1.5cm lecture 9 [1ex] efficient computation of heterogeneous …€¦ · benjamin moll princeton...
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![Page 1: 1.5cm Lecture 9 [1ex] Efficient Computation of Heterogeneous …€¦ · Benjamin Moll Princeton University,Fall 2016 October 17,2016 1. Motivation • Quantitative DSGEmodelscore](https://reader035.vdocuments.mx/reader035/viewer/2022071004/5fc1cf12c4053526630c564d/html5/thumbnails/1.jpg)
Lecture 9Efficient Computation of Heterogeneous Agent
Models with Aggregate Shocks
ECO 521: Advanced Macroeconomics I
Benjamin Moll
Princeton University, Fall 2016
October 17, 20161
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Motivation
• Quantitative DSGE models core of macroeconomic policy analysis
• Rely on representative agent assumptions
• Recent work argues micro heterogeneity important for policyanalysis
1. aggregate dynamics depend on distribution2. care about distributional implications
• But then distribution is a state variable
• Quantitative DSGE analysis infeasible with current methods
• Today: tell you about project that tries to make progress on this(joint with SeHyoun Ahn, Greg Kaplan, Tom Winberry)
2
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Motivation
• Quantitative DSGE models core of macroeconomic policy analysis
• Rely on representative agent assumptions
• Recent work argues micro heterogeneity important for policyanalysis
1. aggregate dynamics depend on distribution2. care about distributional implications
• But then distribution is a state variable
• Quantitative DSGE analysis infeasible with current methods
• Today: tell you about project that tries to make progress on this(joint with SeHyoun Ahn, Greg Kaplan, Tom Winberry)
2
![Page 4: 1.5cm Lecture 9 [1ex] Efficient Computation of Heterogeneous …€¦ · Benjamin Moll Princeton University,Fall 2016 October 17,2016 1. Motivation • Quantitative DSGEmodelscore](https://reader035.vdocuments.mx/reader035/viewer/2022071004/5fc1cf12c4053526630c564d/html5/thumbnails/4.jpg)
Motivation
• Quantitative DSGE models core of macroeconomic policy analysis
• Rely on representative agent assumptions
• Recent work argues micro heterogeneity important for policyanalysis
1. aggregate dynamics depend on distribution2. care about distributional implications
• But then distribution is a state variable
• Quantitative DSGE analysis infeasible with current methods
• Today: tell you about project that tries to make progress on this(joint with SeHyoun Ahn, Greg Kaplan, Tom Winberry)
2
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Motivation
• Quantitative DSGE models core of macroeconomic policy analysis
• Rely on representative agent assumptions
• Recent work argues micro heterogeneity important for policyanalysis
1. aggregate dynamics depend on distribution2. care about distributional implications
• But then distribution is a state variable
• Quantitative DSGE analysis infeasible with current methods
• Today: tell you about project that tries to make progress on this(joint with SeHyoun Ahn, Greg Kaplan, Tom Winberry)
2
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Motivation
• Quantitative DSGE models core of macroeconomic policy analysis
• Rely on representative agent assumptions
• Recent work argues micro heterogeneity important for policyanalysis
1. aggregate dynamics depend on distribution2. care about distributional implications
• But then distribution is a state variable
• Quantitative DSGE analysis infeasible with current methods
• Today: tell you about project that tries to make progress on this(joint with SeHyoun Ahn, Greg Kaplan, Tom Winberry)
2
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What We Do
• Develop general + efficient method to compute het agent models
1. Compute steady state using global approximations
• exploit advantages of continuous time (Achdou et al. 2015)
2. Compute aggregate dynamics using local approximations
• linear in entire distribution (Campbell 1998, Reiter 2009)• reduce dimensionality using SVDs (Reiter 2009)
