Download Real Business Cycle Theory

Real Business Cycle Theory
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Introduction to DSGE models
Dynamic Stochastic General Equilibrium (DSGE) models have
become the main tool for the analysis of macroeconomic ‡uctuations
and monetary policy.
The relationships between macroeconomic variables are derived by
microfounding the behavior of economic agents.
Rational expectations and microfoundations make such a kind of
models robust to the Lucas critique.
Exogenous shocks a¤ect the economy and are responsible of cyclical
‡uctuations.
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Real Business Cycle (RBC) Theory
A …rst approach to DSGE models was designed by Kydland and
Prescott (1982), who introduced the RBC theory.
RBC models are characterized by the following ingredients:
There is a representative agent who has rational expectations.
The evolution of the aggregate variables is described by …rst-order
conditions of intertemporal problems solved by …rms and households.
Firms have no market power, i.e., perfect competition and frictionless
markets.
Prices are fully ‡exible.
Model evaluation relies on parameters calibration and simulation.
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RBC basic concepts
Cyclical ‡uctuations are e¢ cient: business cycle is the results of the
optimal response of the economy to exogenous shocks. Technology
shock is the main driving force of the business cycle and a¤ects real
variables. As a consequence, stabilization policies are not required
and might be also counterproductive.
TFP is not only the main source of long-term growth, but it now
involves economic ‡uctuations.
Monetary neutrality: monetary policy is ine¤ective also in the
short-run, i.e., no real e¤ects. It only a¤ects nominal variables (e.g.,
price level).
A question: are RBC assumptions and outcomes supported by the
empirical evidence?
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Some empirical evidence
Nominal rigidities: empirical studies based on micro data …nd that
prices realignments are infrequent.
Monetary policy non-neutrality: evidence from SVAR suggests that
monetary policy a¤ects real variables. Liquidity e¤ect is observed.
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A Classical Monetary Model
Basic (strong) hypotheses: prices are fully ‡exible and there is perfect
competition in all markets.
Money serves as unit of account.
The economy is composed by households and …rms.
Households solve an intertemporal optimization problem by
maximizing their utility.
Firms maximize pro…ts taking account of the constraint given by the
production function.
No capital accumulation, no …scal sector, closed economy.
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Households
The representative households maximizes a utility function choosing
how much to consume, the quantity of labor to supply and assets
purchasing:
∞
E0 ∑ βt U (Ct , Nt )
t =0
Assumptions on utility function:
∂2 U (C ,N )
∂U (C ,N )
t
t
t
t
Uc ,t
> 0, Ucc ,t
∂C t
∂C t2
consumption is positive and non-increasing.
∂U (C t ,N t )
∂2 U (C t ,N t )
Un,t
0, Unn,t
∂N t
∂N t2
labor is negative and decreasing.
0, marginal utility of
0, marginal utility of
Budget constraint (in nominal terms):
Pt Ct + Qt Bt
Bt
1
+ W t Nt
Tt
No-Ponzi game condition is assumed in order to prevent households
from excessive borrowing.
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Households’problem solution
The following form is assumed for the utility function:
!
1 +γ
Ct1 σ
Nt
U=
1 σ 1+γ
Constrained optimization problem:
max U (Ct , Nt ) + λt (Bt
C t ,N t ,B t
1
+ W t Nt
Tt
Qt Bt )
Pt Ct
First order conditions:
Wt
Pt
Qt
P t +1
Pt
=
Un,t
Uc ,t
λ t +1
= βEt
λt
=)
=)
Wt
= Ctσ Ntγ
Pt
"
Qt = βEt
Ct +1
Ct
σ
Pt
Pt +1
#
= Πt +1 is the gross in‡ation rate
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Firms
The representative …rm has the following production function:
Yt = At Nt1
α
The …rm seeks to maximize his pro…t under the constraint given by
the production function, taking prices and wages as given
max Pt Yt
Y t ,N t
Wt Nt + ψt (At Nt1
α
Yt )
Optimality conditions (real wage equal to marginal product of labor)
Pt
= ψt
Wt
= (1
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α) ψt At Nt
α
=)
Wt
= (1
|
Pt
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α) At Nt α
{z
}
MPL t
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The log-linear economy
Euler equation
c t = Et c t + 1
where it =
qt and ρ =
1
(it
σ
Et π t + 1
ρ)
log β.
Labor supply
wt
pt = σct + γnt
Labor demand
wt
TIP: why it =
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pt = log (1
α ) + at
αnt
qt ?
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Asset prices and interest rate
Let assume that a riskless asset pays a gross interest equal to It .
Therefore, investing 100$ the asset will pay you 100xIt $ after one
year.
Let now assume that a riskless asset pays you 1$ after one year. How
much should it cost?
