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1. ML-5
7/22/11
Heterogeneous Agents Models
Jesús Fernández-Villaverde
University of Pennsylvania
July 11, 2011
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 1 / 53
2. Introduction
Introduction
Often, we want to deal with model with heterogeneous agents.
Examples:
1 Heterogeneity in age: OLG models.
2 Heterogeneity in preferences: risk sharing.
3 Heterogeneity in abilities: job market.
4 Heterogeneity in policies: progressive marginal tax rates.
Why General Equilibrium?
1 It imposes discipline: relation between β and r is endogenous.
2 It generates an endogenous consumption and wealth distribution.
3 It enables meaningful policy experiments.
The following slides borrow extensively from Dirk Krueger’ lecture
s
notes.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 2 / 53
3. Models without Aggregate Uncertainty
Models without Aggregate Uncertainty I
Continuum of measure 1 of individuals, each facing an income
‡uctuation problem.
Labor income: wt yt .
∞
Same labor endowment process fyt gt =0 , yt 2 Y = fy1 , y2 , . . . yN g.
Labor endowment process follows stationary Markov chain with
transitions π (y 0 jy ).
Law of large numbers: π (y 0 jy ) also the deterministic fraction of the
population that has this transition.
Π: stationary distribution associated with π, assumed to be unique.
At period 0 income of all agents, y0 , is given. Population distribution
given by Π.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 3 / 53
4. Models without Aggregate Uncertainty
Models without Aggregate Uncertainty II
Total labor endowment in the economy at each point of time
¯
L= ∑ y Π (y )
y
Probability of event history y t , given initial event y0
π t (y t jy0 ) = π (yt jyt 1) . . . π (y1 jy0 )
Note use of Markov structure.
Substantial idiosyncratic uncertainty, but no aggregate uncertainty.
Thus, there is hope for stationary equilibrium with constant w and r .
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 4 / 53
5. Models without Aggregate Uncertainty
Models without Aggregate Uncertainty III
Preferences
∞
u ( c ) = E0 ∑ β t U ( ct )
t =0
Budget constraint
ct + at +1 = wt yt + (1 + rt )at
Borrowing constraint
at + 1 0
Initial conditions of agent (a0 , y0 ) with initial population measure
Φ0 (a0 , y0 ).
Allocation: fct (a0 , y t ), at +1 (a0 , y t )g.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 5 / 53
6. Models without Aggregate Uncertainty
Models without Aggregate Uncertainty IV
Technology
Yt = F (Kt , Lt )
with standard assumptions.
Capital depreciates at rate 0 < δ < 1.
Aggregate resource constraint
Ct + Kt +1 (1 δ)Kt = F (Kt , Lt )
The only asset in economy is the physical capital stock. No
state-contingent claims (a form of incomplete markets).
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 6 / 53
7. Models without Aggregate Uncertainty
Sequential Markets Competitive Equilibrium I
De…nition
Given Φ0 , a sequential markets competitive equilibrium is allocations for
households fct (a0 , y t ), at +1 (a0 , y t )g allocations for the representative
∞ ∞
…rm fKt , Lt gt =0 , prices fwt , rt gt =0 such that:
1 Given prices, allocations maximize utility subject to the budget
constraint and subject to the nonnegativity constraints on assets and
consumption.
rt = Fk (Kt , Lt ) δ
wt = FL (Kt , Lt )
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 7 / 53
8. Models without Aggregate Uncertainty
Sequential Markets Competitive Equilibrium II
De…nition (cont.)
2. For all t
Z
Kt + 1 = ∑ at +1 (a0 , y t )π (y t jy0 )d Φ0 (a0 , y0 )
y t 2Y t
Z
Lt ¯
= L= ∑ yt π (y t jy0 )d Φ0 (a0 , y0 )
y t 2Y t
Z
∑ ct (a0 , y t )π (y t jy0 )d Φ0 (a0 , y0 )
y t 2Y t
Z
+ ∑ at +1 (a0 , y t )π (y t jy0 )d Φ0 (a0 , y0 )
y t 2Y t
= F (Kt , Lt ) + (1 δ ) Kt
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 8 / 53
9. Models without Aggregate Uncertainty
Recursive Equilibrium
Individual state (a, y ).
