Download Local and Global Determinacy of the Equilibrium Paths in Nonlinear

Local and Global Determinacy of the Equilibrium Paths in
Nonlinear Macroeconomic Models
Anna Agliari
Dept. of Economic and Social Sciences
Catholic University - Piacenza (Italy)
Pavia, September 27, 2011
Outline
1
Framework and Motivation
The Framework
Determinacy
The Present Talk
2
The Woodford's Model
Towards the Nonlinear Model
The Analysis of the Model
3
The OLG Model with Credit Market Imperfections
The Model
Local and Global Analysis
Framework and Motivation
The Framework
Rational Expectation Equilibrium Models
Endogeneous Growth. Real Business Cycles. Monetary Policy.
The state of the economy at time t is inuenced by the decisions the
agents will make in the future periods.
The past history of the system may not guarantee the uniquess of the
bounded Rational Expectations Equilibrium (REE) paths, due to
multiple values of the non-predetermined variable consistent with
them.
In such case, the model exhibits indeterminacy and it explains why
economies with similar initial conditions evolve in dierent ways or
why the same economic policies perform dierently in the same
environment
Framework and Motivation
The Framework
Determinacy in Linear Models.
Assume a unique non-predetermined variable.
Only one xed point exist, either stable, repelling or saddle.
If it is asymptotically stable, then it is globally asymptotically stable
(indeterminacy)
Otherwise, all the trajectories are unbounded, but the steady state and
its stable manifold (determinacy)
Blanchard-Khan Conditions
In a linear model, the indeterminacy occurs when the dimension of the
stable set of a steady state is higher than the number of the predetermined
variables.
Framework and Motivation
The Framework
What About REE Nonlinear Models?
It is well known that
when the model is nonlinear, the local analysis of the steady states of the
perfect foresight equilibrium model may lead to misleading conclusions.
We shall show that
The multiplicity of invariant sets causes global indeterminacy, even if
they are all locally determinate
the occurrence of global bifurcations may be associated with global
indeterminacy around a locally determined steady state.
Framework and Motivation
Determinacy
Local and Global Indeterminacy
Nonlinear model one-step forward looking
Local Indeterminacy
For a given predetermined variable, there exist multiple equilibrium paths of
the linearized model converging to the same steady state or uctuating
around it. The Blanchard-Khan conditions allows us to show if there exists
a continuum of values of the control variables that put the system onto the
stable set.
Global Indeterminacy
There exist multiple steady states or invariant sets (like cycles, closed
invariant curves, chaotic sets) and there are multiple bounded trajectories
converging to them. In case of global indeterminacy, dierent choices of
the non-predetermined variables might imply dierent long run behavior
and the initial conditions do not necessarily determine the steady state to
which the economy will eventually converge.
Framework and Motivation
The Present Talk
Plan of the Talk
Two models will be considered
Woodford's Model
From a Linear Model to a Nonlinear One
Multiple Steady States cause Global Indeterminacy.
OLG Model with Credit Market Imperfection
Description of the Model
Global Bifurcations associated with Global Indeterminacy
Woodford Model
Towards the Nonlinear Model
The Basic Setting.
A New Keynesian model under sticky prices
We consider a
cashless
economy where
households supply labour and purchase goods for consumption;
rms hire labor, produce and sell dierentiated products in a
monopolistically competitive goods market à la Dixit&Stiglitz;
both households and rms behave optimally;
rational expectations framework;
CRRA utility functions and linear production function (Walsh)
Woodford Model
Towards the Nonlinear Model
Equilibrium Conditions.
