3. Model development and Model examples
3.1 Fundamental equations of the system, restriction to the essential information
To develop a model we need basic equations of the system under consideration. These laws
often contain processes which are not relevant and can be neglected.
Example for constructing a simple ocean model
The hydrodynamic equations for an incompressible fluid like the ocean could be written as
follows
v Velocity vector
ρ Mass density
T Temperature
1
∂ 2 v p Pressure
2
= − ∇p − gk + ν h ∇ h v + ν v 2 , νh Eddy viscosity (horizontal)
∂z
ρ
νv Eddy viscosity (vertical)
α Thermal expansion coefficient
Continuity equation ∇ ⋅ v = 0 ,
κ h Eddy conductivity (horizontal)
ρ = ρ 0 (1 + α (T − T0 )),
Equation of state
κ v Eddy conductivity (vertical)
g Gravity acceleration
2
∂T
∂ T
2
f Coriolis-parameter
Temperatur e equation
+ v ⋅ ∇T = κ h ∇ h T + κ v 2 ,
k Vertical unit vector
∂t
∂z
∂v
+ ∇ ⋅ ( vv ) + fk × v
Momentum equation
∂t
CM-98
We can, however, discard the equation of state and the temperature equation when the density
variations within the fluid are nearly negligible due to small temperature contrasts.
Then, we end up with the equations
∂v
1
∂2v
2
+ ∇ ⋅ ( vv ) + fk × v = − ∇p − gk + ν h ∇ h v + ν v 2 , ∇ ⋅ v = 0 ,
ρ
∂t
∂z
in which ρ = const.
We can simplify the system further if the geometrical properties of the flow are special. These
are often idealizations which may help to better understand the dynamics.
Example
Wind driven ocean model in a rectangular basin
(SpOzMo, Julia Dellnitz 2000).
The fluid is bounded by rigid planes at z=-D
and z=0 (simple model for an ocean with depth D).
Then the vertical integration of the continuity equation yields
( v h denotes horizontal velocity vector)
D
∇ ⋅ ∫ v h dz = − w z = 0 + w z = − D = 0 ,
0
since vertical velocity w vanishes at the boundaries.
CM-99
Figure: Sketch of the model domain
Therefore, the vertically averaged horizontal flow vh is nondivergent and we can describe the
vertically averaged horizontal flow by a streamfunction ψ so that
 ∂ψ ∂ψ 
vˆ =  −
,
,0 
 ∂y ∂x 
0
1
with vˆ =
v h dz.
∫
D −D
Vertically averaging of the horizontal momentum equation leads to (note that ∇ h = ∇ in 2D)
0
1 0
 1
τ − τb
∂vˆ
1
+ ∇ ⋅  ∫ v h v h dz  + (wv h z = 0 − wv h z =. H ) + fk × vˆ = −
∇ ∫ pdz + ν h ∇ 2 vˆ + t
,
∂t
D 444
ρD − D
ρD
 D −D
 1
424444
3
=0
where τ t = ν v ∂v h / ∂z z = 0 denotes the horizontal stress at the top and τ b = −ν v ∂v h / ∂z z = − H the
horizontal stress at the bottom of the ocean. The horizontal momentum flux vhvh is usually
small in ocean dynamics. A compromise is to neglect only vertical anomalies so that the
equation becomes
0
τt − τb
∂vˆ
1
2
+ ∇ ⋅ ( vˆ vˆ ) + fk × vˆ = −
∇ ∫ pdz + ν h ∇ vˆ +
,
∂t
ρD − D
ρD
CM-100
So, the components of the momentum equation read
∂  ∂ψ
I
−
∂t  ∂y
∂  ∂ψ
II

∂t  ∂x
∂ψ


ˆ
+
∇
⋅
−
v


∂y


 + ∇ ⋅  vˆ ∂ψ



 ∂x
0

∂ψ
∂  1
∂ψ

2


ν
−
f
pdz
=
−
+
∇
−

 ρD ∫
 h  ∂y
x
x
∂
∂




−D
0

 − f ∂ψ = − ∂  1
2  ∂ψ

ν
pdz
+
∇
vˆ 

h
∫


∂y
∂y  ρ D − D

 ∂x

 τ tx − τ bx
,
+
ρD

 + τ ty − τ by .

