A Model with Costly-State Veri…cation
Jesús Fernández-Villaverde
University of Pennsylvania
December 19, 2012
Jesús Fernández-Villaverde (PENN)
Costly-State
December 19, 2012
1 / 47
A Model with Costly-State Veri…cation
Tradition of …nancial accelerator of Bernanke, Gertler, and Gilchrist
(1999), Carlstrom and Fuerst (1997), and Christiano, Motto, and
Rostagno (2009).
Elements:
1
Information asymmetries between lenders and borrowers)costly state
veri…cation (Townsend, 1979).
2
Debt contracting in nominal terms: Fisher e¤ect.
3
Changing spreads.
We will calibrate the model to reproduce some basic observations of the
U.S. economy.
Jesús Fernández-Villaverde (PENN)
Costly-State
December 19, 2012
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Flowchart of the Model
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Households
Representative household:
E0
∞
∑ βt e d
mt
pt
u (ct , lt ) + υ log
t
t =0
dt is an intertemporal preference shock with law of motion:
dt = ρ d dt
1
+ σd εd ,t , εd ,t
N (0, 1).
Why representative household? Heterogeneity?
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Asset Structure
The household saves on three assets:
1
Money balances, mt .
2
Deposits at the …nancial intermediary, at , that pay an uncontingent
nominal gross interest rate Rt .
3
Arrow securities (net zero supply in equilibrium).
Therefore, the household’s budget constraint is:
ct +
at
mt +1
+
= wt lt + Rt
pt
pt
1
at 1
mt
+
+ Tt + zt + tret
pt
pt
where:
tret = (1
Jesús Fernández-Villaverde (PENN)
γ e ) nt
Costly-State
we
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Optimality Conditions
The …rst-order conditions for the household are:
e d t u1 ( t ) = λ t
Rt
Π t +1
u2 (t ) = u1 (t ) wt
λt = βEt
λ t +1
Asset pricing kernel:
SDFt = Et β
λ t +1
λt
and standard non-arbitrage conditions.
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The Final Good Producer
Competitive …nal producer with technology
yt =
Z 1
0
ε 1
ε
ε
ε 1
yit di
.
Thus, the input demand functions are:
yit =
pit
pt
pt =
Z 1
Price level:
Jesús Fernández-Villaverde (PENN)
0
ε
yt
pit1 ε di
Costly-State
8i,
1
1 ε
.
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Intermediate Goods Producers
Continuum of intermediate goods producers with market power.
Technology:
yit = e zt kitα
1 α
1 lit
where
zt = ρz zt
1
+ σz εz ,t , εz ,t
N (0, 1)
Cost minimization implies:
mct =
Jesús Fernández-Villaverde (PENN)
1 α
1 α wt1 α rtα
1 α
α
e zt
kt 1
α wt
=
lt
1 α rt
1
Costly-State
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Sticky Prices
Calvo pricing: in each period, a fraction 1 θ of …rms can change their
prices while all other …rms keep the previous price.
Then, the relative reset price Πt = pt /pt satis…es:
εgt1 = (ε
1)gt2
gt1 = λt mct yt + βθEt Πtε +1 gt1+1
gt2 = λt Πt yt + βθEt Πtε +11
Πt
Π t +1
gt2+1
Given Calvo pricing, the price index evolves as:
1 = θΠtε
Jesús Fernández-Villaverde (PENN)
1
+ (1
Costly-State
θ ) Πt 1
ε
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Capital Good Producers I
Capital is produced by a perfectly competitive capital good producer.
Why?
It buys installed capital, xt , and adds new investment, it , to generate
new installed capital for the next period:
xt +1 = xt + 1
S
it
it
it
1
where S [1] = 0, S 0 [1] = 0, and S 00 [ ] > 0.
Alternative:
1
Adjustment cost in capital.
2
Time to build.
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Costly-State
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Capital Good Producers II
Technology illiquidity.
Importance of irreversibilities?
The period pro…ts of the …rm are:
qt
xt + 1
S
it
it
it
qt xt
1
i t = qt
1
S
it
it
it
it
1
where qt is the relative price of capital.
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Costly-State
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Capital Good Producers III
Discounted pro…ts:
E0
∞
λt
∑ βt λ0
qt
1
S
t =0
it
it 1
it
it
Since this objective function does not depend on xt , we can make it
equal to (1 δ) kt 1 .