The Demand Side is represented by the Euler condition for optimal
consumption
Pt −σ
Y
Yt = β (1 + it )Et
Pt+1 t+1
The Optimal Price set by rms able to adjust their price
PT θ
Pt
pt∗
= µ ∞T =t
θ −1
Pt
PT
Et ∑ α T −t β T −t YT1−σ ϕT
Pt
T =t
∞
Et ∑ α T −t β T −t YT1−σ ϕT
Price Aggregation à la Calvo
1−θ
Pt1−θ = (1 − α) (pt∗ )1−θ + αPt−1
Woodford Model
Towards the Nonlinear Model
The Woodford's Linear Model
Loglinearizing around the target steady state
Let xt be the output gap and πt the ination rate, then
The Demand Side
xt = Et xt+1 −
1
iˆt − Et πt+1 + ut
σ
The New Keynesian Philips Curve
πt = κxt + β Et πt+1
BUT
Without loglinearization we are not able to make explicit the model
Woodford Model
Towards the Nonlinear Model
Our assumptions
To preserve the nonlinearity of the model
New measures expressing the quantities of the state variables
corresponding to ination and output gap.
Gross Ination Rate
Πt+1 =
Pt+1
Pt
the ratio between output and its value in exible price
gt+1 =
Yt+1
Yf
Firms and householders perform their choices taking into account the
information available at time t in order to shape their expectations at
time t + 1
We consider an
one-step forward-looking
model
Woodford Model
Towards the Nonlinear Model
The Nonlinear Model
Object of our study
Let Lt =
pt∗
Pt
The Demand Side becomes
Et
−σ
gt+1
Πt+1
!
=
β gtσ
1
(1 + it )
The Optimal Price equation becomes
Lt
σ +η
σ +η
1−σ
1−σ θ
Lt − gt
gt
= αβ Et gt+1 Πt+1 gt+1 −
Πt+1
The Price Aggregation becomes
Lt =
1
α
−
1−α 1−α
1
Πt
1−θ !1−θ
Woodford Model
Towards the Nonlinear Model
The Taylor Rule
Interest rate specication
The monetary policy is represented by a rule for setting the nominal
interest rate.
The alternative we favor is to model policy-maker behavior by specifying an
interest rate rule conditioned on values that are observable by policy
makers in real time
The Taylor Rule
it=
1
− 1 + ϕg (gt − 1) + ϕΠ (Πt − 1) + νt
β
Woodford Model
The Analysis of the Model
The Perfect Foresight Dynamics
Under perfect foresight assumption, the model becomes:

σ

Πt+1 = β (1 + it ) ggσt
t+1
1−σ
σ +η
θ −1
 αβ Πt+1
Lt − Πt+1 gt+1
= − ggt1−σ Lt − gtσ +η
t+1
1
1−θ
1
α
−
Πθ −1
, σ > 0 and η > 0 are the
1−α 1−α t
elasticities of the utility functions, θ > 1 is the price elasticity,
α ∈ (0, 1) is the fraction of rms not able to update the price,
β ∈ (0, 1) is the discount rate.
where Lt =
Combined with the Taylor Rule, it gives a two dimensional model
where the outputgap g and the gross ination rate Π are non
predetermined variables.
Woodford Model
The Analysis of the Model
Local Determinacy
The equilibrium E ∗ = (1, 1) is a steady state of the model for any
choice of the parameters of our model and we refer to it as the target
equilibrium.
Proposition
The stationary target equilibrium E ∗ = (1, 1) is locally determinate if
αβ ϕg (2αβ − β + 1) − (1 − β ϕΠ ) (σ + η) (1 − α) (αβ + 1) > 0
Proof
Apply Blanchard and Khan conditions.
Woodford Model
The Analysis of the Model
A Particular case
θ = 2 and σ + η =
1
2
As stated in Proposition 1, the equilibrium E ∗ is determinate if
αβ ϕg (2αβ − β + 1) − (1 − β ϕΠ ) (1 − α) (αβ + 1) > 0
When
αβ ϕg (2αβ − β + 1) − (1 − β ϕΠ ) (1 − α) (αβ + 1) = 0
a local bifurcation, either
transcritical
or
pitchfork
, occurs.
Woodford Model
The Analysis of the Model
Transcritical Bifurcation
Theoretical results
Proposition
If 1 − β ϕΠ > αβ ϕg then the target equilibrium E ∗ coexists with one and
only one feasible steady state P∗ .
Consequently, when 1 − β ϕΠ > αβ ϕg the only possible bifurcation of
the target equilibrium is a transcritical one, at which the repelling
node E ∗ becomes a saddle merging with P∗ .
