Dynamic Models in Economics
Stefano BOSI
University of Evry
CIMPA 2014, Tlemcen
Stefano BOSI (University of Evry)
Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Basic models in dynamic economics
The Ramsey models (1928)
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Basic models in dynamic economics
The Ramsey models (1928)
The Overlapping Generations (OG) models (1947, 1958, 1965)
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Dynamic Models in Economics
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Ramsey model in CT
The Ramsey model in continuous time
max
Z ∞
e
θt
0
u (ct ) dt
subject to the budget constraints (market economy)
k̇t + δkt + ct
rt kt + wt lt
or to the resource constraints (central planner)
k̇t + δkt + ct
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f (kt )
Dynamic Models in Economics
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Ramsey model in DT
The Ramsey model in discrete time
∞
max ∑ βt u (ct )
t =0
subject to the budget constraints (market economy)
kt + δkt + ct
kt +1
rt kt + wt lt
or to the resource constraints (central planner)
kt +1
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kt + δkt + ct
Dynamic Models in Economics
f (kt )
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Overlapping Generations (OG) models
The program
max U (ct , dt +1 )
subject to the budget constraints (market economy)
Kt + 1
+ ct
Nt
dt +1
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wt lt
Rt +1
Dynamic Models in Economics
Kt + 1
Nt
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Extensions
Ramsey + heterogeneity
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Dynamic Models in Economics
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Extensions
Ramsey + heterogeneity
Ramsey + imperfections
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Extensions
Ramsey + heterogeneity
Ramsey + imperfections
OG + heterogeneity
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Extensions
Ramsey + heterogeneity
Ramsey + imperfections
OG + heterogeneity
OG + imperfections
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Dynamic Models in Economics
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Uni…ed Field Theory
Bridge Ramsey and OG
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Uni…ed Field Theory
Bridge Ramsey and OG
Bridge Ramsey in CT and in DT
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Bridging Ramsey and OG
Introduce a degree of altruism
Ut
= u (ct , dt +1 ) + αnt Ut +1
= α T Ut + T
T
1
∏
T
nt +s + u (ct , dt +1 ) +
s =0
1
∑
s =1
Let
nt = n and lim αT Ut +T
T !∞
T
s
αs u (ct +s , dt +s +1 ) ∏ nt
τ =1
1
∏ nt +s = 0
s =0
Ramsey-like objective:
∞
Ut =
∑ (αn)s u (ct +s , dt +s +1 )
s =0
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Issues
Are models in CT equivalent to models in DT?
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Issues
Are models in CT equivalent to models in DT?
Equivalence of steady state and (local) dynamics ?
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Issues
Are models in CT equivalent to models in DT?
Equivalence of steady state and (local) dynamics ?
How to pass from CT to DT?
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Issues
Are models in CT equivalent to models in DT?
Equivalence of steady state and (local) dynamics ?
How to pass from CT to DT?
Is CT a pertinent representation for economic transaction?
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Issues
Are models in CT equivalent to models in DT?
Equivalence of steady state and (local) dynamics ?
How to pass from CT to DT?
Is CT a pertinent representation for economic transaction?
Is an economic variable in CT backward or forward?
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Forward and backward variables: an example
Taylor rules:
i
it
it
Stefano BOSI (University of Evry)
= ρ (π )
= ρ (π t )
= ρ ( π t +1 )
Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Forward and backward variables: an example
Taylor rules:
i
it
it
= ρ (π )
= ρ (π t )
= ρ ( π t +1 )
The monetary authority adjusts the nominal interest rate in response
to observed or expected in‡ation rates.
Stefano BOSI (University of Evry)
Dynamic Models in Economics
CIMPA 2014, Tlemcen
10 / 65
Forward and backward variables: an example
Taylor rules:
= ρ (π )
= ρ (π t )
= ρ ( π t +1 )
i
it
it
The monetary authority adjusts the nominal interest rate in response
to observed or expected in‡ation rates.
Past in‡ation is a weighted average of all the observed past in‡ations:
πt
Stefano BOSI (University of Evry)
θ
Z t
∞
π s e θ (s
Dynamic Models in Economics
t)
ds
CIMPA 2014, Tlemcen
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Forward and backward variables: an example
Taylor rules:
= ρ (π )
= ρ (π t )
= ρ ( π t +1 )
i
it
it
The monetary authority adjusts the nominal interest rate in response
to observed or expected in‡ation rates.
Past in‡ation is a weighted average of all the observed past in‡ations:
πt
θ
Z t
∞
π s e θ (s
t)
ds
The higher θ, the higher the weight of the recent in‡ations with
respect to the remote ones.
Stefano BOSI (University of Evry)
Dynamic Models in Economics
CIMPA 2014, Tlemcen
10 / 65
Forward and backward variables: an example
Taylor rules:
= ρ (π )
= ρ (π t )
= ρ ( π t +1 )
i
it
it
The monetary authority adjusts the nominal interest rate in response
to observed or expected in‡ation rates.
Past in‡ation is a weighted average of all the observed past in‡ations:
πt
θ
Z t
∞
π s e θ (s
t)
ds
The higher θ, the higher the weight of the recent in‡ations with
respect to the remote ones.
Take the limit for θ ! +∞ to de…ne π backward.
Stefano BOSI (University of Evry)
Dynamic Models in Economics
CIMPA 2014, Tlemcen
10 / 65
Forward and backward variables: an example
Taylor rules:
= ρ (π )
= ρ (π t )
= ρ ( π t +1 )
i
it
it
The monetary authority adjusts the nominal interest rate in response
to observed or expected in‡ation rates.
Past in‡ation is a weighted average of all the observed past in‡ations:
πt
θ
Z t
∞
π s e θ (s
t)
ds
The higher θ, the higher the weight of the recent in‡ations with
respect to the remote ones.
Take the limit for θ ! +∞ to de…ne π backward.
A similar formula holds forward.
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Passing from CT to DT
Delay equations.
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Passing from CT to DT
Delay equations.
Discretizations.
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Di¤erence-di¤erential equations
How to mix CT and DT?
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Dynamic Models in Economics
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Di¤erence-di¤erential equations
How to mix CT and DT?
Delay equations.
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Dynamic Models in Economics
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Di¤erence-di¤erential equations
How to mix CT and DT?
Delay equations.
Example.
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Di¤erence-di¤erential equations
How to mix CT and DT?
Delay equations.
Example.
Ak technology with vintage capital:
y (t ) = A
Z t
i (z ) dz
t T
Stefano BOSI (University of Evry)
Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Di¤erence-di¤erential equations
How to mix CT and DT?
Delay equations.
Example.
Ak technology with vintage capital:
y (t ) = A
Z t
i (z ) dz
t T
i (z ) represents investment at time z (vintage z).
Stefano BOSI (University of Evry)
Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Di¤erence-di¤erential equations
How to mix CT and DT?
Delay equations.
Example.
Ak technology with vintage capital:
y (t ) = A
Z t
i (z ) dz
t T
i (z ) represents investment at time z (vintage z).
Machines depreciate suddenly after T > 0 units of time.
Stefano BOSI (University of Evry)
Dynamic Models in Economics
CIMPA 2014, Tlemcen
12 / 65
Di¤erence-di¤erential equations
How to mix CT and DT?
