B.1
Equilibrium unemployment rates
Let Ut,i, Nt,i, Ri,tand Pt,i denote respectively the number of unemployed workers of age
iat the beginning of period t, the number of employed workers, the number of retirees, and the total labor force (note that Pt,i = Nt,i+ Ut,i+ Rt,i, ∀t, i). Unemployment rates at
each age obey the following laws of motion: Ut,1 = (1 − π6)Pt−1,6+ π1λNt−1,1
new unemployed workers
+ π1[φ(s1)F1(w1) + (1 − φ(s1))]Ut−1,1
surviving unemployed workers and for i = 2, 3, 4,
Ut,i = (1 − πi−1)[φ(si−1)Fi−1(wi−1) + (1 − φ(si−1))]Ut−1,i−1
new unemployed workers coming from age i − 1
+ (1 − πi−1)Nt−1,i−1[λ + (1 − λ) max{0, Gi−1(wi) − Gi−1(wi−1)}]
new unemployed workers coming from age i − 1
+ πiλNt−1,i
new unemployed workers
+ πi[φ(si)Fi(wi) + (1 − φ(si))]Ut−1,i
surviving unemployed workers
where Gi(w)denotes the fraction of age iemployed workers at wage w or less. The right
hand side of these equations is the sum of new unemployed workers of age i and the survivors at age i of workers who were unemployed at the end of the period, which correspond to the workers who reject offers that are less than the optimal reservation wage (wi).
Given that the size of the population is a constant, denoted P , one can define stationary equilibrium unemployment rates by age. These constant levels of unemployment rates, denoted ui = Ui/P , are defined by:
u1 =
(1 − π6)p6+ π1λp1
1 − π1[φ(s1)F1(w1) + (1 − φ(s1))] + λπ1
and for i = 2, 3, 4,
(1 − πi[φ(si)Fi(wi) + (1 − φ(si))] + πiλ)ui
= (1 − πi−1)[φ(si−1)Fi−1(wi−1) + (1 − φ(si−1))]ui−1
−[λ + (1 − λ) max{0, Gi−1(wi) − Gi−1(wi−1)}]ui−1
+(1 − πi−1)[λ + (1 − λ) max{0, Gi−1(wi) − Gi−1(wi−1)}]pi−1+ πiλpi
equilibrium rates of retired workers are given by: if Vr 5 < V5u =⇒ U5,t > 0 and, R5,t = 0 if Vr 5 < V5u =⇒ U5,t = 0 and, R5,t = π5[R5,t−1+ λN5,t−1 +(1 − π4)[λ + (1 − λ) max{0, G4(w5) − G4(w4)}]N4,t−1 +(1 − π4)[φ(s4)F4(w4) + 1 − φ(s4)]U4,t−1 R6,t = π6R6,t−1+ (1 − π5)P5
At steady state, these equations imply that the rate of retired workers (Ri/P ) is given
by: if Vr 5 < V5u r5 = 0 if Vr 5 > V5u r5 = (1 − π4)[φ(s4)F4(w4) + (1 − φ(s4)) − λ − (1 − λ) max{0, G4(w5) − G4(w4)}] 1 − π5(1 − λ) u4 +λπ5p5+ (1 − π4)[λ + (1 − λ) max{0, G4(w5) − G4(w4)}]p4 1 − π5(1 − λ) r6 = (1 − π5)p5 1 − π6
After solving this system of equations, one can deduce the aggregate equilibrium unem- ployment rates: u = iui/(1 −
iri)and the equilibrium rate of retirees r =
iri.
Equilibrium unemployment rates by age are defined as ui = ui/pi.
B.2
Equilibrium wage distributions
As the demographic structure affects leads to a non stationary reservation wage, we com- pute Gi,t(w) the fraction of age i employed workers at wage w or less at time t in order to
determine equilibrium unemployment rate. These functions are derived from the following equilibrium flows:
(p1 − u1,t)G1,t(w) = π1
(1 − λ)(p1− u1,t−1)G1,t−1(w) + φ1,t−1(F1(w) − F1(w1))u1,t−1
where φ1,t−1 ≡ φ(s1,t−1). At steady state, this equation implies:
[1 − π1(1 − λ)](p1− u1)G1(w) = φ1(F1(w) − F1(w1))u1
⇔ G1(w) =
φ1u1
[1 − π1(1 − λ)](p1− u1)
For age i = 2, 3, 4, the dynamics of the fraction of age i employed workers at wage w or less at time t is given by:
(pi− ui,t)Gi,t(w)
= πi
(1 − λ)(pi− ui,t−1)Gi,t−1(w) + φi,t−1(Fi(w) − Fi(wi))ui,t−1
+(1 − πi−1)
φi−1,t−1(Fi−1(w) − Fi−1(wi−1))ui−1,t−1
+(pi−1− ui−1,t−1)Gi−1,t−1(w)[(1 − λ)(1 − max{0, Gi−1(wi) − Gi−1(wi−1)})]
For employees, the transition between age i − 1 and age i could lead to a voluntary quit if the wage accepted at age i−1 is lower than the reservation age at age i. This fraction of vol- untary quits is measured by (1−πi−1)(pi−1−ui−1,t−1)Gi−1,t−1(w)(1−λ) max{0, Gi−1(wi)−
Gi−1(wi−1)}. At steady state, we then obtain:
[1 − πi(1 − λ)](pi− ui)Gi(w)
= (1 − πi−1)[(1 − λ)(1 − max{0, Gi−1(wi) − Gi−1(wi−1)})](pi−1− ui−1)Gi−1(w)
+uiπiφi(Fi(w) − Fi(wi)) + ui−1(1 − πi−1)φi−1(Fi−1(w) − Fi−1(wi−1))
For age i = 5, the dynamics of the fraction of age i employed workers at wage w or less at time t is given by:
(pi− ri,t)Gi,t(w)
= πi(1 − λ)(pi− ri,t)Gi,t(w) + (1 − πi−1)
φi−1,t−1(Fi−1(w) − Fi−1(wi−1))ui−1,t−1
+(pi−1− ui−1,t−1)Gi−1,t−1(w)[(1 − λ)(1 − max{0, Gi−1(wi) − Gi−1(wi−1)})]
At steady state, we then obtain:
(1 − πi−1(1 − λ))(pi − ri,t)Gi,t(w)
= (1 − πi−1)
φi−1(Fi−1(w) − Fi−1(wi−1))ui−1
+(pi−1− ui−1)Gi−1(w)[(1 − λ)(1 − max{0, Gi−1(wi) − Gi−1(wi−1)})]
Notice that the equilibrium rates of unemployment by age depend on the equilibrium wage distributions, because some workers decide to quit their jobs. Moreover, the equilibrium wage distribution for a worker of age i is a function of the equilibrium wage distribution of workers of age i−1. Finally, one can define the aggregate equilibrium wage distribution as follows:
(p − u)G(w) =
i
(pi− ui)Gi(w)
where the the aggregate participation rate p is defined by p = 1 − r, with r the rate of retired workers.
B.3
Government budget constraints
Social security is financed by a proportional tax on labor income levied on all working people. For the sake of simplicity, we assume that pensions and non-employment incomes
are not linked to individuals’earning histories. For each period, the social security budget
is balanced. Then non-employment incomes are financed by levying taxes on workers: τb 5 i=1 (pi− ui− ri) j
wi,jdGi,j(wi,j) = 4
i=1
uibi
Similarly, for pensions, we have: τp 5 i=1 (pi− ui− ri) j
wi,jdGi,j(wi,j) = 6
i=5
ripi
The computation of the equilibrium allows us to find the tax rates τb and τp that balance