First-order condition of this problem is:
qt
1
S
it
it
S
1
0
it
it
it
1
it
λ t +1
it +1
qt + 1 S 0
+ βEt
λt
it
1
it +1
it
2
=1
and the law of motion for capital is:
kt = (1
Jesús Fernández-Villaverde (PENN)
δ) kt
1
+ 1
Costly-State
S
it
it
it
1
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Entrepreneurs I
Entrepreneurs use their (end-of-period) real wealth, nt , and a nominal
bank loan bt , to purchase new installed capital kt :
qt kt = nt +
bt
pt
The purchased capital is shifted by a productivity shock ω t +1 :
1
2
3
Lognormally distributed with CDF F (ω ) and
Parameters µω,t and σω,t
Et ω t +1 = 1 for all t.
Therefore:
1
2
Et ω t +1 = e µω,t +1 + 2 σω,t +1 = 1 ) µω,t +1 =
1 2
σ
2 ω,t +1
This productivity shock is a stand-in for more complicated processes
such as changes in demand or the stochastic quality of projects.
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Entrepreneurs II
The standard deviation of this productivity shock evolves:
log σω,t = (1
ρσ ) log σω + ρσ log σω,t
1
+ η σ εσ,t , εσ,t
N (0, 1).
The shock t + 1 is revealed at the end of period t right before
investment decisions are made. Then:
log σω,t
log σω = ρσ (log σω,t 1 log σω ) + η σ εσ,t
bω,t = ρσ σ
bω,t 1 + η σ εσ,t
)σ
More general point: stochastic volatility.
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Entrepreneurs III
The entrepreneur rents the capital to intermediate goods producers,
who pay a rental price rt +1 .
Also, at the end of the period, the entrepreneur sells the undepreciated
capital to the capital goods producer at price qt +1 .
Therefore, the average return of the entrepreneur per nominal unit
invested in period t is:
Rtk+1 =
Jesús Fernández-Villaverde (PENN)
pt + 1 rt + 1 + qt + 1 ( 1
pt
qt
Costly-State
δ)
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Debt Contract
Costly state veri…cation framework.
For every state with associated Rtk+1 , entrepreneurs have to either:
1
Pay a state-contingent gross nominal interest rate Rtl +1 on the loan.
2
Or default.
If the entrepreneur defaults, it gets nothing: the bank seizes its revenue,
although a portion µ of that revenue is lost in bankruptcy.
Hence, the entrepreneur will always pay if it ω t +1
Rtl +1 bt
=
ω t +1 where:
ω t +1 Rtk+1 pt qt kt
If ω t +1 < ω t +1 , the entrepreneur defaults, the bank monitors the
entrepreneur and gets (1 µ) of the entrepreneur’s revenue.
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Costly-State
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Zero Pro…t Condition
The debt contract determines Rtl +1 to be the return such that banks
satisfy its zero pro…t condition in all states of the world:
+(1
|
µ)
[1
|
Z ω t +1
0
F (ω t +1 , σω,t +1 )] Rtl +1 bt
{z
}
Revenue if loan pays
ωdF (ω, σω,t +1 ) Rtk+1 pt qt kt = st Rt bt
| {z }
{z
} Cost of funds
Revenue if loan defaults
st = 1 + e s +est is a spread caused by the cost of intermediation such
that:
e
st = ρs e
st 1 + σs εs ,t , εs ,t
N (0, 1).
For simplicity, intermediation costs are rebated to the households in a
lump-sum fashion.
External …nance premium.
Jesús Fernández-Villaverde (PENN)
Costly-State
December 19, 2012
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Optimality of the Contract
This debt contract is not necessarily optimal.
However, it is a plausible representation for a number of nominal debt
contracts that we observe in the data.
Also, the nominal structure of the contract creates a Fisher e¤ect
through which changes in the price level have an impact on real
investment decisions.
Importance of working out the optimal contract.
Jesús Fernández-Villaverde (PENN)
Costly-State
December 19, 2012
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Characterizing the Contract I
De…ne share of entrepreneurial earnings accrued to the bank:
Γ (ω t +1 , σω,t +1 ) = ω t +1 (1
F (ω t +1 , σω,t +1 )) + G (ω t +1 , σω,t +1 )
where:
G (ω t +1 , σω,t +1 ) =
Z ω t +1
0
ωdF (ω, σω,t +1 )
Thus, we can rewrite the zero pro…t condition of the bank as:
Rtk+1
[Γ (ω t +1 , σω,t +1 )
st Rt
µG (ω t +1 , σω,t +1 )] qt kt =
bt
pt
which gives a schedule relating Rtk+1 and ω t +1 .