ρD

Applying ∂/∂x(II)- ∂/∂y(I) gives the so-called vorticity equation
∇ ⋅ [k × ( τ t − τ b )]
∂ 2
∂ψ ∂∇ 2ψ ∂ψ ∂∇ 2ψ
∂ψ
(∇ ψ ) −
+
+β
= ν h ∇ 2 (∇ 2ψ ) −
.
∂t
∂y ∂x
∂x ∂y
∂x
ρD
where β=∂f/∂y is the so-called Beta-parameter.
Consequently, we have simplified the original system with four unknown three-dimensional
fields to a system with only one two-dimensional field, namely, the streamfunction
ψ = ψ ( x, y , t ).
CM-101
3.2 Low order model formulation (Galerkin method)
For a low order model we represent the model fields in terms of suitable orthogonal functions,
e.g.
∞
ψ = ∑ ψ k (t ) f k ( x, y ).
k =1
This can be used to convert the model equation, e. g.
∂ψ
= Fψ (ψ ).
∂t
to an infinite set of ordinary differential equations by the operation
∫∫ ..... f dxdy ≡
k
f k ..
A
This would lead due to the orthogonality (i.e. < fj | fk >=δjk) to the following dynamical system
dψ k
= f k Fψ
dt
for k = 1,....., ∞
Of course we are not able to solve an infinite number of equations. Therefore, we must
truncate the expansion at a finite k. For dynamical system analysis it is appropriate to choose
rather a low value of truncation number k.
CM-102
Application to the wind-driven ocean model:
Nonlinear vorticity
advection
Planetary vorticity advection
Bottom friction
∂ 2
(∇ ψ ) + ∂ψ ∂∇ ψ − ∂ψ ∂∇ ψ + β ∂ψ
∂t
∂x ∂y
∂y ∂x
∂x
2
Wind stress
2
δ
1
= ∇ ⋅ (k × τ t ) − E ∇ 2ψ + ν h ∇ 4ψ ,
2D
ρD
Horizontal
Diffusion
where D is the depth of the ocean, β the planetary vorticity gradient (meridional derivative of
Coriolis-parameter), τt the wind stress vector and δE the bottom friction parameter.
The stream function has to fulfil free-slip boundary conditions:
ψ = 0 at x = 0 , L x and y = 0 ,L y
(no flow across the boundary)
∇ 2ψ = 0 at x = 0 ,L x and y = 0 ,L y
(no drag at the boundary)
CM-103
With the Fourier expansion
 πk
ψ = ∑ ∑ψ k ,l (t ) sin 
l =1 k =1
 Lx
Nl N k
  πl 
x  sin 
y ,


  Ly 
the boundary conditions are automatically fulfilled. Applying the aforementioned technique
leads to a dynamical system with Nk×Nl ordinary differential equations for a given wind stress.
Truncation at Nl=2 and Nk=2, neglect of bottom friction and a windstress field of the form
 πk
1
−
∇ ⋅ (k × τ t ) ∝ sin 
ρD
 Lx
  πl 
x  sin 
y


  Ly 
lead to the Veronis model (Veronis 1963)
In this model γ denotes the
γ
dA
4
dB
8
=−
B − εA + ,
=
A + AC − εB amplitude of windstress and ε
dt
3π
2
dt 15π
a viscosity parameter.
dC
8
dD
1
=−
D − AB − εC ,
=
C − εD .
dt
15π
dt 3π
CM-104
Some results from numerical simulations at higher truncations (from Dellnitz 2000)
Streamfunction of the steady state
Bifurcation Diagram (Re Reynolds number)
CM-105
3.3 Box model formulation
A box model predicts only integrated quantities. Therefore, one shifts from local budget
equations to global budget equations. The disadvantage of this method is the inaccurate
description of the impact of local anomalies on the global budget. In most cases it has to be
described in an empirical way.
a)
The zero dimensional energy balance model (EBM)
The simplest models in climate physics are the so-called zero-dimensional climate models.
These are indeed box-models in which the complete atmosphere-ocean system has been
aggregated into one box. Such a box model is used to predict the mean temperature T of the
climate system and is able to describe ice-albedo feedbacks.
Absorbed radiation Ri
T
T
Emitted radiation Ro
Ro = ε (T )sT 4
Ri = Q0 [1 − A(T )]
Figure: Schematic illustration of a zero-dimensional climate model (adapted from Fraedrich 2001)
CM-106
The governing equation of the EBM is
C
[ ( )] ( )
dT
= Q0 1 − A T − ε T sT 4
dt
Notation:
Q0 =342W/m2
s =5.67×10-8W/m2/K4
A
ε
T
C
Global mean solar radiative input (present day value)
Stefan-Boltzmann constant
Planetary albedo
Greyness of the system
Annually averaged temperature of the global earth-atmosphere system
Heat capacity of the system
The equation expresses the approximate radiation balance between absorbed radiation
Ri = Q0 [1 − A(T )]
and emitted radiation
Ro = ε (T )sT 4 = R ↑ (0)
CM-107
Any slight imbalance between Ri and Ro leads to a change in the temperature of the system. In
the simple model the albedo A and emissivity ε have to be expressed (parameterized) as a
function of the temperature T .
It is plausible that the change of ice influences the albedo. This leads to the ice albedo
feedback (Budyko and Sellers). A simple parameterization is given by
 Amax
2
Tu − T 2