Such a bifurcation may occur only if
(1 − α) (1 + αβ ) < 2αβ − β + 1
otherwise E ∗ is locally indeterminate.
Woodford Model
The Analysis of the Model
Transcritical Bifurcation
An example
We x
α = 0.75, β = 0.99,
ϕΠ = 1.01,
ϕg decreasing from
ϕm = 1.3468 ∗ 10−4 ,
value at which
1 − β ϕΠ = αβ ϕg
At ϕg = ϕgtr =
3.9244 · 10−5 a
transcritical
bifurcation occurs
When ϕg < ϕm the model exhibits global indeterminacy, otherwise
global determinacy may exists.
Woodford Model
The Analysis of the Model
Pitchfork Bifurcation
Theoretical Results
Proposition
The target equilibrium E ∗ = (1, 1) undergoes a pitchfork bifurcation if
αβ ϕg (2αβ − β + 1) + (β ϕΠ − 1) (1 − α) (αβ + 1) = 0
α + β (α + 1) (2α − 1) + β 2 α 3α 2 − 3α + 1 = 0
hold.
Woodford Model
The Analysis of the Model
Pitchfork Bifurcation
A numerical example
We x
α = 0.332, β = 0.98048,
ϕΠ = 1.01
ϕm = 0.029845 is the
value at which
1 − β ϕΠ = αβ ϕg
At
ϕg = ϕgpit = 0.03940947 a
pitchfork bifurcation
occurs.
ϕm < ϕg < ϕ pit two locally determinate stationary equilibria
∗
exists, even if the target equilibrium E is locally indeterminate.
When
Woodford Model
The Analysis of the Model
Comments
Performing the analysis in a nonlinear framework, the conditions for a
unique steady-state depend on the intertemporal substitution elasticity
of consumption and the elasticity of disutility of labor.
The degree of nominal rigidity of prices enters the condition that may
guarantee local and, under some restrictions, global determinacy of
the model: this means that the structure of the market has an
inuence on the stability conditions.
Global indeterminacy may exist even if the target equilibrium appears
to be locally determinate.
Heteroclinic connections may exist between the determinate and
indeterminate equilibria, made up by the stable manifold of a saddle.
In such a case, around a locally determinate stationary
equilibrium, multiple bounded REE paths exist that converge to
.
an indeterminate one
OLG Model
The Model
The OLG Model
Basic assumptions
The time is discrete
In each period t = 0, 1, ..., there are two generations alive, young and
old agents
Each generation consists of a continuum of homogeneous agents with
unit mass
There is one consumption commodity produced in each period by a
large number of identical rms using capital and labor as inputs
Produced commodity can be either consumed or invested in capital,
which becomes available in the next period
Capital depreciates fully within a period
OLG Model
The Model
Consumption Good
Technology
We assume a constant return to scale production function and a
competitive factor market.
Production Function
Output per worker is yt = f (kt ), where f : R+ → R+ is the production
function in intensive form and satises the standard neoclassical
assumptions. Kt and Lt are the aggregate supplies of physical capital and
labor respectively, and kt = Kt /Lt is the capital per worker.
Factor Rewards
f 0 (kt ) is the rate of return on one unit of capital
wt = W (kt ) := f (kt ) − kt f 0 (kt ) is the wage rate
OLG Model
The Model
Young Agents' Behavior
How to Convert Wage into Consumption Good
Each young agent is endowed with l > 0 unit of labor elastically supplied to
rms, do not consume and save their entire wage income. To nance her
consumption when old
she can lend the
entire wage income
at the competitive
credit market at the
rate of return rt+1
and her second
period consumption
is
i
ct+1
= lti wt rt+1
she can run a non-divisible investment
project which converts one unit of
consumption good in period t into one
unit of capital in period t + 1,
borrowing 1 − sti .
Each project generates f 0 (kt+1 ) units
of consumption goods and her second
period consumption is
i
ct+1
= f 0 (kt+1 ) − (1 − lti wt )rt+1 .