• Apply to textbook RBC + income heterogeneity (Krusell-Smith 98)• slides, notes, and Matlab codes available at
http://www.princeton.edu/~moll/PHACTproject.htm
• Have also implemented other applications• Khan & Thomas (2008), HANK models• ultimately: medium-scale DSGE (cost of extra agg states ≈ 0)
3
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What We Do
• Develop general + efficient method to compute het agent models1. Compute steady state using global approximations
• exploit advantages of continuous time (Achdou et al. 2015)
2. Compute aggregate dynamics using local approximations
• linear in entire distribution (Campbell 1998, Reiter 2009)• reduce dimensionality using SVDs (Reiter 2009)
• Apply to textbook RBC + income heterogeneity (Krusell-Smith 98)• slides, notes, and Matlab codes available at
http://www.princeton.edu/~moll/PHACTproject.htm
• Have also implemented other applications• Khan & Thomas (2008), HANK models• ultimately: medium-scale DSGE (cost of extra agg states ≈ 0)
3
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What We Do
• Develop general + efficient method to compute het agent models1. Compute steady state using global approximations
• exploit advantages of continuous time (Achdou et al. 2015)2. Compute aggregate dynamics using local approximations
• linear in entire distribution (Campbell 1998, Reiter 2009)• reduce dimensionality using SVDs (Reiter 2009)
• Apply to textbook RBC + income heterogeneity (Krusell-Smith 98)• slides, notes, and Matlab codes available at
http://www.princeton.edu/~moll/PHACTproject.htm
• Have also implemented other applications• Khan & Thomas (2008), HANK models• ultimately: medium-scale DSGE (cost of extra agg states ≈ 0)
3
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What We Do
• Develop general + efficient method to compute het agent models1. Compute steady state using global approximations
• exploit advantages of continuous time (Achdou et al. 2015)2. Compute aggregate dynamics using local approximations
• linear in entire distribution (Campbell 1998, Reiter 2009)• reduce dimensionality using SVDs (Reiter 2009)
• Apply to textbook RBC + income heterogeneity (Krusell-Smith 98)• slides, notes, and Matlab codes available at
http://www.princeton.edu/~moll/PHACTproject.htm
• Have also implemented other applications• Khan & Thomas (2008), HANK models• ultimately: medium-scale DSGE (cost of extra agg states ≈ 0)
3
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What We Do
• Develop general + efficient method to compute het agent models1. Compute steady state using global approximations
• exploit advantages of continuous time (Achdou et al. 2015)2. Compute aggregate dynamics using local approximations
• linear in entire distribution (Campbell 1998, Reiter 2009)• reduce dimensionality using SVDs (Reiter 2009)
• Apply to textbook RBC + income heterogeneity (Krusell-Smith 98)• slides, notes, and Matlab codes available at
http://www.princeton.edu/~moll/PHACTproject.htm
• Have also implemented other applications• Khan & Thomas (2008), HANK models• ultimately: medium-scale DSGE (cost of extra agg states ≈ 0)
3
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What We Do
• Develop general + efficient method to compute het agent models1. Compute steady state using global approximations
• exploit advantages of continuous time (Achdou et al. 2015)2. Compute aggregate dynamics using local approximations
• linear in entire distribution (Campbell 1998, Reiter 2009)• reduce dimensionality using SVDs (Reiter 2009)
• Apply to textbook RBC + income heterogeneity (Krusell-Smith 98)• slides, notes, and Matlab codes available at
http://www.princeton.edu/~moll/PHACTproject.htm
• Have also implemented other applications• Khan & Thomas (2008), HANK models• ultimately: medium-scale DSGE (cost of extra agg states ≈ 0) 3
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Plan For Today
1. Model
2. Solution method
3. Results
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Households
max{ct}t≥0
E0∫ ∞0
e−ρtu(ct)dt s.t.