Delay equations.
Example.
Ak technology with vintage capital:
y (t ) = A
Z t
i (z ) dz
t T
i (z ) represents investment at time z (vintage z).
Machines depreciate suddenly after T > 0 units of time.
The depreciation rate depends on delayed investment, which shows the
vintage capital nature of the model.
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Dynamic Models in Economics
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Discretizations
How to discretize continuous-time models?
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Dynamic Models in Economics
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Discretizations
How to discretize continuous-time models?
Euler-Taylor discretizations.
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Dynamic Models in Economics
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Discretizations
How to discretize continuous-time models?
Euler-Taylor discretizations.
But equivalent models can behave di¤erently.
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Dynamic Models in Economics
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Discretizations
How to discretize continuous-time models?
Euler-Taylor discretizations.
But equivalent models can behave di¤erently.
Other representations (other bases in the function space).
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Dynamic Models in Economics
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Our paper
Focus on Euler.
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Dynamic Models in Economics
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Our paper
Focus on Euler.
Focus on (local) bifurcations.
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Our paper
Focus on Euler.
Focus on (local) bifurcations.
Does the discretization (approximation) degree play a role in (local)
dynamics?
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Our paper
Focus on Euler.
Focus on (local) bifurcations.
Does the discretization (approximation) degree play a role in (local)
dynamics?
To plot a CT phase diagram, one usually uses Euler discretization.
Stefano BOSI (University of Evry)
Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Our paper
Focus on Euler.
Focus on (local) bifurcations.
Does the discretization (approximation) degree play a role in (local)
dynamics?
To plot a CT phase diagram, one usually uses Euler discretization.
But, use this method to compare bifurcations.
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Continuous time
System:
ẋ = f (x )
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Continuous time
System:
ẋ = f (x )
Pick a regular sequence of time values:
(tn )n∞=0 = (nh)n∞=0
xn
x (tn )
where h is the discretization step.
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Discretizations
Reconstruct the path from xn to xn +1 component by component
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Discretizations
Reconstruct the path from xn to xn +1 component by component
Focus on the ith component of the vector x 2 Rm and integrate the
time derivative on the right or on the left:
xin +1
xin +1
xin
xin
Stefano BOSI (University of Evry)
=
=
Z nh +σ
nh
Z nh +h
nh +τ
=
ẋi dt
σ =h
=
ẋi dt
τ =0
Z nh +σ
nh
Z nh +h
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nh +τ
fi (x (t )) dt
σ =h
fi (x (t )) dt
τ =0
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Discretizations (cont.)
De…ne
xin +1
xin +1
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xin
xin
= ϕi ( σ )
= ψi ( τ )
Z nh +σ
nh
Z nh +h
nh +τ
Dynamic Models in Economics
fi (x (t )) dt
fi (x (t )) dt
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Discretizations (cont.)
De…ne
xin +1
xin +1
xin
xin
= ϕi ( σ )
= ψi ( τ )
Z nh +σ
nh
Z nh +h
nh +τ
fi (x (t )) dt
fi (x (t )) dt
Approximate ϕ and ψ with a polynomial (Taylor).
∑
(h
0 ) p (p )
ϕi (0)
p!
∑
(0
h ) p (p )
ψi (h )
p!
q
xin +1
xin
= ϕi (h )
p =0
q
xin +1
xin
= ψi (0)
p =0
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Discretizations (cont.)
De…ne
xin +1
xin +1
xin
xin
= ϕi ( σ )
= ψi ( τ )
Z nh +σ
nh
Z nh +h
nh +τ
fi (x (t )) dt
fi (x (t )) dt
Approximate ϕ and ψ with a polynomial (Taylor).
∑
(h
0 ) p (p )
ϕi (0)
p!
∑
(0
h ) p (p )
ψi (h )
p!
q
xin +1
xin
= ϕi (h )
p =0
q
xin +1
xin
= ψi (0)
p =0
The most popular approximation is the Euler discretization : q = 1.
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Dynamic Models in Economics
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Backward-looking discretization
First-order discretization.
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Dynamic Models in Economics
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Backward-looking discretization
First-order discretization.
RHS derivative.
x (t + h ) x (t )
h
x (tn + h) x (tn )
h
xn +1 xn
h
xn +1
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Dynamic Models in Economics
ẋ (t )
f (x (tn ))
f (xn )
xn + hf (xn )
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Backward-looking discretization
First-order discretization.
RHS derivative.
x (t + h ) x (t )
h
x (tn + h) x (tn )
h
xn +1 xn
h
xn +1
ẋ (t )
f (x (tn ))
f (xn )
xn + hf (xn )
The smaller the discretization step h, the more accurate the
approximation.
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Dynamic Models in Economics
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Forward-looking discretization
LHS derivative.
x (t )
x (tn )
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x (t h )
h
x (tn h)
h
xn xn 1
h
xn +1
ẋ (t )
f (x (tn ))
f (xn )
xn + hf (xn +1 )
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One-dimensional Taylor discretizations (backward-looking)
1th order (Euler): xn +1
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xn + f (xn ) h.
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One-dimensional Taylor discretizations (backward-looking)
1th order (Euler): xn +1
2nd order: xn +1
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xn + f (xn ) h.
xn + f (xn ) h + f (xn ) f 0 (xn ) h2 /2.
Dynamic Models in Economics
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One-dimensional Taylor discretizations (backward-looking)
1th order (Euler): xn +1
2nd order: xn +1
xn + f (xn ) h.
xn + f (xn ) h + f (xn ) f 0 (xn ) h2 /2.
0
2
3rd
h order: xn +1 xn + f (xn ) h +i f (xn ) f (xn ) h /2 +
f (xn ) f 0 (xn )2 + f (xn )2 f 00 (xn ) h3 /6.
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Dynamic Models in Economics
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One-dimensional Taylor discretizations (forward-looking)
1th order: xn +1
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xn + f (xn +1 ) h.
Dynamic Models in Economics
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One-dimensional Taylor discretizations (forward-looking)
1th order: xn +1
xn + f (xn +1 ) h.
2nd order: xn +1
xn + f (xn +1 ) h
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f (xn +1 ) f 0 (xn +1 ) h2 /2.
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Two-dimensional case (backward looking)
1th order: xn +1
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xn + hf (xn ).
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Two-dimensional case (backward looking)
1th order: xn +1
2nd order: xn +1
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xn + hf (xn ).
h
i
2
xn + hI + h2 J0 (xn ) f (xn ).
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Two-dimensional case (forward looking)
1th order: xn +1
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xn + hf (xn +1 ).
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Two-dimensional case (forward looking)
1th order: xn +1
2nd order: xn +1
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xn + hf (xn +1 ).
h
i
2
xn + hI h2 J0 (xn +1 ) f (xn +1 ).
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Hybrid discretization
A hybrid discretization is backward-looking for some components of
vector x and forward-looking for other components.
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Dynamic Models in Economics
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Steady state
Systems.
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Dynamic Models in Economics
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Steady state
Systems.
Continuous time: ẋ = f (x ).
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Dynamic Models in Economics
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Steady state
Systems.
Continuous time: ẋ = f (x ).
Euler-Taylor discretization: xn +1 = xn + hf (xn ) or
xn +1 = xn + hf (xn +1 ).
