Supplemental Material (not for publication) Persistent vs. Permanent Income Shocks in the Buffer-Stock Model Jeppe Druedahl Thomas H. Jørgensen May, A Additional Figures and Tables Figure A.: Wealth and Consumption Age Profiles (estimating ρ, ρ =, φ =.). (a) A t, α =.99 (b) A t, α =.9 (c) A t, α =.97 α =.99, true ρ α =.9, true ρ α =.97, true ρ (d) C t, α =.99 (e) C t, α =.9 (f) C t, α =.97.5.5.5.5 α =.99, true ρ.5 α =.9, true ρ.5 α =.97, true ρ.5.5 Notes: Figure A. shows profiles of 5. simulated households..5 A
Figure A.: MPC and MPCP Age Profiles (estimating ρ, ρ =, φ =.).. (a) MPC, α =.99 (b) MPC, α =.9 (c) MPC, α =.97 α =.99, true ρ α =.9, true ρ α =.97, true ρ 5 5 5 (d) MPCP, α =.99 (e) MPCP, α =.9 (f) MPCP, α =.97 9 α =.99, true ρ 9 α =.9, true ρ 9 α =.97, true ρ Notes: Figure A. shows profiles of 5. simulated households. Figure A.: Wealth and Consumption Age Profiles (estimating φ, ρ =, φ =.). (a) A t, α =.99 (b) A t, α =.9 (c) A t, α =.97 α =.99, true φ α =.9, true φ α =.97, true φ (d) C t, α =.99 (e) C t, α =.9 (f) C t, α =.97.5.5.5.5 α =.99, true φ.5 α =.9, true φ.5 α =.97, true φ.5.5 Notes: Figure A. shows profiles of 5. simulated households..5 A
Figure A.: MPC and MPCP Age Profiles (estimating ρ, ρ =, φ =.). (a) MPC, α =.99 (b) MPC, α =.9 (c) MPC, α =.97 α =.99, true φ α =.9, true φ α =.97, true φ 5 5 5 (d) MPCP, α =.99 (e) MPCP, α =.9 (f) MPCP, α =.97 9 α =.99, true φ 9 α =.9, true φ 9 α =.97, true φ Notes: Figure A. shows profiles of 5. simulated households. Table A.: Robustness. Estimating ρ using medians (ρ =, φ =.). BIAS(ˆρ) DIFF(MPC) DIFF(MPCP) α =.99 α =.97 α =.99 α =.97 α =.99 α =.97 baseline -. -5. -. -.5.5. k = -7.7 -. -. -..5. k = 5 -.7 -. -. -..5.7 k = 55... -... Notes: See notes to table. and 5.. A
Table A.: Robustness. Estimating φ using medians (ρ =, φ =.). BIAS( ˆφ) DIFF(MPC) DIFF(MPCP) α =.99 α =.97 α =.99 α =.97 α =.99 α =.97 baseline.5. -. -.5.. k =..5 -. -.9..5 k = 5 7.. -. -.7.. k = 55 -. -9.9. -... Notes: See notes to table. and 5.. A
B Model: Details and Solution Algorithm B. Generalized income process In the model we solve, we use a somewhat more general formulation of the income process allowing for both persistent and permanent shocks simultaneously: t T R : Y t = Γ t L t P t ξ t (B.) P t = G t P t ψ t (B.) L t = L α t η t (B.) log ψ t N (.5 σ ψ, σ ψ) (B.) log η t N (.5 ση, ση) (B.5) ( ) e α t α αt σ α η if α < Γ t = (B.) if α = t > T R : Y t = κ Γ TR L TR P TR (B.7) A t λ t L t P t (B.) B. The Normalization Factor Γ t Note the useful result that if X = e Z, Z N ( µ, σ ) (B.9) then and E [X] = e µ+ σ X αt = ( e Z) α t = e αtz = e K, K ( N ( α t µ, α t σ )) (B.) (B.) Given our assumption of η t e N(µη,σ η) (B.) µ η σ η A5
we thus have E [Y t ] = E [Γ t L t P t ] Γ t = E [L t ] E [P t ] = E [ η α t η αt η αt η t ] L P = e µη+ σ η e αµ η+ α σ e α µ η+ α σ η e α t µ η+ α(t ) σ η L P = e (+α+α + +α t )µ η e (+α+α + +α (t ) )α σ η = e α Γ t = e t α µη+ α t α σ η ( ) α t α αt σ α η (B.) B. Equalizing Income Variances The variance of the permanent income component in period k is given by V [log P k ] = k V [log P ] + V [log ψ j ] j= (B.) = V [log P ] + k σ ψ The variance of the persistent income component in period k is given by V [log L k ] = α k V [log L ] + = α k V [log L ] + k α (k j) V [log η k ] j= k α (k j) V [log η k ] j= (B.5) = α k V [log L ] + ( α (k ) + + α + α + α ) σ η = α k V [log L ] + αk α σ η To ensure that the variances are identical in period k, we set α k α σ η = k σ ψ σ η = k α α k σ ψ and we assume V [log P ] = V [log L ] = in our simulations. A
B. Normalization Relying on the homogeneity in P t, the model can be written in ratio form (a t m t = Mt P t etc.) as = At P t, v t (L t, m t ) = c ρ t max c t ρ + βe [ t (Gt+ ψ t+ ) ρ v t+ (L t+, m t+ ) ] (B.) s.t. a t = m t c t m t+ = R a t + Λ t+ ξ t+ L t+ G t+ ψ t+ L t+ = L α t η t+ a t λ t Denoting the optimal consumption choice by c t (L t, m t ), we have that the first order condition imply [ (Gt+ c t (L t, m t ) ρ βr E t ψ t+ c t+ (L t+, m t+ ) ) ρ ] (B.7) which holds with equality if a t (L t, m t ) m t c t (L t, m t ) > λ t and otherwise c t = m t + λ t B.5 Accuracy of the Numerical Solution The model is solved using the endogenous-grid-point method (EGM) proposed by Carroll (). The grids of a t ( nodes) and L t ( nodes) are chosen exogenously, and the expectation is taken using Gauss-Hermite nodes for ξ t+, η t+, and ψ t+ and by relying on standard bi-linear interpolation of the next period consumption function. The left panel of figure B. shows the profile of the aver Euler error based on simulated data from the numerically solved model. Particularly, we simulate N = 5, individuals for T = time periods and calculate the aver Euler error for a given as E t = N N βr (C it /C it ) ρ. (B.) i= Binding credit constraints would generally imply an aver Euler error below one; to mitigate this, the right panel of figure B. shows the aver Euler error conditioning on lagged wealth being above some threshold such that the households are not credit constrained. As seen, the aver Euler errors fluctuates around and are very close to A7
one, and, importantly, do not differ significantly across the two considered levels of α. Figure B.: Aver Euler Error Age Profiles (a) All (b) A t >...5 σ ψ >, σ η = σ ψ =, σ η >, α =. σ ψ =, σ η >, α =.9..5 σ ψ >, σ η = σ ψ =, σ η >, α =. σ ψ =, σ η >, α =.9.9.9.99.99 5 5 Notes: Figure B. shows aver Euler errors (see equation B.) for 5. simulated households. References Carroll, C. D. (): The method of endogenous gridpoints for solving dynamic stochastic optimization problems, Economics Letters, 9(),. A