a ( x , m ) = b i m i x i , a m i = b i x i i 9 / 48 T HE S PARSE - - PowerPoint PPT Presentation

BEHAVIORAL MACROECONOMICS

Xavier Gabaix Harvard Computing in Economics and Finance– June 2018

1 / 48

INTRODUCTION

◮ Normally, (1) we economists use a simplified model of the

world, knowing that the model is not literally the true world

◮ (2) But agents in our models are assumed to comprehend the

full complexity of their world.

◮ In this paper: (1) applies, but (2) is replaced by (2’): Agents

also have a simplified sub-model of their world.

◮ This paper models (2’), and its consequences ◮ Language: “sparsity”

◮ A vector m ∈ R1,000,000 is sparse if most entries are 0 ◮ The agent pays attention to few dimensions of the world

(endogenously)

◮ His attention vector is sparse ◮ He has a low-dimensional (sparse) submodel of the world 2 / 48

RELATED LITERATURE

◮ Behavioral economics and finance

◮ A lot of the literature is about modeling tastes or beliefs ◮ There is less on the modeling of rationality itself: O’Donoghue

Rabin, k−level models, Crawford et al., Camerer Ho and Chuong, J´ ehiel et al., MacLeod, Hong and Stein, Koszegi and Szeidl, Schwartzstein, Fuster, H´ ebert and Laibson, Bordalo, Gennaioli and Shleifer...

◮ Finance / Macro:

◮ Inattention: Sims 03, Gabaix and Laibson 02, 06, Mankiw Reis

02, Reis 06, Chetty, Kroft Looney 09, Angeletos La’O 10, Ma´ ckowiak and Wiederholt 10, 16, Masatlioglu and Ok 10, Veldkamp 11, Matejka and Sims 11, Caplin, Dean and Martin 11, Woodford 12, Alvarez Lippi, Paciello 13, Koszegi Szeidl 13, Abel, Eberly and Panageas 13, Greenwood Hanson 14, Croce, Lettau Ludvigson 15, Kueng 16, Angeletos and Lian ’17, Farhi Werning ’17

◮ Early behavioral models: Campbell Mankiw 89 3 / 48

RELATED LITERATURE

◮ Learning: Sargent 93, Evans and Honkapohja 01 ◮ Approximation: Krusell Smith 98, Judd 98, Ahn, Kaplan, Moll,

Winberry, Wolf 17

◮ Sparsity in statistics and OR: Tibshirani 96 (Lasso), Cand`

es and Tao 06, Donoho 06...

◮ Difficulty of behavioral dynamic programming: Laibson ’97, O’

Donoghue Rabin ’01, Harris and Laibson 01, 13.

◮ Developments of new tools in computational macro: ◮ Hansen-Sargent ◮ Bounded Rationality: Radner, Sargent 93, Rubinstein 98,

Al-Najjar 09, Bolton and Faure-Grimaud 10, Tirole 11, Gul, Pesendorder Strzalecki 12...

◮ Behavioral Public Finance: Chetty, Kroft Looney 09, Congdon

Mullainathan Schwartzstein 12, Bernheim Rangel 09, Farhi Gabaix 15.

4 / 48

PROGRAM: SEEKING UNIFIED BR IN ECONOMICS

• 1. Micro: “A sparsity-based model of bounded rationality” (QJE

2014): Fairly general and simple device, smax

a

u (a, x) subject to b (a, x) ≥ 0 Basic consumer theory: Walrasian demand, Hicksian demand, Slutsky matrix. Competitive equilibrium: Arrow-Debreu, Edgeworth boxes...

• 2. Macro: “Behavioral Macroeconomics via Sparse Dynamic

programming”: Consumption, RBC

• 3. “A Behavioral New Keynesian model”: monetary and fiscal

policy

• 4. Public economics: “Optimal taxation with behavioral agents”

(with E. Farhi) Ongoing work:

• 5. Finance: in the works.
• 6. Game theory: “Some game theory with sparsity-based

bounded rationality”. Sparse Nash equilibrium

5 / 48

CONCLUSION: MICRO BEHAVIOR FOR MACRO

• 1. Agents react more to near, rather than future, shocks.
• 2. Pervasive failure of the high-frequency Euler equation (Carroll

97)

• 3. High MPC to tax rebates, especially amongst misoptimizing

households (Parker 15, Kueng 16).