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Costly-State
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Characterizing the Contract II
Now, de…ne the ratio of loan over wealth:
$t =
bt /pt
qt kt nt
qt kt
=
=
nt
nt
nt
1
and we get
Rtk+1
[Γ (ω t +1 , σω,t +1 )
st Rt
µG (ω t +1 , σω,t +1 )] (1 + $t ) = $t
that is, all the entrepreneurs, regardless of their level of wealth, will have
the same leverage, $t .
A most convenient feature for aggregation.
Balance sheet e¤ects.
Jesús Fernández-Villaverde (PENN)
Costly-State
December 19, 2012
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Problem of the Entrepreneur
Maximize its expected net worth given the zero-pro…t condition of
bank:
8
R tk+1
<
Γ (ω t +1 , σω,t +1 )) +
R t (1
h k
i
max Et
R
$t
t +1
: η
$t ,ω t +1
ω
,
σ
µG
ω
,
σ
Γ
)
(
)]
[
(
t
+
1
ω,t
+
1
t
+
1
ω,t
+
1
t st R t
1 +$
t
After a fair amount of algebra:
Et
Rtk+1
(1
Rt
Γ (ω t +1 , σω,t +1 )) = Et η t
the
9
=
;
nt
qt kt
where the Lagrangian multiplier is:
ηt =
st Γω (ω t +1 , σω,t +1 )
Γω (ω t +1 , σω,t +1 ) µGω (ω t +1 , σω,t +1 )
This expression shows how changes in net wealth have an e¤ect on the
level of investment and output in the economy.
Jesús Fernández-Villaverde (PENN)
Costly-State
December 19, 2012
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Death and Resurrection
At the end of each period, a fraction γe of entrepreneurs survive to the
next period and the rest die and their capital is fully taxed.
They are replaced by a new cohort of entrepreneurs that enter with
initial real net wealth w e (a transfer that also goes to surviving
entrepreneurs).
Therefore, the average net wealth nt is:
nt = γ e
1
(1
Πt
µG (ω t , σω,t )) Rtk qt
1 kt 1
st
1 Rt 1
bt
pt
1
+ we
1
The death process ensures that entrepreneurs do not accumulate enough
wealth so as to make the …nancing problem irrelevant.
Jesús Fernández-Villaverde (PENN)
Costly-State
December 19, 2012
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The Financial Intermediary
A representative competitive …nancial intermediary.
We can think of it as a bank but it may include other …nancial …rms.
Intermediates between households and entrepreneurs.
The bank:
1
Lends to entrepreneurs a nominal amount bt at rate Rtl +1 ,
2
But recovers only an (uncontingent) rate Rt because of default and the
(stochastic) intermediation costs.
3
Thus, the bank pays interest Rt to households.
Jesús Fernández-Villaverde (PENN)
Costly-State
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The Monetary Authority Problem
Conventional Taylor rule:
Rt
=
R
Rt 1
R
γR
Πt
Π
γ Π (1 γR )
yt
y
γy (1 γR )
exp (σm mt )
through open market operations that are …nanced through lump-sum
transfers Tt .
The variable Π represents the target level of in‡ation (equal to in‡ation
in the steady-state), y is the steady state level of output, and R = Πβ
the steady state nominal gross return of capital.
The term εmt is a random shock to monetary policy distributed
according to N (0, 1).
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Costly-State
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Aggregation
Using conventional arguments, we …nd expressions for aggregate
demand and supply:
yt = ct + it + µG (ω t , σω,t ) (rt + qt (1 δ)) kt 1
1
yt = e zt ktα 1 lt1 α
vt
ε
R1
where vt = 0 ppitt
di is the ine¢ ciency created by price dispersion.
By the properties of Calvo pricing, vt evolves as:
vt = θΠtε vt
1
+ (1
θ ) Πt
ε
.
We have steady state in‡ation Π. Hence, vbt 6= 0 and monetary policy
has an impact on the level and evolution of measured productivity.