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Steady state
Systems.
Continuous time: ẋ = f (x ).
Euler-Taylor discretization: xn +1 = xn + hf (xn ) or
xn +1 = xn + hf (xn +1 ).
Steady states.
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Dynamic Models in Economics
CIMPA 2014, Tlemcen
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Steady state
Systems.
Continuous time: ẋ = f (x ).
Euler-Taylor discretization: xn +1 = xn + hf (xn ) or
xn +1 = xn + hf (xn +1 ).
Steady states.
ẋ = 0 ) f (x ) = 0.
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Dynamic Models in Economics
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Steady state
Systems.
Continuous time: ẋ = f (x ).
Euler-Taylor discretization: xn +1 = xn + hf (xn ) or
xn +1 = xn + hf (xn +1 ).
Steady states.
ẋ = 0 ) f (x ) = 0.
xn +1 = xn ) f (x ) = 0.
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Dynamic Models in Economics
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Steady state
Systems.
Continuous time: ẋ = f (x ).
Euler-Taylor discretization: xn +1 = xn + hf (xn ) or
xn +1 = xn + hf (xn +1 ).
Steady states.
ẋ = 0 ) f (x ) = 0.
xn +1 = xn ) f (x ) = 0.
The steady state is the same whatever the discretization degree h,
whatever the discretization order, whatever the kind of discretization
(backward, forward or hybrid)
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Dynamic Models in Economics
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Steady state
Systems.
Continuous time: ẋ = f (x ).
Euler-Taylor discretization: xn +1 = xn + hf (xn ) or
xn +1 = xn + hf (xn +1 ).
Steady states.
ẋ = 0 ) f (x ) = 0.
xn +1 = xn ) f (x ) = 0.
The steady state is the same whatever the discretization degree h,
whatever the discretization order, whatever the kind of discretization
(backward, forward or hybrid)
Focus for instance on the order for a discretization in backward
looking.
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Dynamic Models in Economics
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Steady state
Systems.
Continuous time: ẋ = f (x ).
Euler-Taylor discretization: xn +1 = xn + hf (xn ) or
xn +1 = xn + hf (xn +1 ).
Steady states.
ẋ = 0 ) f (x ) = 0.
xn +1 = xn ) f (x ) = 0.
The steady state is the same whatever the discretization degree h,
whatever the discretization order, whatever the kind of discretization
(backward, forward or hybrid)
Focus for instance on the order for a discretization in backward
looking.
2
2nd: x = x + hf (x ) + f (x ) f 0 (x ) h2 ) f (x ) = 0.
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Dynamic Models in Economics
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Steady state
Systems.
Continuous time: ẋ = f (x ).
Euler-Taylor discretization: xn +1 = xn + hf (xn ) or
xn +1 = xn + hf (xn +1 ).
Steady states.
ẋ = 0 ) f (x ) = 0.
xn +1 = xn ) f (x ) = 0.
The steady state is the same whatever the discretization degree h,
whatever the discretization order, whatever the kind of discretization
(backward, forward or hybrid)
Focus for instance on the order for a discretization in backward
looking.
2
2nd: x = x + hf (x ) + f (x ) f 0 (x ) h2 ) f (x ) = 0.
2
3rd:h x = x + hf (x ) + h2 f (x ) f 0i(x ) +
h3
6
f (x ) f 0 (x )2 + f (x )2 f 00 (x ) ) f (x ) = 0.
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Dynamic Models in Economics
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Topological equivalence: stability issue
Theorem
(backward-looking discretization) Consider h > 0.
(1) Let the steady state be a sink in continuous time (…gure 1).
(1.1) If T02 < 4D0 , then the steady state is a sink in discrete time if
h < hH 1 and a source if hH 1 < h.
(1.2) If T02 > 4D0 , then the steady state is a sink if 0 < h < hF 1 , a saddle
if hF 1 < h < hF 2 and source if hF 2 < h.
(2) If the steady state is a saddle in continuous time, then the steady state
is a saddle in discrete time if 0 < h < hF 2 and source if hF 2 < h (…gure 2).
(3) If the steady state is a source in continuous time, then the source
property is preserved whatever h > 0 (…gure 3).The system generically
undergoes a Hopf bifurcation at hH 1 and ‡ip bifurcations at hFi , i = 1, 2.
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Dynamic Models in Economics
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Topological equivalence: stability issue
Backward-looking discretization:
2 .5
2 .5
0
-2 .5
2 .5
0
0
2 .5
-2 .5
0
0
2 .5
-2 .5
0
2 .5
-2 .5
-2 .5
-2 .5
Fig. 1: Sink in CT
Fig. 2: Saddle in CT
Fig. 3: Source in CT
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Topological equivalence: stability issue
Backward-looking discretization:
2 .5
2 .5
0
-2 .5
2 .5
0
0
2 .5
-2 .5
0
0
2 .5
-2 .5
0
2 .5
-2 .5
-2 .5
-2 .5
Fig. 1: Sink in CT
Fig. 2: Saddle in CT
Fig. 3: Source in CT
Similar results hold in the case of a forward-looking or hybrid
discretization.
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Topological equivalence: elementary bifurcations
DT: an eigenvalue λ (p ) crosses the unit circle at p = p , bifurcation
value.
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Topological equivalence: elementary bifurcations
DT: an eigenvalue λ (p ) crosses the unit circle at p = p , bifurcation
value.
Saddle-node bifurcation: λ (p ) = +1.
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Topological equivalence: elementary bifurcations
DT: an eigenvalue λ (p ) crosses the unit circle at p = p , bifurcation
value.
Saddle-node bifurcation: λ (p ) = +1.
Hopf bifurcation: jλ (p )j = ja (p ) ib (p )j = 1 with b (p ) 6= 0.
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Topological equivalence: elementary bifurcations
DT: an eigenvalue λ (p ) crosses the unit circle at p = p , bifurcation
value.
Saddle-node bifurcation: λ (p ) = +1.
Hopf bifurcation: jλ (p )j = ja (p ) ib (p )j = 1 with b (p ) 6= 0.
Flip bifurcation: λ (p ) = 1.
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Topological equivalence: elementary bifurcations
DT: an eigenvalue λ (p ) crosses the unit circle at p = p , bifurcation
value.
Saddle-node bifurcation: λ (p ) = +1.
Hopf bifurcation: jλ (p )j = ja (p ) ib (p )j = 1 with b (p ) 6= 0.
Flip bifurcation: λ (p ) = 1.
CT: the eigenvalue real part crosses zero: µ (p ) = α (p )
with α (p ) = 0.
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i β (p )
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Topological equivalence: elementary bifurcations
DT: an eigenvalue λ (p ) crosses the unit circle at p = p , bifurcation
value.
Saddle-node bifurcation: λ (p ) = +1.
Hopf bifurcation: jλ (p )j = ja (p ) ib (p )j = 1 with b (p ) 6= 0.
Flip bifurcation: λ (p ) = 1.
CT: the eigenvalue real part crosses zero: µ (p ) = α (p )
with α (p ) = 0.
i β (p )
Saddle-node bifurcation when β (p ) = 0.
