Natural Resource Economics
Academic year: 2016-2017
Prof. Luca Salvatici
[email protected]
Lesson 15
(In)stability of equilibrium
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Genuine Saving and GDP
Presentation by Francesca Insabato: November
30th
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Outline
1. Comparative dynamics
2. The analytical properties of FOCs system determines
the (in)stability of the steady state solution:
• stable
• (totally) unstable
• stationary
• (partially) unstable (“saddle path”)
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Comparative dynamics: phase
diagram (m, K)
Isoclines:
m
2 pg K
 d

m  d 2 pK K
10 p g
 d
What is going to happen if:
• pK increases?
• d decreases?
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Phase diagram (m, K)
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Stability in first-order
differential equations
Dynamic stability in y(t) = y(t-1) can be determined by
substitution starting from the initial condition y(0) = a0, as
follows:
y(1) =  y(0) =  a
y(2) =  y(1) = 2a
y(3) =  y(2) =  3a
….
y(t) = t a
  1  lim t  yt  y *
<1: dynamic stability condition
If it is not met, the path is dynamically unstable
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Characteristic equation
The solution of a system of linear differential
equations:
.
m *  a1m *  a2 K *
.
K *  b1m *  b2 K *
Is based on the solution of a 2° order “Characteristic
2
equation”:
af  bf  c  0
Stability depends on the values (real or imaginary of
the solutions (f1, f2)): eigenvalues (characteristic
values)
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Linearization
What if the system of differential equations is
not linear?
f ( x0) f ' ( x0)
f ' ' ( x0)
2
f ( x) 

( x  x0) 
( x  x0)  .......
0!
1!
2!
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Eigenvalues and eigenvectors
General structure of the solution:
f1t
m  g 11e
*
f1t
K  g 21e
*
f2 t
 g 12 e
f2 t
 g 22 e
g = eigenvectors; f = eigenvalues
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Eigenvalues computation
Bg  fg 
B  fI g
 0
10 
I  
 identity matrix

01
 a1a2 
B  

b
b
 1 2 
Trivial solution: g=0
Non-trivial solution only if the determinant of the
eigenvectors (g) coefficients matrix (B-fI) is equal to
zero (linearly independent columns)
10
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Solutions of the
characteristic equation
The characteristic equation is obtained from the
determinant
a1  f a2 
B  fI  


b1 b2  f  
f 2   a1  b2 f  a1b2  a2b1   0
Solutions:
 b  b 2  4ac
f1, f 2 
2a
 b  i 4ac  b
f1, f 2 
Imaginary solutions:
2a
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Example
Coefficients:
1
B
1
4

1
Eigenvalues:
Characteristic equation:
4 
1  f
 1

1

f


f  2f  3  0
2
Solutions:
2  4  12
f1 , f2 
  1, 3
2
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Solution
• First eigenvalue:
• Second eigenvalue:
g 11  4g 21  g 11
g 11  g 21  g 21
g 1   2, 1
g 12  4g 22  3g 12
g 12  g 22  3g 22
g 2  2, 1
 t   2e  2e
t
Solution:
K t   e  e
t
3t
3t
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Stability
• Real solutions: f1 < f2 < 0
If the solutions coincide: linear paths
• Imaginary solutions:
real part < 0
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Total stability
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Stable focus
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Instability
• Real solutions: f1 > f2 > 0
If the solutions coincide: linear paths
• Imaginary solutions:
Real part > 0
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Total instability
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Unstable focus
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Stationarity and partial instability
• Stationarity: imaginary solutions with a = 0
• Partial instability (saddle path): f1 > 0 > f2
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Phase diagram (m, K)
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Isoclines
m  m   d   2 p g K  10 p g
.
m
.
K 
 dK
2 pK
Bg  fg 
B
 fI g  0
10 
I  
 identity matrix

01
(d   )  f
1
B  fI 
2 pk
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 d f
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Characteristic equation and
eigenvalues
Characteristic equation:
𝑝𝑔
𝜙 − 𝛿𝜙 − 𝑑 𝑑 + 𝛿 −
=0
𝑝𝑘
Eigenvalues:
𝑝
𝑔
2
𝛿+ 𝛿 +4 𝑑 𝑑+𝛿 +
𝑝𝑘
𝜙1 =
> 0 > 𝜙2
2
𝑝𝑔
2
𝛿− 𝛿 +4 𝑑 𝑑+𝛿 +
𝑝𝑘
=
2
2
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OECD Trade News: Support to fossil
fuels remains high
(http://www.oecd.org/site/tadffss/
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