at = wtzt + rtat − ct
zt ∈ {zℓ, zh} Poisson with intensities λℓ, λh
at ≥ a
• ct : consumption• u: utility function, u′ > 0, u′′ < 0.• ρ: discount rate• rt : interest rate• a > −∞: borrowing limit e.g. if a = 0, can only save
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Firms
• Aggregate production function
Yt = eZtKαt N
1−αt
• Perfect competitition in factor markets
wt = (1− α)YtNt, rt = α
YtKt− δ
• Market clearing
Kt =
∫agt(a, z)dadz,
Nt =
∫zgt(a, z)dadz ≡ 1
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Equilibrium• Aggregate state: (gt , Zt)⇒ absorb into time subscript t
• recursive notation w.r.t. individual states only• Et is expectation w.r.t. aggregate states only fully recursive
ρvt(a, z) =maxcu(c) + ∂avt(a, z)(wtz + rta − c)
+ λz(vt(a, z′)− vt(a, z)) +
1
dtEt [dvt(a, z)],
(HJB)
∂tgt(a, z) =− ∂a[st(a, z)gt(a, z)]− λzgt(a, z) + λz ′gt(a, z ′), (KF)
dZt = −νZtdt + σdWt ,
wt = (1− α)eZtKαt ,
rt = αeZtKα−1t − δ,
Kt =
∫agt(a, z)dadz
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Equilibrium• Aggregate state: (gt , Zt)⇒ absorb into time subscript t
• recursive notation w.r.t. individual states only• Et is expectation w.r.t. aggregate states only fully recursive
ρvt(a, z) =maxcu(c) + ∂avt(a, z)(wtz + rta − c)
+ λz(vt(a, z′)− vt(a, z)) +
1
dtEt [dvt(a, z)],
(HJB)
∂tgt(a, z) =− ∂a[st(a, z)gt(a, z)]− λzgt(a, z) + λz ′gt(a, z ′), (KF)
dZt = −νZtdt + σdWt ,
wt = (1− α)eZtKαt ,
rt = αeZtKα−1t − δ,
Kt =
∫agt(a, z)dadz
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Solution Method
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Extension of Standard Linearization Method
1. Compute non-linear approximation to non-stochastic steady state
2. Compute first-order Taylor expansion around steady state
3. Solve linear stochastic differential equation
Background on linearization methods:• Deterministic models
• Chapter 6.3 of Stokey-Lucas-Prescott• http://www.princeton.edu/~moll/ECO503Web/Lecture4_ECO503.pdf
• Stochastic models• Sims (2001) “Solving Linear Expectations Models”• these notes http://www.robertopancrazi.com/LN3_solving_lrem.pdf
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Linearization of Continuous-Time RBC model
• Optimality conditions in RBC model
Et [dΛt ] = Λt(ρ+ δ − αeZtKα−1t
)dt
dKt =
(eZtKαt − δKt − Λ
− 1γ
t
)dt
dZt = −νZtdt + σdWt
• We have:
control variable = Λtendog state variables = Kt
exog state variables = Zt
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Linearization of Continuous-Time RBC model
• Can write system asEt [dΛt ]dKtdZt
= f (Λt , Kt , Zt)dt +00σ
dWtwith f : R3 → R3
• Since Et [dWt ] = 0, above system implies:
Et
dΛtdKtdZt
= f (Λt , Kt , Zt)dt
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Linearization of Continuous-Time RBC model
1. Compute non-stochastic steady state (Λ, K,Z = 0) by hand
2. Compute first-order Taylor expansion of f (Λt , Kt , Zt)3. Diagonalize matrix B, hope same number of stable eigenvalues as
state variables (2 in this model)
• if so, set control variables orthogonal to unstableeigenvectors, get policy function
Λt = DKKt +DZZt
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Linearization of Continuous-Time RBC model
1. Compute non-stochastic steady state (Λ, K,Z = 0) by hand
2. Compute first-order Taylor expansion of f (Λt , Kt , Zt)
Et
dΛtdKtdZt
= f (Λt , Kt , Zt)dt
3. Diagonalize matrix B, hope same number of stable eigenvalues asstate variables (2 in this model)
• if so, set control variables orthogonal to unstableeigenvectors, get policy function
Λt = DKKt +DZZt
12
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Linearization of Continuous-Time RBC model
1. Compute non-stochastic steady state (Λ, K,Z = 0) by hand
2. Compute first-order Taylor expansion of f (Λt , Kt , Zt)
Et
d ΛtdKtdZt
=BΛΛ BΛK BΛZBKΛ BKK BKZ0 0 −ν
︸ ︷︷ ︸
B
ΛtKtZt
dt
3. Diagonalize matrix B, hope same number of stable eigenvalues asstate variables (2 in this model)
• if so, set control variables orthogonal to unstableeigenvectors, get policy function
Λt = DKKt +DZZt
12
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Linearization of Continuous-Time RBC model
1. Compute non-stochastic steady state (Λ, K,Z = 0) by hand
2. Compute first-order Taylor expansion of f (Λt , Kt , Zt)
Et
d ΛtdKtdZt
=BΛΛ BΛK BΛZBKΛ BKK BKZ0 0 −ν
︸ ︷︷ ︸
B
ΛtKtZt
dt
