Heterogeneous Agent New Keynesian Modelsmoll/HANK_PI.pdfHANK: Heterogeneous Agent New Keynesian models • Framework for quantitative analysis of aggregate shocks and macroeconomic

HANKHeterogeneous Agent New Keynesian Models

Greg KaplanBen Moll

Gianluca Violante

Princeton Initiative 2016

HANK: Heterogeneous Agent New Keynesian models

• Framework for quantitative analysis of aggregate shocks andmacroeconomic policy

• Three building blocks

1. Uninsurable idiosyncratic income risk2. Nominal price rigidities3. Assets with different degrees of liquidity

• Today: Transmission mechanism for conventional monetary policy

1

How monetary policy works in RANK

• Total consumption response to a drop in real rates

C response = direct response to r︸︷︷︸>95%

+ indirect effects due to Y︸︷︷︸<5%

• Direct response is everything, pure intertemporal substitution

• However, data suggest:

1. Low sensitivity of C to r

2. Sizable sensitivity of C to Y

3. Micro sensitivity vastly heterogeneous, depends crucially onhousehold balance sheets

2

How monetary policy works in RANK

• Total consumption response to a drop in real rates

C response = direct response to r︸︷︷︸>95%

+ indirect effects due to Y︸︷︷︸<5%

• Direct response is everything, pure intertemporal substitution

• However, data suggest:

1. Low sensitivity of C to r

2. Sizable sensitivity of C to Y

3. Micro sensitivity vastly heterogeneous, depends crucially onhousehold balance sheets

2

How monetary policy works in HANK

• Once matched to micro data, HANK delivers realistic:

• wealth distribution: small direct effect

• MPC distribution: large indirect effect (depending on ∆Y )

C response = direct response to r︸︷︷︸ + indirect effects due to Y︸︷︷︸RANK: >95% RANK: <5%

HANK: <1/3 HANK: >2/3

• Overall effect depends crucially on fiscal response, unlike in RANKwhere Ricardian equivalence holds

3






HANK: <1/3 HANK: >2/3


3






HANK: <1/3 HANK: >2/3


3

Why does this difference matter?

Suppose Central Bank wants to stimulate C

RANK view:

• sufficient to influence the path for real rates {rt}

• household intertemporal substitution does the rest

HANK view:

• must rely heavily on GE feedbacks to boost hh labor income

• through fiscal policy reaction and/or an investment boom

• responsiveness of C to i is, to a larger extent, out of CB’s control

4

Literature and contribution

Combine two workhorses of modern macroeconomics:• New Keynesian models Gali, Gertler, Woodford

• Bewley models Aiyagari, Bewley, Huggett

Closest existing work:1. New Keynesian models with limited heterogeneity

Campell-Mankiw, Gali-LopezSalido-Valles, Iacoviello, Bilbiie, Challe-Matheron-Ragot-Rubio-Ramirez,

Broer-Hansen-Krusell-Öberg

• micro-foundation of spender-saver behavior

2. Bewley models with sticky pricesOh-Reis, Guerrieri-Lorenzoni, Ravn-Sterk, Gornemann-Kuester-Nakajima, DenHaan-Rendal-Riegler,

Bayer-Luetticke-Pham-Tjaden, McKay-Reis, McKay-Nakamura-Steinsson, Huo-RiosRull, Werning, Luetticke

• assets with different liquidity Kaplan-Violante

• new view of individual earnings risk Guvenen-Karahan-Ozkan-Song

• Continuous time approach Achdou-Han-Lasry-Lions-Moll 5

HANK

6

Building blocks

Households• Face uninsured idiosyncratic labor income risk• Consume and supply labor• Hold two assets: liquid and illiquid

Firms• Monopolistically competitive intermediate-good producers• Quadratic price adjustment costs à la Rotemberg (1982)

Government• Issues liquid debt, spends, taxes

Monetary Authority• Sets nominal rate on liquid assets based on a Taylor rule

7

Households

max{ct ,ℓt ,dt}t≥0

E0∫ ∞

0

e−(ρ+λ)tu(ct , ℓt)dt s.t.