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Topological equivalence: elementary bifurcations
DT: an eigenvalue λ (p ) crosses the unit circle at p = p , bifurcation
value.
Saddle-node bifurcation: λ (p ) = +1.
Hopf bifurcation: jλ (p )j = ja (p ) ib (p )j = 1 with b (p ) 6= 0.
Flip bifurcation: λ (p ) = 1.
CT: the eigenvalue real part crosses zero: µ (p ) = α (p )
with α (p ) = 0.
i β (p )
Saddle-node bifurcation when β (p ) = 0.
Hopf bifurcation when β (p ) 6= 0.
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Topological equivalence: elementary bifurcations
DT: an eigenvalue λ (p ) crosses the unit circle at p = p , bifurcation
value.
Saddle-node bifurcation: λ (p ) = +1.
Hopf bifurcation: jλ (p )j = ja (p ) ib (p )j = 1 with b (p ) 6= 0.
Flip bifurcation: λ (p ) = 1.
CT: the eigenvalue real part crosses zero: µ (p ) = α (p )
with α (p ) = 0.
i β (p )
Saddle-node bifurcation when β (p ) = 0.
Hopf bifurcation when β (p ) 6= 0.
No room for ‡ip.
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Topological equivalence: elementary bifurcations
DT: an eigenvalue λ (p ) crosses the unit circle at p = p , bifurcation
value.
Saddle-node bifurcation: λ (p ) = +1.
Hopf bifurcation: jλ (p )j = ja (p ) ib (p )j = 1 with b (p ) 6= 0.
Flip bifurcation: λ (p ) = 1.
CT: the eigenvalue real part crosses zero: µ (p ) = α (p )
with α (p ) = 0.
i β (p )
Saddle-node bifurcation when β (p ) = 0.
Hopf bifurcation when β (p ) 6= 0.
No room for ‡ip.
One or two-dimensional analysis (Center Manifold Theorem).
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Two-dimensional systems
Two-dimensional systems (no loss of generality).
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Two-dimensional systems
Two-dimensional systems (no loss of generality).
Continuous time:
ẋ1 = f1 (x1 , x2 )
ẋ2 = f2 (x1 , x2 )
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Two-dimensional systems
Two-dimensional systems (no loss of generality).
Continuous time:
ẋ1 = f1 (x1 , x2 )
ẋ2 = f2 (x1 , x2 )
Euler-Taylor discretization:
x1n +1
x1n + hf1 (x1n , x2n )
g1 (x1n , x2n )
x2n +1
x2n + hf2 (x1n , x2n )
g2 (x1n , x2n )
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Two-dimensional systems
Two-dimensional systems (no loss of generality).
Continuous time:
ẋ1 = f1 (x1 , x2 )
ẋ2 = f2 (x1 , x2 )
Euler-Taylor discretization:
x1n +1
x1n + hf1 (x1n , x2n )
g1 (x1n , x2n )
x2n +1
x2n + hf2 (x1n , x2n )
g2 (x1n , x2n )
Jacobian matrices.
"
J0
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∂f1
∂x1n
∂f2
∂x1n
∂f1
∂x2n
∂f2
∂x2n
#
and J1
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"
∂g 1
∂x1n
∂g 2
∂x1n
∂g 1
∂x2n
∂g 2
∂x2n
#
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Two-dimensional systems
Two-dimensional systems (no loss of generality).
Continuous time:
ẋ1 = f1 (x1 , x2 )
ẋ2 = f2 (x1 , x2 )
Euler-Taylor discretization:
x1n +1
x1n + hf1 (x1n , x2n )
g1 (x1n , x2n )
x2n +1
x2n + hf2 (x1n , x2n )
g2 (x1n , x2n )
Jacobian matrices.
"
J0
∂f1
∂x1n
∂f2
∂x1n
∂f1
∂x2n
∂f2
∂x2n
#
and J1
"
∂g 1
∂x1n
∂g 2
∂x1n
∂g 1
∂x2n
∂g 2
∂x2n
#
Linearity:
J1 = I + hJ0
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Two-dimensional systems
Two-dimensional systems (no loss of generality).
Continuous time:
ẋ1 = f1 (x1 , x2 )
ẋ2 = f2 (x1 , x2 )
Euler-Taylor discretization:
x1n +1
x1n + hf1 (x1n , x2n )
g1 (x1n , x2n )
x2n +1
x2n + hf2 (x1n , x2n )
g2 (x1n , x2n )
Jacobian matrices.
"
J0
∂f1
∂x1n
∂f2
∂x1n
∂f1
∂x2n
∂f2
∂x2n
#
and J1
"
∂g 1
∂x1n
∂g 2
∂x1n
∂g 1
∂x2n
∂g 2
∂x2n
#
Linearity:
J1 = I + hJ0
I , identity, J0 does not depend on h, J1 depends linearly on h.
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Local analysis
Trace and determinant of J0 : (T0 , D0 ).
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Local analysis
Trace and determinant of J0 : (T0 , D0 ).
Trace and determinant of J1 : (T1 , D1 ).
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Local analysis
Trace and determinant of J0 : (T0 , D0 ).
Trace and determinant of J1 : (T1 , D1 ).
CT and Euler-Taylor characteristic polynomials
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P0 (λ)
λ2
T 0 λ + D0
P1 (λ)
2
T 1 λ + D1
λ
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Local analysis
Trace and determinant of J0 : (T0 , D0 ).
Trace and determinant of J1 : (T1 , D1 ).
CT and Euler-Taylor characteristic polynomials
P0 (λ)
λ2
T 0 λ + D0
P1 (λ)
2
T 1 λ + D1
λ
Links:
T1 = 2 + hT0
D1 = 1 + h (T0 + hD0 ) = T1
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1 + h 2 D0
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Saddle-node equivalence
Theorem
A saddle-node bifurcation generically occurs in CT if and only if it arises
under a Euler-Taylor discretization, whatever the discretization step h.
Sketch of the proof: D0 = 0 , D1 = T1
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1 whatever h.
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Saddle-node equivalence
Theorem
A saddle-node bifurcation generically occurs in CT if and only if it arises
under a Euler-Taylor discretization, whatever the discretization step h.
Sketch of the proof: D0 = 0 , D1 = T1
1 whatever h.
Even a rough discretization (large h) preserves the bifurcation at the
same parameter value.
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Hopf equivalence
Conditions for Hopf bifurcation in ET "tend" to conditions in CT as
the distance h between the systems vanishes.
Theorem
Assume f (x, p ) 2 C 2 and limh !0+ jdet J0 /Hp j < ∞, where the ratio is
evaluated in the steady state corresponding to a Hopf bifurcation value pH
in discrete time (under a positive discretization degree: h > 0). A Hopf
bifurcation in continuous time generically requires T0 = 0 and D0 0. In
discrete time, a Hopf bifurcation needs T0 = hD0 and D0 T02 /4: these
conditions "converge" to the corresponding conditions in continuous time
(i.e. the right-hand sides hD0 and T02 /4 go to zero) as h goes to zero.
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What about the ‡ip bifurcation?
Saddle-node persists at the same parameter value.
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What about the ‡ip bifurcation?
Saddle-node persists at the same parameter value.
Hopf persists in a neighborhood of the parameter value.
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What about the ‡ip bifurcation?