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Equilibrium Conditions I
The …rst-order conditions of the household:
e d t u1 ( t ) = λ t
Rt
g
λt = βEt fλt +1
Π t +1
u2 (t ) = u1 (t ) wt
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Equilibrium Conditions II
The …rst-order conditions of the intermediate …rms:
εgt1 = (ε
1)gt2
gt1 = λt mct yt + βθEt Πtε +1 gt1+1
gt2 = λt Πt yt + βθEt Πtε +11
kt
mct =
Jesús Fernández-Villaverde (PENN)
1
=
1
1 α
1
1
α
α
Costly-State
Πt
Π t +1
gt2+1
wt
lt
α rt
1 α wt1 α rtα
α
e zt
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Equilibrium Conditions III
Price index evolves:
1
1 = θΠtε
+ (1
θ ) Πt 1
ε
Capital good producers:
qt
1
+ βEt
it
it
S0
1
λ t +1
it + 1
qt + 1 S 0
λt
it
kt = (1
Jesús Fernández-Villaverde (PENN)
S
δ) kt
1
+ 1
Costly-State
it
it
it
1
it
1
2
it + 1
=1
it
it
S
it
it 1
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Equilibrium Conditions IV
Entrepreneur problem:
Rtk+1 = Πt +1
rt +1 + qt +1 (1
qt
δ)
Rtk+1
qt kt nt
[Γ (ω t +1 , σω,t +1 ) µG (ω t +1 , σω,t +1 )] =
st Rt
qt kt
k
R
Et t +1 (1 Γ (ω t +1 , σω,t +1 )) =
Rt
nt
1 F (ω t +1 , σω,t +1 )
Et s t
1 F (ω t +1 , σω,t +1 ) µω t +1 Fω (ω t +1 , σω,t +1 ) qt kt
Rtl +1 bt = ω t +1 Rtk+1 pt qt kt
bt
qt kt = nt +
pt
nt = γ e
1
(1
Πt
Jesús Fernández-Villaverde (PENN)
µG (ω t , σω,t )) Rtk qt
Costly-State
1 kt 1
st
1 Rt 1
bt
pt
1
+ we
1
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Equilibrium Conditions V
The government follows its Taylor rule:
Rt
=
R
Rt 1
R
γR
Πt
Π
γ Π (1 γR )
yt
y
γy (1 γR )
exp (σm mt )
Market clearing
yt = ct + it + µG (ω t , σω,t ) (rt + qt (1 δ)) kt
1
yt = e zt ktα 1 lt1 α
vt
ε
vt = θΠt vt 1 + (1 θ ) Πt ε
Jesús Fernández-Villaverde (PENN)
Costly-State
1
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Equilibrium Conditions VI
Stochastic processes:
dt = ρ d dt
+ σd εd ,t
zt = ρz zt 1 + σz εz ,t
1
st = 1 + e s +est
log σω,t = (1
Jesús Fernández-Villaverde (PENN)
e
st = ρ s e
st
+ σs εs ,t
ρσ ) log σω + ρσ log σω,t
1
Costly-State
1
+ η σ εσ,t
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Calibration
Utility function:
lt1 +ϑ
1+ϑ
ψ: households work one-third of their available time in the steady state
and ϑ = 0.5, inverse of Frisch elasticity.
Technology:
u (ct , lt ) = log ct
α
0.33
δ
0.023
ψ
S 00 [1]
14.477
ε
8.577
Entrepreneur:
µ
0.15
σω
2.528
we
n
n k
2
s
25bp.
For the Taylor rule, Π = 1.005, γR = 0.95, γΠ = 1.5, and γy = 0.1 are
conventional values.
For the stochastic processes, all the autoregressive are 0.95.
Jesús Fernández-Villaverde (PENN)
Costly-State
December 19, 2012
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Computation
We can …nd the deterministic steady state.
We linearize around this steady state.
We solve using standard procedures.
Alternatives:
1
Non-linear solutions.
2
Estimation using likelihood methods.
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How Can We Use the Model?
Christiano, Motto, and Rostagno (2003): Great depression.
Christiano, Motto, and Rostagno (2008): Business cycle ‡uctuations.
Fernández-Villaverde and Ohanian (2009): Spanish crisis of 2008-2010.
Fernández-Villaverde (2010): …scal policy.
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Costly-State
December 19, 2012
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Figure: IRFs of Output to Di¤erent Fiscal Policy Shocks
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