• 4. Agents accumulate a too small buffer of savings, as they’re

(partially) myopic to the risk of income fluctuations (Lusardi 11)

• 5. Agents start saving “too late” for retirement (falls at age 45;

controversial)

• 6. Very low sensitivity to interest rates (Hall 88) (so rat IES

should be low). However, people seem OK with non-smooth consumption profiles (so rat IES should be high).

• 7. When choosing their portfolio, agents are pay less attention to

the hedging demand motive.

• 8. Plain introspection: we don’t solve the full macro equilibrium

in our head

6 / 48

CONCLUSION: AGGREGATE MACRO

• 1. The Lucas critique has zero (or only partial) bite
• 2. RBC model: has more internal propagation + less sensitivity

to the interest rate

• 3. The agent is a hybrid of Lucas neoclassical agent and a

present-looking old Keynesian agent.

3.1 Like Lucas agent: Has general methodology + sensitivity to important parameters 3.2 Like old Keynesian agent: Has more common-sense behavior

So this may be a useful synthesis.

• 4. Macro policy (“A behavioral New Keynesian model”)

4.1 Forward guidance by central bank is less powerful with behavioral agents 4.2 Fiscal policy is more powerful, as they’re non Ricardian 4.3 Economy is more stable, even at the ZLB

7 / 48

INTRODUCTION

ROADMAP

◮ The static smax operator, from “A sparsity-based model of

bounded rationality” (2014)

◮ Application 1: Behavioral version of Varian ◮ Application 2: Measuring inattention in health care plans

◮ Sparse dynamic programming

◮ Application 1: Consumption-savings problems ◮ Application 2: Finite-horizon life-cycle model ◮ Application 3: RBC (not today)

◮ New Keynesian model

◮ Monetary & Fiscal policy ◮ ZLB, paradoxes 8 / 48

ATTENTION-AUGMENTED DECISION UTILITY

◮ Primitive u (a, x) ◮ Form attention-augmented decision utility: u (a, x, m)

u (a, x, m) := u (a, m1x1, ..., mnxn) = u (a, m ⊙ x) with m ⊙ x := (mixi)i=1...n

◮ This gives a (x, m) := arg maxa u (a, x, m). ◮ Sensitivity of action to attention:

ami := ∂a (m, x) ∂mi = −u−1

aa · uami

evaluated at m = md, ad = a

• x, md

.

◮ Example 1: if u (a, x) = − 1 2(a − ∑i bixi)2, then

u (a, x, m) = − 1

2 (a − ∑i bimixi)2,

a (x, m) = ∑

i

bimixi, ami = bixi

9 / 48

THE SPARSE BR ALGORITHM: MOTIVATION

◮ Agent will take action a (x, m) = arg maxa u (a, x, m) and

experience utility v (x, m) = u (a (x, m) , x)

◮ So it is sensible to allocate attention m as:

max

m E [v (x, m)] − C (m) ◮ Smax agent simplifies the problem:

◮ replaces utility by linear-quadratic approximation, and removes

correlations in x

◮ Chooses optimal attention m in that simplified model ◮ Chooses optimal action with a (x, m) action with correct utility.

◮ First part: with ι = (1, ..., 1) =full attention,

E [v (x, m)] = v (ι) − 1 2 (m − ι)′ Λ (m − ι) + o

• E x2

◮ Also, take C (m) = κ ∑i mα i . Agent does:

max

m∈[0,1]n −1

2 ∑

i

(1 − mi)2 Λii − κ ∑

i

mα

i

10 / 48

SPARSE MAX: QUICK VERSION

◮ Proposition: One solves smaxa;m u (a, x, m) as follows.

Step 1: Choose attention: m∗

i = Aα(E

• a′

miuaaami

• /κ)

Step 2: Choose action: as = arg maxa u (a, x, m∗).

