A Model with Explicit Solution
Jesús Fernández-Villaverde
University of Pennsylvania
December 2, 2012
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
1 / 34
Motivation
Brunnermeier and Sannikov, 2012.
Full equilibrium dynamics of an economy with …nancial frictions:
1
Nonlinearity: model will respond very di¤erently to small and large shocks.
2
Volatility paradox: lower values of exogenous risk may lead to higher
levels of endogenous risk.
Features:
1
Continuous time.
2
We have more productive but less patient agents borrowing from less
productive but more patient agents. Financial frictions di¢ cult the ‡ow of
funds between both groups.
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
2 / 34
Preferences
Continuum of in…nitely lived, risk-neutral agents:
1
Experts, I = [0, 1]:
E0
Z
e
ρt
dct
where ct is cumulative consumption until time t.
We impose dct
0.
2
Households, J = [0, 1]:
E0
Z
e rt dc t
where c t is cumulative consumption until time t.
We do not impose dc t
0 (negative consumption can be thought as
additional labor e¤ort): hence r is risk-free rate.
Assumption : r < ρ.
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
3 / 34
Technology I
Experts with e¢ ciency units of capital kt produce output:
yt = akt
Experts can invest:
dkt = (φ (ιt )
δ) kt dt + σkt dZt
where
1
2
3
4
ιt is investment rate per unit of capital.
φ (ιt ) is an investment technology with adjustment costs (φ (0) = 0,
φ0 ( ) = 0, and φ00 ( ) < 0).
We do not impose ιt > 0. Concavity of φ ( ) imposes large costs to
disinvestment.
dZt is a Brownian motion.
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
4 / 34
Technology II
Households with e¢ ciency units of capital k t produce output:
y t = ak t
where a < a.
Households can invest:
dk t = (φ (ιt )
δ) k t dt + σk t dZt
where δ > δ.
We take output as numeraire.
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
5 / 34
Price of Capital
Capital can be traded at price qt , which evolves as:
dqt = µqt qt dt + σqt qt dZt
Boundaries for price of capital:
q = max
ι
q = max
ι
Jesús Fernández-Villaverde (PENN)
r
r
a ι
φ (ι) + δ
a ι
φ (ι) + δ
A Model with Explicit Solution
December 2, 2012
6 / 34
Returns I
Thus, the value of the capital hold by an expert generates:
1
Capital gains:
d (kt qt )
= φ ( ιt )
kt qt
2
δ + µqt + σσqt dt + σ + σqt dZt
Dividend
a
ιt
qt
3
Total return:
drtk =
a
ιt
qt
δ + µqt + σσqt dt + σ + σqt dZt
+ φ ( ιt )
Similarly, return for a household:
dr kt =
a
ιt
qt
Jesús Fernández-Villaverde (PENN)
+ ( φ ( ιt )
δ + µqt + σσqt ) dt + (σ + σqt ) dZt
A Model with Explicit Solution
December 2, 2012
7 / 34
Returns II
Two components of risk on returns:
1
2
σdZt is exogenous risk caused by stochastic process for capital e¢ ciency.
σqt dZt is endogenous risk caused by …nancial frictions. Without them,
σqt = 0 because qt = q.
Even if experts are risk-neutral with respect to consumption, they exhibit
risk-averse behavior because the return to investment is time-varying.
Experts su¤er losses when they want to buy more capital: its price is
lower.
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
8 / 34
Budget Constraint
Because of an agency problem, experts must retain 100 percent of
equity and …nance the rest of their investment with risk-free debt.
Experts:
dnt
= xt drtk + (1
nt
xt ) rdt
dct
nt
where nt
0 is net wealth, a fraction xt
0 invested in capital and a
fraction 1 xt in the risk-free asset. In general, xt > 1. The solvency
constrain: nt
0.
Households:
with x t
dnt
= x t dr kt + (1
nt
x t ) rdt
dc t
nt
0.
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
9 / 34
Equilibrium I
De…nition
For any initial endowments of capital
Z
I
k0i di +
n
Z
J
k0i , k j0 ; i 2 I, j 2 J
o
such that:
k i0 dj = K0
an equilibrium is described by a group of stochastic processes on the …ltered
probability space de…ned by the Brownian
motion
n
o fZt , t 0g: the price
j
i
process for capital fqt g, net worths nt , nt
0 , capital holdings
n
o
n
o
kti , k jt
0 , investment decisions ιit , ιjt , and consumption choices
n
o
0, dc jt of individual agents i 2 I, j 2 J such that:
dcti
1
2
Given prices, all experts and households maximize.
