Discrete- and Continuous-time Models in
Economics
Some Methodological Considerations
Alfredo Medio
Department of Statistics
University of Udine
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
What we will not discuss
Grand questions about time
Philosophy: What is the “true” nature of time
Physics: Is the physical world better described by
continuous– or discrete– time theories?
Economics: What is “the best” way to treat time in
economic models?
Psychology (?): How is it that the discussion about discrete
versus continuous–time generates so much heat among
economists while very few of us object to treating
economic variables, most of which are obviously discrete
quantities, as real numbers?
etc.
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
What we will not discuss
Grand questions about time
Philosophy: What is the “true” nature of time
Physics: Is the physical world better described by
continuous– or discrete– time theories?
Economics: What is “the best” way to treat time in
economic models?
Psychology (?): How is it that the discussion about discrete
versus continuous–time generates so much heat among
economists while very few of us object to treating
economic variables, most of which are obviously discrete
quantities, as real numbers?
etc.
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
What we will not discuss
Grand questions about time
Philosophy: What is the “true” nature of time
Physics: Is the physical world better described by
continuous– or discrete– time theories?
Economics: What is “the best” way to treat time in
economic models?
Psychology (?): How is it that the discussion about discrete
versus continuous–time generates so much heat among
economists while very few of us object to treating
economic variables, most of which are obviously discrete
quantities, as real numbers?
etc.
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
What we will not discuss
Grand questions about time
Philosophy: What is the “true” nature of time
Physics: Is the physical world better described by
continuous– or discrete– time theories?
Economics: What is “the best” way to treat time in
economic models?
Psychology (?): How is it that the discussion about discrete
versus continuous–time generates so much heat among
economists while very few of us object to treating
economic variables, most of which are obviously discrete
quantities, as real numbers?
etc.
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
What we will not discuss
Grand questions about time
Philosophy: What is the “true” nature of time
Physics: Is the physical world better described by
continuous– or discrete– time theories?
Economics: What is “the best” way to treat time in
economic models?
Psychology (?): How is it that the discussion about discrete
versus continuous–time generates so much heat among
economists while very few of us object to treating
economic variables, most of which are obviously discrete
quantities, as real numbers?
etc.
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
What we will not discuss
Grand questions about time
Philosophy: What is the “true” nature of time
Physics: Is the physical world better described by
continuous– or discrete– time theories?
Economics: What is “the best” way to treat time in
economic models?
Psychology (?): How is it that the discussion about discrete
versus continuous–time generates so much heat among
economists while very few of us object to treating
economic variables, most of which are obviously discrete
quantities, as real numbers?
etc.
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Time in economic models
Choice and consequences
Let assume that the decision about treatment of time is
taken (on the basis of theory, chance, or simply because of
the mathematics we know better). The following questions
then arise:
What are the implications of our choice of time
representation in economic models? Are there any results
in our analysis that crucially depend on that choice?
More specifically: Is the choice of the period time-length
important, especially if we take the discrete-time option?
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Time in economic models
Choice and consequences
Let assume that the decision about treatment of time is
taken (on the basis of theory, chance, or simply because of
the mathematics we know better). The following questions
then arise:
What are the implications of our choice of time
representation in economic models? Are there any results
in our analysis that crucially depend on that choice?
More specifically: Is the choice of the period time-length
important, especially if we take the discrete-time option?
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Time in economic models
Choice and consequences
Let assume that the decision about treatment of time is
taken (on the basis of theory, chance, or simply because of
the mathematics we know better). The following questions
then arise:
What are the implications of our choice of time
representation in economic models? Are there any results
in our analysis that crucially depend on that choice?
More specifically: Is the choice of the period time-length
important, especially if we take the discrete-time option?
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Time in economic models
Choice and consequences
Let assume that the decision about treatment of time is
taken (on the basis of theory, chance, or simply because of
the mathematics we know better). The following questions
then arise:
What are the implications of our choice of time
representation in economic models? Are there any results
in our analysis that crucially depend on that choice?
More specifically: Is the choice of the period time-length
important, especially if we take the discrete-time option?
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Zooming in further: time representation and chaos
Some interesting questions
Computational chaos (cf. E. Lorenz): could complex
dynamics be the result of an incorrect discrete
representation of phenomena?
