(1 + r) Comme l allocation optimale doit vérifier f = n + δ et r = n, on a τ t.q. 1 α α β (n + δ) n = 1 + τ 1 + β
17/66 La répartition est-elle optimale? Oui si r M < n : sous-accumulation. Dans ce cas, les volume des cotisations futures dépassera le rendement (intérêt et principal) obtenu en plaçant les cotisations présentes sur le marché financier. Le système de retraite par répartition domine alors celui fondé sur la capitalisation. En France, on observe r > n. Depuis 2000, 0.38% < n < 0.574% et 5.2% < r < 6%. Changer de système : trop coûteux car il y a un transition générations sacrifiées.
18/66 Partie II : Retraite Chapitre II : Choix de départ à la retraite
19/66 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.
20/66 The extensive margin of the labor supply 100 Taux d emploi sur le cycle de vie hommes 2006 % 90 80 70 60 50 40 30 20 10 0 Fra Bel Can DanRUEU Esp Fin All Ita PB 15 24 ans 25 54 ans 55 59 ans 60 64 ans Rang des pays déterminé par le taux d emploi des 60 64 ans Sue Jap
21/66 Retirement Choice and SS Taxes on labor incomes have also an impact on the retirement age decision 0.7 0.65 Bel 0.6 Fra P B Ita Taux de non emploi des 55 65 ans 0.55 0.5 0.45 0.4 0.35 R U Esp Can E U Sué All 0.3 0.25 Jap 0.2 0 1 2 3 4 5 6 7 8 9 10 Taxe implicite sur la prolongation d activité (indice)
22/66 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.
23/66 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 : s.t. { T max c(t),r T 0 0 e rt c(t)dt = T e δt U(c(t))dt + R 0 e rt wdt R } e δt vdt
Retirement Choice 4/66 The FOC are : U (c(t)) = λe (r δ)t ve δr = λwe rr } v }{{} Opportunity Cost = U (c(r))w } {{ } 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 : R 0 e rt dt c = w T 0 e rt dt = w 1 e rr 1 e rt The optimal age of retirement is then given by : v = U ( w 1 e rr 1 e rt ) w
5/66 Retirement Choice and SS Assume now that there is a Social Security system. The agent budgetary constraint becomes T 0 The FOC are : e rt c(t)dt = R 0 e rt (1 τ)wdt + U (c(t)) = λe (r δ)t [ ve δr = λ (1 τ)we rr p(r)e rr + v }{{} Opportunity Cost T R T [ = U (c(r)) (1 τ)w p(r) + R e rt p(r)dt ] e rt p (R)dt T R 0 ] e rt p (R)dt } {{ } For a given U (c(r)), the marginal return can be taxed
6/66 Retirement Choice and SS We can then define this tax on the continued activity tax : TI = τw }{{} Effective Tax + p(r) } {{ } Implicit Tax T R e rt p (R)dt 0 } {{ } Bonus The SS system is actuarially fair if and only if τw + p(r) = T R 0 implying no distortion on the labor supply e rt p (R)dt v = U (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 rr p(r)e rr because τw 1 e rt 1 e rt = 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 = (1 τ)w v = U (c) ( w 1 e rrss 1 e rt ) [(1 τ)w p] 7/66 If there is no bonus when the agent delays her retirement age R SS < R
28/66 Les surcotes : vers un système actuariellement neutre Individu d âge z décidant de reproter son départ à l âge z + 1. beneficier d une surcote, notée λ(z + 1). La CNAV doit alors financer sur une periode allant de l âge z + 1 à 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, à l âge z, cet individu paie à la CNAV une cotisation égale à τw(z). En outre, la CNAV lui aurait versé une pension de pens(z) = [λ(z) + 0, 5]w ref (z) si l agent était parti à la retraite à l âge z.
29/66 Les surcotes : vers un système actuariellement neutre Financierement, la CNAV est indifferente entre ces deux options si la surcote verifie : [ ] T (z+1) [λ(z+1)+0,5]w s ref (z+1) T (z) t,t+1 [λ(z) + 0, 5]w ref (z) i (1+r) i = τw(z) (1 + r) i i Table: Taux annuels de décotes et de surcotes actuariellement neutres r = 3% r = 4% Age décote surcote décote 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%
0/66 Pourquoi la surcote actuarielle génère-t-elle des inégalités? Les inégalités face à la mort : entre un ouvrier et un cadre la différence est de 6 années à 60 ans. La SS doit verser une surcote sur 2 périodes pour le cadre, 1 période pour l ouvrier Recettes : R i = (λ + τ)w i, pour i = c, o. Cadre : D c = 2δ c w c δ c = (λ + τ)/2. Ouvrier : D o = δ o w o δ o = (λ + τ). La prime incitative doit être très grande pour un ouvrier car son risque est de la toucher pendant peu de temps.