Initial net worths satisfy n0i = k0i q0 and ni0 = k i0 q0 for all i 2 I, j 2 J.
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
10 / 34
Equilibrium II
De…nition
3. Markets clear:
Z
I
dcti di +
dKt =
Z
I
Z
Z
J
dc it dj =
φ ιit
Jesús Fernández-Villaverde (PENN)
I
kti di +
Z
I
δ kti di +
Z
J
a
Z
J
k it dj = Kt
ιit kti di +
φ ιit
A Model with Explicit Solution
Z
J
δ k jt dj
a
ιjt k jt dj
dt
dt + σKt dZt
December 2, 2012
11 / 34
Excess Returns I
First, to maximize experts return with respect to ιt , from
drtk =
a
ιt
qt
+ ( φ ( ιt )
set:
φ 0 ( ιt ) =
δ + µqt + σσqt ) dt + (σ + σqt ) dZt
1
) ι t = ι ( qt )
qt
Similarly, for a household:
φ 0 ( ιt ) =
1
) ι t = ι ( qt )
qt
Thus:
ι t = ι t = ι ( qt )
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
12 / 34
Excess Returns II
Now, de…ne expected excess returns:
Et drtk
dt
Et dr kt
dt
r
=
r
=
Jesús Fernández-Villaverde (PENN)
a
a
ι ( qt )
+ φ (ι (qt ))
qt
ι (qt )
+ φ (ι (qt ))
qt
A Model with Explicit Solution
δ + µqt + σσqt
r
δ + µqt + σσqt
r
December 2, 2012
13 / 34
Optimal Strategies I
Consider a feasible strategy for the experts fxt , d ζ t g with payo¤:
θ t nt = Et
Z ∞
e
ρ (s t )
t
dcs
where dct = nt d ζ t
Then, for a price process for capital fqt g that generates a …nite
maximal payo¤, a feasible strategy is optimal if and only if:
max
s.t.
Jesús Fernández-Villaverde (PENN)
ζt 0
xbt 0,d b
nt d b
ζ t + Et (θ t nt )
dnt
=b
xt drtk + (1 b
xt ) rdt
nt
Et e ρt θ t nt ! 0
A Model with Explicit Solution
db
ζt
December 2, 2012
14 / 34
Optimal Strategies II
Moreover, for
d θt
= µtθ dt + σtθ dZt
θt
a feasible strategy is optimal if and only if:
1
2
3
θt
1 and d ζ t > 0 only when θ t = 1.
µtθ = ρ r .
Either:
xt > 0 and
or
xt = 0 and
4
Ee
ρt θ
t nt
Et drtk
dt
Et drtk
dt
r=
σ + σqt σtθ (desire to leverage)
r
σ + σqt σtθ (‡ight to quality)
! 0.
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
15 / 34
Optimal Strategies III
For the households,
Et dr kt
dt
with equality if
1
1
ψt =
Kt
r
Z
J
0
k it dj > 0
It can be veri…ed that, in equilibrium,
ψt qt Kt > Nt =
Jesús Fernández-Villaverde (PENN)
Z
I
A Model with Explicit Solution
nti di
December 2, 2012
16 / 34
Wealth Distribution I
Aggregate wealth:
Nt =
qt Kt
Z
I
nti di
Nt =
Z
J
nit dj
Hence, experts’s wealth share:
ηt =
Jesús Fernández-Villaverde (PENN)
Nt
2 [0, 1]
qt Kt
A Model with Explicit Solution
December 2, 2012
17 / 34
Wealth Distribution II
Law of motion:
d ηt
ηt
ψt
=
+
ηt
2
dr k rdt (σ + σqt ) dt
ηt
a ι ( qt )
+ (1 ψt ) (δ δ) dt d ζ t
qt
If ψt > 0 ) xt = 0 and we have
r=
Et drtk
+ (σ + σqt ) σtθ ) rdt = Et drtk + (σ + σqt ) σtθ dt
dt
Then:
dr k
rdt = dr k
Et drtk
= (σ + σqt ) dZt
= (σ + σqt ) dZt
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
(σ + σqt ) σtθ dt
(σ + σqt ) σtθ dt
σtθ dt
December 2, 2012
18 / 34
Wealth Distribution III
We can substitute
d ηt
ηt
ψt
ηt
(σ + σqt ) dZt σtθ dt (σ + σqt ) dt
ηt
a ι ( qt )
+
+ (1 ψt ) (δ δ) dt d ζ t
qt
!