Observational chaos: could dynamic complexity arise from
a correct but discontinuous observation of phenomena?
Synchronization chaos: could chaos be the result of
incorrect representation of lags?
Many years ago (1991) Segio Invernizzi and I had
something to say on the third question. In the limited time
allotted to my presentation today I will share with you some
ideas concerning the first two questions
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Zooming in further: time representation and chaos
Some interesting questions
Computational chaos (cf. E. Lorenz): could complex
dynamics be the result of an incorrect discrete
representation of phenomena?
Observational chaos: could dynamic complexity arise from
a correct but discontinuous observation of phenomena?
Synchronization chaos: could chaos be the result of
incorrect representation of lags?
Many years ago (1991) Segio Invernizzi and I had
something to say on the third question. In the limited time
allotted to my presentation today I will share with you some
ideas concerning the first two questions
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Zooming in further: time representation and chaos
Some interesting questions
Computational chaos (cf. E. Lorenz): could complex
dynamics be the result of an incorrect discrete
representation of phenomena?
Observational chaos: could dynamic complexity arise from
a correct but discontinuous observation of phenomena?
Synchronization chaos: could chaos be the result of
incorrect representation of lags?
Many years ago (1991) Segio Invernizzi and I had
something to say on the third question. In the limited time
allotted to my presentation today I will share with you some
ideas concerning the first two questions
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Zooming in further: time representation and chaos
Some interesting questions
Computational chaos (cf. E. Lorenz): could complex
dynamics be the result of an incorrect discrete
representation of phenomena?
Observational chaos: could dynamic complexity arise from
a correct but discontinuous observation of phenomena?
Synchronization chaos: could chaos be the result of
incorrect representation of lags?
Many years ago (1991) Segio Invernizzi and I had
something to say on the third question. In the limited time
allotted to my presentation today I will share with you some
ideas concerning the first two questions
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Zooming in further: time representation and chaos
Some interesting questions
Computational chaos (cf. E. Lorenz): could complex
dynamics be the result of an incorrect discrete
representation of phenomena?
Observational chaos: could dynamic complexity arise from
a correct but discontinuous observation of phenomena?
Synchronization chaos: could chaos be the result of
incorrect representation of lags?
Many years ago (1991) Segio Invernizzi and I had
something to say on the third question. In the limited time
allotted to my presentation today I will share with you some
ideas concerning the first two questions
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Time-interval as a parameter
The optimal growth model: one–dimensional case
This fundamental dynamical question has been treated by
economists both in continuous and in discrete–time
models. Very seldom do we find in the literature any
explanation for the choice between different
representations of time
The continuous–time model
R∞
sup t=0 V [x(t), ẋ(t)]e−rt
(x, ẋ) ∈ T ⊂ X × Y , x(0) = x0
(1)
Policy function
ẋ = f (x),
Alfredo Medio et al.
x(0) = x0
(2)
Discrete- and Continuous-time Models in Economics
Time-interval as a parameter
The optimal growth model: one–dimensional case
This fundamental dynamical question has been treated by
economists both in continuous and in discrete–time
models. Very seldom do we find in the literature any
explanation for the choice between different
representations of time
The continuous–time model
R∞
sup t=0 V [x(t), ẋ(t)]e−rt
(x, ẋ) ∈ T ⊂ X × Y , x(0) = x0
(1)
Policy function
ẋ = f (x),
Alfredo Medio et al.
x(0) = x0
(2)
Discrete- and Continuous-time Models in Economics
Time-interval as a parameter
The optimal growth model: one–dimensional case
This fundamental dynamical question has been treated by
economists both in continuous and in discrete–time
models. Very seldom do we find in the literature any
explanation for the choice between different
representations of time
The continuous–time model
R∞
sup t=0 V [x(t), ẋ(t)]e−rt
(x, ẋ) ∈ T ⊂ X × Y , x(0) = x0
(1)
Policy function
ẋ = f (x),
Alfredo Medio et al.
x(0) = x0
(2)
Discrete- and Continuous-time Models in Economics
Time-interval as a parameter
The optimal growth model: one–dimensional case
This fundamental dynamical question has been treated by
economists both in continuous and in discrete–time
models. Very seldom do we find in the literature any
explanation for the choice between different
representations of time
The continuous–time model
R∞
sup t=0 V [x(t), ẋ(t)]e−rt
(x, ẋ) ∈ T ⊂ X × Y , x(0) = x0
(1)
Policy function
ẋ = f (x),
Alfredo Medio et al.