◮ ami = −u−1 aa uami, and uaa evaluated at (a, m) =

• ad, md

with ad = arg maxa u

• a, md, x
• ◮ Aα (v) := arg minm 1

2 (1 − m)2 |v| + κmα

σ2 A0(σ2) 1 2 3 4 5 6 1 σ2 A1(σ2) 1 2 3 4 5 6 1 σ2 A2(σ2) 1 2 3 4 5 6 1 11 / 48

APPLICATION

◮ Quadratic example: u (a, x, m) = − 1 2 (a − ∑i bimixi)2,

ar =

106

∑

i=1

bixi : Non-Sparse Action as = ∑

i

bimixi : Sparse action mi = Aα

• b2

i σ2 xi /κ

• ◮ Sims ’03: no source-dependent inattention

aSims = m∑

i

bixi + η

◮ Others (Woodford, Ma´

ckowiak and Wiederholt) have source-dependent inattention

◮ smaxa;m u (a, x, m) applies easily beyond the linear-quadratic

framework

12 / 48

SPARSE MAX WITH CONSTRAINTS: MOTIVATION

◮ Example: Consumption problem: maxc u (c) s.t. p · c ≤ w

pi = pd

i + xi

ps

i = pd i + mixi ◮ How to satisfy the budget constraint? ◮ The “trade-off” intuition holds with perceived prices:

u′

c1

u′

c2

= ps

1

ps

2 ◮ So u′ (c) = λps. To satisfy budget: pick λ such that.

p · c (λ) = w.

13 / 48

SPARSE MAX WITH CONSTRAINTS

smax

a

u (a, x) subject to b (a, x) ≥ 0 Define u (a, x, m) := u (a, m ⊙ x), b (a, x, m) := b (a, m ⊙ x), L (a, x, m) := u (a, x, m) + λd · b (a, x, m), with λd multiplier when m = 0. Definition (Sparse max with constraints) Step 1. Allocate attention: with ami := −L−1

aa Lami

mi = A(−E [amiLaaami ] /κ) Step 2: Find action. Solve in a, λ: ua (a, x, m∗) + λba (a, x, m∗) = 0 (perceived model m for FOC) b (a, x, ι) = 0 (actual model ι for budget) with ι = (1, ..., 1) =full rationality.

14 / 48

BASIC CONSUMPTION PROBLEM: MARSHALLIAN

DEMAND

◮ I use s for sparse, r for rational / traditional. ◮ Proposition. Demand is:

cs (p, w) = cr ps, w ′ where w ′ solves p · cr (ps, w ′) = w.

◮ Quasi-linear utility: Take u (c) = v (c1, ..., cn−1) + cn, with v

• concave. Demand for good i < n is independent of wealth

and is: cs

i (p) = cr i (ps) . ◮ If traditional demand is linear in wealth:

cs (p, w) = cr (ps, w) p · cr (ps, 1)

◮ Budget is w = $100. I have $105 in my supermarket cart.

◮ Demand linear in wealth: So, I lower my consumption by 5%. ◮ General case: I lower it by −dc = ∂cs

∂w (ps, w) × $5

15 / 48

SPARSITY MODEL WITH BUDGET CONSTRAINT

◮ Cobb-Douglas preferences: u (c) = ∑n i=1 αi ln ci,

cs

i (p, w) = αi

ps

i

w ∑j αj

pj ps

j

◮ CES demand: u (c) = ∑i c1−1/η i

/ (1 − 1/η) cs

i (p, w) = (ps i )−η

w ∑j pj

• ps

j

−η

16 / 48

SLUTSKY MATRIX

◮ Slutsky matrix: Sij := ∂ci ∂pj + ∂ci ∂w cj = ∂ci ∂pj compensated ◮ Important as it encodes: (i) elasticities (ii) welfare losses ◮ Symmetry in the trad. model (Sr ij = Sr ji, i.e., with

compensated consumptions: ∂ci

∂pj = ∂cj ∂pi ): intuition is not

• bvious, so perhaps regular people won’t satisfy it:

“This is a rather nonintuitive result” (Varian 1992). “Symmetry is not easy to interpret in plain economic terms. As emphasized by Samuelson (1947), it is beyond what one would derive without the help of mathematics.” (Mas-Colell, Whinston and Green 1995). "The fact that the partial derivatives are identical [...] is quite amazing [...] I am unable to give a good intuitive explanation...." (Kreps 2012).