bt = rb(bt)bt + wztℓt−dt − χ(dt , at)− ct +Γ− T (wztℓt + Γ)

at= raat + dt

zt = some Markov processbt ≥ −b, at ≥ 0

• ct : non-durable consumption • dt : illiquid deposits• bt : liquid assets • χ: transaction cost function• zt : individual productivity • T : labor income tax/transfer• ℓt : hours worked • Γ: income from firm ownership• at : illiquid assets • no housing – see working paper

8

Households


E0∫ ∞

0


bt = rb(bt)bt + wztℓt−dt − χ(dt , at)− ct +Γ− T (wztℓt + Γ)

at = raat + dt


• ct : non-durable consumption • dt : illiquid deposits (≷ 0)• bt : liquid assets • χ: transaction cost function• zt : individual productivity • T : labor income tax/transfer• ℓt : hours worked • Γ: income from firm ownership• at : illiquid assets • no housing – see working paper

8

Households


E0∫ ∞

0


bt = rb(bt)bt + wztℓt−dt − χ(dt , at)− ct −T (wztℓt + Γ) + Γ

at = raat + dt


• ct : non-durable consumption • dt : illiquid deposits (≷ 0)• bt : liquid assets • χ: transaction cost function• zt : individual productivity • T : income tax/transfer• ℓt : hours worked • Γ: income from firm ownership• at : illiquid assets • no housing – see working paper

8

Households


E0∫ ∞0


bt = rb(bt)bt + wztℓt − dt − χ(dt , at)− ct − T (wztℓt + Γ) + Γ

at = raat + dt


• ct : non-durable consumption • dt : illiquid deposits (≷ 0)• bt : liquid assets • χ: transaction cost function• zt : individual productivity • T : income tax/transfer• ℓt : hours worked • Γ: income from firm ownership• at : illiquid assets • no housing – see working paper

8

Households

• Adjustment cost function

χ(d, a) = χ0 |d |+ χ1∣∣∣∣ d

max{a, a}

∣∣∣∣χ2 max{a, a}• Linear component implies inaction region• Convex component implies finite deposit rates

9

Households

• Adjustment cost function

χ(d, a) = χ0 |d |+ χ1∣∣∣∣ d

max{a, a}

∣∣∣∣χ2 max{a, a}• Linear component implies inaction region• Convex component implies finite deposit rates

• Recursive solution of hh problem consists of:1. consumption policy function c(a, b, z ;w, r a, rb)2. deposit policy function d(a, b, z ;w, r a, rb)3. labor supply policy function ℓ(a, b, z ;w, r a, rb)⇒ joint distribution of households µ(da, db, dz ;w, r a, rb)

9

Firms

Representative competitive final goods producer:

Y =

(∫ 10

yε−1ε

j dj

) εε−1

⇒ yj =(pjP

)−εY

Monopolistically competitive intermediate goods producers:

• Technology: yj = Zkαj n1−αj ⇒ m = 1Z

(rα

)α ( w1−α

)1−α• Set prices subject to quadratic adjustment costs:

Θ

(p

p

)=θ

2

(p

p

)2Y

Exact NK Phillips curve:(r a −

Y

Y

)π =

ε

θ(m − m) + π, m = ε−1

ε

10

Determination of illiquid return, distribution of profits

• Illiquid assets = part capital, part equitya = k + qs

• k : capital, pays return r − δ• s: shares, price q, pay dividends ωΠ = ω(1−m)Y

• Arbitrage:ωΠ+ q

q= r − δ := r a

• Remaining (1− ω)Π? Scaled lump-sum transfer to hh’s:

Γ = (1− ω)z

zΠ

• Set ω = α (capital share)⇒ neutralize countercyclical markupstotal illiquid flow = rK + ωΠ = αmY + ω(1−m)Y = αYtotal liquid flow = wL+ (1− ω)Π = (1− α)Y

11





q= r − δ := r a


Γ = (1− ω)z

zΠ


11





q= r − δ := r a


Γ = (1− ω)z

zΠ


11





q= r − δ := r a


Γ = (1− ω)z

zΠ


11

Monetary authority and government

• Taylor rulei = rb + ϕπ + ϵ, ϕ > 1

with rb := i − π (Fisher equation), ϵ = innovation (“MIT shock”)

• Progressive tax on labor income:

T (wzℓ+ Γ) = −T + τ × (wzℓ+ Γ)

• Government budget constraint (in steady state)

G − rbBg =∫T dµ

• Transition? Ricardian equivalence fails⇒ this matters!