Aggregate state variable Φ(a, y ).
A = [0, ∞) : set of possible asset holdings.
Y : set of possible labor endowment realizations.
P (Y ) is power set of Y .
B(A) is Borel σ-algebra of A.
Z =A Y and B(Z ) = P (Y ) B(A).
M: set of all probability measures on the measurable space
M = (Z , B(Z )).
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 9 / 53
10. Models without Aggregate Uncertainty
Household Problem in Recursive Formulation
v (a, y ; Φ) = max u (c ) + β
c 0,a 0 0
∑
0
π (y 0 jy )v (a 0 , y 0 ; Φ 0 )
y 2Y
s.t. c + a0 = w (Φ)y + (1 + r (Φ))a
Φ0 = H (Φ)
Function H : M ! M is called the aggregate “law of motion”.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 10 / 53
11. Models without Aggregate Uncertainty
De…nition
A RCE is value function v : Z M ! R, policy functions for the
household a0 : Z M ! R and c : Z M ! R, policy functions for the
…rm K : M ! R and L : M ! R, pricing functions r : M ! R and
w : M ! R and law of motion H : M ! M s.t.
1 v , a0 , c are measurable with respect to B(Z ), v satis…es Bellman
equation and a0 , c are the policy functions, given r () and w ()
2 K , L satisfy, given r () and w ()
r (Φ) = FK (K (Φ), L(Φ)) δ
w (Φ) = FL (K (Φ), L(Φ))
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 11 / 53
12. Models without Aggregate Uncertainty
De…nition (cont.)
3. For all Φ 2 M
Z
K 0 (Φ0 ) = K (H (Φ)) = a0 (a, y ; Φ)d Φ
Z
L( Φ ) = yd Φ
Z Z
c (a, y ; Φ)d Φ + a0 (a, y ; Φ)d Φ = F (K (Φ), L(Φ)) + (1 δ )K ( Φ )
4. Aggregate law of motion H is generated by π and a0 .
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 12 / 53
13. Models without Aggregate Uncertainty
Transition Functions I
De…ne transition function QΦ : Z B(Z ) ! [0, 1] by
π (y 0 jy ) if a0 (a, y ; Φ) 2 A
QΦ ((a, y ), (A, Y )) = ∑ 0 else
y 0 2Y
for all (a, y ) 2 Z and all (A, Y ) 2 B(Z )
QΦ ((a, y ), (A, Y )) is the probability that an agent with current
assets a and income y ends up with assets a0 in A tomorrow and
income y 0 in Y tomorrow.
Hence
Φ0 (A, Y ) = (H (Φ)) (A, Y )
Z
= QΦ ((a, y ), (A, Y ))Φ(da dy )
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 13 / 53
14. Models without Aggregate Uncertainty
Stationary RCE I
De…nition
A stationary RCE is value function v : Z ! R, policy functions for the
household a0 : Z ! R and c : Z ! R, policies for the …rm K , L, prices
r , w and a measure Φ 2 M such that
1 v , a0 , c are measurable with respect to B (Z ), v satis…es the
household’ Bellman equation and a0 , c are the associated policy
s
functions, given r and w .
2 K , L satisfy, given r and w
= Fk ( K , L )
r δ
w = FL ( K , L )
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 14 / 53
15. Models without Aggregate Uncertainty
Stationary RCE II
De…nition (cont.)
3.