17 / 48

SLUTSKY NON-SYMMETRY

◮ Sij := ∂ci ∂pj + ∂ci ∂w cj = ∂ci ∂pj compensated ◮ Prop. At default price,

Ss

ij = Sr ijmj

Hence the sparse Slutsky matrix is not symmetric in general. Non-salient prices imply small columns.

◮ So a sparse agent can’t be represented by a rational agent, in

general (even an agent with adjustment costs).

◮ Example: 1 =good, 2 =shipping charges (cf Allcott Wozny

12, Brown, Hossain and Morgan 10, Chetty et al. 09, Ellison & Ellison 09, Gabaix and Laibson 06). Say m1 = 0.99, m2 ≃ 0 (“don’t see shipping charge”). 0 ≃

• ∂cgood

∂pshipping

• <
• ∂cshipping

∂pgood

• ◮ Income effects are preserved: ∂cs

i

∂w = ∂cr

i

∂w ◮ Prior theory of non-symmetric Slutsky: Browning and

Chiappori (1998): based on intra-household bargaining.

18 / 48

RECOVERING ATTENTION FROM THE SLUTSKY

MATRIX

◮ Sij := ∂ci ∂pj compensated ◮ Ss ij = Sr ijmj ◮ Proposition. We can recover mj (up to a multiplicative

constant), with ∑i γi = 1: mj = ∏

i

Sij Sji γi m

◮ Also, Sr ij = Ss ijm−1 j

should be symmetric.

◮ Development in Abaluck and Adams (’17): Estimate attention

to health care plans, from deviations from Slutsky symmetry

19 / 48

CONSUMER AND EQUILIBRIUM THEORY

◮ Paper re-does consumer theory, equilibrium theory

• 1. Slutsky matrices: non-symmetry, not negative-semi definite
• 2. Expenditure function, Hicksian demand
• 3. Roy’s identity, Shephard’s lemma, WARP
• 4. Nominal illusion
• 5. Producer theory
• 6. Equilibrium theory: Arrow-Debreu, 1st and 2nd welfare

theorems

• 7. Edgeworth boxes: Phillips curve in each Edgeworth box

◮ Get a theory of what predictions of basic micro are robust to

BR (e.g. sign predictions: “if the price goes up, demand goes down”), and what are not (like symmetry of Slutsky matrix; absence of nominal illusion).

20 / 48

SPARSE DYNAMIC PROGRAMMING

MOTIVATING EXAMPLE: PERMANENT-INCOME PROBLEM

◮ max(ct)t≥0 E ∑t βtc1−γ t

/ (1 − γ) s.t.: wt+1 = (1 + r + rt) (wt − ct) + y + yt

• rt+1 = ρr

rt + εr

t+1,

• yt+1 = ρy

yt + εy

t+1 ◮ What’s c (zt), zt := (wt, ˆ

rt, ˆ yt)?

◮ Want to capture: people “do not think” about the interest ˆ

rt.

◮ We anchor on the default model:

wt+1 = (1 + r) (wt − ct) + y with policy cd

t = ¯ rwt+ ¯ y R

, value function V d (wt).

21 / 48

SPARSE DYNAMIC PROGRAMMING

MOTIVATING EXAMPLE: PERMANENT-INCOME PROBLEM

◮ Agent has a “simplifiable subjective model”:

wt+1 = F w (ct, zt, m) = (1 + r + mr rt) (wt − ct) + y + my yt

• yt+1 = F y (ct, zt, m) := ρy (m)

yt + mσy εy

t+1,

ρy (m) := mρy ρy +

• 1 − mρy
• ρd

y

and same for rt+1 as for yt+1: F r := ρr (m) rt + mεr εr

t+1, ◮ The agent allocates attention:

m =

• my, mρy , mσy , mr, mρr , mσr
• ◮ More generally F z = (F w, F r, F y), mental model is:

zt+1 = F z (at, zt, m)

◮ Often, use

F zi (a, z, m) = mρi ⊙ F zi a, mi ⊙ z +

• 1 − mi ⊙ zi,d

+

• 1 − mρi
• zi,d

22 / 48

SPARSE DYNAMIC PROGRAMMING: BASIC VERSION

◮ Value function under world parametrized by m (maybe with

terminal condition zT ∈ FT ) V (z0, m) = max

(at) E[ T

∑

t=0

βtu (at, zt, m)] s.t. zt+1 = F z (at, zt, εt+1, m)