12

Summary of market clearing conditions

• Liquid asset marketBh + Bg = 0

• Illiquid asset marketA = K + q

• Labor marketN =

∫zℓ(a, b, z)dµ

• Goods market:

Y = C + I + G + χ+Θ+ borrowing costs

13

Solution Method

14

Solution Method (from Achdou-Han-Lasry-Lions-Moll)

• Solving het. agent model = solving PDEs1. Hamilton-Jacobi-Bellman equation for individual choices2. Kolmogorov Forward equation for evolution of distribution

• Many well-developed methods for analyzing and solving these• simple but powerful: finite difference method• codes: http://www.princeton.edu/~moll/HACTproject.htm

• Apparatus is very general: applies to any heterogeneous agentmodel with continuum of atomistic agents

1. heterogeneous households (Aiyagari, Bewley, Huggett,...)

2. heterogeneous producers (Hopenhayn,...)

• can be extended to handle aggregate shocks (Krusell-Smith,...)

15

http://www.princeton.edu/~moll/HACTproject.htm

Computational Advantages relative to Discrete Time

1. Borrowing constraints only show up in boundary conditions• FOCs always hold with “=”

2. “Tomorrow is today”• FOCs are “static”, compute by hand: c−γ = Vb(a, b, y)

3. Sparsity• solving Bellman, distribution = inverting matrix• but matrices very sparse (“tridiagonal”)• reason: continuous time⇒ one step left or one step right

4. Two birds with one stone• tight link between solving (HJB) and (KF) for distribution• matrix in discrete (KF) is transpose of matrix in discrete (HJB)• reason: diff. operator in (KF) is adjoint of operator in (HJB) 16

Parameterization

17

Three key aspects of parameterization

1. Measurement and partition of asset categories into: 50 shades of K

• Liquid (cash, bank accounts + government/corporate bonds)• Illiquid (equity, housing)

2. Income process with leptokurtic income changes income process

• Nature of earnings risk affects household portfolio

3. Adjustment cost function and discount rate adj cost function

• Match mean liquid/illiquid wealth and fraction HtM

• Production side: standard calibration of NK models• Standard separable preferences: u(c, ℓ) = log c − 12ℓ2

18









18









18









18

Model matches key feature of U.S. wealth distribution

Data ModelMean illiquid assets (rel to GDP) 2.920 2.920Mean liquid assets (rel to GDP) 0.260 0.268Poor hand-to-mouth 10% 9%Wealthy hand-to-mouth 20% 18%

19

Wealth distributions: Liquid wealth

0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1Liquid wealth Lorenz curve

Model2004 SCF

• Top 10% share: SCF 2004: 86%, Model: 73%• Top 1% share: SCF 2004: 47%, Model: 16%• Gini coefficient: SCF 2004: 0.98, Model: 0.85

20

Wealth distributions: Illiquid wealth

0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2Illiquid wealth Lorenz curve

Model2004 SCF

• Top 10% share: SCF 2004: 70%, Model: 87%• Top 1% share: SCF 2004: 33%, Model: 40%• Gini coefficient: SCF 2004: 0.81, Model: 0.82

21

Model generates high and heterogeneous MPCs

0 200 400 600 800 1000Amount of transfer ($)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7Fraction of lump sum transfer consumed

↑

Quarterly MPC out of $500 = 18%

One quarterTwo quartersOne year

0400

0.1

0.2

300 20

0.3

0.4

Quarterly MPC $500

10

Illiquid Wealth ($000)

200

0.5

Liquid Wealth ($000)