A(T ) =  Amin + ( Amax − Amin ) 2
2
T
−
T
u
l

Amin

for T < Tl
for Tl ≤ T ≤ Tu
for T > Tu
where Amax is the maximum albedo (snow ball earth), Amin the minimum albedo (ice free planet),
Tl the lower temperature threshold and Tu the upper temperature threshold. The realistic
temperature range is between Tl and Tu so that one may only wish to consider the second case
in the albedo formula, namely A( T )=A0-b T 2 (Fraedrich 1978).
The greyness of the system depends upon the greenhouse effect. With increasing temperature
the water vapour amount increases and, therefore, also the greenhouse effect. This reduces
the emissivity of the system since a bigger part of emitted surface radiation heats the
atmosphere that emits radiation back to the surface. A simple parameterization (Fraedrich
1978) is given by
ε (T ) = ε 0 - κT 2
CM-108
The graph shows the dependency of ε and A on temperature T . We use the following
parameters according to Fraedrich (1978): A0=1.2, b=10-5 1/K2, ε0=0.87 and κ =3x10-6 1/K2.
For a realistic temperature range (250K< T <320K) albedo and greyness have physically valid
values. Present day values (assuming T = 288.6K) are: A=0.284 and ε=0.62.
CM-109
Fixed albedo and fixed greyness
For fixed present day albedo and greyness the analysis is simple. Then, the dynamical system
reduces to:
C
dT
= Q0 [1 − A] − εsT 4
dt
The equilibrium becomes
Te = 4
Q0 [1 − A]
εs
Therefore, no bifurcation occurs. The linearization
gives
C
Attractor
dT ′
= −4εsTe 3T ′
dt
The stability of the equilibrium is obvious since the
growth rate
σ = −4εsTe 3 /C
is negative.
CM-110
Figure: Heating as a function of temperature:
Net heating (red line), absorbed radiation
(green line) and emitted radiation (blue line).
Fixed greyness (only ice-albedo feedback)
For fixed greyness the system equation becomes
C
dT
= Q0 [1 − A0 + bT 2 ] − εsT 4
dt
Now, the equilibria result from the biquadratic equation
bQ0 2 Q0
[ A0 − 1] = 0
T −
T +
εs
εs
4
and the physically valid equilibria become
Repellor
Attractor
2
Te1, 2
bQ0
bQ0  Q0

=
± 
[ A0 − 1]
 −
2εs
εs
 2εs 
Since A0>1 no solution occurs for
4εs
Q0 < 2 [ A0 − 1]
b
Figure: Heating as a function of temperature:
Net heating (red line), absorbed radiation
(green line) and emitted radiation (blue line).
CM-111
At the critical solar input we obtain for the equilibrium temperature and the albedo
Tec =
bQ0
A −1
= 2 0
= 200K
2εs
b
,
Ac = A0 − bTec = 2 − A0 = 0.8
2
The critical albedo is quite large but possibly below Amax (snowball earth). Note that with the full
albedo formula we possibly have two additional equilibria:
Tes = 4
Q0 [1 − Amax ]
for Tes <
εs
A0 − Amax
b
(Snowball Earth)
Teh = 4
Q0 [1 − Amin ]
for Teh >
εs
A0 − Amin
b
(Ice free planet)
and
The first solution represents a snowball earth and the other a warm planet without any ice and
snow. These additional equilibria are stable if they exist.
CM-112
Linearizing the governing equation yields
dT ′
C
= (2Q0 bTe − 4εsTe 3 )T ′
dt
Inserting the equilibrium temperature gives the growth rate
2
2


bQ
T
bQ
Te


0 
e
0 


σ = 2Q0 b − 2bQ0 m 4 
 − εsQ0 [ A0 − 1]  = m4 
 − εsQ0 [ A0 − 1]

