info document - Paris School of Economics

After retirement, individuals receive a pension and there is no human capital investment.The retirement pension depends on the earnings one period before retirement. Sincemarkets are complete and there is no uncertainty during retirement, a riskless bond issufficient for smoothing consumption. Therefore, the problem of a retired agent at ages > R can be written asW R (a, y R−1 , m; s) = max[u(c, ¯n) + βW R (a ′ , y R−1 , m; s + 1) ] (12)c,a ′s.t(1 + ¯τ c )c + qa ′ = (1 − ¯τ n (y s ))y s + a + T ry s = Ω(y R−1 , m)Since the pre-retirement earnings y R−1 depends on the state of the individual at ageR − 1, i.e. it is given by y R−1 = y(h, a, m; ɛ, , R − 1), the problem above implicitly dependson pre-retirement state (ɛ, a, h, m; R − 1). However, we do not need to keep track of all thevariables since y R−1 and m are the only variables that matters for pension. 9 However, in thedefinition of equilibrium, we need to identify the fraction of agents in state (h, a, m; ɛ, R−1)to determine the total social security payments.Definition 1 The stationary equilibrium for this economy is a set of equilibrium decisionsrules c(x), n(x), Q(x), i(x) and a ′ (ɛ ′ , x); and value function functions for V (x) and W R (x)for working and retirement periods respectively, where x = (h, a, m; ɛ, s, j) (notice the inclusionof j into this vector); and a time-invariant measure Λ(x) such that1. Given prices P H and P L , the labor income tax function ¯τ(y), consumption tax ¯τ c andgovernment policy functions Φ and Ω; individuals solve problems in (11) and (12).2. Prices are determined competitively.3. Aggregate amount of raw labor and human capital are given by∫H = ɛh(x)(1 − i(x))dΛ(x)x∫L = ɛl(1 − i(x))dΛ(x)x9 Basically, the value function of the agent the year before retirement is V (ɛ, a, h, m; R − 1) =max c,n,Q,a ′ (ɛ ′ ) u(c, n) + βEW R (a ′ (ɛ ′ ), y(ɛ, a, h, m; R − 1), m; R) subject to constraints of problem (11). Itshould be clear that individuals choose the same amount of Arrow securities for all states (which is equivalentto holding a riskless bond) since there is no uncertainty in future income.21