◮ Define current value function:

v (a, z, m) := u (a, z, m) + βE [V (F z (a, z, ε, m))]

◮ Bellman equation: V (z, m) = maxa v (a, z, m) ◮ Sparse policy with exogenous attention m,

as (z, m) := arg max

a

v (a, z, m)

◮ Traditional policy (with m = ι in rational model)

ar (z) = as (z, ι)

◮ Sparse policy with endogenous attention:

as (z) := arg smax

a;m|md v (a, z, m)

23 / 48

BR PERMANENT INCOME

◮ Proposition: In the sparse model, ct = cd t + ˆ

cs

t + O

• x2

, with cd

t = rwt+ ¯ y R

and ˆ cs

t

= Es

t[∑ τ≥t

br (wt) mr ˆ rτ + bymy ˆ yτ Rτ−t+1 ] br (wt) :=

r R (wt − ¯

y) − ψcd R , by := r R where Es

t is transition function under the subjective model. ◮ Rational policy: particular case with mr = my = 1

24 / 48

CALIBRATION

◮ With cd (w) = rw+ ¯ y R

, Br (wt) =

br R−ρr , By =

¯ r R

R−ρy ,

ln cs = ln cd (wt) + Brmr rt + Bymy yt + O

• x2

mx = A

• σ2

x B2 x |vcc| /κ

• for x =

r, y

◮ Set κ = −ucc

• cd2κ2. ¯

κ = 3% means “Pay attention to ˆ r iff a 1 std deviation of ˆ r would change c by more than 3%”

◮ Sensible comparative statics: mrincreases with σ2 r ◮ Useful: matches the low sensitivity to the interest rate (in a

rational model, needs low IES), while people accept

25 / 48

BR LIFE-CYCLE: MODIGLIANI-BRUMBERG (1954)

◮ Agent works (income ¯

y) for t ∈ [0, L), and retires (income yt = ¯ y + y) for t ∈ t ∈ [L, T)

◮ No discounting:

max

(ct)0≤t<T T−1

∑

t=0

u (ct) s.t.

T−1

∑

t=0

ct ≤ w0 +

T−1

∑

t=0

yt

t

10 20 30 40 50 60 80 85 90 95 100 Income Rational Consumption 26 / 48

BR LIFE-CYCLE: MODIGLIANI-BRUMBERG (1954)

◮ Ω0 := w0 + ∑T−1 τ=0 yτ = w0 + T ¯

y − x with x = − (T − L) y > 0

◮ Rational agent. Plans ct = Ω0 T ,

c0 = w0 − x T + ¯ y

◮ Same at future dates t ≤ L

ct = wt − x T − t + ¯ y

◮ Value function:

V r (wt, x, t) = (T − t) u wt − x T − t + ¯ y

• 27 / 48

BR LIFE-CYCLE: BEHAVIORAL

v (ct, wt, x, t) := u (ct) + V r (wt + ¯ y − ct, x, t + 1)

◮ Rational agent:

ct = arg max

ct v (ct, wt, x, t) ◮ Behavioral agent:

ct = arg smax

ct;mt v (ct, wt, mtx, t)

i.e. ct (mt) = wt − mtx T − t + ¯ y and evaluated at mt = 0 mt = A

• −vt

cc

∂ct ∂m 2

|m=0

1 κt

• ◮ Calibration: take κt =
• u′′

cd

t

• ¯

κ2.

28 / 48

BR LIFE-CYCLE

t

10 20 30 40 50 60 ct 80 85 90 95 100

Consumption

Fully Rational Moderately Behavioral Very Behavioral 29 / 48

EVIDENCE

• 1. Expenditure declines after the age of 45 (Aguiar and Hurst

2013).

• 2. There is a fall in expenditure

2.1 At retirement (Bernheim, Skinner and Weinberg 01). 2.2 At end of unemployment benefits (Ganong and Noel 17).

• 3. People say that they plan for retirement late or not at all:

23% of the 18-29 year old say that have “figured out how much they need to save for retirement”, while 51% of the 45-59 year old say they have done so (Lusardi 2011).