C
 2 
 2 

C
Obviously, Te1 is stable and Te 2 is unstable.
Therefore, the bifurcation at 200K is a
saddle node bifurcation.
The figure on the right displays the equilibria
for the model that takes a maximum albedo
(Amax=0.9) and a minimum albedo (Amin=0.1)
into account. Then, a further saddle node bifurcation
includes a new branch for a stable snowball earth
solution. However, the snowball earth cannot exist for
Q0>εs(A0-Amax)2/(1-Amax)/b2=316.4W/m2. Therefore, this Figure: Equilibria of the conceptual climate
model state is impossible for present day conditions.
model including the ice-albedo feedback.
CM-113
Fixed albedo (only water vapor feedback)
For fixed albedo the system equation becomes
C
dT
= Q0 [1 − A] − (ε 0 − κT 2 )sT 4
dt
Now, the equilibria result from the sixth-order polynomial equation
ε 0 4 Q0
T − T +
[1 − A] = 0
κ
κs
6
Attractor
Repellor
Whether any two positive equilibria exist depends on the
6
4
minimum of the curve f (T ) = T − ε 0 / κT that is given by
ε 2ε
2ε
1 2ε
f (Tmin ) =  0  − 0  0  = −  0  at Tmin =
23 κ 
3 κ  κ 3 κ 
3
2
3
2 ε0
3κ
Two solutions occur for f (Tmin ) < −Q0 (1 − A) / κs . This yields
κs
 2 ε 0  = 858W/m2
Q0 <


