Lecture1 Stanford 235 Web Nomovie(1)

8/17/2019 Lecture1 Stanford 235 Web Nomovie(1)

http://slidepdf.com/reader/full/lecture1-stanford-235-web-nomovie1 1/48

Heterogeneous Agent Models

in Continuous Time, I

ECO 235: Advanced Macroeconomics

Benjamin Moll

Princeton

StanfordFebruary 10, 2014

1 / 4 8



What my lectures are about

• Do income and wealth distribution matter for the

macroeconomy?• Timely topic: in U.S.

• high inequality

• big recession, slow recovery

• Not crazy to ask: is there a link?

• is inequality holding back the recovery?

• did inequality contribute to recession in first place?

• these questions have received much attention, e.g. some

recent musings by Stiglitz, Krugman, Taylor:http://opinionator.blogs.nytimes.com/2013/01/19/inequality-is-holding-back-the-recovery/

http://krugman.blogs.nytimes.com/2013/01/20/inequality-and-recovery/

http://online.wsj.com/news/articles/SB10001424127887324094704579064712302845646

• But important question much more generally...2 / 4 8

http://opinionator.blogs.nytimes.com/2013/01/19/inequality-is-holding-back-the-recovery/





http://opinionator.blogs.nytimes.com/2013/01/19/inequality-is-holding-back-the-recovery/




• Do income and wealth distribution matter for the

macroeconomy?• One influential answer by Krusell and Smith (1998): No

• Rationale: “approximate aggregation”

• in standard heterogeneous agent model with aggregate shocks,

“macroeconomic aggregates can be almost perfectly describedusing only the mean of the wealth distribution.”

• reason: savings policy functions approximately linear in wealth(see their Figure 2) ⇒ higher order moments don’t matter

• more technical contribution: state variable = ∞-dimensional

distribution, solution: approximate with first I moments

• But some open questions:

• Is this a general result? Or specific to this class of models?

• How solve models where “approx. aggregation” doesn’t hold?

• ...3 / 4 8




• Progress report on preliminary project, joint work with Yves

Achdou, Jean-Michel Lasry and Pierre-Louis Lions• Our goal: develop continuous time methods for analyzing

and computing heterogeneous agent models, so as to reaptheir technical advantages

• Lecture 1 (today): “warm-up”, mainly pedagogical• redo textbook heterogeneous agent model of

Aiyagari (1994), Bewley (1986), Huggett (1993)

• some new theoretical results

•

how to do computations• Lecture 2 (Wednesday):

• “Wealth Distribution and the Business Cycle”

• framework to think about questions on previous slides

•

better way (IMHO) of doing Krusell and Smith (1998)4 / 4 8



Generality of Apparatus

• Both models can be characterized by a system of two PDEs

1 Hamilton-Jacobi-Bellman equation for individual choices

2 Kolmogorov Forward equation for evolution of distribution

• A “Mean Field Game” (Lions and Lasry, 2007)

• Apparatus is extremely general: any heterogeneous agentmodel with a continuum of atomistic agents can be written

like this

5 / 4 8



Develop computational tools

• Main difference to existing cont. time lit (Luttmer (2007),...):handle models for which closed form solutions do not exist

• Codes (for now only lecture 1) available fromhttp://www.princeton.edu/~ moll/HACTproject.htm

6 / 4 8

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






Dynamics of Wealth Distribution (Lec. 2)

movie here

7 / 4 8



Dynamics of Wealth Distribution (Lec. 2)

P r o d u

c t i v

i t y,

z D e n s i t y ,

g ( a , z , t )

Wealth,a

8 / 4 8



Background: HJB Equations

9 / 4 8



Hamilton-Jacobi-Bellman Equation: Some

“History”

(a) William Hamilton (b) Carl Jacobi (c) Richard Bellman• Aside: why called “dynamic programming”?• Bellman: “Try thinking of some combination that will possibly

give it a pejorative meaning. It’s impossible. Thus, I thought dynamic programming was a good name. It was something not even a Congressman could object to. So I used it as anumbrella for my activities.”http://en.wikipedia.org/wiki/Dynamic_programming#History 10/48

http://en.wikipedia.org/wiki/Dynamic_programming#History

http://en.wikipedia.org/wiki/Dynamic_programming#History



Hamilton-Jacobi-Bellman Equations

• Pretty much all deterministic optimal control problems in

continuous time can be written as

V (x 0) = maxu (t )∞t =0

∞0

e −ρt h (x (t ) , u (t )) dt

subject to the law of motion for the state

x (t ) = g (x (t ) , u (t )) and u (t ) ∈ U

for t ≥ 0, x (0) = x 0 given.