The present value of 1$ today is
Thus, the asset price Qt =
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1
It ;
1
It .
in log terms qt =
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it .
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Equilibrium
Market clearing condition
yt = ct
Labor market clearing
σyt + γnt = log (1
α ) + at
αnt
Asset market clearing
yt = Et yt +1
1
(it
σ
Et π t + 1
ρ)
Aggregate production function
yt = at + (1
α) nt
Technology shock follows an AR(1) process
at = $ a at
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1
+ εat
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Model solution (1)
We insert the production function into the labor market clearing,
obtaining
σ [ at + ( 1
σ (1
α) nt ] + γnt
α) nt + γnt + αnt
nt
where Ψna =
(1 σ )
,
σ(1 α)+γ+α
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ϑn =
= log (1 α) + at
+
= log (1 α) + (1
+
= Ψna at + ϑn
αnt
σ ) at
log (1 α)
σ(1 α)+γ+α
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Model solution (2)
The production function becomes
yt
yt
yt
where Ψya =
= at + (1 α) (Ψna at + ϑn )
+
(1 α ) (1 σ ) + σ (1 α ) + γ + α
=
at + ( 1
σ (1 α ) + γ + α
+
= Ψya at + ϑy
1 +γ
σ(1 α)+γ+α
and ϑy = (1
α ) ϑn .
From the Euler equation the real interest rate, rt = it
becomes
rt
rt
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α ) ϑn
Et π t + 1 ,
= σEt f∆yt +1 g + ρ
+
= σΨya Et f∆at +1 g + ρ
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Model solution (3)
The equilibrium real wage, ω t = wt
ωt
ωt
ωt
where Ψwa =
= log (1 α) + at
+
= log (1 α) + at
+
= Ψwa at + ϑw
σ+γ
σ(1 α)+γ+α
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and ϑw =
pt , is given by:
αnt
α (Ψna at + ϑn )
[σ(1 α)+γ] log (1 α)
.
σ (1 α)+γ+α
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Model dynamics
The equilibrium dynamics of the real variables (nt , yt , rt , ω t ) is
a¤ected ONLY by technology, which is the real driving force of the
system. Thus, nt , yt , rt , ω t exhibit ‡uctuations that are optimal
response to innovations in at .
Policy neutrality: real variables are determined independently of
monetary policy.
Both output and real wage always increase following a positive
technology shock.
The real interest rate increases (decreases) if the change in
technology is permanent (transitory).
E¤ect on employment is ambigous and depends on σ:
for σ < 1 substitution e¤ect on labor supply prevails over the negative
wealth e¤ect generated by a smaller marginal utility of consumption
(nt "). For σ > 1 the opposite happens (nt #);
for σ = 1 substitution and wealth e¤ect exactly cancel out and
employment remain unchanged.
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Nominal variables determination
In order to determine the nominal variables we need to specify how
the monetary policy is conducted.
We consider three possible solutions:
an exogenous path for the nominal interest rate;
a simple in‡ation-based interest rate rule
an exogenous path for the money supply
In all cases we make use of the Fisher equation:
rt = it
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Et π t + 1
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1) An exogenous path for the nominal interest rate
Let suppose that nominal interest rate follows an exogenous
stationary process it , where it has a mean value equal to ρ, consistent
with zero steady state in‡ation. Moreover, rt is determined
independently of the monetary policy and Et π t +1 = it rt .
Expected in‡ation is determined by the Fisher equation, while current
in‡ation is not. Thus, any path for the price level that satis…es the
following relation is consistent with equilibrium:
pt +1 = pt + ii
rt + ξ t +1
where ξ t +1 is a sunspot shock, i.e., a shock unrelated to economic
fundamentals.
An equilibrium where nonfundamentals factors can generate
‡uctuations in economic variables is de…ned as indeterminate
equilibrium. This framework leads to price level indeterminacy.
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Source of indeterminacy
Let assume that money supply evolves as
mt = pt + yt
ηit
As a consequence, money supply inherits the indeterminacy of pt .
In other words, the central bank …xes the interest rate and let money
be determined endogenously. But since we have undetermined prices,
money is undetermined as well.
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2) A simple in‡ation-based interest rate rule
The monetary authority sets nominal interest rate according to
it = ρ + φπ π t
where φπ 0 measures the response of the nominal interest rate to
in‡ation.
This interest rate rule determines the nominal variables and tells that
the Central Bank responds to in‡ation pressure by raising the nominal
interest rate.