1/66 La sécurité sociale n est pas une compagnie d assurance La SS ne peut pas connaître les risques de mort de tous les individus Les individus sont mieux informés sur leur santé personnelle. Elle propose donc une surcote moyenne, calculée sur la durée de vie espérée 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 à ces incitations. Ce système n est donc pas efficient il faut laisser aux assureurs le soin de gérer l assurance vie.
32/66 Bilan : Quelle type d incitation choisir? La surcote actuarielle en rente des inégalités dues à l inobservabilité du risque individuel de mort. Pas très lisible pour les agents Deux autres solutions plus efficaces et plus lisibles La surcote avec sortie en capital : à l âge de retraite, versemment d un capital qui peut alors géré par un assureur. Le cumul emploi-retraite : plus favorable pour les agents contraints financièrement.
33/66 La sortie en capital Table: Surcotes en capital mesurées en années de salaires nets r = 3% r = 4% Age de départ 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 calculées pour trois proportions différentes du salaire moyen (0,5, 1 et 2). Le ratio de remplacement pour ces différents niveaux de salaire est respectivement de 0,64, 0,51 et 0,45 (OCDE 2006).
34/66 La réforme de 2003 : Un modèle 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.
35/66 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 differ across labor abilities.
6/66 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 ).
37/66 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 at the beginning of the MRA.
8/66 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}
39/66 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 supply l units of labor input (l(e) = l) at a wage rate w. Consistently with empirical evidence, we will consider that the unemployment state is an absorbing state until retirement. Workers become retired as employed or unemployed.
0/66 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, d 150) 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 = 1 N N 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.
41/66 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]
42/66 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.
3/66 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 { } u(c, 1 l(φ)) V w (a, φ, γ, ξ) = max c 0 + β { φ ξ P(φ φ, γ, ξ)p(ξ γ, ξ)v w (a, φ, γ, ξ ) (1 + g)a = (1 + r)a + y(φ, γ, ξ) [1 Θ w (y(φ, γ, ξ))] c I et u subject to a 0 44/66 V w denotes the value function of workers. P(φ φ, γ, ξ) is the probability that a worker of labor market status φ, ability γ and age ξ becomes type φ the next period P(ξ γ, ξ) is the probability that a worker of ability γ and age ξ becomes age ξ 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 u by workers when employed. I e = 1 if φ = e, and zero otherwise.
45/66 Conditional on being alive (1 π M (γ)), workers become one year older ξ = ξ + 1. V r (a, φ, γ, ξ + 1) is the expected utility to be retired. V w (a, e, γ, Y ) : value of a new-born worker (ξ = Y ), employed (φ = e), with ability γ linked to γ (father) by P(γ γ). The young individual inherits the estate of his deceased father. η [0, 1] measures the father s concern for his offspring s well being. Behaviors : Households from ERA to MRA V w (a, φ, γ, ξ) u(c, [ 1 l(φ)) = max c 0 + β (1 πm (γ)) max [V w (a, φ, γ, ξ + 1), V r (a ], φ, γ, ξ + 1)] +π M (γ)η γ P(γ γ)v w (a, e, γ {, Y ) (1 + g)a = (1 + r)a + y(φ, γ, ξ) [1 Θ w (y(φ, γ, ξ))] c I et u subject to a 0
46/66 Behaviors : Retired Households ( V r (a, φ, γ, ξ) = max c 0 {u(c, 1) + β (1 πm (γ)) V r (a, φ, γ, ξ) +π M (γ)η { γ P(γ γ)v w (a, e, γ, Y ) (1 + g)a = (1 + r)a + ω(φ, γ, ξ) c subject to a 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.
47/66 Equilibrium 1 The exogenous Markov processes are summarized by s : Φ Γ Ξ S (φ, γ, ξ) s(φ, γ, ξ) with P(s s) = Pr{s t+1 = s s t = s} The steady state equilibrium is characterized by Worker and retiree choices {c w (a, s), c r (a, s), a w (a, s), a 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)
48/66 Equilibrium 2 Households solve value functions. Factor prices are competitive. Λ(a, s), associated with (A(a, s), P(s s)) : Λ(a, s ) = Λ(a, s)p(s s) s {a:a =A(a,s)} where A(a, s) is such that : [ ] { Ψ(a, s)a A(a, s) = w (a, s) 1 if Vw (a, s) V r (a, s) +[1 Ψ(a, s)]a r with Ψ(a, s) = (a, s) 0 otherwise Factor inputs are aggregated over individuals : L = Ψ(a, s)lµ(s) K = Λ(a, s)a(a, s) s a s a The payroll tax rate θ adjusts to balance the SS budget : Ψ(a, s)θy(s) = (1 Ψ(a, s))ω GR (s)) s a s a
Equilibrium : numerical solutions 49/66 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 ] : tv 1. tv m = max R 1. R m + β(π 1) v 1.. v m where is the Kronecker product and Π is the transition matrix that governs life-cycle, employment opportunities and altruism.