0100-10

0

22

Description Value Target / SourcePreferencesλ Death rate 1/180 Av. lifespan 45 yearsγ Risk aversion 1φ Frisch elasticity (GHH) 0.5ψ Disutility of labor 27 Av. hours worked equal to 1/3ρ Discount rate (pa) 4.7% Internally calibrated

Productionε Demand elasticity 10 Profit share 10 %α Capital share 0.33δ Depreciation rate (p.a.) 10%θ Price adjustment cost 100 Slope of Phillips curve, ε/θ = 0.1

Governmentτ Proportional labor tax 0.25T Lump sum transfer (rel GDP) 0.075 40% hh with net govt transferg Govt debt to annual GDP 0.26 government budget constraint

Monetary Policyϕ Taylor rule coefficient 1.25rb Steady state real liquid return (pa) 2%

Housingrh Net housing return (pa) 1.5% Kaplan and Violante (2014)

Illiquid Assetsr a Illiquid asset return (pa) 6.5% Equilibrium outcome

Borrowingrborr Borrowing rate (pa) 8.4% Internally calibrated

b Borrowing limit -0.42 1 × quarterly labor inc

23

Results

24

Transmission of monetary policy shock to C

Innovation ϵ < 0 to the Taylor rule: i = rb + ϕπ + ϵ

• All experiments: ϵ0 = −0.0025, i.e. −1% annualized

0 5 10 15 20Quarters

-1.5

-1

-0.5

0

0.5

Deviation(ppannual)

Taylor rule innovation: εLiquid return: rb

Inflation: π


-0.5

0

0.5

1

1.5

2

2.5

Deviation(%

)

OutputConsumptionInvestment

25


Innovation ϵ < 0 to the Taylor rule: i = rb + ϕπ + ϵ

• All experiments: ϵ0 = −0.0025, i.e. −1% annualized


-1.5

-1

-0.5

0

0.5

Deviation(ppannual)

Taylor rule innovation: εLiquid return: rb

Inflation: π


-0.5

0

0.5

1

1.5

2

2.5

Deviation(%

)

OutputConsumptionInvestment

25


dC0 =

∫ ∞0

∂C0

∂rbtdrbt dt︸︷︷︸

direct

+

∫ ∞0

[∂C0∂r atdr at +

∂C0∂wtdwt +

∂C0∂TtdTt

]dt︸︷︷︸

indirect

26


dC0 =

∫ ∞0

∂C0


direct

+

∫ ∞0


∂C0∂wtdwt +

∂C0∂TtdTt

]dt︸︷︷︸

indirect

26


dC0 =

∫ ∞0

∂C0

∂rbtdrbt dt +

∫ ∞0


∂C0∂wtdwt +

∂C0∂TtdTt

]dt

✓Intertemporal substitution and income effects from rb ↓


-0.1

0

0.1

0.2

0.3

0.4

0.5

Deviation(%

)

Total ResponseDirect: rb

27


dC0 =

∫ ∞0

∂C0

∂rbtdrbt dt +

∫ ∞0


∂C0∂wtdwt +

∂C0∂TtdTt

]dt

✓Portfolio reallocation effect from r a − rb ↑


-0.1

0

0.1

0.2

0.3

0.4

0.5

Deviation(%

)


Indirect: ra

28


dC0 =

∫ ∞0

∂C0

∂rbtdrbt dt +

∫ ∞0


∂C0∂wtdwt +

∂C0∂TtdTt

]dt

✓Labor demand channel from w ↑


-0.1

0

0.1

0.2

0.3

0.4

0.5

Deviation(%

)


Indirect: ra

Indirect: w +Γ

29


dC0 =

∫ ∞0

∂C0

∂rbtdrbt dt +

∫ ∞0


∂C0∂wtdwt +

∂C0∂TtdTt

]dt

✓Fiscal adjustment: T ↑ in response to ↓ in interest payments on B


-0.1

0

0.1

0.2

0.3

0.4

0.5

Deviation(%

)


Indirect: ra

Indirect: w +Γ

Indirect: T

30


dC0 =

∫ ∞0

∂C0


19%

+

∫ ∞0


∂C0∂wtdwt +

∂C0∂TtdTt

]dt︸︷︷︸

81%


-0.1

0

0.1

0.2

0.3

0.4

0.5

Deviation(%

)


Indirect: ra

Indirect: w +Γ

Indirect: T

31

Monetary transmission across liquid wealth distribution

• Total change = c-weighted sum of (direct + indirect) at each b

32

Why small direct effects?

