The Method of Moderation For Solving Dynamic Stochastic Optimization - PowerPoint PPT Presentation

The Method of Moderation For Solving Dynamic Stochastic Optimization Problems Christopher Carroll 1 Kiichi Tokuoka 2 Weifeng Wu 3 1 Johns Hopkins University and NBER ccarroll@jhu.edu 2 International Monetary Fund ktokuoka@imf.org 3 Fannie Mae weifeng wu@fanniemae.com June 2012 http://econ.jhu.edu/people/ccarroll/papers/ctwMoM/

The Problem The Method of Moderation Normalization A Refinement Sneak Preview An Extension The Method of Endogenous Gridpoints Conclusions References The Basic Problem at Date t � T − t � � β n u ( c max E t c c t + n ) , (1) n =0 y = p p t θ t (2) y y t p R t = R - constant interest factor = 1 + r p p p t +1 = Γ t +1 p p p t - permanent labor income dynamics (3) N ( − σ 2 θ / 2 , σ 2 log θ t + n ∼ θ ) - lognormal transitory shocks ∀ n > 0 . “Bewley” problem if liquidity constraint, Γ t +1 nonstochastic “Friedman/Buffer Stock” problem if also permanent shocks Carroll, Tokuoka, and Wu The Method of Moderation

The Problem The Method of Moderation Normalization A Refinement Sneak Preview An Extension The Method of Endogenous Gridpoints Conclusions References The Basic Problem at Date t � T − t � � β n u ( c max E t c c t + n ) , (1) n =0 y = p p t θ t (2) y y t p R t = R - constant interest factor = 1 + r p p p t +1 = Γ t +1 p p p t - permanent labor income dynamics (3) N ( − σ 2 θ / 2 , σ 2 log θ t + n ∼ θ ) - lognormal transitory shocks ∀ n > 0 . “Bewley” problem if liquidity constraint, Γ t +1 nonstochastic “Friedman/Buffer Stock” problem if also permanent shocks Carroll, Tokuoka, and Wu The Method of Moderation

The Problem The Method of Moderation Normalization A Refinement Sneak Preview An Extension The Method of Endogenous Gridpoints Conclusions References Bellman Equation v t ( m m m t , p p p t ) = max u ( c c t ) + E t [ β v t +1 ( m c m m t +1 , p p p t +1 )] (4) c t c c − m m m ‘market resources’ (net worth plus current income) p − permanent labor income p p where � c 1 − ρ � u ( c ) = (5) 1 − ρ Carroll, Tokuoka, and Wu The Method of Moderation

The Problem The Method of Moderation Normalization A Refinement Sneak Preview An Extension The Method of Endogenous Gridpoints Conclusions References Normalize the Problem u ( c t ) + E t [ β Γ 1 − ρ v t ( m t ) = max t +1 v t +1 ( m t +1 )] (6) c t s.t. = m t − c t a t m t +1 = (R / Γ t +1 ) a t + θ t +1 � �� � ≡R t +1 where nonbold variables are bold ones normalized by p p p : m t = m m m t / p p p t (7) ⇒ c t ( m ) from which we can obtain c t ( m c m m t , p p p t ) = c t ( m m m t / p p p t ) p p (8) c p t Carroll, Tokuoka, and Wu The Method of Moderation

The Problem The Method of Moderation Normalization A Refinement Sneak Preview An Extension The Method of Endogenous Gridpoints Conclusions References My First Notational Atrocity ... in this paper. E t [ β Γ 1 − ρ v t ( a t ) = t +1 v t +1 ( R t +1 a t + θ t +1 )] (9) so FOC is u ′ ( c t ) v ′ = t ( m t − c t ) . (10) or c − ρ v ′ = t ( a t ) (11) t Carroll, Tokuoka, and Wu The Method of Moderation

The Problem The Method of Moderation Normalization A Refinement Sneak Preview An Extension The Method of Endogenous Gridpoints Conclusions References My First Notational Atrocity ... in this paper. E t [ β Γ 1 − ρ v t ( a t ) = t +1 v t +1 ( R t +1 a t + θ t +1 )] (9) so FOC is u ′ ( c t ) v ′ = t ( m t − c t ) . (10) or c − ρ v ′ = t ( a t ) (11) t Carroll, Tokuoka, and Wu The Method of Moderation

The Problem The Method of Moderation Normalization A Refinement Sneak Preview An Extension The Method of Endogenous Gridpoints Conclusions References My First Notational Atrocity ... in this paper. E t [ β Γ 1 − ρ v t ( a t ) = t +1 v t +1 ( R t +1 a t + θ t +1 )] (9) so FOC is u ′ ( c t ) v ′ = t ( m t − c t ) . (10) or c − ρ v ′ = t ( a t ) (11) t Carroll, Tokuoka, and Wu The Method of Moderation

The Problem The Method of Moderation Normalization A Refinement Sneak Preview An Extension The Method of Endogenous Gridpoints Conclusions References Concave Consumption Function, Target Wealth Ratio Carroll, Tokuoka, and Wu The Method of Moderation