3. Diagonalize matrix B, hope same number of stable eigenvalues asstate variables (2 in this model)
• if so, set control variables orthogonal to unstableeigenvectors, get policy function
Λt = DKKt +DZZt
12
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Linearization of Heterogeneous Agent Model
1. Compute non-linear approximation to non-stochastic steady state
• finite difference method from Achdou et al. (2015)• steady state reduces to sparse matrix equations• borrowing constraint absorbed into boundary conditions
2. Compute first-order Taylor expansion around steady state
3. Solve linear stochastic differential equation
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Linearization of Heterogeneous Agent Model
1. Compute non-linear approximation to non-stochastic steady state
• finite difference method from Achdou et al. (2015)• steady state reduces to sparse matrix equations• borrowing constraint absorbed into boundary conditions
2. Compute first-order Taylor expansion around steady state
3. Solve linear stochastic differential equation
13
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Step 1: Compute Non-Stochastic Steady State
ρv(a, z) =maxcu(c) + ∂av(a, z)(wz + ra − c)
+ λz(v(a, z′)− v(a, z))
(HJB SS)
0 =− ∂a[s(a, z)g(a, z)]− λzg(a, z) + λz ′g(a, z ′) (KF SS)
w = (1− α)Kαt , r = αKα−1 − δ,
K =
∫ag(a, z)dadz
(PRICE SS)
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Step 1: Compute Non-Stochastic Steady State
ρvi ,j =u(ci ,j) + ∂avi ,j(wzj + rai − ci ,j)+ λj(vi ,−j − vi ,j)
(HJB SS)
0 =− ∂a[s(a, z)g(a, z)]− λzg(a, z) + λz ′g(a, z ′) (KF SS)
w = (1− α)Kαt , r = αKα−1 − δ,
K =
∫ag(a, z)dadz
(PRICE SS)
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Step 1: Compute Non-Stochastic Steady State
ρv =u (v) + A (v;p) v (HJB SS)
0 =− ∂a[s(a, z)g(a, z)]− λzg(a, z) + λz ′g(a, z ′) (KF SS)
w = (1− α)Kαt , r = αKα−1 − δ,
K =
∫ag(a, z)dadz
(PRICE SS)
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Step 1: Compute Non-Stochastic Steady State
ρv =u (v) + A (v;p) v (HJB SS)
0 =A (v;p)T g (KF SS)
w = (1− α)Kαt , r = αKα−1 − δ,
K =
∫ag(a, z)dadz
(PRICE SS)
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Step 1: Compute Non-Stochastic Steady State
ρv =u (v) + A (v;p) v (HJB SS)
0 =A (v;p)T g (KF SS)
p =F (g) (PRICE SS)
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Linearization of Heterogeneous Agent Model
1. Compute non-linear approximation to non-stochastic steady state
• finite difference method from Achdou et al. (2015)• steady state reduces to sparse matrix equations• borrowing constraint absorbed into boundary conditions
2. Compute first-order Taylor expansion around steady state
• automatic differentiation: exact numerical derivatives• wrote efficient Matlab implementation for sparse systems• important: different slopes at different point in state space
3. Solve linear stochastic differential equation
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Linearization of Heterogeneous Agent Model
1. Compute non-linear approximation to non-stochastic steady state
• finite difference method from Achdou et al. (2015)• steady state reduces to sparse matrix equations• borrowing constraint absorbed into boundary conditions
2. Compute first-order Taylor expansion around steady state
• automatic differentiation: exact numerical derivatives• wrote efficient Matlab implementation for sparse systems• important: different slopes at different point in state space
3. Solve linear stochastic differential equation
15
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Step 2: Linearize Discretized System
• Discretized system with aggregate shocks
ρvt = u (vt) + A (vt ;pt) vt +1
dtEtdvt
dgtdt= A (vt ;pt)
T gt
pt = F (gt;Zt)
dZt = −νZtdt + σdWt
•
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Step 2: Linearize Discretized System
• Discretized system with aggregate shocks
ρvt = u (vt) + A (vt ;pt) vt +1
dtEtdvt
dgtdt= A (vt ;pt)
T gt
pt = F (gt;Zt)
dZt = −νZtdt + σdWt
• Key: same general form as RBC model earlier
Et
dvtdgtdZt
= f (vt ,gt , Zt)dt,vtgtZt
= control
endog stateexog state
Dimensionality: if 2 income types, 500 wealth grid points, thenboth vt and gt are 1000× 1⇒ [vt ,gt , Zt ]′ is 2001× 1
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Step 2: Linearize Discretized System
• Discretized system with aggregate shocks
ρvt = u (vt) + A (vt ;pt) vt +1
dtEtdvt