50/66 Numerical solutions : Howard improvement algorithm To make a guess to an initial feasible policy function a = g(a, s), To compute the n n matrix I h, for h = 1,..., m, defined by I h (a, a ) = 1 if a = g(a, s) and I h (a, a ) = 0 otherwise, Using the definition of R j (i, h), to evaluate the vectors [v 1,..., v m ] implied by using that policy forever : v 1. v m = R 1. R m + β Π 11I 1... Π 1mI 1....... Π m1i m... Π mmi m 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. These operation are repeated to convergence. v 1. v m
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 π M 0.0410 a 0.0483 0.0538 a : Each year, an H-ability agent faces a 4.10 percent probability of dying 51/66 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 b : A young H agent s wage is 2.14 higher than a young L agent s wage
52/66 Calibration : intergenerational ability transition matrix Table: Intergenerational change in ability Son s Ability (t + 1) H M L Father s Ability (t) 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
53/66 Calibration : unemployment risk Table: Unemployment replacement rate (ρ u ) and employment risk (π u ) 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
54/66 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 See the other PDF file on retirement decisions.
55/66 The actuarially fair scheme The actuarially-fair adjustment λ (γ, ξ) is such : 1 π M (γ) i=0 (λ (γ, ξ) + 0.5)w ref (γ, ξ) (1 + r) i } {{ } Pension paid by Social Security if the individual retires at age ξ = (1 π M (γ)) 1 π M (γ) i=1 (λ (γ, ξ + 1) + 0.5)w ref (γ, ξ + 1) (1 + r) i } {{ } Pension paid by Social Security if the individual retires at age ξ+1 θ w(γ, ξ) } {{ } taxes collected on wages during age ξ
56/66 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.
57/66 Partie II : Retraite Chapitre III : Retraite et emploi des seniors
8/66 Extension : Job Search and Retirement Decision Let V e i (w) be the value of the optimization problem for a worker of age C i and paid w, Vi 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 e i (w), V u i } for i = 1,.., 5
59/66 Extension : Job Search and Retirement Decision Bellman equations can be written as : for i = 1, 2, 3 and if the worker is employed V e i (w) = u((1 τ p τ b )w, T h) + β {π i [(1 λ i )V i (w) + λ i V u i ] +(1 π i )[(1 λ i+1 )V i+1 (w) + λ i+1 V u i+1] } (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 60/66 V u i = max {u((1 τ p )b i, T s i ) s i [ ] +β {π i φ(s i ) V i (w)df i (w) + (1 φ(s i ))Vi u [ +(1 π i ) φ(s i+1 ) V i+1 (w)df i+1 (w) +(1 φ(s i+1 ))Vi+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. F i (w) is the age-specific wage offer distribution.
61/66 Extension : Job Search and Retirement Decision for i = 4 V e 4 (w) = u((1 τ p τ b )w, T h) + β {π 4 [(1 λ 4 )V 4 (w) + λ 4 V u 4 ] +(1 π 4 )[(1 λ 5 ) max{v 5 (w), V r 5 } + λ 5 max{v u 5, V r 5 }]} V4 u = max {u((1 τ p )b 4, T s 4 ) s 4 [ ] +β {π 4 φ(s 4 ) V 4 (w)df 4 (w) + (1 φ(s 4 ))V4 u [ φ(s5 ) max{v +(1 π 4 ) 5 (w), V5 r }df ]}} 5(w) +(1 φ(s 5 )) max{v5 u, V 5 r }
2/66 Extension : Job Search and Retirement Decision for i = 5 V e 5 (w) = u((1 τ p τ b )w, T h) +β {π 5 [(1 λ 5 ) max{v 5 (w), V5 r } + λ 5 max{v5 u, V5 r }] +(1 π 5 )V6 r6 } V5 u = max {u((1 τ p )b 5, T s 5 ) s 5 [ +β {π 5 φ(s 5 ) max{v 5 (w), V5 r }df 5 (w) +(1 φ(s 5 )) max{v5 u, V5 r }] + (1 π 5 )V6 r6 }} V5 r = u(p 5, T ) + β { π 5 V5 r + (1 π 5 )V6 r5 } where p 5 denotes the retiree s pension at age 5.
63/66 Extension : Job Search and Retirement Decision for i = 6 6 = u(p 5, T ) + β { π 6 V6 r5 } 6 = u(p 6, T ) + β { π 6 V6 r6 } V r5 V r6 Benchmark : the pension is not increased by additional years of working beyond the full pension rate : p 6 = p 5. V e does not increase if the agent decides to postpone retirement (huge tax on continued activity). Policy : an actuarially fair increase in pension (p 6 > p 5) 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.
64/66 Extension : Job Search and Retirement Decision The optimal decision for search intensity is given by : for i = 1, 2, 3, 4 u 2((1 τ p )b i, T s i ) = φ (s i )βπ i ([ ] ) V i (w)df i (w) Vi 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.
65/66 Extension : Job Search and Retirement Decision for i = 5 u 2((1 τ p )b 5, T s 5 ) = φ (s 5 )βπ 5 ( [ max[v5 (w), V r 5 ]df 5(w) ] max[v u 5, V r 5 ] ) 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.
66/66 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.