Z
K = a0 (a, y )d Φ
Z
L( Φ ) = yd Φ
Z Z
c (a, y )d Φ + a0 (a, y )d Φ = F (K , L) + (1 δ )K
4. For all (A, Y ) 2 B(Z )
Z
Φ(A, Y ) = Q ((a, y ), (A, Y ))d Φ
where Q is transition function induced by π and a0 .
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 15 / 53
16. Models without Aggregate Uncertainty
Example: Discrete State Space
Suppose A = fa1 , . . . , aM g. Then Φ is M N 1 column vector and
Q = (qij ,kl ) is M N M N matrix with
qij ,kl = Pr (a0 , y 0 ) = (ak , yl )j(a, y ) = (ai , yl )
Stationary measure Φ satis…es matrix equation
Φ = Q T Φ.
Φ is (rescaled) eigenvector associated with an eigenvalue λ = 1 of
QT .
Q T is a stochastic matrix and thus has at least one unit eigenvalue.
If it has more than one unit eigenvalue, continuum of stationary
measures.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 16 / 53
17. Models without Aggregate Uncertainty
Existence, Uniqueness, and Stability
Existence (and Uniqueness) of Stationary RCE boils down to one
equation in one unknown.
Asset market clearing condition
Z
K = K (r ) = a0 (a, y )d Φ Ea(r )
By Walras’law forget about goods market.
¯ ¯
Labor market equilibrium L = L and L is exogenously given.
Capital demand of …rm K (r ) is de…ned implicitly as
¯
r = Fk ( K ( r ) , L ) δ
Existence is usually easy to show.
Uniqueness is more complicated.
Stability is not well-understood.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 17 / 53
18. Models without Aggregate Uncertainty
Computation
1 Fix an r 2 ( δ, 1/β 1).
2 For a …xed r , solve household’ recursive problem. This yields a value
s
0
function vr and decision rules ar , cr .
3 0
The policy function ar and π induce Markov transition function Qr .
4 Compute the unique stationary measure Φr associated with this
transition function.
5 Compute excess demand for capital
d (r ) = K (r ) Ea(r )
If zero, stop, if not, adjust r .
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 18 / 53
19. Models without Aggregate Uncertainty
Qualitative Results
Complete markets model: r CM = 1/β 1.
This model: r < 1/β 1.
Overaccumulation of capital and oversaving (because of precautionary
reasons: liquidity constraints, prudence, or both).
Question: How big a di¤erence does it make?
Policy implications?
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 19 / 53
20. Models without Aggregate Uncertainty
Calibration I
CRRA with values σ = f1, 3, 5g.
r CM = 0.0416 (β = 0.96).
Cobb-Douglas production function with α = 0.36.
δ = 8%.
Earning pro…le:
1
log(yt +1 ) = θ log(yt ) + σε 1 θ2 2
ε t +1
s.t.
corr (log(yt +1 ), log(yt )) = θ
Var (log(yt +1 )) = σ2
ε
Consider θ 2 f0, 0.3, 0.6, 0.9g and σε 2 f0.2, 0.4g.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 20 / 53
21. Models without Aggregate Uncertainty
Calibration II
Discretize, using Tauchen’ method.
s
Set N = 7.
Since log(yt ) 2 ( ∞, ∞) subdivide in intervals
( ∞, 5σ ) [ 5σ , 3 [ 3 σε , 5 σε ) 5
[ 2 σε , ∞)
2 ε 2 ε 2 σε ) ... 2 2
State space for log-income: “midpoints”
Y log = f 3σε , 2σε , σε , 0, σε , 2σε , 3σε g
Matrix π: …x si = log(y ) 2 Y log today and the conditional probability
of sj = log(y 0 ) 2 Y log tomorrow is
2
(x θsi )
Z 2σy
e
π (log(y 0 ) = sj j log(y ) = si ) = dx
Ij (2π )0.5 σy
1
2
where σy = σε 1 θ2 .