◮ There are rational explanations for 1, 2.1: impatience + credit

constraints (Gourinchas and Parker 03, Kaplan and Violante 14).

◮ Hyperbolic: still see full wealth Ω0 = w0 + ∑T−1 τ=0 yτ, with no

“cognitive discounting.” Plain hyperbolic (with no risk) doesn’t give any drop at retirement, just a gentle continuous decline.

30 / 48

MORALE FROM LIFECYCLE EXAMPLE

◮ Sparse agents consume their wealth patiently, like patient

rational agents, but are myopic for small things.

◮ Agents consume wealth w patiently, as a rational agent. ◮ They’re myopic about the future small shocks ◮ The deterministic steady state is rational, it’s the deviation

from it that’s behavioral

◮ ...contra models with credit constraints or, hand-to-mouth

behavior.

◮ The Euler equation fails. The Euler equation holds under the

BR-perceived world, but not under the actual world.

◮ Agents react more to “near” shocks than to “distant” shocks

31 / 48

COMPUTATIONAL IMPLEMENTATION

V (z0) = max

(at) ∑ t≥0

βtu (at, zt) s.t. zt+1 = F z (at, zt)

• 1. Choose “default” model, where many variables are set to

constants (e.g. all variables but wealth wt)

• 2. Calculate a proxy value function V p (z). E.g.

V p (w, x) = V r (w, x) + O

• x2

(e.g. V (w, x) = V d (w) + ∑i Bi (w) xi + O

• x2

)

• 3. Put “m’s” in front of stochastic variables (except default

variables): F zi (a, z, m), u (a, z, m)

• 4. Do

a (z, V p) := arg smax

a;m|md{u (a, z, m) + βE [V p (F z (a, z, m))]} ◮ This way, you can simulate the whole policy / life forward.

32 / 48

PREVIEW OF “A BEHAVIORAL NEWKEYNESIAN MODEL”

◮ Paper re-does Gali’s textbook, Chapter 2-5, with behavioral

agents.

◮ Microfoundations like above gives in Gen.Eq.

xt = MEt [xt+1] + bddt − σ (it − Etπt+1 − rn

t ) (IS curve)

πt = βMf Et [πt+1] + κxt (Phillips curve)

◮ xt is output gap, πt inflation, it nominal rate, rn t the natural

rate, dt the deficit

◮ Traditional model has M = Mf = 1, bd = 0 ◮ Here M, Mf < 1, bd > 0

33 / 48

INDIVIDUAL PROBLEM

max

(ct,Nt)t≥0

U = E

∞

∑

t=0

βtu (ct, Nt) , u (c, N) = c1−γ 1 − γ − N1+φ 1 + φ, kt+1 = (1 + r + ˆ r (X t)) (kt + ¯ y + ˆ y (Nt, X t) − ct) , X t+1 = G (X t, ǫt+1) , ˆ y (Nt, X t) = ω (X t) Nt + yf (X t) − ¯ y, with X t = (de-meaned) state vector, and yf (X t) firms’ profits.

◮ Behavioral agent maximizes in a subjective model with

“cognitive discounting” X t+1 = ¯ mG (X t, ǫt+1) with ¯ m ∈ [0, 1]. Rational case: ¯ m = 1.

◮ So behavioral agent maximizes U with perceived law of

motions

34 / 48

INDIVIDUAL PROBLEM: COGNITIVE DISCOUNTING

◮ Linearize: X t+1 = ¯

m (ΓX t + ǫt+1), EBR

t

[X t+1] = ¯ mΓX t, EBR

t

[X t+k] = ¯ mkΓkX t, EBR

t

[X t+k] = ¯ mkEt [X t+k] .

◮ As linearizing, ˆ

r (X t) = brX t for some coefficient br: EBR

t

[ ˆ r (X t+k)] = ¯ mkEt [ ˆ r (X t+k)] .

◮ This is “cognitive discounting”.