• ρ ≥ 0: discount rate

• x ∈ X ⊆ Rm: state vector

• u ∈ U ⊆ Rn: control vector

• h : X × U →R

: instantaneous return function11/48



Example: Neoclassical Growth Model

V (k 0) = maxc (t )∞t =0

∞0

e −ρt U (c (t ))dt

subject to˙k (t ) = F (k (t ))− δ k (t ) − c (t )

for t ≥ 0, k (0) = k 0 given.

• Here the state is x = k and the control u = c

• h(x , u ) = U (u )• g (x , u ) = F (x ) − δ x − u

12/48



Generic HJB Equation

• How to analyze these optimal control problems? Here:

“cookbook approach”

• Result: the value function of the generic optimal control

problem satisfies the Hamilton-Jacobi-Bellman equation

ρV (x ) = maxu ∈U

h(x , u ) + V ′(x ) · g (x , u )

• In the case with more than one state variable m > 1,V ′(x ) ∈ Rm is the gradient of the value function.

13/48



Example: Neoclassical Growth Model

• “cookbook” implies:

ρV (k ) = maxc

U (c ) + V ′(k )[F (k ) − δ k − c ]

• Proceed by taking first-order conditions etc

U ′(c ) = V ′(k )

• Derivation from discrete time Bellman equation

14/48



Poisson Uncertainty

• Easy to extend this to stochastic case. Simplest case:

two-state Poisson process

• Example: RBC Model. Production is At F (k t ) where

At ∈ {A1, A2} Poisson with intensities λ1, λ2

• Result: HJB equation is

ρV i (k ) = maxc

U (c )+V ′i (k )[Ai F (k )−δ k −c ]+λi [V j (k ) − V i (k )]

for i = 1, 2, j = i .

• Derivation similar as before.

15/48



Textbook Heterogeneous Agent Model

16/48



Textbook Heterogeneous Agent Model

• Aiyagari (1994), Bewley (1986), Huggett (1993)

• Start with simplest of these: Huggett (1993)

•

wealth = unproductive bonds• idiosyncratic risk = exogenous endowment shocks

• Later: very easily generalized to Aiyagari (1994)

• wealth = capital

• idiosyncratic risk = labor income

17/48



Huggett Model: Households

• are heterogeneous in their wealth a and income z , solve

maxc t

E0

∞0

e −ρt u (c t )dt s.t.

dat = [z t + r t at − c t ]dt

z t ∈ {z 1, z 2} Poisson with intensities λ1, λ2

at ≥ a

• c t : consumption

• u : utility function, u ′ > 0, u ′′ < 0.

• ρ: discount rate

• r t : interest rate

• a > −∞: borrowing limit e.g. if a = 0, can only save

• later: carries over to z t = general diffusion process.

19/48



EquilibriumInterest rate such that bonds in zero net supply:

0 =S (r (t )) = ag 1(a, t )da + ag 2(a, t )da (EQ)

ρv i (a, t ) = maxc

u (c ) + ∂ av i (a, t )[z i + r (t )a − c ] (HJB)

+ λi [v j (a, t )

−v i (a, t )] + ∂ t v i (a, t ),

∂ t g i (a, t ) = − ∂ a[s i (a, t )g i (a, t )] − λi g i (a, t ) + λ j g j (a, t ) (KFE),

s i (a, t ) =z i + r (t )a − c i (a, t ), c i (a, t ) = (u ′)−1(∂ av i (a, t ))

• Given initial condition g i ,0(a), the two PDEs (HJB) and

(KFE) together with (EQ) fully characterize equilibrium.

• Notation: for any function f , ∂ x f means ∂ f

∂ x

• Derivation of (KFE)20/48



Borrowing Constraints?• Q: where is borrowing constraint a ≥ a in (HJB)?

• A: “in” boundary condition

• Result: v i must satisfy

∂ av i (a, t ) ≥ u ′(z i + r (t )a), i = 1, 2 (BC)

• Derivation:

• the FOC still holds at the borrowing constraint

u ′(c i (a, t )) = ∂ av i (a, t ) (FOC)

• for borrowing constraint not to be violated, need

s i (a, t ) = z i + r (t )a − c i (a, t ) ≥ 0 (∗)• (FOC) and (∗) ⇒ (BC).

• Can derive (BC) more rigorously from concept of constrained

viscosity solution of (HJB). But complicated apparatus notreally needed (given smoothness of v i ).