Saddle-node persists at the same parameter value.
Hopf persists in a neighborhood of the parameter value.
Flip disappears in CT.
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Critical discretization step
There exists a critical discretization degree h under which ‡ip
disappears?
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Critical discretization step
There exists a critical discretization degree h under which ‡ip
disappears?
Do we loose the ‡ip, when ET approaches CT?
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Critical discretization step
There exists a critical discretization degree h under which ‡ip
disappears?
Do we loose the ‡ip, when ET approaches CT?
Does h increase with the discretization order?
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Critical discretization step
There exists a critical discretization degree h under which ‡ip
disappears?
Do we loose the ‡ip, when ET approaches CT?
Does h increase with the discretization order?
Cases with and without h > 0.
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Dynamic Models in Economics
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Critical discretization step
There exists a critical discretization degree h under which ‡ip
disappears?
Do we loose the ‡ip, when ET approaches CT?
Does h increase with the discretization order?
Cases with and without h > 0.
Su¢ cient conditions for a lower bound.
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Critical discretization step
There exists a critical discretization degree h under which ‡ip
disappears?
Do we loose the ‡ip, when ET approaches CT?
Does h increase with the discretization order?
Cases with and without h > 0.
Su¢ cient conditions for a lower bound.
Hint: bounded eigenvalues.
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Critical discretization step
There exists a critical discretization degree h under which ‡ip
disappears?
Do we loose the ‡ip, when ET approaches CT?
Does h increase with the discretization order?
Cases with and without h > 0.
Su¢ cient conditions for a lower bound.
Hint: bounded eigenvalues.
Bounded parameter support.
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Critical discretization step
There exists a critical discretization degree h under which ‡ip
disappears?
Do we loose the ‡ip, when ET approaches CT?
Does h increase with the discretization order?
Cases with and without h > 0.
Su¢ cient conditions for a lower bound.
Hint: bounded eigenvalues.
Bounded parameter support.
Unbounded support.
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Flip bifurcation
Order one.
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Flip bifurcation
Order one.
Dimension one: 1 + hfx (x (p ) , p ) =
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1.
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Flip bifurcation
Order one.
Dimension one: 1 + hfx (x (p ) , p ) = 1.
Dimension two: D1 = T1 1 or D0 h2 + 2T0 h + 4 = 0.
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Flip bifurcation
Order one.
Dimension one: 1 + hfx (x (p ) , p ) = 1.
Dimension two: D1 = T1 1 or D0 h2 + 2T0 h + 4 = 0.
Order two.
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Flip bifurcation
Order one.
Dimension one: 1 + hfx (x (p ) , p ) = 1.
Dimension two: D1 = T1 1 or D0 h2 + 2T0 h + 4 = 0.
Order two.
Dimension one:
2 = hfx (x (p ) , p ) +
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i
h2 h
fx (x (p ) , p )2 + f (x (p ) , p ) fxx (x (p ) , p )
2
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Discretization threshold (order one).
Theorem
Let f 2 C 1 and Y
fx (X (P )), where X (P ) is the graph of the steady
states correspondence. If ∞ < inf Y , there exists a nonempty
discretization range (0, h ) with h
j 2/ inf Y j, where no ‡ip
bifurcation arises.
Corollary
Let f 2 C 1 and Z
fx (S P ), where S is the domain of x. If
∞ < inf Z , then there exists a discretization range (0, h ) with
h
j 2/ inf Z j with no ‡ip bifurcation.
Lower boundedness required.
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Dynamic Models in Economics
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Discretization threshold (order two)
Theorem
Let f 2 C 3 on S
P and f , fx , fxx be bounded over X (P ). Then there
exists a nonempty discretization range (0, h ), where generically no ‡ip
bifurcation arises.
Theorem
If P is compact and, for every (x, p ) 2 S P, f 2 C 2 and fx is nonzero,
then there are no ‡ip bifurcations in the interval (0, h ).
Hint. Compactness + IFT.
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Discretization threshold (order two)
Theorem
Let f 2 C 3 on S
P and f , fx , fxx be bounded over X (P ). Then there
exists a nonempty discretization range (0, h ), where generically no ‡ip
bifurcation arises.
Theorem
If P is compact and, for every (x, p ) 2 S P, f 2 C 2 and fx is nonzero,
then there are no ‡ip bifurcations in the interval (0, h ).
Hint. Compactness + IFT.
Order three: same lines.
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Solow models
Continuous time (original system):
K̇t
L̇t
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= sF (Kt , Lt )
= n0 Lt
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δKt
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Solow models
Continuous time (original system):
K̇t
L̇t
= sF (Kt , Lt )
= n0 Lt
δKt
Discrete time:
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Kt + 1
Kt
Lt +1
Lt
= sF (Kt , Lt )
= n1 Lt
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δKt
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Solow: intensive laws
Solow
continuous time
intensive law
k̇t = sf (kt ) (δ + n0 ) kt
discrete time
kt +1 =
lin. discr. OS
kn +1
lin. discr. RF
kn +1
quadr. discr. OS
quadr. discr. RF
kn +1
kn +1
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1 δ
1 +n 1 kt
+
1 hδ
1 +hn 0 kn
s
1 +n 1 f
+
hs
1 +hn 0 f
kn + h [sf (kn )
kn +
2
h + h2
(kt )
(kn )
(δ + n0 ) kn ]
2
[sf 0 (kn ) δ] [sf (kn ) δkn ]+ h2 sn 0 [f (kn ) kn f 0 (kn )
1 +hn 0 +
kn + [sf (kn )
(hn 0 )2
2
(δ + n0 ) kn ] h +
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h2
2
[sf 0 (kn )
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(
Steady state in Solow models
The steady state:
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Steady state in Solow models
The steady state:
in continuous time,
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Steady state in Solow models
The steady state:
in continuous time,
in discrete time,
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Steady state in Solow models
The steady state:
in continuous time,
in discrete time,
under a linear or quadratic discretization of the original system,
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Steady state in Solow models
The steady state:
in continuous time,
in discrete time,
under a linear or quadratic discretization of the original system,
under a linear or quadratic discretization of the intensive law,
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Steady state in Solow models
The steady state:
in continuous time,
in discrete time,
under a linear or quadratic discretization of the original system,
under a linear or quadratic discretization of the intensive law,
is always given by
f (k )
n+δ
=
k
s
(with n0 = n1
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n).
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Local bifurcations
Solow
continuous time
discrete time
bifurcations
no
no
linear discretization original system
hF =
2
(1 a )(δ+n 0 )
2
(1 a )(δ+n 0 )
linear discretization intensive law
hF =
quadratic discretization original system
quadratic discretization intensive law
no ‡ip
no ‡ip
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Solow with negative externalities
Richer dynamics (‡ip).
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Solow with negative externalities
Richer dynamics (‡ip).
Original systems (two-dimensional dynamics).
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Solow with negative externalities
Richer dynamics (‡ip).
Original systems (two-dimensional dynamics).
Continuous time:
K̇t
= sAKta Lt1
L̇t
= n0 Lt
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α
h
m
(Kt /Lt )1
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a
i
δKt
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Solow with negative externalities
Richer dynamics (‡ip).
Original systems (two-dimensional dynamics).