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 21 / 53
22. Models without Aggregate Uncertainty
Calibration III
Find the stationary distribution of π, hopefully unique, by solving
Π = πT Π
log
˜
Take Y = e Y
3 σε 2 σε
˜
Y = fe ,e ,e σε
, 1, e σε , e 2 σε , e 3 σε g
Compute average labor endowment y = ∑y 2Y y Π(y ).
¯ ˜
Normalize all states by y
¯
Y = f y1 , . . . , y7 g
e 3 σε e 2 σε e σε 1 e σε e 2 σε e 3 σε
= f , , , , , , g
y¯ y
¯ y y y
¯ ¯ ¯ y
¯ y
¯
Then:
∑ y Π (y ) = 1
y 2Y
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 22 / 53
23. Models without Aggregate Uncertainty
Results
¯
Cobb-Douglas production function and L = 1 we have Y = K α and
1
r + δ = αK α
s is the aggregate saving rate:
αY αδ αδ
r +δ = = )s=
K s r +δ
Benchmark of complete markets: r CM = 4.16% and s = 23.7%.
Keeping σ and σε …xed, an increase in θ leads to more precautionary
saving and more overaccumulation of capital.
Keeping θ and σε …xed, an increase in σ leads to more precautionary
saving and more overaccumulation of capital
Keeping σ and θ …xed, an increase in σε leads to more precautionary
saving and more overaccumulation of capital.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 23 / 53
24. Transition Dynamics
Unexpected Aggregate Shocks and Transition Dynamics
Hypothetical thought experiment:
Economy is in stationary equilibrium, with a given government policy.
Unexpectedly government policy changes. Exogenous change may be
either transitory or permanent.
Want to compute transition path induced by the exogenous change,
from the old stationary equilibrium to a new stationary equilibrium.
Example: permanent introduction of a capital income tax at rate τ.
Receipts are rebated lump-sum to households as government transfers.
Key: assume that after T periods the transition from old to new
stationary equilibrium is completed.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 24 / 53
25. Transition Dynamics
Algorithm I
1 Fix T .
2 Compute stationary equilibrium Φ0 , v0, r0 , w0 , K0 associated with
τ = τ 0 = 0.
3 Compute stationary equilibrium Φ∞ , v∞ , r∞ , w∞ , K∞ associated with
τ ∞ = τ. Assume:
ΦT , vT , rT , wT , KT = Φ∞ , v∞ , r∞ , w∞ , K∞
4 ˆ ˆ
Guess sequence fKt gT=11 Note that K1 is determined by decisions at
t
ˆ ¯
time 0, K1 = K0 , and Lt = L0 = L is …xed. Also:
ˆ ¯
wt = F L ( K t , L )
ˆ
ˆ ¯
rt = FK (Kt , L) δ
ˆ
ˆ ˆ ˆ
Tt = τ t rt Kt .
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 25 / 53
26. Transition Dynamics
Algorithm II
5 ˆ ˆ ˆ t
Since we know vT (a, y ) and frt , wt , Tt gT=11 , we can solve for
T 1
fvt , ct , at +1 gt =1 backwards.
ˆ ˆ ˆ
6 With fat +1 g de…ne transition laws fΓt gT=11 . Given Φ0 = Φ1 from
ˆ ˆ
t
the initial stationary equilibrium, iterate forward:
Φ t +1 = Γ t ( Φ t )
ˆ ˆ ˆ
for t = 1, . . . , T 1.
R
7 With fΦt gT=1 , compute At = ad Φt for t = 1, . . . , T .
ˆ t ˆ ˆ
8 Check whether:
ˆ
max At ˆ
Kt < ε
1 t <T
ˆ
If yes, go to 9. If not, adjust your guesses for fKt gT=11 in 4.t
9 ˆ
Check whether AT KT < ε. If yes, we are done and should save
fvt , at +1 , ct , Φt , rt , wt , Kt g. If not, go to 1. and increase T .