2(1 − A)  3 κ 
3
Figure: Heating as a function of temperature:
Net heating (red line), absorbed radiation
(green line) and emitted radiation (blue line).
CM-114
Linearizing this equation gives
dT ′
C
= (6κsT 5 − 4ε 0 Q0 sT 3 ) T ′
dt
The growth rate increases without limit at high temperatures leading to the so-called runaway
greenhouse effect (the ocean evaporates).
It has to be noted that the greyness cannot become
negative but already zero greyness would lead to a
2
runaway greenhouse effect. Therefore, ε 0 > κT
must be fulfilled for a stable climate. However,
this is the case for T < 538.5K and the equilibrium
temperatures are within this range.
It is obvious from the figure showing the heating terms
(previous slide) that the upper equilibrium is unstable
and the lower one stable.
Figure: Equilibria of the conceptual climate
model including water vapour
feedback.
CM-115
Full model (ice-albedo and water vapor feedback)
The system equation including temperature-dependent albedo and greyness becomes
C
dT
= Q0 [1 − A0 + bT 2 ] − (ε 0 − κT 2 )sT 4
dt
In the full model up to three equilibria having positive temperatures can arise. However, taking
the physically necessary upper and lower limits of albedo into account even up to four
equilibria can be found. The stability of these equilibria can be determined by calculating the
derivative of the right hand side of the system equation with respect to temperature.
The figure on the right shows the equilibria of the full
model. In the model a transition of the present day
climate to a snowball earth is more likely than the
transition to a runaway greenhouse. However, the
simple model may include incorrect parameter
assumptions and exclude many processes that may
counteract a climate changes (e.g. clouds). Therefore,
the model should rather be considered as a roughand-ready theory for climate dynamics without making a
statement for real changes in the future. For the latter
purpose complex climate models are appropriate.
CM-116
Runaway
greenhouse
Icefree
Planet
Snowball
Earth
Moderate
Climate
b) Stommels two-box-Model of the meridional overturning circulation
The ocean is divided into a polar and an equatorial box. The model predicts the temperature
and salinity average of each box (see figure adapted from Dijkstra 2005).
heat flux
CT (Tea-Te)
fresh water flux
CS (Sea-Se)
heat flux
CT (Tpa-Tp)
fresh water flux
CS (Spa-Sp)
volume flux Ψ~αT(Te-Tp)+ αS(Sp-Se)
temperature Te
temperature Tp
salinity Se
salinity Sp
volume V
volume V
CM-117
We obtain the following governing equations of the system
V
dT p
= CT (T p − T p ) + Ψ (Te − T p ) ,
a
dt
dT
a
V e = CT (Te − Te ) + Ψ (T p − Te ),
dt
dS p
a
= CS (S p − S p ) + Ψ (Se − S p ) ,
V
dt
dS e
a
V
= Cs (Se − Se ) + Ψ (S p − Se )
dt
where CT and CS are relaxation constants and the volume flux Ψ is proportional to the density
difference ∆ρ between the boxes
Ψ =γ
∆ρ
ρ0
[
]
= γ α T (Te − T p ) + α S ( S p − S e ) ,
where αT and αS are the thermal expansion and haline contraction coefficients, respectively.
CM-118
Therefore, we can reduce the system to two equations for the temperature difference ∆T=Te-TP
and salinity difference ∆S=Se-SP between the boxes:
d∆T
= C 'T ( ∆T a − ∆T ) − γ ' α T ∆T − α S ∆S ∆T ,
dt
d∆S
= C ' S ( ∆S a − ∆S ) − γ ' α T ∆T − α S ∆S ∆S ,
dt
where C´T=C T /V, C´S=C S /V and γ'=2γ/V. Nondimensionalization (see Dijkstra 2005) leads to
the following dynamical system:
dT
= η1 − T (1 + T − S )
~
dt
dS
II ~ = η2 -S (η3 + T − S )
dt
I
where
~
t = C 'T t , T = (γ 'α T / C 'T )∆T , S = (γ ' α T / C 'T )∆S
η1 = (γ 'α T / C 'T )∆T a , η 2 = (γ 'α T / C 'T )∆S a , η 3 = C 'S / C 'T
CM-119
Equilibria
From I and II we obtain
0 = η1 − Te (1 + Te − S e ) ⇔ Te =
η1
1 + Te − S e
0 = η 2 − S e (η 3 + Te − S e ) ⇔ S e =
η2
η 3 + Te − S e
Combining both leads to
Te − S e =
η1
1 + Te − S e
-
η2
η 3 + Te − S e
Setting η3=1 (relaxation time scales for heat and salinity are identical) we get the equilibrium for
the nondimensional overturning mass flux
− 1 / 2 + 1 / 4 + η1 − η 2 for η1 > η 2 TH state (sinking in the north)
1 / 2 − 1 / 4 + η 2 − η1 for η1 < η 2 SA state (sinking in the south)
ψ~e = Te − S e = 
CM-120
We see that the sign of the overturning circulation changes sign when η1= η2. In this case the
thermal expansion and the saline compression are compensating each other.
↑
ψ
η1 -η2→
Figure: Overturning strength in Stommels box model as a function of χ-η2 (for η3= 1).
CM-121
Equilibria
T, S
η1=0.25
η3=0.3
T, S
η1=0.25
η3=3.0
η2
η2
TH+SA
SA
η2
TH
η1
Figure: Temperature difference (T) and salinity difference (S) in Stommels box model as a function
of η2 for a) η1=0.25,η3= 0.3 and b) η1=0.25,η3= 3.0. c) displays a regime diagram as a function of η1 and η2
for η3= 0.3. Note that multiple equilibria arise due to the imperfection η3≠1. (adapted from Dijkstra 2005).
CM-122
3.4 Population dynamics: The Malthusian model
Malthus developed a theory for population growth in 1798. It is also a theory for economic
growth in the pre-industrial era.
The evolution of the human population N (number of humans) is governed by the birth rate B
and death rate D. Population growth requires that the birth rate is larger than the death rate.
The governing equation becomes
dN
= N [B − D ) ]
dt
To obtain a simple model the rates B and D should be parameterized as a function of N itself.
They could also depend upon other variables but these are prescribed as exogenous
parameters in the present model. The Malthusian model assumes that the birth rate B is
independent of the population. In modern societies this is possibly not realistic as, e.g., the
birth rate decreases in urban areas with large populations. However, it should be acceptable in
the pre-industrial era when family planning was still not established. The death rate D is on the
other hand a decreasing function of wealth (consumption C per capita) so that
B = const.,
D = D (C / N ) ,
dD
<0
d (C / N )
CM-123
A parameterization for the death rate like in the figure below is given by
cD
D=
C/N
Population
decay
Population
growth
Figure: Relative population tendency, birth rate B and death rate D as a function of per capita
consumption.
CM-124
To solve the problem one has to parameterize the consumption as a function of population. In
Malthus model it is assumed that a fixed area of land A exists and that N labourers cultivate the
area. The yield Y per time unit of this cultivation is parameterized by the following production
function
 N 
Y ( A, N ) = c A A arctan 