Continuous time:
h
m
Kt + 1
= Kt + sAKta L1t
α
Lt +1
= (1 + n1 ) Lt
K̇t
= sAKta Lt1
L̇t
= n0 Lt
α
(Kt /Lt )1
a
i
δKt
Discrete time:
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h
m
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(Kt /Lt )1
a
i
δKt
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Laws of motion
Solow + extern.
continuous time
intensive law
k̇t
sAkta m
discrete time
kt +1 =
lin. discr. OS
kn +1
lin. discr. RF
kn +1
quadr. discr. RF
kn +1
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a
1 (δ+sA )
1 +n 1 kt
+
(δ + n0 ) kt
sA
a
1 +n 1 mkt
1 h (δ+sA )
hsA
a
1 +hn 0 kn + 1 +hn 0 mkn
kn + h sAkna m kn1 a
kn +1 =
quadr. discr. OS
kt1
kn +
2
h + h2
[
asAkna 1 m
kn + [msAkna
(n0 + δ) kn
2
(δ+sA )] [sAkna m (δ+sA )kn ]+ h2 n 0 (
1 +hn 0 +
(hn 0 )2
2
(δ + n0 + sA) kn ] h +
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h 2 [amsAk na
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Steady state in Solow models with negative externalities
The steady state:
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Steady state in Solow models with negative externalities
The steady state:
in continuous time,
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Steady state in Solow models with negative externalities
The steady state:
in continuous time,
in discrete time,
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Steady state in Solow models with negative externalities
The steady state:
in continuous time,
in discrete time,
under a linear or quadratic discretization of the original system,
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Dynamic Models in Economics
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Steady state in Solow models with negative externalities
The steady state:
in continuous time,
in discrete time,
under a linear or quadratic discretization of the original system,
under a linear or quadratic discretization of the intensive law,
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Dynamic Models in Economics
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Steady state in Solow models with negative externalities
The steady state:
in continuous time,
in discrete time,
under a linear or quadratic discretization of the original system,
under a linear or quadratic discretization of the intensive law,
is always given by
k1 =
Stefano BOSI (University of Evry)
sA
m
n + δ + sA
Dynamic Models in Economics
1
1 a
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Local bifurcations in Solow models with negative
externalities
Solow + externalities
continuous time
bifurcations
no
discrete time
1
s
AF
linear discretization OS
1
s
AF
1
s
linear discretization RF
AF =
quadratic discretization OS
quadratic discretization RF
no ‡ip
no ‡ip
Stefano BOSI (University of Evry)
h
h
2
1 a
(1 + n1 )
2 1 +hn 0
1 a
h
2
(1 a )h
Dynamic Models in Economics
n0
(n1 + δ)
i
(n0 + δ)
i
δ
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Cass-Koopmans
Continuous time:
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R∞
0
βt u (ct ) dt.
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Cass-Koopmans
Discrete time:
R∞
βt u (ct ) dt.
0
∞
∑ t = 0 β t u ( ct ) .
Continuous time:
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CK in continuous time
System with multiplier:
k̇t
λ̇t
= f (kt ) δkt
=
λt f 0 (kt )
ct ( λ t )
δ
where λt = βt u 0 (ct ).
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CK in discrete time
System with consumption:
kt +1 kt
u 0 ( ct )
u 0 ( ct + 1 )
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= f (kt ) δkt ct
β t +1
=
1 + f 0 (kt +1 )
βt
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δ
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First-order discretization
The DT system is recovered under a hybrid discretization.
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First-order discretization
The DT system is recovered under a hybrid discretization.
What hybridization?
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First-order discretization
The DT system is recovered under a hybrid discretization.
What hybridization?
Backward-looking discretization of the resource constraint.
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First-order discretization
The DT system is recovered under a hybrid discretization.
What hybridization?
Backward-looking discretization of the resource constraint.
Forward-looking discretization of the Euler equation.
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First-order discretization
The DT system is recovered under a hybrid discretization.
What hybridization?
Backward-looking discretization of the resource constraint.
Forward-looking discretization of the Euler equation.
The resource constraint behaves as in Solow:
kt +h
Stefano BOSI (University of Evry)
kt
h [f (kt )
Dynamic Models in Economics
δkt
ct ]
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First-order discretization
The DT system is recovered under a hybrid discretization.
What hybridization?
Backward-looking discretization of the resource constraint.
Forward-looking discretization of the Euler equation.
The resource constraint behaves as in Solow:
kt +h
kt
h [f (kt )
δkt
ct ]
h = 1 ) discrete-time form.
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First-order discretization
The DT system is recovered under a hybrid discretization.
What hybridization?
Backward-looking discretization of the resource constraint.
Forward-looking discretization of the Euler equation.
The resource constraint behaves as in Solow:
kt +h
kt
h [f (kt )
δkt
ct ]
h = 1 ) discrete-time form.
The added value is Euler.
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Discretizations of Euler equations
Equivalent equations:
λ̇t
µ̇t
=
λt f 0 (kt ) δ
= µt ρt + δ f 0 (kt )
where µt = u 0 (ct ).
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The equivalence issue
Euler equations:
Euler
looking
λ-type
F
B
µ-type
F
B
discretization
µt
µ t +h
µt
µ t +h
µt
µ t +h
µt
µ t +h
β t +h
βt
(1 + h [f 0 (kt +h )
β t +h
1
βt 1 h [f 0 (k t ) δ ]
1 + h [f 0 (kt +h ) δ]
1
β β
1 h [f 0 (k t ) δ]+ t β t +h
δ])
βt
β t +h
β t +h
t
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The equivalence issue
Euler equations:
Euler
looking
λ-type
F
B
µ-type
F
B
discretization
µt
µ t +h
µt
µ t +h
µt
µ t +h
µt
µ t +h
β t +h
βt
(1 + h [f 0 (kt +h )
β t +h
1
βt 1 h [f 0 (k t ) δ ]
1 + h [f 0 (kt +h ) δ]
1
β β
1 h [f 0 (k t ) δ]+ t β t +h
δ])
βt
β t +h
β t +h
t
If h = 1: λ-type + forward-looking Euler-Taylor = discrete time.
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Second-order discretization
For simplicity, hybrid.
BL resource constraint + FL λ-type Euler.
kt +h
µt
µ t +h
kt
h [f (kt )
β t +h
βt
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δkt
[f 0 (kt ) δ](f (kt ) δkt ct [1 +ε(ct )])
2
2
h 2 ([f 0 (kt +h ) δ] [f (kt +h ) δkt +h ct +
δ] +
2
ct ] + h 2
1 + h [f 0 (kt +h )
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Second-order discretization
Could we think a DT second-order Cass-Koopmans model?
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Second-order discretization
Could we think a DT second-order Cass-Koopmans model?
Set h = 1:
kt +1
kt
µt
µ t +1
Stefano BOSI (University of Evry)
f (kt )
β t +1
βt
δkt
ct +
1 + f 0 (kt +1 )
[f 0 (kt ) δ](f (kt ) δkt ct [1 +ε(ct )])
2
[f 0 (kt +1 ) δ]2 [f (kt +1 ) δkt +1 ct +1 ]
δ+
2
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Steady state
MGR: f 0 (k ) = ρ + δ under constant ρt =
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ln β.