35 / 48

BEHAVIORAL IS CURVE: DERIVATION IN BASIC CASE

◮ Traditional model: Et

• βRt
• ct+1

ct

−γ = 1 gives ˆ ct = Et [ ˆ ct+1] − 1 γR ˆ rt

◮ Behavioral agent: EBR t

[βRt

• c(X t+1,kt+1)

c(X t,kt)

−γ ] = 1. Also, agent correctly forecasts kt+1 = 0. So, EBR

t

• βRt

c (X t+1) c (X t) −γ = 1. i.e. ˆ c (X t) = EBR

t

[ ˆ c (X t+1)] − 1 γR ˆ rt. = ¯ mEt [ ˆ ct (X t+1)] − 1 γR ˆ rt, i.e., with M = ¯ m, ˆ ct = MEt [ ˆ ct+1] − σ ˆ rt

36 / 48

DISCOUNTED EULER EQUATION

◮ With M := ¯ m R−rmY ,

xt = MEt [xt+1] − σ ˆ rt.

◮ Iterate:

xt = −σEt ∑

τ≥t

Mτ−t ˆ rτ.

◮ In rational model, M = 1, so

xt = −σEt ∑

τ≥t

ˆ rτ.

◮ Hence, interest rate in 1000 periods has same impact as

interest rate today: odd. “Forward guidance puzzle” (McKay, Nakamura Steinnson ’15).

37 / 48

PREVIEW OF BEHAVIORAL NEW KEYNESIAN MODEL

• 1. Behavioral version of the work-horse model used for policy
• 2. Monetary policy is less powerful (esp. forward guidance)
• 3. Helicopter drops of money / Fiscal policy is more powerful
• 4. Optimal joint fiscal+monetary policy.
• 5. Taylor principle strongly modified. Equilibrium is determinate

(even with rigid monetary policy): stable economy at the ZLB.

• 6. The ZLB is much less costly.
• 7. Optimal policy

7.1 Do “helicopter drops of money” at the ZLB→First Best 7.2 “Price-level targeting” is not optimal any more

• 8. Resolution of neo-Fisherian paradoxes

◮ Empirical support for main features of model

38 / 48

BEHAVIORAL RBC MODEL

◮ Standard model: max(ct,lt) ∑ βtu (ct, lt)

Kt+1 = eζtK α

t L1−α t

+ (1 − δ) Kt =: f (Kt, Lt, ζ) ζt+1 = ρζζt + εt+1

◮ Zt = (Kt, ζt) ◮ In equilibrium, C (Z), L (Z), and

R (Z) := fK (K (Z) , L(Z), ζ(Z)) , w (Z) := fL (K (Z) , L(Z), ζ(Z))

◮ Behavioral: for p (Z) = (R (Z) , w (Z)) and pd the average

value pi (Z, mi) = mipi (Z) + (1 − mi) pd

i

39 / 48

RBC: HOUSEHOLD’S PROBLEM

The household’s problem (with given attention m for now). V (k, Z, m) = max

a=(c,l) u (c, l) + βE

• V
• k′, Z ′, m
• s.t.

(1) k′ = R (Z, m) k + w (Z, m) l − c K ′ = F (K, L (Z)) − C (Z) ζ′ = ρζ + ε This return the policy function a (k, Z, m).

40 / 48

BEHAVIORAL RECURSIVE COMPETITIVE EQUILIBRIUM

A behavioral recursive competitive equilibrium is a set of decision rules a (k, Z, m), aggregate per capita action A (Z) = (C (Z) , L (Z)), factor prices p (Z) such that those function satisfy:

• 1. The household problem (1)
• 2. Firms maximize profits: R (Z) = fK (K (Z) , L (Z) , ζ (Z)),

w (Z) = fL (K (Z) , L (Z) , ζ (Z))

• 3. Consistency between micro and macro:

A (Z) = a (K (Z) , Z, m (Z)), where, with the notations of problem (1) a (k, Z, m (Z)) = arg smax

a;m

• u (a) + βE
• V
• k′, Z ′, m
• 41 / 48

OPERATIONALIZING THAT

◮ 3 methods to solve such a problem in practice

• 1. Value function iteration
• 2. Perturbation methods
• 3. FOC and linearization in Dynare.

◮ Method 3. is the simplest, so I do that.