The Problem The Method of Moderation Normalization A Refinement Sneak Preview An Extension The Method of Endogenous Gridpoints Conclusions References The Method of Endogenous Gridpoints Define vector of end-of-period asset values � a For each a [ j ] compute v ′ t ( a [ j ]) Each of these v ′ t [ j ] corresponds to a unique c [ j ] via FOC: c [ j ] − ρ v ′ = t ( a [ j ]) (12) � � − 1 /ρ v ′ c [ j ] = t ( a [ j ]) (13) But the DBC says = m t − c t (14) a t m [ j ] = a [ j ] + c [ j ] (15) So computing v ′ t at a vector of � a values (easy) has produced for us the corresponding � c and � m values at virtually no cost! From these we can interpolate to construct ` c t ( m ). Carroll, Tokuoka, and Wu The Method of Moderation

The Problem The Method of Moderation Normalization A Refinement Sneak Preview An Extension The Method of Endogenous Gridpoints Conclusions References The Method of Endogenous Gridpoints Define vector of end-of-period asset values � a For each a [ j ] compute v ′ t ( a [ j ]) Each of these v ′ t [ j ] corresponds to a unique c [ j ] via FOC: c [ j ] − ρ v ′ = t ( a [ j ]) (12) � � − 1 /ρ v ′ c [ j ] = t ( a [ j ]) (13) But the DBC says = m t − c t (14) a t m [ j ] = a [ j ] + c [ j ] (15) So computing v ′ t at a vector of � a values (easy) has produced for us the corresponding � c and � m values at virtually no cost! From these we can interpolate to construct ` c t ( m ). Carroll, Tokuoka, and Wu The Method of Moderation

The Problem The Method of Moderation Normalization A Refinement Sneak Preview An Extension The Method of Endogenous Gridpoints Conclusions References The Method of Endogenous Gridpoints Define vector of end-of-period asset values � a For each a [ j ] compute v ′ t ( a [ j ]) Each of these v ′ t [ j ] corresponds to a unique c [ j ] via FOC: c [ j ] − ρ v ′ = t ( a [ j ]) (12) � � − 1 /ρ v ′ c [ j ] = t ( a [ j ]) (13) But the DBC says = m t − c t (14) a t m [ j ] = a [ j ] + c [ j ] (15) So computing v ′ t at a vector of � a values (easy) has produced for us the corresponding � c and � m values at virtually no cost! From these we can interpolate to construct ` c t ( m ). Carroll, Tokuoka, and Wu The Method of Moderation

The Problem The Method of Moderation Normalization A Refinement Sneak Preview An Extension The Method of Endogenous Gridpoints Conclusions References The Method of Endogenous Gridpoints Define vector of end-of-period asset values � a For each a [ j ] compute v ′ t ( a [ j ]) Each of these v ′ t [ j ] corresponds to a unique c [ j ] via FOC: c [ j ] − ρ v ′ = t ( a [ j ]) (12) � � − 1 /ρ v ′ c [ j ] = t ( a [ j ]) (13) But the DBC says = m t − c t (14) a t m [ j ] = a [ j ] + c [ j ] (15) So computing v ′ t at a vector of � a values (easy) has produced for us the corresponding � c and � m values at virtually no cost! From these we can interpolate to construct ` c t ( m ). Carroll, Tokuoka, and Wu The Method of Moderation

The Problem The Method of Moderation Normalization A Refinement Sneak Preview An Extension The Method of Endogenous Gridpoints Conclusions References The Method of Endogenous Gridpoints Define vector of end-of-period asset values � a For each a [ j ] compute v ′ t ( a [ j ]) Each of these v ′ t [ j ] corresponds to a unique c [ j ] via FOC: c [ j ] − ρ v ′ = t ( a [ j ]) (12) � � − 1 /ρ v ′ c [ j ] = t ( a [ j ]) (13) But the DBC says = m t − c t (14) a t m [ j ] = a [ j ] + c [ j ] (15) So computing v ′ t at a vector of � a values (easy) has produced for us the corresponding � c and � m values at virtually no cost! From these we can interpolate to construct ` c t ( m ). Carroll, Tokuoka, and Wu The Method of Moderation

The Problem The Method of Moderation Normalization A Refinement Sneak Preview An Extension The Method of Endogenous Gridpoints Conclusions References The Method of Endogenous Gridpoints Define vector of end-of-period asset values � a For each a [ j ] compute v ′ t ( a [ j ]) Each of these v ′ t [ j ] corresponds to a unique c [ j ] via FOC: c [ j ] − ρ v ′ = t ( a [ j ]) (12) � � − 1 /ρ v ′ c [ j ] = t ( a [ j ]) (13) But the DBC says = m t − c t (14) a t m [ j ] = a [ j ] + c [ j ] (15) So computing v ′ t at a vector of � a values (easy) has produced for us the corresponding � c and � m values at virtually no cost! From these we can interpolate to construct ` c t ( m ). Carroll, Tokuoka, and Wu The Method of Moderation