ˆ ˆ ˆ ˆ ˆ ˆ ˆ
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 26 / 53
27. Transition Dynamics
Welfare Consequences of the Policy Reform I
Previous procedure determines aggregate variables such as
rt , wt , Φt , Kt , decision rules cr , at +1 , and value functions vt .
We can use v0 , v1 , and vT to determine the welfare consequences
from the reform.
Suppose that
c1 σ
U (c ) =
1 σ
Optimal consumption allocation in initial stationary equilibrium, in
∞
sequential formulation, fcs gs =0 .
∞
c1 σ
v0 (a, y ) = E0 ∑ 1t σ
s =0
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 27 / 53
28. Transition Dynamics
Welfare Consequences of the Policy Reform II
De…ne g implicitly as:
∞
ct1 σ
v1 (a, y ) = v0 (a, y ; g ) = (1 + g )1 σ
E0 ∑
s =0 1 σ
= (1 + g )1 σ
v0 (a, y )
Then: 1
v1 (a, y ) 1 σ
g (a, y ) = 1
v0 (a, y )
Steady state welfare consequences:
1
vT (a, y ) 1 σ
gss (a, y ) = 1
v0 (a, y )
g (a, y ) and gss (a, y ) may be quite di¤erent.
Example: social security reform.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 28 / 53
29. Models with Aggregate Uncertainty
Aggregate Uncertainty and Distributions as State Variables
Why complicate the model? Want to talk about economic
‡uctuations and its interaction with idiosyncratic uncertainty.
But now we have to characterize and compute entire recursive
equilibrium: distribution as state variable.
In…nite-dimensional object.
Very limited theoretical results about existence, uniqueness, stability,
goodness of approximation
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 29 / 53
30. Models with Aggregate Uncertainty
The Model I
Aggregate production function:
Yt = st F (Kt , Lt )
Let
st 2 f sb , sg g = S
with sb < sg and conditional probabilities π (s 0 js ).
Idiosyncratic labor productivity yt correlated st .
yt 2 fyu , ye g = Y
where yu < ye stands for the agent being unemployed and ye stands
for the agent being employed.
Probability of being unemployed is higher during recessions than
during expansions.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 30 / 53
31. Models with Aggregate Uncertainty
The Model II
Probability of individual productivity tomorrow of y 0 and aggregate
state s 0 tomorrow, conditional on states y and s today:
π (y 0 , s 0 jy , s ) 0
π is 4 4 matrix.
Law of large numbers: idiosyncratic uncertainty averages out and only
aggregate uncertainty determines Πs (y ), the fraction of the
population in idiosyncratic state y if aggregate state is s.
Consistency requires:
∑
0
π (y 0 , s 0 jy , s ) = π (s 0 js ) all y 2 Y , all s, s 0 2 S
y 2Y
π (y 0 , s 0 jy , s )
Πs 0 (y 0 ) = ∑ π (s 0 js )
Πs (y ) for all s, s 0 2 S
y 2Y
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 31 / 53
32. Models with Aggregate Uncertainty
Recursive Formulation
Individual state variables (a, y ).
Aggregate state variables (s, Φ).
Recursive formulation:
v (a, y , s, Φ) = max fU (c ) + β
0 c ,a 0
∑ ∑
0 0
π (y 0 , s 0 jy , s )v (a 0 , y 0 , s 0 , Φ 0 )
y 2Y s 2S
s.t. c + a0 = w (s, Φ)y + (1 + r (s, Φ))a
Φ0 = H (s, Φ, s 0 )
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 32 / 53
33. Models with Aggregate Uncertainty
De…nition
A RCE is value function v : Z S M ! R, policy functions for the
household a0 : Z S M ! R and c : Z S M ! R, policy
functions for the …rm K : S M ! R and L : S M ! R, pricing
functions r : S M ! R and w : S M ! R and an aggregate law of
motion H : S M S ! M such that
1 v , a0 , c are measurable with respect to B(S ), v satis…es the
household’ Bellman equation and a0 , c are the associated policy
s
functions, given r () and w ()
2 K , L satisfy, given r () and w ()
r (s, Φ) = FK (K (s, Φ), L(s, Φ)) δ
w (s, Φ) = FL (K (s, Φ), L(s, Φ))
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 33 / 53
34. Models with Aggregate Uncertainty
De…nition (cont.)