OLG Model
The Model
Credit Market
Imperfections Captured by the Parameter λ
The young agents would borrow and run an investment project if
Protability Constraint
rt+1 ≤ f 0 (kt+1 )
An entrepreneur can hide a portion 1 − λ , λ ∈ (0, 1], of their revenue from
nanciers. The young agents are able to borrow and start the investment
project if
Borrowing Constraint
(1 − sti )rt+1 ≤ λ f 0 (kt+1 )
The parameter λ allows us to investigate the aggregate implication of the
credit market imperfection.
OLG Model
The Model
Equilibrium in the Labor Market
i ) 7→ ci
i
Utility function: (lti , ct+1
t+1 − θ u(lt )
Optimization problem of
young lender i
max lti wt rt+1 − θ u(lti )
lti ∈[0,l]
Optimization problem of young
entrepreneur i
max f 0 (kt+1 ) − (1 − lti wt )rt+1 − θ u(lti )
lti ∈[0,l]
Independently of whether they become borrowers or lenders, all young
agents will supply the same amount of labor
Equilibrium
Denoted by W −1 the inverse of the wage function
Ls (wt , rt+1 ) = u0
−1 wt rt+1 Kt
= −1
= Ld (wt , Kt )
θ
W (wt )
OLG Model
The Model
Equilibrium in the Credit Market
Rate of return
Individual and aggregate savings are the same, sti = st .
If st ∈ [0, 1 − λ ) then
rt+1 =
λ
f 0 (kt+1 ) < f 0 (kt+1 )
1 − st
and all the agents would strictly prefer to borrow in the credit market
and run an investment project.
If st ∈ [1 − λ , 1) then
rt+1 = f 0 (kt+1 ) ≤
λ
f 0 (kt+1 )
1 − st
and the young agents are indierent between becoming a borrower or
a lender.
The rate of return depends not only on the marginal product of capital but
even on the aggregate saving.
OLG Model
The Model
Perfect Foresight Dynamics
State variables Kt and Lt
Capital and Labor market clearing conditions imply
The Map
M:

 Kt+1 = S(Kt , Lt )
 L =
t+1
S(Kt ,Lt )
ξ (Kt ,Lt )
where
h
i

0
0 −1 1−S(K,L) θ Lu (L)

if S(K, L) < 1 − λ
 (f )
λ
S(K,L)
ξ (K, L) :=
h 0 i

Lu (L)

( f 0 )−1 θS(K,L)
if S(K, L) ≥ 1 − λ .
OLG Model
Local and Global Analysis
Assumptions
To assure the existence of interior steady states
Assumption 1
Let f be such that
- the function k 7→ W k(k) is strictly decreasing and satises boundary
conditions
W (k)
W (k)
lim
< 1 < lim
= W 0 (0);
k↑∞
k↓0
k
k
- the function k 7→ ρ(k) = k f 0 (k) is non-decreasing.
Assumption 2
Let the parameter pair (θ , l) satisfy
θ > θmin :=
ρ(k∗ )
u0 ( k1∗ )
and l >
1
k∗
OLG Model
Local and Global Analysis
Assumptions
A technical one, to have at most three steady states
Let ε(l) :=
u0 (l)
lu00 (l)
denote the elasticity of labor supply (elasticity of (u0 )−1 ).
Assumption 3
Let u be such that l 7→ ε(l) is non-decreasing.
OLG Model
Local and Global Analysis
Existence of the Steady States
Four dierent regions
Region A: a unique steady
state S∗ = (k∗ L1∗ , L1∗ )
0.6
0.5
Region C: a unique steady
state Q∗ = (k∗ L3∗ , L3∗ )
0.4
0.3
Region B: three steady
states, S∗ , Q∗ , and E ∗ ,
where E ∗ = (k∗ L2∗ , L2∗ )
0.2
0.1
0.0
0.0
0.2
0.4
0.6
Parameter space (λ , θ )
0.8
1.0
Region D: two steady
states, S∗ and E ∗ , since
Q∗ is unfeasible.
