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Mohamed Rehan M S – rehan@iitj.ac.in
Dynamics of Unemployment andInflation
An extended Phillips – NAIRU model
Dynamics of Unemployment and Inflation
Mohamed Rehan M S
Center of Excellence in Systems Science
Indian Institute of Technology, Jodhpur13th November 2013
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Mohamed Rehan M S – rehan@iitj.ac.in
Historical inverse relationship between the rateof unemployment and the rate of inflation inan economy
First observed by William Phillips
Cornerstone for Keynesian Economic Policies
Trivia: Seven Nobel Prizes have been awarded for work critical of
the Phillips curve!!!
The Phillips Curve
Dynamics of Unemployment and Inflation 2
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The Phillips Curve
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Original model deemed to simplistic
The relationship exists only in the short run
Newer models have other factors included in
them, most popular amongst them beingExpectations Augmented Phillips Curves
Imbibes concept of NAIRU (Non AcceleratingInflation Rate of Unemployment) – a level ofunemployment below which inflation rises
Phillip’s Curve today
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Comparison of various Phillips Curves
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Multiple mathematical models for NAIRUPhillip’s Curves
All models carry the two basic principles
Inflation is inversely proportional to Unemploymentin the short run
In the long run, monetary policy cannot affectunemployment
Mathematical Formulation
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Ut+
= Ut i L U
t o
tU
tU = unemployment
i = inflow rate of labor
L = Labor force
o = outflow rate of labor
ot =
+ (−)
Jt = Jobs created at time t
d = Fraction of currently employed people
looking for better jobs
Derivation of our model
Dynamics of Unemployment and Inflation 7
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Jt = J
s γ m π
t γ = scaling factor
m = money growth rate in the market
πt = Inflation rate at time t
Combining all three equations we get
Ut+= Ut i L Ut Js γ m πt
Ut d 1 Ut
Ut = f(Ut, πt)
Derivation of our model (contd.)
Dynamics of Unemployment and Inflation 8
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πt =
(πt
e
,−
)πt
e = expected rate of inflation
w,t = real bargained wage
wp = real price determined wage
= price change factor
wp
= 1 μ y
w,t = 1 1 b Ut y
Derivation of our model (contd.)
Dynamics of Unemployment and Inflation 9
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πt+
e = aπt 1 a π
t
e
Substituting all previous equations we obtain
πt =1
δ(πt
e μ (1 b)t
1 μ )
and solving it for πte
πte = δπt
μ (1 b)t
1 μ
Derivation of our model (contd.)
Dynamics of Unemployment and Inflation 10
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Finally, we obtain
πt+ =1
δ[
μ
1 μ aπt 1 a δπt
μ 1 b Ut
1 μ
1 b
1 μUt+]
Which can be expressed as
πt+ =1
δ[
μ
1 μ aπt 1 a δπt
μ 1 b Ut
1 μ
1 b
1 μf(Ut, πt)]
In other words, πt+ = g(Ut, πt)
Derivation of our model (contd.)
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Ut+= Ut i L Ut
Js γ m πt
Ut d 1 Ut Ut
πt+ =1
δ[
μ
1 μ aπt 1 a δπt
μ 1 b Ut
1 μ
1 b
1 μUt+]
a = 0.5 μ= 0.04
b = 0.5 δ= 2
d = 0.01 γ= 0.5m = 0.03 L= 1
Assumptions for Numerical Analysis
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Fixed point obtained at (0.0224,0.03)
Stability analysis for various values of i
i=0.12, λ =-0.8239,0.7356, stable
i=0.16, λ =-1.4104,0.7395, unstable
Stability lost at i=0.131992
Numerical Analysis
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Bifurcation Diagram - Unemployment
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Bifurcation Diagram - Inflation
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Chaotic Attractor
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As t tends to infinity, any value is either
Eventually periodic to the fixed point, or
Sucked into an orbit, which may be
• Periodic (i=0.18, all periods possible)
• Chaotic Attractor
Out of 50000 iterations, not a single point isrepeated, when it goes to a chaotic orbit!!!
Chaotic Attractor (Contd.)
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Sensitive dependence studied using two initialpoints (0.2,0.15) & (0.2,0.15+10-10)
The points diverge from each other with time,implying that the system shows sensitivedependence on initial conditions
As t tends to infinity, both points converge toentirely different orbits
Sensitive Dependence
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( )
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Sensitive Dependence (Contd.)
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S (C )
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Sensitive Dependence (Contd.)
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B f A
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Basin of Attraction
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C l
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The NAIRU model exhibits chaotic behaviorafter a certain value of inflow of people tounemployment (approx. 0.18).
Hence large layoffs, population bursts, heavy
migration etc., can have serious impacts onthe economy, and de-stabilize it by leading itto chaos
Existence of chaotic orbits implies that there isno stable tradeoff
Conclusions
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C l i (C d )
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The chaotic attractor resembles the originalPhillips Curve with a negative slope
However, the curve is composed of unstabledisequilibria, and reaffirms that there is no
feasible inflation-unemployment tradeoff
Long periods of high inflation rates cannotoccur. This should be kept in mind while
designing monetary policy
Conclusions (Contd.)
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F P
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Conduct analysis for different parameters, oneby one.
Conduct analysis for all parameters varyingtogether and interacting with each other
Instead of fixed values of various parameters,include their values from the actual time series.Try to develop a prediction technique for
values in non-chaotic regionImprove model with other exogenous inputs
Future Prospects
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R f
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Mohamed Rehan M S – rehan@iitj.ac.in
Day, Richard H. "Complex Economic Dynamics Vol. I. An
introduction to dynamical systems and market mechanisms."(1994).
Day, Richard Hollis. Complex economic dynamics: An
introduction to Macroeconomic dynamics. Vol. 2 . Vol. 2. Mit
Press, 1999.Ferri, Piero, Edward Greenberg, and Richard H. Day. "ThePhillips curve, regime switching, and the NAIRU." Journal of
Economic Behavior & Organization 46, no. 1 (2001): 23-37.
Neugart, Michael. "Complicated dynamics in a flow model ofthe labor market." Journal of Economic Behavior &
Organization 53, no. 2 (2004): 193-213.
References
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QUESTIONS???
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THANK YOU
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