3. For all Φ 2 M and all s 2 S
Z
K (H (s, Φ)) = a0 (a, y , s, Φ)d Φ
Z
L(s, Φ) = yd Φ
Z Z
c (a, y , s, Φ)d Φ + a0 (a, y , s, Φ)d Φ
= F (K (s, Φ), L(s, Φ)) + (1 δ)K (s, Φ)
4. The aggregate law of motion H is generated by the exogenous
Markov process π and the policy function a0 .
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 34 / 53
35. Models with Aggregate Uncertainty
Transition Function and Law of Motion
De…ne QΦ,s ,s 0 : Z B(Z ) ! [0, 1] by
π (y 0 , s 0 jy , s ) if a0 (a, y , s, Φ) 2 A
QΦ,s ,s 0 ((a, y ), (A, Y )) = ∑ 0 else
y 0 2Y
Aggregate law of motion
Φ0 (A, Y ) = H (s, Φ, s 0 ) (A, Y )
Z
= QΦ,s ,s 0 ((a, y ), (A, Y ))Φ(da dy )
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 35 / 53
36. Models with Aggregate Uncertainty
Computation of the Recursive Equilibrium I
Key computational problem: aggregate wealth distribution Φ is an
in…nite-dimensional object.
Agents need to keep track of the aggregate wealth distribution to
forecast future capital stock and thus future prices. But for K 0 need
entire Φ since Z
K = a0 (a, y , s, Φ)d Φ
0
If a0 were linear in a, with same slope for all y 2 Y , exact aggregation
would occur and K would be a su¢ cient statistic for K 0 .
Trick: Approximate the distribution Φ with a …nite set of moments.
Let the n-dimensional vector m denote the …rst n moments of the
asset distribution
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 36 / 53
37. Models with Aggregate Uncertainty
Computation of the Recursive Equilibrium II
Agents use an approximate law of motion
m0 = Hn (s, m )
Agents are boundedly rational: moments of higher order than n of the
current wealth distribution may help to more accurately forecast
prices tomorrow.
We choose the number of moments and the functional form of the
function Hn .
Krusell and Smith pick n = 1 and pose
log(K 0 ) = as + bs log(K )
for s 2 fsb , sg g. Here (as , bs ) are parameters that need to be
determined.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 37 / 53
38. Models with Aggregate Uncertainty
Computation of the Recursive Equilibrium III
Household problem
( )
v (a, y , s, K ) = max
0 c ,a 0
U (c ) + β ∑ ∑ 0 0 0 0 0
π (y , s jy , s )v (a , y , s , K ) 0
y 0 2Y s 0 2S
s.t. c + a0 = w (s, K )y + (1 + r (s, K ))a
log(K 0 ) = as + bs log(K )
Reduction of the state space to a four dimensional space
(a, y , s, K ) 2 R Y S R.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 38 / 53
39. Models with Aggregate Uncertainty
Algorithm I
1 Guess (as , bs ).
2 Solve households problem to obtain a0 (a, y , s, K ).
3 Simulate economy for large number of T periods for large number N
of households:
Start with initial conditions for the economy (s0 , K0 ) and for each
i
household (a0 , y0 ).i
Draw random sequences fst gT=1 and fyti gT=1,i =1 and use
t t
,N
a 0 (a, y , s, K ) and perceived law of motion for K to generate sequences
of fat gT=1,i =1 .
i
t
,N
Aggregate:
1 N i
N i∑ t
Kt = a
=1
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 39 / 53
40. Models with Aggregate Uncertainty
Algorithm II
4 Run the regressions
log(K 0 ) = αs + βs log(K )
to estimate (αs , βs ) for s 2 S.