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Local dynamics
At the SS.
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Dynamic Models in Economics
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Local dynamics
At the SS.
µt stationary,
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Dynamic Models in Economics
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Local dynamics
At the SS.
µt stationary,
λt = βt µt decreasing.
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Local dynamics
At the SS.
µt stationary,
λt = βt µt decreasing.
Focus on µ-type CT:
k̇t
µ̇t
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= f (kt ) δkt u 0 1 (µt )
= µt ρt + δ f 0 (kt )
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Local dynamics
At the SS.
µt stationary,
λt = βt µt decreasing.
Focus on µ-type CT:
k̇t
µ̇t
= f (kt ) δkt u 0 1 (µt )
= µt ρt + δ f 0 (kt )
Hybrid discretization:
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Local dynamics
At the SS.
µt stationary,
λt = βt µt decreasing.
Focus on µ-type CT:
k̇t
µ̇t
= f (kt ) δkt u 0 1 (µt )
= µt ρt + δ f 0 (kt )
Hybrid discretization:
BL resource constraint,
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Local dynamics
At the SS.
µt stationary,
λt = βt µt decreasing.
Focus on µ-type CT:
= f (kt ) δkt u 0 1 (µt )
= µt ρt + δ f 0 (kt )
k̇t
µ̇t
Hybrid discretization:
BL resource constraint,
+ FL µ-type Euler.
kt +h
kt
µt
µ t +h
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h [f (kt ) δkt ct ]
β t +h
1 + h f 0 (kt +h )
βt
Dynamic Models in Economics
δ
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Continuous time
Jacobian matrix:
J0 =
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ρ
µ
AB c ε
cε
µ
0
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Continuous time
Jacobian matrix:
J0 =
ρ
µ
AB c ε
cε
µ
0
A, B, ε > 0 ) saddle-path stability.
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Continuous time
Jacobian matrix:
J0 =
ρ
µ
AB c ε
cε
µ
0
A, B, ε > 0 ) saddle-path stability.
ρ = 0: Ramsey (bliss point).
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Euler-Taylor Jacobian
Jacobian matrix:
J1
Stefano BOSI (University of Evry)
"
1 + hρ
hA
2
hB
1 + h1 +AB
hρ
Dynamic Models in Economics
#
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Euler-Taylor Jacobian
Jacobian matrix:
J1
"
1 + hρ
hA
2
hB
1 + h1 +AB
hρ
#
Whatever the discretization step,
Stefano BOSI (University of Evry)
Dynamic Models in Economics
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Euler-Taylor Jacobian
Jacobian matrix:
J1
"
1 + hρ
hA
2
hB
1 + h1 +AB
hρ
#
Whatever the discretization step,
D1 = β
h
> 1 and
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Dynamic Models in Economics
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Euler-Taylor Jacobian
Jacobian matrix:
J1
"
1 + hρ
hA
2
hB
1 + h1 +AB
hρ
#
Whatever the discretization step,
D1 = β h > 1 and
D1 = T1 1 h2 AB/ (1 + hρ) < T1
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Dynamic Models in Economics
1
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Euler-Taylor Jacobian
Jacobian matrix:
J1
"
1 + hρ
hA
2
hB
1 + h1 +AB
hρ
#
Whatever the discretization step,
D1 = β h > 1 and
D1 = T1 1 h2 AB/ (1 + hρ) < T1
) saddle.
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1
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Cass-Koopmans model with externalities
Positive externalities of public good in production and utility
functions.
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Dynamic Models in Economics
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Cass-Koopmans model with externalities
Positive externalities of public good in production and utility
functions.
We generalize Zhang (2000) in continuous time.
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Dynamic Models in Economics
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Cass-Koopmans model with externalities
Positive externalities of public good in production and utility
functions.
We generalize Zhang (2000) in continuous time.
Discrete-time version provided.
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Dynamic Models in Economics
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Cass-Koopmans model with externalities
Positive externalities of public good in production and utility
functions.
We generalize Zhang (2000) in continuous time.
Discrete-time version provided.
We recover the continuity property of the Hopf bifurcation.
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CT Ramsey (1928)
Ramsey argued against discounting utility of future generations as
being "ethically indefensible".
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CT Ramsey (1928)
Ramsey argued against discounting utility of future generations as
being "ethically indefensible".
This "ethical" undiscounted utility functional in Ramsey (1928) is
replaced in the Cass-Koopmans (1965) by a weighted average of
future felicities (discounting).
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CT Ramsey (1928)
Ramsey argued against discounting utility of future generations as
being "ethically indefensible".
This "ethical" undiscounted utility functional in Ramsey (1928) is
replaced in the Cass-Koopmans (1965) by a weighted average of
future felicities (discounting).
Utility functional
Z ∞
0
[ u ( ct )
u (c )] dt
subject to the resource constraint
k̇t + δkt + ct
f (kt )
given the initial endowment k0 .
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CT Ramsey (1928) (cont.)
ε (c )
u 0 (c ) / [u 00 (c ) c ] > 0 denotes the elasticity of
intertemporal substitution.
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CT Ramsey (1928) (cont.)
ε (c )
u 0 (c ) / [u 00 (c ) c ] > 0 denotes the elasticity of
intertemporal substitution.
Hamiltonian:
Ht = u ( c t )
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u (c ) + µt [f (kt )
Dynamic Models in Economics
δkt
ct ]
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CT Ramsey (1928) (cont.)
ε (c )
u 0 (c ) / [u 00 (c ) c ] > 0 denotes the elasticity of
intertemporal substitution.
Hamiltonian:
Ht = u ( c t )
u (c ) + µt [f (kt )
δkt
ct ]
Dynamic system
k̇t
µ̇t
= f (kt ) δkt
=
µt f 0 (kt )
c ( µt )
(1)
δ
(2)
+ TC.
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DT Ramsey (1928)
The planner maximizes the intertemporal utility
∞
∑ [ u ( ct )
u (c )]
t =0
subject to the sequence of resource constraints:
kt +1
kt + δkt + ct
f (kt )
where c is the bliss point de…ned above.
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DT Ramsey (1928) (cont.)
The optimal dynamical system is:
kt +1
Stefano BOSI (University of Evry)
kt
µt
µ t +1
= f (kt )
δkt
= 1 + f 0 (kt +1 )
Dynamic Models in Economics
c ( µt )
(3)
(4)
δ
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DT Ramsey (1928) (cont.)
The optimal dynamical system is:
kt +1
kt
µt
µ t +1
= f (kt )
δkt
= 1 + f 0 (kt +1 )
c ( µt )
(3)
(4)
δ
Question: could the discrete-time system (3)-(4) can be recovered
through a discretization of the continuous-time system (1)-(2)?
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DT Ramsey (1928) (cont.)
The optimal dynamical system is:
kt +1
kt
µt
µ t +1
= f (kt )
δkt
= 1 + f 0 (kt +1 )
c ( µt )
(3)
(4)
δ
Question: could the discrete-time system (3)-(4) can be recovered
through a discretization of the continuous-time system (1)-(2)?
Yes, if hybrid discretization.
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DT Ramsey (1928) (cont.)