x(0) = x0
(2)
Discrete- and Continuous-time Models in Economics
The discretized model
Discretization
t
= n∆t
ẋ(n∆t) ≈ x[(n+1)∆t]−x(n∆t)
∆t
β(∆t) = (1 + r ∆t)−1/∆t ,







(3)
lim∆t→0 =
e−r
P
n
sup ∞
n=0 V∆t (xn,∆t , xn+1,∆t )β(∆t)
(xn,∆t , xn+1,∆t ) ∈ T 0 ⊂ X × X
xn,∆t ≈ x(n∆t)
−x V∆t (x, y ) = V x, y∆t
(4)
The policy function
xn+1,∆t = h∆t (xn,∆t )
Alfredo Medio et al.
(5)
Discrete- and Continuous-time Models in Economics
The discretized model
Discretization
t
= n∆t
ẋ(n∆t) ≈ x[(n+1)∆t]−x(n∆t)
∆t
β(∆t) = (1 + r ∆t)−1/∆t ,







(3)
lim∆t→0 =
e−r
P
n
sup ∞
n=0 V∆t (xn,∆t , xn+1,∆t )β(∆t)
(xn,∆t , xn+1,∆t ) ∈ T 0 ⊂ X × X
xn,∆t ≈ x(n∆t)
−x V∆t (x, y ) = V x, y∆t
(4)
The policy function
xn+1,∆t = h∆t (xn,∆t )
Alfredo Medio et al.
(5)
Discrete- and Continuous-time Models in Economics
The discretized model
Discretization
t
= n∆t
ẋ(n∆t) ≈ x[(n+1)∆t]−x(n∆t)
∆t
β(∆t) = (1 + r ∆t)−1/∆t ,







(3)
lim∆t→0 =
e−r
P
n
sup ∞
n=0 V∆t (xn,∆t , xn+1,∆t )β(∆t)
(xn,∆t , xn+1,∆t ) ∈ T 0 ⊂ X × X
xn,∆t ≈ x(n∆t)
−x V∆t (x, y ) = V x, y∆t
(4)
The policy function
xn+1,∆t = h∆t (xn,∆t )
Alfredo Medio et al.
(5)
Discrete- and Continuous-time Models in Economics
The discretized model
Discretization
t
= n∆t
ẋ(n∆t) ≈ x[(n+1)∆t]−x(n∆t)
∆t
β(∆t) = (1 + r ∆t)−1/∆t ,







(3)
lim∆t→0 =
e−r
P
n
sup ∞
n=0 V∆t (xn,∆t , xn+1,∆t )β(∆t)
(xn,∆t , xn+1,∆t ) ∈ T 0 ⊂ X × X
xn,∆t ≈ x(n∆t)
−x V∆t (x, y ) = V x, y∆t
(4)
The policy function
xn+1,∆t = h∆t (xn,∆t )
Alfredo Medio et al.
(5)
Discrete- and Continuous-time Models in Economics
A basic question
Conditions on the time–interval ∆t
What are the conditions on ∆t for the discretized model to be
“correct” in the sense of reproducing some fundamental
dynamical properties of the continuous–time one?
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Correct approximation
In the one–dimensional case, the solution to the
continuous–time model is dynamically trivial. More
specifically, if φ(t, x) is the solution to equation(2), the
time–one map φ1 (x) ≡ φ(1, x) is monotone.
In this context, we will say that the discretized model is a
good approximation of the continuous–time one only if the
discrete–time policy function h∆t is monotone
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Correct approximation
In the one–dimensional case, the solution to the
continuous–time model is dynamically trivial. More
specifically, if φ(t, x) is the solution to equation(2), the
time–one map φ1 (x) ≡ φ(1, x) is monotone.
In this context, we will say that the discretized model is a
good approximation of the continuous–time one only if the
discrete–time policy function h∆t is monotone
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Correct approximation
In the one–dimensional case, the solution to the
continuous–time model is dynamically trivial. More
specifically, if φ(t, x) is the solution to equation(2), the
time–one map φ1 (x) ≡ φ(1, x) is monotone.