21/48



Stationary Equilibrium

• A scalar r and functions (v 1, v 2, g 1, g 2) satisfying

ρv i (a) = maxc

u (c ) + v ′i (a)[z i + ra − c ] + λi [v j (a)− v i (a)]

0 =− d

da[s i (a)g i (a)]− λi g i (a) + λ j g j (a)

0 =S (r ) ≡ ag 1(a)da + ag 2(a)da

where s i (a) = z i + ra − c i (a)

22/48



Stationary Eq.: Theoretical Results

1 tight characterization of speed at which households hit

borrowing constraint– note: similarity to stopping time problems

2 intuitive formula linking equilibrium interest rate and numberof borrowing constrained households

• characterize interest rate effects of permanent credit crunch(Guerrieri and Lorenzoni, 2012)

3 analytic Taylor approximation of equilibrium interest rate

if shocks are “small”

4 unfortunately not yet: conditions for uniqueness of stationaryequilibrium

23/48



Savings Behavior at Borrowing Constraint

Proposition

Assume r < ρ and a > −z 1/r , z 1 < z 2. The solution to (HJB) has following properties:

1 Type z 1 with wealth a is constrained and hence s 1(a) = 0.

2 Taylor expansion of (s 1(a))2 around borrowing constraint is

s 1(a) ∼ −ν √

a − a, ν =

2

(ρ−r )u ′(c 1)+λ1[u ′(c 1)−u ′(c 2)]−u ′′(c 1) > 0.

2c 1γ

Therefore wealth of worker who keeps z 1 converges to

borrowing constraint in finite time at speed governed by ν :

a(t ) − a ∼ (√

a0 − a − ν t )2

Note: similarity to stopping time problems

3 Dirac point mass of type z 1 individuals at constraint. Proof 24/48




Wealth, a

S a v i n g s , s

i ( a )

a

s1

(a)

s2(a)

25/48




Wealth, a

D e n s i t i e

s , g i

( a )

a

g1(a)

g2(a)

Note: in numerical solution, Dirac mass = finite spike in density 26/48



vs. Discrete Time: Kinked Savings

Wealth, at

S a v i

n g s , a t +

1

a

• constraint binds also on interior of state space

• distribution typically has spikes away from constraint

27/48



Savings and Wealth Distribution given r

Wealth, a

S a v i n g s ,

s i

( a

)

a

s1(a)

s2(a)

Wealth, a

D e n s i t i e s

, g i

( a

)

a

g1(a)

g2(a)

28/48



Increase in r from r L to r H > r L

Wealth, a

S a v i n g s ,

s i

( a

)

a

s1(a;r

L)

s2(a;r

L)

s1(a;r

H)

s2(a;r

H)

Wealth, a

D e n s i t i e s

, g i

( a

)

a

g1(a;r

L)

g2(a;r

L)

g1(a;r

H)

g2(a;r

H)

29/48



Stationary Equilibrium

−0.15 −0.1 −0.05 0 0.05−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

r = ρ

S (r )

a = a r

S (r )

Recall S (r ) =

ag 1(a)da +

ag 2(a)da

30/48



Existence and Uniqueness?

• Existence: can show

limr ↑ρ

S (r ) = ∞, limr ↓−∞

S (r ) = a

⇒ existence guaranteed.

• Uniqueness? More tricky

• need to show S ′(r ) ≥ 0.

• satisfied in all numerical examples.

• sufficient condition: ∂ s i (a)/∂ r ≥ 0

31/48



Interest Rate & Borrowing Constraints

• Well-known feature of Aiyagari-Bewley-Huggett models: r < ρ

• In Aiyagari extension F ′

(K ) = r + δ ⇒ “overaccumulation,”K > K full insurance

• But by how much?

• e.g. is r < 0 so that ZLB may bind (in extension with nominal

rigidities as in Guerrieri and Lorenzoni, 2012)

Proposition

Assume a > −z 1/r. The stationary interest rate r satisfies

(ρ − r )U ′ = u ′(c 1)λ1M 1 > 0,

where U ′ = ∞a

[u ′(c 1(a))g 1(a) + u ′(c 2(a))g 2(a)]da and M 1 > 0 is

Dirac mass of type z 1 at borrowing constraint.

Proof

32/48

I i i



Intuition• At individual level

u ′(c i (a)) MC of self-insurance

= (1 + (r −

ρ)dt )u ′(c i (a)) + Et [du ′(c i (a))] MB of self-insurance

• At aggregate level (cross-sectional average in stationary dist.)