dgtdt= A (vt ;pt)
T gt
pt = F (gt;Zt)
dZt = −νZtdt + σdWt
• Key: same general form as RBC model earlier
Et
dvtdgt0
dZt
= f (vt ,gt ,pt , Zt)dt,vtgtptZt
=
controlendog state
pricesexog state
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Step 2: Linearize Discretized System
• Discretized system with aggregate shocks
ρvt = u (vt) + A (vt ;pt) vt +1
dtEtdvt
dgtdt= A (vt ;pt)
T gt
pt = F (gt;Zt)
dZt = −νZtdt + σdWt
• Linearize using automatic differentiation (code: @myAD)
Et
d vtd gt0
dZt
=Bvv 0 Bvp 0
Bgv Bgg Bgp 0
0 Bpg I BpZ0 0 0 −ν
︸ ︷︷ ︸
B
vtgtptZt
dt
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Linearization of Heterogeneous Agent Model
1. Compute non-linear approximation to non-stochastic steady state• finite difference method from Achdou et al. (2015)• steady state reduces to sparse matrix equations• borrowing constraint absorbed into boundary conditions
2. Compute first-order Taylor expansion around steady state• automatic differentiation: exact numerical derivatives• wrote efficient Matlab implementation for sparse systems• important: different slopes at different point in state space
3. Solve linear stochastic differential equation
• moderately-sized systems⇒ standard methods work fine• large systems⇒ first reduce dimensionality using SVDs
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Linearization of Heterogeneous Agent Model
1. Compute non-linear approximation to non-stochastic steady state• finite difference method from Achdou et al. (2015)• steady state reduces to sparse matrix equations• borrowing constraint absorbed into boundary conditions
2. Compute first-order Taylor expansion around steady state• automatic differentiation: exact numerical derivatives• wrote efficient Matlab implementation for sparse systems• important: different slopes at different point in state space
3. Solve linear stochastic differential equation• moderately-sized systems⇒ standard methods work fine• large systems⇒ first reduce dimensionality using SVDs
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Step 3: Solve Linear System
• Usual strategy: diagonalize + hope same number of stableeigenvalues as state variables (I × J + 1 in this model)
• if so, set control variables orthogonal to unstableeigenvectors, get policy function
vt = Dggt +DZZt
• Works for moderately-sized systems (100-500 wealth grid points)
• Diagonalization prohibitively expensive for large systems(500 - 10,000 wealth grid points)
• ⇒ reduce dimensionality using SVDs (Reiter 2009)• background: review article by Antoulas (2005) “An overview of
approximation methods for large-scale dynamical systems”• also see book by Antoulas (2005) “Approximation of Large
Scale Dynamical Systems”
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Step 3: Solve Linear System
• Usual strategy: diagonalize + hope same number of stableeigenvalues as state variables (I × J + 1 in this model)
• if so, set control variables orthogonal to unstableeigenvectors, get policy function
vt = Dggt +DZZt
• Works for moderately-sized systems (100-500 wealth grid points)
• Diagonalization prohibitively expensive for large systems(500 - 10,000 wealth grid points)
• ⇒ reduce dimensionality using SVDs (Reiter 2009)• background: review article by Antoulas (2005) “An overview of
approximation methods for large-scale dynamical systems”• also see book by Antoulas (2005) “Approximation of Large
Scale Dynamical Systems” 18
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Results
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Performance with 200 wealth grid points
(a) Without dim reduction (b) With dim reduction
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Performance with 600 wealth grid points
(a) Without dim reduction (b) With dim reduction
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Aggregate Dynamics
IRF to 1-quarter TFP shock, entire distribution (n = 400 components)
(a) Consumption
20 40 60 80 100 120 140 160 180 2000
0.2
0.4
0.6
0.8
1
1.2
(b) Investment
20 40 60 80 100 120 140 160 180 2001
0
1
2
3
4
5
6
7
22
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Aggregate Dynamics and Dimensionality Reduction
IRF to 1-quarter TFP shock, reduced distribution (n = 95 components)
(a) Consumption
20 40 60 80 100 120 140 160 180 2000
0.2
0.4
0.6
0.8
1
1.2
(b) Investment
20 40 60 80 100 120 140 160 180 2001
