AREA DIAS PERDIDOS / ACCIDENTE %

and thereby introduces a high degree of nonlinearity - non-differentiabilities in the Pt

Dt-dimension - into the consumption policy function CD

(j)_{(·) and the} associated shadow tax τ(j)_{(·). While linear interpolation between two grid} points yields very accurate approximations of these functions for most Pt

Dt val- ues, this is generally not true close to the boundaries of the inaction regions, if these boundaries are not elements of our discretized state space.

Including the Pt

Dt boundaries of the inaction region into the discretized state space poses two challenges: First, the exact locations of these boundaries are not known a priori, but depend on the optimal solution. Therefore, the Pt

Dt grid is required to change in every iteration. We describe in the sequel how we use an adaptive grid point choice to ensure that our best guess for the inaction region boundaries is always part of the Pt

Dt grid. Second, these boundaries are not independent of other states, but vary with (St−1,W_Dtt, mt). Hence, the _DPt_t

2.12. APPENDIX

also to be dependent on other state variables.50 _{We clarify below how we} interpolate our policy to states not contained in the discretized state space.

Adaptive grid points: Since the non-differentiability problem only oc-

curs in the Pt

Dt-dimension, we fix a vector (St−1,

Wt

Dt, mt) in the sequel. First, we observe, that the interior of the inaction region in the Pt

Dt-dimension can be identified by the shadow tax function τ(_·): The optimal consumption (or, equivalently, stock holding) policy does not change in a neighborhood of the current value of Pt Dt, if and only ifτ St−1,_DPt_t,W_Dtt, mt ∈(₋τ , τ). Since in such cases St−1 = St, the same relationship must hold for the function ˜τ defined

on the alternative ”state space” (St,_DPt_t,W_Dtt, mt). In our solution algorithm, we

solve for this function ˜τ by solving equation (2.19) under the assumption that consumption satisfies the no trade relationship (2.18) and set it to τ, when- ever its value exceeds τ and to ₋τ, whenever its value is less than ₋τ. The boundaries of the inaction region are therefore given for those values of Pt

Dt, for which no trade consumption defined by (2.18) and τ(_tj+1) _{∈ {−}τ , τ_} solve equation (2.19). This yields two equations

S_t∗+ Wt Dt ∗−γ (1_±τ) = M(S_t∗, P D ± , Wt Dt ∗ , m∗_t)

which we solve for the adapted grid points P_D± in each iteration of the above algorithm.51 _{We make sure, that in our algorithm not only the functions}

CD(j)_(·),τ(j)_{(·) and T D}(j)_{, but also these adapted grid points converge. The} present approach is similar to the approach proposed in Brumm and Grill (2014). The latter cover the discretized state space with simplices and look for ‘just binding’ constraints on each edge of these simplices. We only look at edges that are orthogonal to the (St−1,W_Dt_t, mt)-hyperplane, which is computationally

more efficient within the present setup.

Interpolation: We fix the set of initial grid points GS, GW D, GP D, Gm

for the state space. Our discretized state space is, however, not given by the product GS×GW D ×GP D ×Gm, but instead by

GS×GW D×GP D×Gm

∪ {(S, W D, P D+(S, W D, m), m)_|(S, W D, m)_∈GS×GW D×Gm}

∪ {(S, W D, P D−(S, W D, m), m)|(S, W D, m)∈GS×GW D×Gm}

The standard linear interpolation method on a Cartesian product of one- dimensional grids is therefore augmented as follows: for a given query point

50_{Including all inaction boundaries for any combination of (S}

t−1,W_Dtt, mt) into a common Pt

Dt grid creates a computationally prohibitively large number of discretization points.

51 Note, that P D + and P D − are functions of ((S ∗ t, Wt Dt ∗ , m∗

t), although this is sup-

(Sq, W Dq, P Dq, mq), we first search for indicesi, j, k, such that Sq ∈[Si, Si+1], W Dq ∈[W Dj, W Dj+1] and mq ∈[mk, mk+1] and then linearly interpolate the policy in the P D-dimension for each combination (S, W D, m)_{∈ {}Si, Si+1} × {W Dj, W Dj+1} × {mk, mk+1} using as a PD grid the intersection of the dis- cretized state space with the line parallel to theP D-axis that crosses (S, W D, m). This yields eight interpolated policy valuesCDu,v,w with (u, v, w)∈ {i, i+ 1}×

{j, j+ 1_{} × {}k, k+ 1_} of the function

(S, W D, m)_7→CD(S, W D, P Dq, m)

at the chosen closest (S, W D, m)-grid points. We then use ordinary three- dimensional linear interpolation to obtain the interpolated policy value for

CD(Sq, W Dq, P Dq, mq), i.e. CDinterp(Sq, W Dq, P Dq, mq) = X u=i,i+1 X v=j,j+1 X w=k,k+1 |Sq−Su||W Dq−W Dv||mq−mw| (Si+1−Si)(W Dj+1−W Dj)(mk+1−mk) CDu,v,w

We proceed analogously for linear extrapolation.

2.12.4

Testing for Equality of Gain Estimates