OLG Model
Local and Global Analysis
Local Stability Analysis
Bifurcations
Proposition
If Assumptions 1, 2, and 3 are satsed then
(a) the steady states S∗ and Q∗ , whenever they exist, are always
saddles.
(b) the steady state E ∗ , whenever it exists, can be either a
source, or a sink.
The horizontal line θ = θmin separates the regions where the saddle Q∗
from unfeasible becomes feasible
The branch of the curve ψ1 (λ ) with λ ≤ λc is a saddle-node
bifurcation curve, the crossing of which causes the appearance of the
saddle S∗ and of the node E ∗ (either repelling or attracting)
The curve ψ2 (λ ) is a border collision bifurcation curve, at which two
steady states merge changing their state from virtual to non-virtual.
OLG Model
Local and Global Analysis
Local Determinacy
The steady states S∗ and Q∗ , whenever they exist, are always
determinate.
locally
The steady state E ∗ , whenever it exists, may be locally
indeterminate and, for a given capital stock, there exists a
continuum of perfect foresight trajectories converging to the middle
steady state, eventually uctuating around it.
Due to the nonlinearity of the model, the proof of the existence of a perfect
foresight path in a neighborhood of a steady state does not rule out the
possibility of other bounded trajectories.
The model under consideration may exhibit global indeterminacy, even if
restricted to a neighborhood of a local determinate steady state.
OLG Model
Local and Global Analysis
The Parameterized Economy
Towards the global indeterminacy
Production Function
f (k) = kα
where α ∈ (0, 1) is the capital
share in production
Marginal Disutility of Working
1
u0 (l) = l ε
where ε > 0 is the labor supply
elasticity
Corollary
α
ε
If α < 0.50, ε > 1−2α
, λ < 1+ε
, θ ∈ (ψ1 (λ ), ψ2 (λ )) then the steady state
∗
E exists and is locally indeterminate.
Global indeterminacy
The coexistence of multiple steady states implies that the equilibrium paths
are globally indeterminate, even if the steady states are all locally
determinate.
OLG Model
Local and Global Analysis
Saddle-Node and BC Bifurcations
Heteroclinic connection involving E ∗
After the SNB, global indeterminacy around
E∗
ε = 0.5;θ = 0.111
2
0.20
2
S (K , L ) = 1 − λ
0.15
Q*
L
0.10
Ws (Q * )
0.05
E*
0.00
0.0
S*
0
0.1
0.2
0.3
0.4
Parameter space (λ , θ )
Ws (S * )
0
α = 0.33, λ = 0.2
K
1.5
In a neighborhood of E ∗ , L can
be chosen so that the economy
converges
to S∗
0
0
OLG Model
Local and Global Analysis
Saddle-Node and BC Bifurcations
Heteroclinic connection involving E ∗
After the BCB, global determinacy
α = 0.33, λ = 0.2
ε = 0.5;θ = 0.14
2
S (K , L ) = 1 − λ
0.20
0.15
L
E
*
0.10
Q*
0.05
Ws (S
*
0.00
0.0
)
0.1
0.2
0.3
Parameter space (λ , θ )
S*
E ∗ is a virtual xed point
0
0
0.4
K
1.5
OLG Model
Local and Global Analysis
A Heteroclinic Connection Involving Q∗
When θ ∈ (θsn , θbcb ) a heteroclinic bifurcation occurs
Enlargement
ε = 0.5, θ = 0.113
22
1.55
1.55
Ws (Q
Q*
Ws (Q( * ) )
LL
L
EE*
S
Q*
Wu (S * )
*
)
E*
*
Ws (S( * ) )
S*
00
00
KK
1.5
1.5
11
0.55
0.55
Ws (S * )
K
0.86
0.86
Around the repelling focus
there are bounded equilibrium paths that can
(b)
(a)
(b)
uctuate around it (a)
before
reaching one of the saddle xed points.