U ′

MC of self-insurance

= (1 + (r − ρ)dt )U ′ + u ′(c 1)λ1M 1dt

MB of self-insurance

,

(ρ− r )U ′ = u ′(c 1)λ1M 1,

• ρ − r measures extra benefit from self-insurance due to

presence of borrowing constraint

• Formula links interest rate and number of borrowingconstrained households. Uses:

1 decompose effects of parameter changes (comp. statics)

2 Taylor approximation for small λ

3 Any ideas for other uses??? Data??? 33/48

C i S i



Comparative Statics

1 Credit crunch, −a ↓ (Guerrieri and Lorenzoni, 2012)

2 Increase in uncertainty, λ1 = λ2 ↑

3 Increase in likelihood of “unemployment”, λ2 ↑

• All exercises: comparison of steady states, ignore transition

dynamics

34/48

C di C h ↓



Credit Crunch, −a ↓

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Borrowing Limit, -a

S t a t i o n a r y I n t e r

e s t R a t e , r

35/48

C dit C h ↓ D iti



Credit Crunch, −a ↓: Decomposition• Recall r = ρ − λ1

u ′(c 1)U ′

M 1, c 1 = z 1 + r a

• Credit crunch has two countervailing effects:

1 more people constrained M 1 ↑⇒ r ↓2 lower indebtedness = less “starvation” u ′(c 1)/U ′ ↓⇒ r ↑

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

No. of Borrowing Constrained, M 1

u ′(c1)/U ′

Borrowing Limit, -a

36/48

U t i t λ ↑



Uncertainty λ ↑

0 0.5 1 1.5 2−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

S t a t i o n a r y I n t e r e s t R a t e , r

Uncertainty, λ

37/48

Uncertainty λ ↑: Decomposition



Uncertainty λ ↑: Decomposition• Recall r = ρ − λ

u ′(c 1)U ′

M 1, c 1 = z 1 + r a

• Increase in uncertainty has three effects:

1 more likely to hit constraint λ ↑⇒ r ↓2 less people constrained M 1 ↓⇒ r ↑3 less inequality U ′ ↓⇒ r ↓ (u ′(c 1) ambiguous)

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

No. of Borrowing Constrained, M 1

Uncertainty, λ

u′

(c1)/U ′

Uncertainty, λ 38/48

Higher likelihood of “unemployment” λ ↑



Higher likelihood of unemployment λ2 ↑

0 0.5 1 1.5 2−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

S t a t i o n a r y I n t e r e s t R a t e , r

Likelihood of Unemployment, λ2

39/48

Small λ Equilibrium



Small λ Equilibrium• Consider limit: λ1, λ2 → 0 while holding constant λ1/λ2

Formally: λ1 = λλ1, λ2 = λλ2, λ

→0

• ⇒ population shares also constant: θ1 = λ2

λ1+λ2, θ2 =

λ1

λ1+λ2

Corollary

The Taylor series approximation of r (λ) around λ = 0 is

r (λ) ∼ ρ − θ1u ′(z 1 + ρa)θ1u ′(z 1 + ρa) + θ2u ′(z 2 + ρa)

λ

where a satisfies θ1a + θ2a = 0, and we used r → ρ, M 1 → θ1.

Implications: interest rate low if

1 high uncertainty (λ)2 large gap between income shocks (z 1/z 2 low)3 low income shocks relatively more likely (high θ1)4 high risk aversion (high γ in u ′(c ) = c −γ )5 loose borrowing constraint (low a)

40/48



Derivation from Discrete-time Bellman



Derivation from Discrete-time BellmanBack

• Time periods of length ∆

• discount factor

β (∆) = e −ρ∆

• Note that lim∆→0 β (∆) = 1 and lim∆→∞ β (∆) = 0.

• Discrete-time Bellman equation:

V (k t ) = maxc t ∆U (c t ) + e −ρ

∆V (k t +∆) s.t.

k t +∆ = ∆[F (k t ) − δ k t − c t ] + k t

42/48

Derivation from Discrete-time Bellman

http://-/?-

http://-/?-



Derivation from Discrete time Bellman

• For small ∆ (will take ∆ → 0), e −ρ∆ = 1 − ρ∆

V (k t ) = maxc t

∆U (c t ) + (1 − ρ∆)V (k t +∆)

• Subtract (1 − ρ∆)V (k t ) from both sides

ρ∆V (k t ) = maxc t ∆U (c t ) + (1 − ∆ρ)[V (k t +∆)− V (k t )]

• Divide by ∆ and manipulate last term

ρV (k t ) = maxc t

U (c t ) + (1

−∆ρ)

V (k t +∆)− V (k t )

k t +∆ − k t

k t +∆ − k t

∆Take ∆ → 0

ρV (k t ) = maxc t

U (c t ) + V ′(k t )k t

43/48

Derivation of Poisson KFE Back



Derivation of Poisson KFE Back

• Work with CDF (in wealth dimension)