5 If the R 2 for this regression is high and (αs , βs ) (as , bs ) stop. An
approximate equilibrium was found.
6 Otherwise, update guess for (as , bs ). If guesses for (as , bs ) converge,
but R 2 remains low, add higher moments to the aggregate law of
motion and/or experiment with a di¤erent functional form for it.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 40 / 53
41. Models with Aggregate Uncertainty
Calibration I
Period 1 quarter.
CRRA utility with σ = 1 (log-utility)
β = 0.994 = 0.96.
α = 0.36.
δ = (1 0.025)4 1 = 9.6%.
Aggregate component: two states (recession, expansion)
S = f0.99, 1.01g ) σs = 0.01
Symmetric transition matrix π (sg jsg ) = π (sb jsb ).
7
Expected time in each state: 8 quarters, hence π (sg jsg ) = 8 and
7 1
π (s 0 js ) = 8
1
8
7
8 8
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 41 / 53
42. Models with Aggregate Uncertainty
Calibration II
Idiosyncratic component: two states (employment and
unemployment):
Y = f0.25, 1g
Unemployed person makes 25% of the labor income of an employed
person.
Transition probabilities:
π (y 0 , s 0 jy , s ) = π (y 0 js 0 , y , s ) π (s 0 js )
Specify four 2 2 matrices π (y 0 js 0 , y , s ).
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 42 / 53
43. Models with Aggregate Uncertainty
Calibration III
Expansion: average time of unemployment equal to 1.5 quarters
1
π (y 0 = yu js 0 = sg , y = yu , s = sg ) =
3
2
π (y 0 0
= ye js = sg , y = yu , s = sg ) =
3
Recession: average time of unemployment equal to 2.5 quarters
π (y 0 = yu js 0 = sb , y = yu , s = sb ) = 0.6
π (y 0 = ye js 0 = sb , y = yu , s = sb ) = 0.4
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 43 / 53
44. Models with Aggregate Uncertainty
Calibration IV
Switch from g to b: probability of remaining unemployed 1.25 higher
π (y 0 = yu js 0 = sb , y = yu , s = sg ) = 0.75
π (y 0 = ye js 0 = sb , y = yu , s = sg ) = 0.25
Switch from b to g : probability of remaining unemployed 0.75 higher
π (y 0 = yu js 0 = sg , y = yu , s = sb ) = 0.25
π (y 0 = ye js 0 = sg , y = yu , s = sb ) = 0.75
Idea: best times for …nding a job are when the economy moves from a
recession to an expansion, the worst chances are when the economy
moves from a boom into a recession.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 44 / 53
45. Models with Aggregate Uncertainty
Calibration V
In recessions unemployment rate is Πsb (yu ) = 10% and in expansions
it is Πsg (yu ) = 4%. Remember:
π (y 0 , s 0 jy , s )
Πs 0 (y 0 ) = ∑ π (s 0 js )
Πs (y ) for all s, s 0 2 S
y 2Y
This gives
π (y 0 = yu js 0 = sg , y = ye , s = sg ) = 0.028
π (y 0 = ye js 0 = sg , y = ye , s = sg ) = 0.972
π (y 0 = yu js 0 = sb , y = ye , s = sb ) = 0.04
π (y 0 = ye js 0 = sb , y = ye , s = sb ) = 0.96
π (y 0 = yu js 0 = sb , y = ye , s = sg ) = 0.079
π (y 0 = ye js 0 = sb , y = ye , s = sg ) = 0.921
π (y 0 = yu js 0 = sg , y = ye , s = sb ) = 0.02
π (y 0 = ye js 0 = sg , y = ye , s = sb ) = 0.98
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 45 / 53
46. Models with Aggregate Uncertainty
Calibration VI
In summary:
0 1
0.525 0.035 0.09375 0.0099
B 0.35 0.84 0.03125 0.1151 C
π=B
@ 0.03125 0.0025 0.292 0.0245 A
C
0.09375 0.1225 0.583 0.8505
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 46 / 53
47. Models with Aggregate Uncertainty
Numerical Results
Model delivers
1 Aggregate law of motion
m0 = Hn (s, m )
2 Individual decision rules
a0 (a, y , s, m )
3 Time-varying cross-sectional wealth distributions
Φ(a, y )
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 47 / 53
48. Models with Aggregate Uncertainty
Aggregate Law of Motion I
Agents are boundedly rational: aggregate law of motion perceived by
agents may not coincide with actual law of motion.