E∗
OLG Model
Local and Global Analysis
Saddle-Node Bifurcation
E ∗ is a stable xed point
After the SNB, global indeterminacy around
S∗
ε = 1;θ = 0.12
ε = 1, λ = 0.2
0.25
0.20
1.8
1.2
0.15
0.10
E
L
*
0.05
0.00
0.0
Ws (S
*
)
Wu (S
*
)
0.1
0.2
0.3
0.4
Parameter space (λ , θ )
S*
0.5
0.3
K
0.6
In the neighborhood of S∗
there also exist innitely many
bounded trajectories
converging
to the middle
0.5
0.3
steady state
OLG Model
Local and Global Analysis
Homoclinic Bifurcation of S∗
A chaotic repellor exists
Homoclinic tangle
ε = 1;θ = 0.128
Around the locally determinate
saddle S∗ it is possible to nd
innitely many bounded
trajectories belonging to either
some repelling cycle or to their
stable sets and the equilibrium
path may also uctuate
chaotically.
The economy can escape from
the low steady state if the
expectations are coordinate.
1.8
L
E*
Ws (S * )
S*
0.5
0.3
Wu (S * )
K
0.85
OLG Model
Local and Global Analysis
Coexistence of Dierent Attractors
ε = 1, θ = 0.13: Two cycles of period 8 appeared via saddle-node bifurcation
1.8
1.8
1.8
1.8
Ws (S * )
L
L
L
E*
E
C1
C1
S*
0.5
0.3
E*
Wu (S * )
S*
0.5
0.3
K
(a)
K
0.85
0.5
0.85 0.3
(a)
Around the saddle cycle N : innitely
many bounded equilibrium paths
converging to the attracting cycle C
0.5
0.3
K
(b)
0.85
0.85
(b)
The saddle S∗ can be still chaotic,
then even a heteroclinic connection
between S∗ and N may exist.
OLG Model
Local and Global Analysis
Comments
We provide a model, based upon strategic complementarities, that can
produce indeterminacy without relying on restrictive assumption about
production and utility functions.
We have shown that the existence of heteroclinic connection and the
occurrence of homoclinic bifurcations may be associated with global
indeterminacy around a locally determined steady state.
Summary
Summary
1
In nonlinear models, the multiplicity of steady states implies global
indeterminacy, even if the steady states are all locally determinate.
2
In nonlinear model, the occurrence of global bifurcations may
cause indeterminacy even if the model is restricted to a small
neighborhood of a locally determinate steady state.
3
Next problem
: what about Neimark-Sacker bifurcation and closed
invariant curves?
Appendix
For Further Reading
For Further Reading I
M. Woodford.
.
Interest and Prices, Foundations of a Theory of Monetary Policy
Princeton University Press, 2003.
C. Walsh.
.
Monetary Theory and Policy
The MIT Press, 2003.
Matsuyama, K.
Financial Market Globalization, Symmetry-Breaking and Endogenous
Inequality of Nations.
Econometrica 72, 85384, 2004.
Appendix
For Further Reading
For Further Reading II
Benhabib, J., S. Schmitt-Grohe and M. Uribe
The Perils of Taylor Rules.
Journal of Economic Theory. 96, 4069, 2001.
Chiappori, P. A. and R. Guesnerie.
Sunspot Equilibria in Sequential Markets Model.
In: Werner Hildenbrand and Hugo Sonnenschein, eds., Handbook of
Mathematical Economics, Vol. 4. Amsterdam: North-Holland, pp.
16831762, 1989.
Grandmont, J.M., P. Pintus and R. de Vilder
Capital-Labor Substitution and Competitive Nonlinear Endogenous
Business Cycles.
Journal of Economic Theory. 80, 1459, 1998.
Appendix
For Further Reading
For Further Reading III
Cornaro, A. and A. Agliari.
Global and local determinacy in a one-step forward looking New
Keynesian model.
Economic Modelling 28, 13541362, 2011.
Agliari, A. and G. Vachadze.
Homoclinic and Heteroclinic Bifurcations in an Overlapping
Generations Model with Credit Market Imperfection.
Computational Economic 38, 241260, 2011.
Agliari, A. and G. Vachadze.
Credit Market Imperfection, Labor Supply Complementarity, and
Global Indeterminacy.
mimeo.