42 / 48

VIA EULER EQUATION

◮ Take log utility (Prescott 1995):

u (c, l) = χ ln c + (1 − χ) ln (1 − l)

◮ Recall

k∆

t := (1 − mr) k ˆ

rt + (1 − mw) ¯ l ˆ wt Behavioral Euler equation is: 1 = Et

• βRs

t

uc,t+1

• ct+1 − βχ ¯

rk∆

t

• uc,t
• i.e.

ˆ ct = Et [ ˆ ct+1] − mr ¯ c ˆ rt ¯ R − βχ ¯ rk∆

t ◮ FOC for labor supply:

( ¯ w + mw ˆ wt) uc (ct, lt) = −ul (ct, lt)

43 / 48

CONSEQUENCES

◮ This model generates internal propagation. For instance, take

mr = mw and fixed labor supply. Then, ˆ Kt+1 = (1 − φK) ˆ Kt + bζt where φK solves φK = 1 ¯ R mr −ψCfKK r + φK So, φ < φrat, and φ → 0 when mr → 0

◮ Conclusion: the speed of mean-reversion of the economy is

φK = min

• φK, φζ
• So, for mr small enough, the economy exhibit “internal

propagation” at rate φK.

◮ Also, when m → 0, then φK → 0: general

44 / 48

“COGNITIVE DISCOUNTING”: AN EXTRA FEATURE

THAT SEEMS USEFUL

◮ When people forecast ˆ

Zt = Zt − Z d, instead of the true law

• f motion

Zt+1 = F Z (Zt, εt+1) then perceived: Zt+1 = ¯ mF Z (Zt, εt+1) i.e. the more distant variables are more dimly perceived: E s

t [ ˆ

rτ] = mr ¯ mτ−tEt [ ˆ rτ]

◮ Behavioral Euler equation: with

Hr

t := mrEt

• ∑τ≥t (mβ)τ−t ˆ

rτ

• ˆ

ct = Et [ ˆ ct+1] − mr ˆ rt ¯ R ¯ c − βχ ¯ rk∆

t

− βχ ¯ r [( ¯ k − ψC/ ¯ r) (1 − ¯ mr) Hr

t+1 + lmax (1 − ¯

mw) Hw

t+1]

45 / 48

MORE BR LEADS TO GREATER PERSISTENCE AND

FLUCTUATIONS

Statistic Case 1 Case 2 Case 3 σc 0.0073 0.0064 0.0091 σk 0.0112 0.0163 0.0583 σy 0.0134 0.0141 0.0234 σR 0.0004 0.0005 0.0013 σ∆c 0.0009 0.0008 0.0002 σ∆k 0.0010 0.0010 0.0012 σ∆y 0.0073 0.0073 0.0073 σ∆R 0.0003 0.0003 0.0003 ρc 0.9927 0.9915 0.9998 ρk 0.9959 0.9980 0.9998 ρy 0.8507 0.8640 0.9510 ρR 0.7827 0.8484 0.9812

• Parameters. Case 1: mR = 1, ¯

m = 1. Case 2: mR = 0.2, ¯ m = 1. Case 3: mR = 0.2, ¯ m = 0.8. All cases: ρζ = 0.8, α = 0.33, β = 0.99, δ = 0.023, χ = 0.99, mw = 1, σ = 0.007.

46 / 48

INTERIM RESULTS ON RBC MODEL

◮ This model generates internal propagation. ◮ Little relation between consumption growth and the interest,

as in the data

◮ “Excess volatility” given TFP shocks ◮ General hope

◮ Allows to talk about new things ◮ It’s actually quite tractable ◮ Hopefully will improve the fit: stay tuned! 47 / 48

CONCLUSION

◮ Fairly general and simple device, smax ◮ Prior paper: Gives a behavioral version of the 2 basic chapters

• f micro:

◮ Basic consumer theory: maxc∈Rn u (c) s.t.

p · c ≤ w : Marshallian demand, Slutsky matrix, expenditure function, Roy identity, Competitive equilibrium theory: Arrow-Debreu, Edgeworth boxes,...

◮ Today: Sparse dynamic programming ◮ Applications: Sparse versions of:

◮ Life-cycle model, infinite and finite horizon ◮ Basic RBC model ◮ New Keynesian model

◮ Ready to be used to analyze policy / analyze datasets where

bounded rationality seems important

◮ New type of computational challenges in the sparse max:

state-dependent optimal attention

48 / 48