Concepto y caracteres del derecho al olvido.

The choice of the exogenous savings grid is crucial for the endogenous grid method. We want a high density of grid points where policy functions have high curvatures. We also want to avoid agents falling off the grids in the forward iteration. So before applying the solution methods described above we develop an agent-specific81savings grid GS = {s

1, s2, . . . , sn}. In this section

we abstract form vivos and transfers, because these periods have to be treated differently and we take care of that in Section B.3.

Determining the lowest savings grid point

We know that A0 = −A_j,tcan be either a situation in which the household chooses zero savings, i.e. she is on the edge of being borrowing constraint, or she is borrowing constraint and would like to choose A0 < −A_j,t. So let us denote the lowest possible savings grid point by A0_l, which is given by A0

l = −Aj,t in period t and compute C(A 0

l) as well as l(A 0

l), assuming we are in the

interior solution. Cash-on-hand X(A0_l) and asset level A(A0_l) will then be determined by:

X = A0+ (1 + τc)C+ (1 − τss)wt, j,qγη(1 − `) + Tt(Yt)

A= X −(1 − τss)wt, j,qγη Rnt

.

(i) A(A0_l) < −A_j,t−1 − for tolerance level . This violates the borrowing constraint making A0_l an invalid savings grid point. Further, this implies that the agent cannot possibly be borrowing constraint in period t, because entering the period with any valid asset level

Avalid ≥ Aj,t−1 > A(A0l) has to lead to savings A

0

(Avalid) > A0l = −Aj,t. Our task now is to

80_{We deal with borrowing constraint households in Section B.5.3.}

81_{Note that agent specific stands in for a combination of age, γ- and η-shock, qualification, innate human capital}

find the lowest valid savings grid point in period t which is the one leading to endogenous assets A= A_j,t−1. We could do so by letting a solver search for A(X(A0))+ A_j,t−1 = 0 under tolerance level , i.e. find

− < A(X(A0))+ A_j,t−1 < 

over A0. However, we could get a problem whenever we receive an asset level 0 < A(X(A0) + A_j,t−1 <  and in the forward iteration for assets Afwd we get 0 < Afwd <

A(X(A0)+ A_j,t−1. In this case we would have to extrapolate. We can avoid that by solving A(X(A0))+ A_j,t−1+  = 0 instead, i.e. find

− < A(X(A0

))+ A_j,t−1+  <  (71)

implying −2 < A(X(A0₎₎+ A

j,t−1 < 0. These asset holdings violate the borrowing con-

straint theoretically, but they are numerically indistinguishable from −A_j,t−1. So we can built the state contingent grid GS = {s

1, s2, . . . , sn} with s1 = A0 received from (71).

(ii) A(A0_l) ≥ A_j,t−1−. This tells us that an household entering period t with asset level A(A0 l)

would choose A0 = −A

j,t, implying it is a valid and interior solution. However, this also

tells us that she is borrowing constraint in period t whenever she enters the period with an asset level A with −A_j,t−1 ≤ A < A(A0

l). So we set s2 = −Aj,t and solve at grid points

GS _{= {s}

2, . . . , sn} the way we are used to. In order to have a solution for the borrowing

constraint situation (which is possible as opposed to CASE 1), we save the following solution on an extra grid point s1= −A_j,t:

Set assets today to the minimum A = −A_j,t−1. This leads to cash-on-hand today X(s1) =

Rnt(−Aj,t−1)+(1−τss)wt, j,qγη. Now solve for consumption and leisure via the intra-temporal

should have X(s1) < X(s2) as

X(s1)= Rnt(−Aj,t−1)+ (1 − τss)wt, j,qγη < RntA(A 0

l)+ (1 − τss)wt, j,qγη = X(s2)

and our case distinction A(A0_l) ≥ A_j,t−1 − . But, although unlikely, due to numerical inaccuracies we have X(s1) < X(s2) compute the solution at x1 = ωx2(s2) for some ω

close to one.

Determining the highest savings grid point

On the one hand snshould be as high as possible, because we do not want to restrict the space

our policy functions live in. On the other hand we want to prevent agents from falling of the grid. That could happen in case we choose a maximum savings grid point sn = ¯A0 that might

lead to cash-on-hand tomorrow of X0_(s

n) > Xn0, where X 0

n is the highest cash-on-hand level in

the endogenous cash-on-hand-grid of t+ 1.

Cash-on-hand t+1 is given by X0= Rk_t+1A0+(1−τss)wt+1, j+1,qγη0. As wages tomorrow differ

with the idiosyncratic income shock η0_{for each shock there is an asset level ¯A}0_(η0_{) satisfying:}

Rn_t+1A¯0(η0)+ (1 − τss)wt+1, j+1,qγη

0 _{= X}0

n.

And as better shocks imply higher wages, i.e. wt+1, j+1,qγ¯η0 > wt+1, j+1,qη0, we have ¯A0( ¯η0) <

¯

A0(η0). Therefore, choosing sn = ¯A0( ¯η0) ensures X0(sn) = Rn_t+1s2+ (1 − τss)wt+1, j+1,qγ¯η0 ≤ X0n.

Formally speaking we choose sn such that sn = minη0{A0(η0)}, or even more precisely, we set it