Plugging the interest rate rule into the Fisher equation, we obtain
rt
ρ = b
rt = φπ π t
φπ π t
Et π t + 1
+
= Et π t +1 + brt
This is a stochastic di¤erence equation: We have two cases:
φπ > 1 (Taylor principle)
φπ < 1
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Forward solution (1)
Actual in‡ation is determined by π t =
E t π t +1
φπ
At time t + 1
π t +1 =
+
πt
=
+
πt
=
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b
rt
φπ
Et + 1 π t + 2
Et b
rt +1
+
φπ
φπ
1
Et
φπ
At time t + 2
π t +2 =
+
Et + 1 π t + 2
Et b
rt +1
+
φπ
φπ
+
b
rt
φπ
Et + 2 π t + 3
Et + 1 b
rt +2
+
φπ
φπ
1
1
Et
Et + 1
φπ
φπ
Et + 2 π t + 3
Et + 1 b
rt +2
+
φπ
φπ
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+
Et b
rt +1
b
rt
+
φπ
φπ
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Forward solution (2)
By the Law of Iterated Expectations (LIE) Et [Et +1 (x )] = Et (x )
h
i
π t = φ1 Et φ1 Et +1 Et +φ2 πt +3 + Et +φ1 brt +2 + Etφbrt +1 + φbrt can be
π
π
π
π
π
π
rewritten as
πt =
Et π t + 3
Et b
rt +1
b
rt
1
Et π t + 3 +
+
+
3
3
2
φπ
φπ
φπ
φπ
Continuing the forward solutions
πt =
1
φjπ+1
j
1
k =0
φkπ+1
Et π t + j + 1 + ∑
... and continuing …nally we get
∞
1
k =0
φkπ+1
πt = ∑
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Et b
rt +k
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Et b
rt +k
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Model solution
If φπ > 1 the di¤erence equation φπ π t = Et π t +1 + b
rt has only one
stationary solution obtained from forward solution
∞
1
k =0
φkπ+1
πt = ∑
Et b
rt +k
In‡ation and, consequently, price level are fully determined by the real
interest rate, which, in turn, is a function of fundamentals.
The Central Bank, by choosing the value of φπ , can determines the
degree of in‡ation volatility.
For φπ < 1 the stationary solution of φπ π t = Et π t +1 + b
rt takes the
following form
π t +1 = φ π π t b
rt + ξ t +1
where ξ t +1 is a sunspot shock.
Thus, any process π t satisfying the previous equation is consistent
with a stationary equilibrium.
Price level, in‡ation and nominal interest rate are undetermined.
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3) An exogenous path for the money supply
We suppose that the monetary authority sets an exogenous path for
the money supply mt .
By combining the money demand mt = pt + yt
equation rt = it Et π t +1 we get
rt =
pt + yt
η
Writing Et π t +1 as Et pt +1
pt =
ηr
η
1+η
mt
ηit and the Fisher
Et π t + 1
pt and rearranging
Et pt +1 +
1
1+η
mt + ut
y
where ut = 1t+η t evolves independently of monetary policy (they are
real variables).
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Price level determination
Assuming η > 0 and solving forward we obtain
pt =
∞
where ut0 = ∑
k =0
∞
η
∑
1 + η k =0
η
1 +η
k
k
η
1+η
Et mt +k + ut0
Et ut +k does not depend on monetary policy.
The previous equation can be rewritten in terms of expected growth
rate of money:
∞
pt = mt + ∑
k =1
η
1+η
k
Et ∆mt +k + ut0
Thus, an arbitrary exogenous path for money supply always
determines the price level uniquely.
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Interest rate determination
Plugging price level equation into money supply we get
yt
ηit
= mt
∞
η
1+η
mt + ∑
k =1
+
it
=
1 ∞
∑
η k =1
η
1+η
k
Et ∆mt +k + ut0
k
Et ∆mt +k +
yt + ut0
η
This implies that also the nominal interest rate is determined uniquely.
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An example
Let assume that money growth evolves according to an AR(1)
process:
∆mt = $m ∆mt 1 + εm
t
Moreover, we assume no real shock and yt = 0, rt = 0. The price
level becomes
η$m
pt = mt +
∆mt
1 + η (1 $m )
Following a monetary shock, given $m > 0 as empirically observed,
the price level response is greater than the money supply increase.
Nominal interest rate goes up in response to a monetary shock
it =
$m
∆mt
1 + η (1 $m )
Shortcomings: these results contrast with empirical evidence (slow
response of prices to monetary shocks and presence of liquidity e¤ect).
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Money in the utility function (MIU)
We assume that real money provides utility to the agents (Hint:
money facilitates transactions on the market).