The optimal dynamical system is:
kt +1
kt
µt
µ t +1
= f (kt )
δkt
= 1 + f 0 (kt +1 )
c ( µt )
(3)
(4)
δ
Question: could the discrete-time system (3)-(4) can be recovered
through a discretization of the continuous-time system (1)-(2)?
Yes, if hybrid discretization.
The discrete-time Ramsey model comes from a …rst-order hybrid
Euler discretization of the continuous-time model, that is a
backward-looking discretization of the resource constraint (1) and a
forward-looking discretization of the Euler equation (2).
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DT Ramsey (1928) (cont.)
Backward-looking linear discretization of the continuous-time resource
constraint (1):
kt +h
kt
h [f (kt )
δkt
c (µt )]
we recover exactly the discrete-time resource constraint (3) with a
unit discretization step (h = 1).
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DT Ramsey (1928) (cont.)
Backward-looking linear discretization of the continuous-time resource
constraint (1):
kt +h
kt
h [f (kt )
δkt
c (µt )]
we recover exactly the discrete-time resource constraint (3) with a
unit discretization step (h = 1).
But the intertemporal arbitrage can not be recovered in
backward-looking. Focus on (2) and apply instead the forward-looking
discretization (??): µt +h µt = hµt +h [f 0 (kt +h ) δ]. We obtain
µt
1 + h f 0 (kt +h ) δ
(5)
µ t +h
which gives exactly the discrete-time Euler equation (4) under a unit
discretization step h = 1.
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DT Ramsey (1928) (cont.)
Backward-looking linear discretization of the continuous-time resource
constraint (1):
kt +h
kt
h [f (kt )
δkt
c (µt )]
we recover exactly the discrete-time resource constraint (3) with a
unit discretization step (h = 1).
But the intertemporal arbitrage can not be recovered in
backward-looking. Focus on (2) and apply instead the forward-looking
discretization (??): µt +h µt = hµt +h [f 0 (kt +h ) δ]. We obtain
µt
1 + h f 0 (kt +h ) δ
(5)
µ t +h
which gives exactly the discrete-time Euler equation (4) under a unit
discretization step h = 1.
The forward-looking discretization of (2) is more adapted to capture
the saving decision, the expected productivity (interest rate) matters
in the current trade-o¤ between consumption and saving.
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DT Ramsey (1928) (cont.)
For all the three dynamical systems (CT, DT, HD) the steady state is
de…ned by:
f 0 (k ) = δ
c
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= c ( µ ) = f (k )
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(6)
δk
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(7)
64 / 65
DT Ramsey (1928) (cont.)
For all the three dynamical systems (CT, DT, HD) the steady state is
de…ned by:
f 0 (k ) = δ
c
(6)
= c ( µ ) = f (k )
δk
(7)
Let J0 be the Jacobian matrix of the continuous time system (1)-(2).
Evaluating derivatives in the steady state gives:
J0 =
where A
εδ (1
Stefano BOSI (University of Evry)
0
µf 00
(k )
1/u 00 (c )
0
α) /α > 0 and B
=
δ (1
Dynamic Models in Economics
0
Ak/µ
Bµ/k
0
(8)
α) /σ > 0.
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DT Ramsey (1928) (cont.)
For all the three dynamical systems (CT, DT, HD) the steady state is
de…ned by:
f 0 (k ) = δ
c
(6)
= c ( µ ) = f (k )
δk
(7)
Let J0 be the Jacobian matrix of the continuous time system (1)-(2).
Evaluating derivatives in the steady state gives:
J0 =
0
µf 00
(k )
1/u 00 (c )
0
=
0
Ak/µ
Bµ/k
0
(8)
where A εδ (1 α) /α > 0 and B δ (1 α) /σ > 0.
The trace and the determinant in continuous time are T0 = 0 and
D0 = AB < 0. D0 < 0 entails the saddle-path stability property.
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DT Ramsey (1928) (cont.)
For all the three dynamical systems (CT, DT, HD) the steady state is
de…ned by:
f 0 (k ) = δ
c
(6)
= c ( µ ) = f (k )
(7)
δk
Let J0 be the Jacobian matrix of the continuous time system (1)-(2).
Evaluating derivatives in the steady state gives:
J0 =
0
µf 00
(k )
1/u 00 (c )
0
=
0
Ak/µ
Bµ/k
0
(8)
where A εδ (1 α) /α > 0 and B δ (1 α) /σ > 0.
The trace and the determinant in continuous time are T0 = 0 and
D0 = AB < 0. D0 < 0 entails the saddle-path stability property.
The associated Jacobian matrix J1 of the hybrid Euler discretization
J1
1
0
hµf 00 (k ) 1
1
1
0
h/u 00 (c )
1
with A and B de…ned asDynamic
above.
Models in Economics
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=
1
hAk/µ
hBµ/k 1 + ABh2
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Conclusions
The DT version of Solow corresponds to a linear discretization of the
original system with a unit discretization step: h = 1.
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Conclusions
The DT version of Solow corresponds to a linear discretization of the
original system with a unit discretization step: h = 1.
The DT version of Ramsey-Cass-Koopmans corresponds to a λ-type
hybrid discretization with h = 1.
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Conclusions
The DT version of Solow corresponds to a linear discretization of the
original system with a unit discretization step: h = 1.
The DT version of Ramsey-Cass-Koopmans corresponds to a λ-type
hybrid discretization with h = 1.
A quadratic discretization is enough to rule out the emergence of
two-period cycles in the Solow model even in presence of negative
externalities.
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Dynamic Models in Economics
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Conclusions
The DT version of Solow corresponds to a linear discretization of the
original system with a unit discretization step: h = 1.
The DT version of Ramsey-Cass-Koopmans corresponds to a λ-type
hybrid discretization with h = 1.
A quadratic discretization is enough to rule out the emergence of
two-period cycles in the Solow model even in presence of negative
externalities.
Robustness of saddle-path stability in a hybrid Euler-Taylor
discretization of (a µ-type) RCK whatever the discretization step.
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Conclusions
The DT version of Solow corresponds to a linear discretization of the
original system with a unit discretization step: h = 1.
The DT version of Ramsey-Cass-Koopmans corresponds to a λ-type
hybrid discretization with h = 1.
A quadratic discretization is enough to rule out the emergence of
two-period cycles in the Solow model even in presence of negative
externalities.
Robustness of saddle-path stability in a hybrid Euler-Taylor
discretization of (a µ-type) RCK whatever the discretization step.
Examples of ‡ip bifurcation in a Solow model with externalities and of
Hopf bifurcations in a RCK model with externalities.
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Conclusions
The DT version of Solow corresponds to a linear discretization of the
original system with a unit discretization step: h = 1.
The DT version of Ramsey-Cass-Koopmans corresponds to a λ-type
hybrid discretization with h = 1.
A quadratic discretization is enough to rule out the emergence of
two-period cycles in the Solow model even in presence of negative
externalities.
Robustness of saddle-path stability in a hybrid Euler-Taylor
discretization of (a µ-type) RCK whatever the discretization step.
Examples of ‡ip bifurcation in a Solow model with externalities and of
Hopf bifurcations in a RCK model with externalities.
These examples illustrate the equivalence theorems.
Stefano BOSI (University of Evry)
Dynamic Models in Economics
CIMPA 2014, Tlemcen
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