ψt η t
q
q
θ
η t ( σ + σt ) σ + σt + σt
dt
=
+ a qι(tqt ) + (1 ψt ) (δ δ)
ψ
ηt
+ t
(σ + σqt ) dZt d ζ t
ηt
=
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
19 / 34
Wealth Distribution IV
By de…ning
µnt =
ψt
ηt
ηt
(σ + σqt ) σ + σqt + σtθ +
η
σt =
ψt
ηt
ηt
ι ( qt )
+ (1
qt
a
ψt ) ( δ
δ)
(σ + σqt )
we get
d ηt
η
= µnt dt + σt dZt
ηt
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
d ζt
December 2, 2012
20 / 34
Markov Equilibria
Search for functions:
qt
θt
ψt
= q (η t )
= θ (η t )
= ψ (η t )
Then, once we know η t , we can get qt , θ t , ψt , and from
d ηt
η
= µnt dt + σt dZt
ηt
d ζt
we get d η t .
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
21 / 34
A Proposition I
Let us suppose that we know η, q (η ) , q 0 (η ) , θ (η ) , θ 0 (η ) .
1
Find ψ 2 η, η +
q 0 (η )
q (η )
such that:
a a
+δ
q (η )
δ + σtθ (σ + σqt ) = 0
where:
η
Jesús Fernández-Villaverde (PENN)
σt
=
σqt
=
σtθ
=
1
η1
(ψ
η) a
(ψ
η)
q 0 (η )
q (η )
q 0 (η ) η
σ η
q (η ) t
θ 0 (η ) η
σ η
θ (η ) t
A Model with Explicit Solution
December 2, 2012
22 / 34
A Proposition II
2
If ψ > 1, set ψ = 1 and recalculate σt , σqt , σtθ .
3
Compute:
η
µnt =
µqt = r
η
σt
a
σ + σqt + σtθ +
ι (q (η ))
q (η )
a
ι (q (η ))
+ (1
q (η )
φ (q (η )) + δ
µtθ = ρ
q 00 (η ) =
2 (µqt q
σtθ (σ + σqt )
q 0 (η ) µnt η )
(η )
η 2
η2
θ 0 (η ) µnt η
2 µtθ θ (η )
η 2
σt
Jesús Fernández-Villaverde (PENN)
δ)
r
σt
θ 00 (η ) =
σσqt
ψ) (δ
A Model with Explicit Solution
η2
December 2, 2012
23 / 34
A Proposition III
4
Use boundary conditions
q 0 (η ) = 0, q (0) = q
θ (η ) = 1, θ 0 (η ) = 0
lim θ (η ) = ∞
where η is the re‡ecting boundary when experts consume.
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
24 / 34
An Algorithm
Set q (0) = q, θ (0) = 1, and θ 0 (0) =small number.
Set qL = 0 and qH =large number.
qL
Guess q 0 (0) = qH +
and solve for q (η ) and θ (η ) until the …rst of the
2
three conditions holds:
1
2
3
q (η ) = q.
θ 0 (η ) = 0.
q 0 (η ) = 0.
If q 0 (η ) = 0, set qL = q 0 (0), otherwise qH = q 0 (0).
Iterate until convergence.
Check that q 0 (η ) and θ 0 (η ) reach 0 at the same point η .
Normalize θ (η ) =
Jesús Fernández-Villaverde (PENN)
θ (η )
θ (η )
to match boundary condition.
A Model with Explicit Solution
December 2, 2012
25 / 34
Calibration
Parameters
ρ
0.06
r
0.05
a
0.1
a
0.05
δ
0.03
δ
0.05
σ
0.4
pφ (ι)
0.1 1 + 20ι
This implies q = 0.5858 and q = 1.3101.
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
26 / 34
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
27 / 34
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
28 / 34
Three Ine¢ ciencies
1
Capital misallocation: for low η t , households manage part of the capital
(ψ < 1).
2
Under investment: ι (qt ) < ι (q ) .
3
Consumption distortions: experts should only consume at time 0.
Note: ine¢ ciencies get worse for low η t .
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
29 / 34
Adverse Feedback Loop
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
30 / 34
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
31 / 34
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
32 / 34
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
33 / 34
Jesús Fernández-Villaverde (PENN)
A Model with Explicit Solution
December 2, 2012
34 / 34