G i (a, t ) = Pr(at ≤ a, z t = z i )

• Over time period of length ∆, wealth evolves as

at +∆ = at + ∆s i (at )

• income switches from z i to z j with probability ∆λi

• Fraction of people with wealth below a evolves as

Pr(at +∆

≤a, z t +∆ = z i ) = (1

−∆λi )Pr(at

≤a−

∆s i (a), z t = z i )

+∆λ j Pr(at ≤ a − ∆s j (a), z t = z j )

• Intuition: if have wealth < a −∆s i (a) at t , have wealth < aat t + ∆

44/48

Derivation of Poisson KFE



Derivation of Poisson KFE

• Subtracting G i (a, t ) from both sides and dividing by ∆

G i (a, t + ∆) − G i (a, t )∆

= G i (a − ∆s i (a), t ) − G i (a, t )∆

− λi G i (a − ∆s i (a), t ) + λ j G j (a − ∆s j (a), t )

• Taking the limit as ∆

→0

∂ t G i (a, t ) = −s i (a)∂ aG i (a, t )− λi G i (a, t ) + λ j G j (a, t )

where we have used that

lim∆→0

G i (a − ∆s i (a), t ) − G (a, t )∆

= limx →0

G (a − x , t ) − G (a, t )x

s i (a)

= −s i (a)∂ aG i (a, t )

• Differentiate w.r.t. a and use g i (a, t ) = ∂ aG i (a, t ) ⇒ (KFE).

45/48

Proof of Proposition Back



Proof of Proposition

• FOC: u ′(c 1(a)) = v ′1(a) and so u ′′(c 1(a))c ′1(a) = v ′′1 (a)

• Envelope condition:

(ρ − r )v ′1(a) = v ′′1 (a)s 1(a) + λ1(v ′2(a)− v ′1(a))

• Combining, using s ′1(a) = r − c ′1(a), and rearranging:

(s ′

1(a) − r )s 1(a) = (r −ρ)u ′(c 1(a))+λ1[u ′(c 2(a))−u ′(c 2(a))]

u ′′(c 1(a))

• If s 1(a) → 0, c 1(a) → c 1, c 2(a) → c 2 as a → a, then

s 1(a)s ′1(a) → (r −ρ)u ′(c 1)+λ1[u ′(c 2)−u ′(c 1)]u ′′(c 1) (∗)

• The Taylor series approximation of (s 1(a))2 around a is

(s 1(a))2 ∼ 2s 1(a)s ′1(a)(a − a)

• Substitute in from (

∗) and take the square root.

46/48




oo o opos t o

• Envelope condition

(ρ − r )v ′i (a) = s i (a)v

′′i (a) + λi (v

′ j (a)− v

′i (a))

• Multiplying the two equations by g 1 and g 2 and integrating:

(ρ − r )

∞

a

(v ′1(a)g 1(a) + v ′

2(a)g 2(a))da

= ∞a

[v ′′1 (a)s 1(a)g 1(a) + v ′′

2 (a)s 2(a)g 2(a)]da

+

∞

a

[v ′2(a)(−λ2g 2(a) + λ1g 1(a)) + v ′

1(a)(−λ1g 1(a) + λ2g 2(a))]da

• Integration by parts: ∞

a

v ′′

1 (a)s 1(a)g 1(a)da = v ′

1(a)s 1(a)g 1(a)

∞

a

−

∞

a

v ′

1(a) d

da[s 1(a)g 1(a)]da

= −v ′

1(a) lima→a

s 1(a)g 1(a) −

∞

a

v ′

1(a) d

da[s 1(a)g 1(a)]da

47/48




p• Hence

(ρ−

r ) ∞

a

(v ′

1

(a)g 1(a) + v ′

2

(a)g 2(a))da

=− v ′1(a) lima→a

s 1(a)g 1(a)

+

∞

a

v ′1(a)

− d

da(s 1(a)g 1(a)) − λ1g 1(a) + λ2g 2(a)

da

+ ∞a

v ′2(a)− d

da(s 2(a)g 2(a)) − λ2g 2(a) + λ1g 1(a)

da

• Second and third lines ≡ 0 from (KFE). First line? (KFE) ⇒0 =

−lima→a

s 1(a)g 1(a)

−λ1M 1

• Hence

(ρ − r )

∞a

(v ′1(a)g 1(a) + v ′2(a)g 2(a))da = v ′1(a)λ1M 1

• Finally, use FOC u ′

(c i (a)) = v ′

i (a). 48/48

Lecture1 Stanford 235 Web Nomovie(1)

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Transcript of Lecture1 Stanford 235 Web Nomovie(1)