In this context, we will say that the discretized model is a
good approximation of the continuous–time one only if the
discrete–time policy function h∆t is monotone
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Monotonicity conditions
Define
mij
Mij
= min{Vij (x, y )} : (x, y ) ∈ X × Y
= max{Vij (x, y )} : (x, y ) ∈ X × Y }
(6)
Recalling the definition of V∆t (x, y )) and Deneckere and
Pelikan (1986) we gather that, for fixed ∆t, h∆t is
non–decreasing if V∆t,12 > 0
Therefore, a sufficient condition for h∆t monotone
non–decreasing is
m12 >
1
M22
∆t
(7)
Corollary
There exists ∆1 t > 0 s.t. if 0 < ∆t < ∆1 t, h∆t is
monotone–non–decreasing and its dynamics is simple
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Monotonicity conditions
Define
mij
Mij
= min{Vij (x, y )} : (x, y ) ∈ X × Y
= max{Vij (x, y )} : (x, y ) ∈ X × Y }
(6)
Recalling the definition of V∆t (x, y )) and Deneckere and
Pelikan (1986) we gather that, for fixed ∆t, h∆t is
non–decreasing if V∆t,12 > 0
Therefore, a sufficient condition for h∆t monotone
non–decreasing is
m12 >
1
M22
∆t
(7)
Corollary
There exists ∆1 t > 0 s.t. if 0 < ∆t < ∆1 t, h∆t is
monotone–non–decreasing and its dynamics is simple
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Monotonicity conditions
Define
mij
Mij
= min{Vij (x, y )} : (x, y ) ∈ X × Y
= max{Vij (x, y )} : (x, y ) ∈ X × Y }
(6)
Recalling the definition of V∆t (x, y )) and Deneckere and
Pelikan (1986) we gather that, for fixed ∆t, h∆t is
non–decreasing if V∆t,12 > 0
Therefore, a sufficient condition for h∆t monotone
non–decreasing is
m12 >
1
M22
∆t
(7)
Corollary
There exists ∆1 t > 0 s.t. if 0 < ∆t < ∆1 t, h∆t is
monotone–non–decreasing and its dynamics is simple
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Monotonicity conditions
Define
mij
Mij
= min{Vij (x, y )} : (x, y ) ∈ X × Y
= max{Vij (x, y )} : (x, y ) ∈ X × Y }
(6)
Recalling the definition of V∆t (x, y )) and Deneckere and
Pelikan (1986) we gather that, for fixed ∆t, h∆t is
non–decreasing if V∆t,12 > 0
Therefore, a sufficient condition for h∆t monotone
non–decreasing is
m12 >
1
M22
∆t
(7)
Corollary
There exists ∆1 t > 0 s.t. if 0 < ∆t < ∆1 t, h∆t is
monotone–non–decreasing and its dynamics is simple
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Observational chaos
A quadratic map with chaotic dynamics
Let us consider the quadratic map
G(yn ) : [−1, 1]
yn+1 = G(yn ) = 1 − 2yn2
(8)
Map (8) is topologically conjugate, via the homeomorphism
y = 2x − 1, to the logistic map G(x) = 4x(1 − x) and has
the exact solution
yn = − cos(2θ0 2n )
(9)
where θ0 is an arbitrary constant depending on initial
conditions
Map (8) is “very chaotic”: topologically conjugate to a full
shift on two symbols (mathematical representation of
repeated tossing of a fair coin!)
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Observational chaos
A quadratic map with chaotic dynamics
Let us consider the quadratic map
G(yn ) : [−1, 1]
yn+1 = G(yn ) = 1 − 2yn2
(8)
Map (8) is topologically conjugate, via the homeomorphism
y = 2x − 1, to the logistic map G(x) = 4x(1 − x) and has
the exact solution
yn = − cos(2θ0 2n )
(9)
where θ0 is an arbitrary constant depending on initial
conditions
Map (8) is “very chaotic”: topologically conjugate to a full
shift on two symbols (mathematical representation of
repeated tossing of a fair coin!)