• Intertemporal substitution: (+) for non-HtM• Income effect: (-) for rich households• Portfolio reallocation: (-) for those with low but > 0 liquid wealth

33

Role of fiscal response in determining total effect

T adjusts G adjusts Bg adjusts(1) (2) (3)

Elasticityof C0 to rb -2.21 -2.07 -1.48Share of Direct effects: 19% 22% 46%

• Fiscal response to lower interest payments on debt:

• T adjusts: stimulates AD through MPC of HtM households

• G adjusts: translates 1-1 into AD

• Bg adjusts: no initial stimulus to AD from fiscal side34

Monetary transmission in RANK and HANK

∆C = direct response to r + indirect GE responseRANK: 90% RANK: 10%HANK: 2/3 HANK: 1/3

• RANK view:

• High sensitivity of C to r : intertemporal substitution

• Low sensitivity of C to Y : the RA is a PIH consumer

• HANK view:

• Low sensitivity to r : income effect of wealthy offsets int. subst.

• High sensitivity to Y : sizable share of hand-to-mouth agents

⇒ Q: Is Fed less in control of C than we thought?

• Work in progress: perturbation methods⇒ estimation, inference

35



• RANK view:



• HANK view:




• Work in progress: perturbation methods⇒ estimation, inference

35



• RANK view:



• HANK view:




• Work in progress: perturbation methods⇒ estimation, inference35

36

Numerical Solution of HJB Equations

37

Finite Difference Methods

• See http://www.princeton.edu/~moll/HACTproject.htm

• Explain using neoclassical growth model, easily generalized toheterogeneous agent models

ρv(k) = maxcu(c) + v ′(k)(F (k)− δk − c)

• Functional forms

u(c) =c1−σ

1− σ , F (k) = kα

• Use finite difference method• Two MATLAB codes

http://www.princeton.edu/~moll/HACTproject/HJB_NGM.m

http://www.princeton.edu/~moll/HACTproject/HJB_NGM_implicit.m

38

http://www.princeton.edu/~moll/HACTproject.htm



Barles-Souganidis

• There is a well-developed theory for numerical solution of HJBequation using finite difference methods

• Key paper: Barles and Souganidis (1991), “Convergence ofapproximation schemes for fully nonlinear second order equationshttps://www.dropbox.com/s/vhw5qqrczw3dvw3/barles-souganidis.pdf?dl=0

• Result: finite difference scheme “converges” to unique viscositysolution under three conditions

1. monotonicity2. consistency3. stability

• Good reference: Tourin (2013), “An Introduction to Finite DifferenceMethods for PDEs in Finance.”

39

https://www.dropbox.com/s/vhw5qqrczw3dvw3/barles-souganidis.pdf?dl=0

Finite Difference Approximations to v ′(ki)

• Approximate v(k) at I discrete points in the state space,ki , i = 1, ..., I. Denote distance between grid points by ∆k .

• Shorthand notationvi = v(ki)

• Need to approximate v ′(ki).• Three different possibilities:

v ′(ki) ≈vi − vi−1∆k

= v ′i ,B backward difference

v ′(ki) ≈vi+1 − vi∆k

= v ′i ,F forward difference

v ′(ki) ≈vi+1 − vi−12∆k

= v ′i ,C central difference

40

Finite Difference Approximations to v ′(ki)

!