0
1
2
3
4
5
6
7
23
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Micro Heterogeneity and Macro Nonlinearities
• Key motivation for studying het agent models: micro heterogeneitymay generate nonlinear dynamics in aggregate variables
• economy’s response to shock may depend on initialdistribution of agents
• economy’s response to shock may depend on size of shock
• Our methodology preserves such aggregate nonlinearities
• true even though it relies on linear approximations• key: different slopes at different points of state space• in contrast to rep agent model: only one slope
24
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Micro Heterogeneity and Macro Nonlinearities
• Key motivation for studying het agent models: micro heterogeneitymay generate nonlinear dynamics in aggregate variables
• economy’s response to shock may depend on initialdistribution of agents
• economy’s response to shock may depend on size of shock
• Our methodology preserves such aggregate nonlinearities
• true even though it relies on linear approximations• key: different slopes at different points of state space• in contrast to rep agent model: only one slope
24
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Micro Heterogeneity and Macro Nonlinearities
IRF to one quarter TFP shock, starting from steady state
(a) Consumption
Quarters since shock20 40 60 80 100 120 140 160 180 200
Per
cent
age
devi
atio
n fro
m s
tead
y st
ate
0
0.2
0.4
0.6
0.8
1
1.2
(b) Investment
Quarters since shock20 40 60 80 100 120 140 160 180 200
Per
cent
age
devi
atio
n fro
m s
tead
y st
ate
1
0
1
2
3
4
5
6
7
25
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Micro Heterogeneity and Macro Nonlinearities
IRF to one quarter TFP shock, starting from recession
(a) Consumption
Quarters since shock20 40 60 80 100 120 140 160 180 200
Per
cent
age
devi
atio
n fro
m s
tead
y st
ate
0
0.2
0.4
0.6
0.8
1
1.2
(b) Investment
Quarters since shock20 40 60 80 100 120 140 160 180 200
Per
cent
age
devi
atio
n fro
m s
tead
y st
ate
1
0
1
2
3
4
5
6
7
26
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Conclusion
• Developed general + efficient methodology to solve heterogeneousagent macro models
• Extension of standard linearization methods• solve for steady state using continuous time• compute Taylor expansion using auto diff w/ sparsity• reduce state space using SVDs• http://www.princeton.edu/~moll/PHACTproject.htm
• Next step: medium-scale DSGE model featuring• het households w/ leptokurtic shocks + asset choice• het firms w/ productivity shocks + fixed costs• sticky prices
27
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Conclusion
• Developed general + efficient methodology to solve heterogeneousagent macro models
• Extension of standard linearization methods• solve for steady state using continuous time• compute Taylor expansion using auto diff w/ sparsity• reduce state space using SVDs• http://www.princeton.edu/~moll/PHACTproject.htm
• Next step: medium-scale DSGE model featuring• het households w/ leptokurtic shocks + asset choice• het firms w/ productivity shocks + fixed costs• sticky prices
27
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Conclusion
• Developed general + efficient methodology to solve heterogeneousagent macro models
• Extension of standard linearization methods• solve for steady state using continuous time• compute Taylor expansion using auto diff w/ sparsity• reduce state space using SVDs• http://www.princeton.edu/~moll/PHACTproject.htm
• Next step: medium-scale DSGE model featuring• het households w/ leptokurtic shocks + asset choice• het firms w/ productivity shocks + fixed costs• sticky prices
27
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Instead: Fully Recursive Notation Back
w(g, Z) = (1− α)eZK(g)α, r(g, Z) = αeZK(g)α−1 − δ (P)
K(g) =
∫ag(a, z)dadz (K)
ρV (a, z, g, Z) = maxcu(c) + ∂aV (a, z, g, Z)[w(g, Z)z + r(g, Z)a − c ]
+ λz [V (a, z′, g, Z)− V (a, z, g, Z)]
+∂ZV (a, z, g, Z)(−νZ) +1
2∂ZZV (a, z, g, Z)σ
2
+
∫δV (a, z, g, Z)
δg(a, z)T [g, Z](a, z)dadz
(∞d HJB)T [g, Z](a, z) = −∂a[s(a, z, g, Z)g(a, z)]− λzg(a, z) + λz ′g(a, z ′)
(KF operator)s(a, z, g, Z) = w(g, Z)z + r(g, Z)a − c∗(a, z, g, Z)
• big problem: distribution g is a state variable (infinite dimensional)• δV/δg(a, z): functional derivative of V wrt g at point (a, z) 28