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Observational chaos
A quadratic map with chaotic dynamics
Let us consider the quadratic map
G(yn ) : [−1, 1]
yn+1 = G(yn ) = 1 − 2yn2
(8)
Map (8) is topologically conjugate, via the homeomorphism
y = 2x − 1, to the logistic map G(x) = 4x(1 − x) and has
the exact solution
yn = − cos(2θ0 2n )
(9)
where θ0 is an arbitrary constant depending on initial
conditions
Map (8) is “very chaotic”: topologically conjugate to a full
shift on two symbols (mathematical representation of
repeated tossing of a fair coin!)
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Observational chaos
A quadratic map with chaotic dynamics
Let us consider the quadratic map
G(yn ) : [−1, 1]
yn+1 = G(yn ) = 1 − 2yn2
(8)
Map (8) is topologically conjugate, via the homeomorphism
y = 2x − 1, to the logistic map G(x) = 4x(1 − x) and has
the exact solution
yn = − cos(2θ0 2n )
(9)
where θ0 is an arbitrary constant depending on initial
conditions
Map (8) is “very chaotic”: topologically conjugate to a full
shift on two symbols (mathematical representation of
repeated tossing of a fair coin!)
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Continuous–time representation
Put t/τ = n, where τ is a multiplier depending on the unit
of measure of time
Then, arbitrary constants apart, y (t) = − cos(2θ0 2t/τ ) is
the exact solution of the differential equation:
ÿ − mẏ + F (t)y = 0
with m =
ln(2)
τ ,
(10)
constant and F (t) = (2θ0 ln(2)/τ )2 22t/τ
Equation (10) is a second–order differential equation,
linear but time–dependent, but can be transformed exactly
into an autonomous equation
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Continuous–time representation
Put t/τ = n, where τ is a multiplier depending on the unit
of measure of time
Then, arbitrary constants apart, y (t) = − cos(2θ0 2t/τ ) is
the exact solution of the differential equation:
ÿ − mẏ + F (t)y = 0
with m =
ln(2)
τ ,
(10)
constant and F (t) = (2θ0 ln(2)/τ )2 22t/τ
Equation (10) is a second–order differential equation,
linear but time–dependent, but can be transformed exactly
into an autonomous equation
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Continuous–time representation
Put t/τ = n, where τ is a multiplier depending on the unit
of measure of time
Then, arbitrary constants apart, y (t) = − cos(2θ0 2t/τ ) is
the exact solution of the differential equation:
ÿ − mẏ + F (t)y = 0
with m =
ln(2)
τ ,
(10)
constant and F (t) = (2θ0 ln(2)/τ )2 22t/τ
Equation (10) is a second–order differential equation,
linear but time–dependent, but can be transformed exactly
into an autonomous equation
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Continuous–time representation
Put t/τ = n, where τ is a multiplier depending on the unit
of measure of time
Then, arbitrary constants apart, y (t) = − cos(2θ0 2t/τ ) is
the exact solution of the differential equation:
ÿ − mẏ + F (t)y = 0
with m =
ln(2)
τ ,
(10)
constant and F (t) = (2θ0 ln(2)/τ )2 22t/τ
Equation (10) is a second–order differential equation,
linear but time–dependent, but can be transformed exactly
into an autonomous equation
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Time transformation
Put
ϕ(t) = 2θ0 2t/τ
(11)
Considering that
dy
dy dϕ
=
dt
dϕ dt
d 2y
dy d 2 ϕ
dϕ 2 d 2 y
=
+
dϕ dt 2
dt
dt 2
dϕ2
We can write
d 2y
+y =0
dϕ
(12)
(13)
(14)
a second–order, linear autonomous differential equation in
the new independent variable ϕ
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Time transformation
Put
ϕ(t) = 2θ0 2t/τ
(11)
Considering that
dy
dy dϕ
=
dt
dϕ dt
d 2y
dy d 2 ϕ
dϕ 2 d 2 y
=
+
dϕ dt 2
dt
dt 2
dϕ2
We can write
d 2y
+y =0
dϕ
(12)
(13)
(14)
a second–order, linear autonomous differential equation in
the new independent variable ϕ
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Time transformation
Put
ϕ(t) = 2θ0 2t/τ
(11)
Considering that
dy
dy dϕ
=
dt
dϕ dt
d 2y
dy d 2 ϕ
dϕ 2 d 2 y
=
+
dϕ dt 2
dt
dt 2
dϕ2
We can write
d 2y
+y =0
dϕ
(12)
(13)
(14)
a second–order, linear autonomous differential equation in
the new independent variable ϕ
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Time transformation
Put
ϕ(t) = 2θ0 2t/τ
(11)
Considering that
dy
dy dϕ
=
dt
dϕ dt
d 2y
dy d 2 ϕ
dϕ 2 d 2 y
=
+
dϕ dt 2
dt
dt 2
dϕ2
We can write
d 2y
+y =0
dϕ
(12)
(13)
(14)
a second–order, linear autonomous differential equation in
the new independent variable ϕ