"# $ # #%$

!# !#%$

!# $

Central

Backward

Forward

41

Finite Difference Approximation

FD approximation to HJB is

ρvi = u(ci) + v′i [F (ki)− δki − ci ] (∗)

where ci = (u′)−1(v ′i ), and v ′i is one of backward, forward, central FDapproximations.Two complications:

1. which FD approximation to use? “Upwind scheme”2. (∗) is extremely non-linear, need to solve iteratively:

“explicit” vs. “implicit method”

My strategy for next few slides:• what works• slides on my website: why it works (Barles-Souganidis)

42

Which FD Approximation?• Which of these you use is extremely important• Best solution: use so-called “upwind scheme.” Rough idea:

• forward difference whenever drift of state variable positive• backward difference whenever drift of state variable negative

• In our example: definesi ,F = F (ki)− δki − (u′)−1(v ′i ,F ), si ,B = F (ki)− δki − (u′)−1(v ′i ,B)

• Approximate derivative as followsv ′i = v

′i ,F1{si ,F>0} + v

′i ,B1{si ,B<0} + v

′i 1{si ,F<0<si ,B}

where 1{·} is indicator function, and v ′i = u′(F (ki)− δki).• Where does v ′i term come from? Answer:

• since v is concave, v ′i ,F < v ′i ,B (see figure)⇒ si ,F < si ,B• if s ′i ,F < 0 < s ′i ,B, set si = 0⇒ v ′(ki) = u′(F (ki)− δki), i.e.

we’re at a steady state.

43





′i ,F1{si ,F>0} + v

′i ,B1{si ,B<0} + v

′i 1{si ,F<0<si ,B}

where 1{·} is indicator function, and v ′i = u′(F (ki)− δki).

• Where does v ′i term come from? Answer:• since v is concave, v ′i ,F < v ′i ,B (see figure)⇒ si ,F < si ,B• if s ′i ,F < 0 < s ′i ,B, set si = 0⇒ v ′(ki) = u′(F (ki)− δki), i.e.

we’re at a steady state.

43





′i ,F1{si ,F>0} + v

′i ,B1{si ,B<0} + v

′i 1{si ,F<0<si ,B}

where 1{·} is indicator function, and v ′i = u′(F (ki)− δki).• Where does v ′i term come from? Answer:

• since v is concave, v ′i ,F < v ′i ,B (see figure)⇒ si ,F < si ,B• if s ′i ,F < 0 < s ′i ,B, set si = 0⇒ v ′(ki) = u′(F (ki)− δki), i.e.

we’re at a steady state.43

Sparsity

• Discretized HJB equation is

ρvi = u(ci) +vi+1 − vi∆k

s+i ,F +vi − vi−1∆k

s−i ,B

• Notation: for any x , x+ = max{x, 0} and x− = min{x, 0}

• Can write this in matrix notation

ρv = u + Av

where A is I × I (I= no of grid points) and looks like...

44

Visualization of A (output of spy(A) in Matlab)

nz = 1360 10 20 30 40 50 60 70

0

10

20

30

40

50

60

70

45

spy(A)

The matrix A

• FD method approximates process for k with discrete Poissonprocess, A summarizes Poisson intensities

• entries in row i :

−s−i ,B∆k︸︷︷︸inflowi−1≥0

s−i ,B∆k−s+i ,F∆k︸︷︷︸

outflowi≤0

s+i ,F∆k︸︷︷︸

inflowi+1≥0

vi−1

vi

vi+1

• negative diagonals, positive off-diagonals, rows sum to zero:• tridiagonal matrix, very sparse

• A depends on v (nonlinear problem). Next: iterative method...

46

Iterative Method

• Idea: Solve FOC for given vn, update vn+1 according tovn+1i − vni∆

+ ρvni = u(cni ) + (v

n)′(ki)(F (ki)− δki − cni ) (∗)

• Algorithm: Guess v0i , i = 1, ..., I and for n = 0, 1, 2, ... follow1. Compute (vn)′(ki) using FD approx. on previous slide.2. Compute cn from cni = (u′)−1[(vn)′(ki)]3. Find vn+1 from (∗).4. If vn+1 is close enough to vn: stop. Otherwise, go to step 1.

• See http://www.princeton.edu/~moll/HACTproject/HJB_NGM.m

• Important parameter: ∆ = step size, cannot be too large (“CFLcondition”).