Only thing to forecast is K 0 . Hence try n = 1.
Converged law of motion:
log(K 0 ) = 0.095 + 0.962 log(K ) for s = sg
log(K 0 ) = 0.085 + 0.965 log(K ) for s = sb
How irrational are agents? Use simulated time series f(st , Kt gT=0 ,
t
divide sample into periods with st = sb and st = sg , and run
log(Kt +1 ) = αj + βj log(Kt ) + εjt +1
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 48 / 53
49. Models with Aggregate Uncertainty
Aggregate Law of Motion II
De…ne
εjt +1 = log(Kt +1 )
ˆ ˆ
αj ˆ
βj log(Kt ) for j = g , b
Then:
!0.5
1 2
σj =
Tj ∑ εjt
ˆ
t 2τ j
2
ˆj
∑t 2 τ j εt
Rj2 = 1 2
∑t 2τj (log Kt +1 ¯
log K )
If σj = 0 for j = g , b (if Rj2 = 1 for j = g , b), then agents do not
make forecasting errors
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 49 / 53
50. Models with Aggregate Uncertainty
Aggregate Law of Motion III
Results
Rj2 = 0.999998 for j = b, g
σg = 0.0028
σb = 0.0036
Maximal forecasting errors for interest rates 25 years into the future is
0.1%.
Corresponding utility losses?
Approximated equilibria may be arbitrarily far away from exact one.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 50 / 53
51. Models with Aggregate Uncertainty
Why Quasi-Aggregation?
If all agents have linear savings functions with same marginal
propensity to save
a0 (a, y , s, K ) = as + bs a + cs y
Then:
Z Z
K0 = a0 (a, y , s, K )d Φ = as + bs ad Φ + cs L
¯
= as + bs K
˜
Exact aggregation: K su¢ cient statistic for Φ for forecasting K 0 .
In this economy: savings functions almost linear with same slope for
y = yu and y = ye .
Only exceptions are unlucky agents (y = yu ) with little assets. But
these agents hold a negligible fraction of aggregate wealth and do not
matter for K dynamics.
Hence quasi-aggregation!!!
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 51 / 53
52. Models with Aggregate Uncertainty
Why is Marginal Propensity to Save Close to 1? I
PILCH model with certainty equivalence and r = 1/β
!
T t
r yt +s
ct = Et ∑ s + at
1+r s =0 (1 + r )
Agents save out of current assets for tomorrow
at + 1 r
= 1 at + G ( y )
1+r 1+r
Thus under certainty equivalence
at + 1 = at + H ( y )
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 52 / 53
53. Models with Aggregate Uncertainty
Why is Marginal Propensity to Save Close to 1? II
In this economy agents are prudent and face liquidity constraints, but
almost act as if they are certainty equivalence consumers. Why?
1 With σ = 1 agents are prudent, but not all that much.
2 Unconditional standard deviation of individual income is roughly 0.2, at
the lower end of the estimates.
3 Negative income shocks (unemployment) are infrequent and not very
persistent.
Jesús Fernández-Villaverde (PENN) Heterogeneous Agents July 11, 2011 53 / 53