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Solution in the transformed time
Arbitrary constants apart, the exact solution of equation
(14) is
y (ϕ(t)) = − cos(ϕ(t)) = −cos(2θ0 2t/τ )
(15)
Q.E.D
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Solution in the transformed time
Arbitrary constants apart, the exact solution of equation
(14) is
y (ϕ(t)) = − cos(ϕ(t)) = −cos(2θ0 2t/τ )
(15)
Q.E.D
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Continuous–time orbits
Orbits of (15)
y (t) = − cos(2θ0 2t/τ )
are sinusoidal oscillations with
rapidly increasing frequency
(and correspondingly rapidly
decreasing period)
In the limit as t → ∞, the
curve y (t) approaches a set
morphologically similar to the
so–called “topologist’s
sin(1/x) curve”
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Continuous–time orbits
Orbits of (15)
y (t) = − cos(2θ0 2t/τ )
are sinusoidal oscillations with
rapidly increasing frequency
(and correspondingly rapidly
decreasing period)
In the limit as t → ∞, the
curve y (t) approaches a set
morphologically similar to the
so–called “topologist’s
sin(1/x) curve”
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Continuous–time orbits
Orbits of (15)
y (t) = − cos(2θ0 2t/τ )
are sinusoidal oscillations with
rapidly increasing frequency
(and correspondingly rapidly
decreasing period)
In the limit as t → ∞, the
curve y (t) approaches a set
morphologically similar to the
so–called “topologist’s
sin(1/x) curve”
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Exact solution sampled at discrete intervals
Exact solution x(t) sampled at discrete intervals ∆t/τ = 1,
yields chaotic dynamics, as described by the map
yn+1 = 1 − 2yn2
Alfredo Medio et al.
(16)
Discrete- and Continuous-time Models in Economics
Connected values xn and orbit x(t)
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
A general result
Observational chaos
1
Both the original, continuous-time system and the derived
system resulting from discrete–time, but exact observation
of the former generate complex orbits
2
Discrete observation of the continuous–time system
generates orbits that are “more chaotic”. More precisely:
the time series of equation (9) have a flat power spectrum,
as for white noise series. If we reduce the length of the
observation period ∆t and get a better approximation to
the continuous–time series, we obtain the so called 1/f
power spectrum, corresponding to a more structured, less
stochastic dynamics
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
A general result
Observational chaos
1
Both the original, continuous-time system and the derived
system resulting from discrete–time, but exact observation
of the former generate complex orbits
2
Discrete observation of the continuous–time system
generates orbits that are “more chaotic”. More precisely:
the time series of equation (9) have a flat power spectrum,
as for white noise series. If we reduce the length of the
observation period ∆t and get a better approximation to
the continuous–time series, we obtain the so called 1/f
power spectrum, corresponding to a more structured, less
stochastic dynamics
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
A general result
Observational chaos
1
Both the original, continuous-time system and the derived
system resulting from discrete–time, but exact observation
of the former generate complex orbits
2
Discrete observation of the continuous–time system
generates orbits that are “more chaotic”. More precisely:
the time series of equation (9) have a flat power spectrum,
as for white noise series. If we reduce the length of the
observation period ∆t and get a better approximation to
the continuous–time series, we obtain the so called 1/f
power spectrum, corresponding to a more structured, less
stochastic dynamics
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Some simple conclusions
Importance of the period time–length ∆t
The common usage of modeling dynamical problems
arising from economics in terms of difference equations
where the time interval that separate agents’ (discrete)
decisions is a constant of unspecified length may obscure
important aspects of the questions at issue
We have discussed some general cases in which the
explicit discussion of ∆t as a model variable may lead to
interesting results
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics
Alfredo Medio et al.
Discrete- and Continuous-time Models in Economics