The households problem under MIU becomes
∞
E0 ∑ β t U
Ct ,
t =0
Mt
, Nt
Pt
Budget constraint
Pt Ct + Qt Bt + Mt
Bt
1
+ Mt
1
+ W t Nt
Tt
We de…ne Λt = Bt 1 + Mt 1 as the total …nancial wealth. Budget
constraint can be rewritten as
Pt Ct + Qt Λt +1 + (1
Qt ) Mt
Λ t + Wt Nt
Tt
Interpretation: (1 Qt ) = 1 exp ( it ) ' it =) opportunity cost
of holding money. (Remeber it = log(Qt ) =) exp (it ) = Qt )
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MIU optimality conditions
Maximizing the utility function under the budget constraint yield the
following FOCs
Wt
Pt
Qt
Um,t
Uc ,t
Un,t
Uc ,t
Uc ,t +1 Pt
= βEt
Uc ,t Pt +1
=
= 1
exp( it )
Model properties depend on the speci…cation of the utility function.
Two cases:
separable in real balances
non-separable in real-balances
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Separable utility
The utility functional form is
U
Mt
Ct ,
, Nt
Pt
=
Ct1 σ
(Mt /Pt )1
+
1 σ
1 ν
ν
1 +γ
Nt
1+γ
!
Given the separability, Uc ,t and Un,t continue to be independent of
real balances. The equilibrium level for output, employment, real
interest rate and real wage is the same of a cashless economy.
The money demand equation is
σ
Mt
= Ctν [1
Pt
exp ( it )]
1
ν
In log-linear terms
σ
ct ηit
ν
Money demand serves only at determining the quantity of money that
the Central Bank supplies. Moreover, money has only nominal e¤ects.
mt
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pt =
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Non-separable utility
The utility functional form is
Mt
, Nt
Ct ,
Pt
U
=
Xt1 σ
1 σ
1 +γ
Nt
1+γ
!
Xt is a composite index of consumption and real balances having the
following form
"
Xt
=
Xt
= Ct1
ϑ ) Ct1
(1
ϑ
Mt
Pt
ν
+ϑ
Mt
Pt
1 ν
#11ν
for ν 6= 1
ϑ
for ν = 1
where ν denotes the elasticity of substitution between consumption
and real balances and ϑ the relative weight of real balances in utility.
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New optimality conditions
Being the utility function non-separable the marginal utilities of
consumption and real balances are
Uc ,t
= (1
ϑ ) Xtν
Um,t
= ϑXtν
σ
σ
Ct
ν
Mt
Pt
ν
Xt +1
Xt
ν σ
The optimality conditions now become
Wt
Pt
Qt
Mt
Pt
γ
Nt Xtσ ν Ctν
(1 ϑ )
"
Ct +1
= βEt
Ct
=
= Ct [1
ν
exp( it )]
1
ν
#
1
ν
ϑ
1
Pt
Pt +1
ϑ
Implications: monetary policy non-neutral as both labor supply and
Euler equation are a¤ected by real balances.
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Optimal monetary policy in a MIU context
The policymaker aims to maximize the utility of the representative
household subject to the resource constraint Ct = At Nt1 α . Formally,
max U
Ct ,
Mt
, Nt
Pt
Optimality conditions:
Un,t
Uc ,t
Um,t
= (1
α)At Nt
α
= 0
Condition Um,t = 0 equals the marginal utility of real balances with
the social marginal cost of producing real balances.
Remember that households choose their optimal real money balances
according to
Um,t
= 1 exp( it )
Uc ,t
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Friedman rule
Um,t = 0 is satis…ed only in the case it = 0 =) Friedman rule. The
opportunity cost of holding money is given by the nominal interest
rate and the latter must be zero in order to equate the social cost of
holding money.
Policy implication: π = ρ < 0 =) moderate de‡ation in the
long-run.
Nonetheless, Friedman rule involves that price level is not determined.
Central Bank can avoid undeterminacy implementing the Taylor
principle in the following rule:
it = φπ (rt
1
+ πt )
Plugging the previous rule into the Fisherian equation leads to the
following di¤erence equation
φπ it = Et it +1
whose only stationary solution is it = 0.
Equilibrium in‡ation is π t = rt .
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References
- Christiano, L.J., M. Eichenbaum and C.L. Evans (1999), “Monetary
policy shocks: What have we learned and to what end?,” Handbook of
Macroeconomics, in: J. B. Taylor & M. Woodford (ed.), volume 1, chapter
2: 65-148.
- Galí, J. (2008), “Monetary Policy, In‡ation and the Business Cycle. An
Introduction to the New Keynesian Framework,” (Chapters 1 and 2).
- King R. e S. Rebelo, (2000), “Resuscitating Real Business Cycles,” in
Woodford, M. and Taylor, J., (eds.), Handbook of Macroeconomics, vol.
1B North-Holland (NBER Working Paper 7534).
- Kydland, F.E. and E.C. Prescott (1982), “Time to Build and Aggregate
Fluctuations,” Econometrica, 50(6): 1345-70.
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