• Pretty inefficient: I need 5,990 iterations (though quite fast)47


Efficiency: Implicit Method

• Efficiency can be improved by using an “implicit method”vn+1i − vni∆

+ ρvn+1i = u(cni ) + (vn+1i )′(ki)[F (ki)− δki − cni ]

• Each step n involves solving a linear system of the form1

∆(vn+1 − vn) + ρvn+1 = u + Anvn+1((ρ+ 1

∆)I− An)vn+1 = u + 1

∆vn

• but An is super sparse⇒ super fast• See http://www.princeton.edu/~moll/HACTproject/HJB_NGM_implicit.m

• In general: implicit method preferable over explicit method1. stable regardless of step size ∆2. need much fewer iterations3. can handle many more grid points

48


Efficiency: Implicit Method

• Efficiency can be improved by using an “implicit method”vn+1i − vni∆

+ ρvn+1i = u(cni ) + (vn+1i )′(ki)[F (ki)− δki − cni ]

• Each step n involves solving a linear system of the form1

∆(vn+1 − vn) + ρvn+1 = u + Anvn+1((ρ+ 1

∆)I− An)vn+1 = u + 1

∆vn

• but An is super sparse⇒ super fast• See http://www.princeton.edu/~moll/HACTproject/HJB_NGM_implicit.m

• In general: implicit method preferable over explicit method1. stable regardless of step size ∆2. need much fewer iterations3. can handle many more grid points 48


Implicit Method: Practical Consideration

• In Matlab, need to explicitly construct A as sparse to takeadvantage of speed gains

• Code has part that looks as followsX = -min(mub,0)/dk;Y = -max(muf,0)/dk + min(mub,0)/dk;Z = max(muf,0)/dk;

• Constructing full matrix – slowfor i=2:I-1

A(i,i-1) = X(i);A(i,i) = Y(i);A(i,i+1) = Z(i);

endA(1,1)=Y(1); A(1,2) = Z(1);A(I,I)=Y(I); A(I,I-1) = X(I);

• Constructing sparse matrix – fastA =spdiags(Y,0,I,I)+spdiags(X(2:I),-1,I,I)+spdiags([0;Z(1:I-1)],1,I,I);

49

Intermediate good firm pricing problem

max{pt}t≥0

∫ ∞0

e−rat

{Πt(pt)−Θt

(ptpt

)}dt s.t.

Π(p) =( pP−m

)( pP

)−εY

m = 1Z

(rα

)α ( w1−α

)1−αΘ(π)=

θ

2π2Y

Back to firms

50

Fifty shades of K

Liquid Illiquid Total

Non-productive

Household depositsnet of revolving debtCorp & Govt bondsBh = 0.26

0.26

Productive Deposits at inv fundBf = −0.48

Indirectly held equityDirectly held equityNoncorp bus equity

net housingnet durables

2.92

K

Total −Bg = 0.26 A = 2.92 3.18

• Quantities are multiples of annual GDP• Sources: Flow of Funds and SCF 2004• Working paper: part of housing, durables = unproductive illiquid assets

back 51

Continuous time earnings dynamics

• Literature provides little guidance on statistical models of highfrequency earnings dynamics

• Key challenge: inferring within-year dynamics from annual data

• Higher order moments of annual changes are informative

• Target key moments of one 1-year and 5-year labor earningsgrowth from SSA data

• Model generates a thick right tail for earnings levels

52

Leptokurtic earnings changes (Guvenen et al)

53

Two-component jump-drift process

• Flow earnings (y = wzl) modeled as sum of two components:

log yt = y1t + y2t

• Each component is a jump-drift with:

• mean-reverting drift: −βyitdt

• jumps with arrival rate: λi , drawn from N (0, σi)

• Estimate using SMM aggregated to annual frequency

• Choose six parameters to match eight moments:

54

Model distribution of earnings changes

Moment Data Model Moment Data ModelVariance: annual log earns 0.70 0.70 Frac 1yr change < 10% 0.54 0.56Variance: 1yr change 0.23 0.23 Frac 1yr change < 20% 0.71 0.67Variance: 5yr change 0.46 0.46 Frac 1yr change < 50% 0.86 0.85Kurtosis: 1yr change 17.8 16.5Kurtosis: 5yr change 11.6 12.1

Transitory component: λ1 = 0.08, β1 = 0.76, σ1 = 1.74

Persistent component: λ2 = 0.007, β2 = 0.009, σ2 = 1.53

02

46

8D

en

sity

-4 -2 0 2 4

1 Year Log Earnings Changes

0.5

11

.52

De

nsity

-5 0 5

5 Year Log Earnings Changes

back55

Adjustment cost function calibration

χ(d, a) = χ0 |d |+ χ1∣∣∣∣ d

max{a, a}

∣∣∣∣χ2 max{a, a}• Choose χ0, χ1, χ2 together with:

• discount rate ρ• borrowing intermediation wedge: κb = rbor − rb

to match five features of the household wealth distribution

Moment Data ModelMean illiquid assets (rel to GDP) 2.920 2.920Mean liquid assets (rel to GDP) 0.260 0.263Poor hand-to-mouth 10% 12%Wealthy hand-to-mouth 20% 17%Fraction borrowing 15% 14%

56

Monetary Policy inBenchmark NK Models

57

Monetary Policy in Benchmark NK Models

Goal:• Introduce decomposition of C response to r change

Setup:• Prices and wages perfectly rigid = 1, GDP=labor =Yt• Households: CRRA(γ), income Yt , interest rate rt

⇒ Ct({rs , Ys}s≥0)• Monetary policy: sets time path {rt}t≥0, special case

rt = ρ+ e−ηt(r0 − ρ), η > 0 (∗)

• Equilibrium: Ct({rs , Ys}s≥0) = Yt• Overall effect of monetary policy

−d logC0dr0

=1

γη

58

Monetary Policy in RANK

• Decompose C response by totally differentiating C0({rt , Yt}t≥0)

dC0 =

∫ ∞0

∂C0∂rtdrtdt︸︷︷︸

direct response to r

+

∫ ∞0

∂C0∂Ytd Ytdt︸︷︷︸

indirect effects due to Y

.

• In special case (∗)

−d logC0dr0

=1

γη

[ η

ρ+ η︸︷︷︸direct response to r

+ρ

ρ+ η︸︷︷︸indirect effects due to Y

].

• Reasonable parameterizations⇒ very small indirect effects, e.g.• ρ = 0.5% quarterly• η = 0.5, i.e. quarterly autocorr e−η = 0.61

⇒η

ρ+ η= 99%,

ρ

ρ+ η= 1%

59

What if some households are hand-to-mouth?

• “Spender-saver” or Two-Agent New Keynesian (TANK) model

• Fraction Λ are HtM “spenders”: Cspt = Yt

• Decomposition in special case (∗)

−d logC0dr0

=1

γη

[(1− Λ)

η

ρ+ η︸︷︷︸direct response to r

+ (1− Λ)ρ

ρ+ η+ Λ︸︷︷︸

indirect effects due to Y

].

• ⇒ indirect effects ≈ Λ = 20-30%

60

What if there are assets in positive supply?

• Govt issues debt B to households sector

• Fall in rt implies a fall in interest payments of (rt − ρ)B

• Fraction λT of income gains transferred to spenders

• Initial consumption restponse in special case (∗)

−d logC0dr0

=1

γη+

λT

1− λB

Y︸︷︷︸fiscal redistribution channel

.

• Interaction between non-Ricardian households and debt in positivenet supply matters for overall effect of monetary policy

61

Comparison to One-Asset Model

Wealth, $ Thousands0 200 400 600 800 1000

ElasticityofConsumption

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Direct Effect: rb

(a) One-Asset Model (b) Two-Asset Model

62

Documents

Heterogeneous Agent New Keynesian Modelsmoll/HANK_PI.pdfHANK: Heterogeneous Agent New Keynesian models • Framework for quantitative analysis of aggregate shocks and macroeconomic