 N0 
where cA is a technology parameter and N0 a reference population. The yield Y increases with
population but the additional yield per additional labourer decreases with N. This law of
diminishing returns finds its reason in the limited amount of agricultural area. The maximum
possible yield is Y=cA Aπ/2. Then, additional labourers cannot increase the yield anymore since
all land areas are cultivated and optimally utilized.
When the market is cleared, that is, the yield of the cultivation is immediately sold and
consumed, we can equate production Y with consumption C
 N 
C = Y ( A, N ) = c A A arctan 

 N0 
CM-125
C=cA Aπ/2
Figure: Consumption as a function of
population. The diminishing returns
lead to a decrease of consumption
per capita with increasing population.
Using these parameterizations we obtain the following differential equation for population
growth


cD N
dN
= N B −
dt
 N 

c
A
arctan


A


 N0 






CM-126
For large N>>N0 this equation approaches the logistic equation
dN
N 

= σ N N 1 −

dt
N

m 
where σN=B and Nm=cAπ/(2cD). In this case the maximum population is N= Nm . However, the
equilibrium is possibly already attained at a smaller value.
The equilibrium is stable because the linearized equation becomes
 dC dN
dN ' 
dD

=  B − De − N e
dt
d (C / N ) N e 
Ne
 123
Ne
=0
 dD
 dC dN N e N e − C e

= −
Ne
 d (C / N ) N e 
Ce
− 2
Ne

  N '
 

  N '

Since C(0)=0 and d 2C/ dN 2<0 (diminishing returns) we have
dC
dN
N e − Ce < 0
Ne
and using the assumption dD/d(C/N) <0 yields a negative growth rate.
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This becomes also clear by looking at the tendencies in a diagram
Attractor
The most important message of the Malthusian model is that the equilibrium is associated with
the relation
B = D (C / N ) ⇒
C / N = const . at the equilibrium state.
Therefore, increase of land area A (e.g. by discovering new land) or technological progress
(increase of cA) does not eventually lead to a higher consumption per capita C/N (wealth)!
Therefore, the wealth cannot be elevated and the humans stay poor since the population
growth compensates the higher food production.
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3.5 Macroeconomic approach in economics
In economics an approach analogous to the box model formulation exists. We may understand
an economic system as a system of many households, firms and banks. A description of the
financial budget for each household or firm is gained by so-called microeconomic models. On
the other hand macroeconomic models only describe the budget of many households, firms
and banks. A simple common approach divides the complete (national) economy into one
aggregate for households, one for firms and one for banks. This gives such a picture:
Notations
Households
H
Interest iB
Profit Π
Wage wL
Consumption C
Firms
F
Saving S
Investment I
Banks
B
Figure: Monetary circuit in a simple macroeconomic model
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H
F
B
S
I
C
Π
K
L
w
i
Households
Firms
Assets (administrated
by the bank)
Saving
Investment
Consumption
Profit
Capital
Labour
Wage per work unit
Interest
The monetary circuit of this macroeconomic model can be summarized by three equations:
dH
= iB + wL + Π − C − S
dt
dF
II
= −iB − wL − Π + C + I
dt
dB
III
=S
dt
I
d
(H + F + B ) = I ≠ 0 !
⇒
dt
Note that the assets B increase monotonically although the savings are transferred in the form
of investments I to the firms and, consequently, B also appears in the form of a credit.
Therefore, money is not conserved in this economic system. Saving S must become zero if an
equilibrium exists. However, this appears illusory as households with large incomes are not
able to spend their money only on consumption. Therefore, such an economy is not
sustainable in the long term.
The model completely covers the growing gap between rich and poor since heterogeneous
households are summarized into one aggregate. On the other hand a mathematical proof that
the inequality between rich and poor rises in such an economic system has been given in the
textbook by Kremer (2012)1.
1
Kremer, J.. 2012: Grundlagen der Ökonomie. Metropolis Verlag, Marburg, 267. The proof is also described in the
following article: Kremer, J., 2010: Dynamic Analysis - Investigating the long-term behavior of Economies.
(http://www.rheinahrcampus.de/fileadmin/prof_seiten/kremer/DynamicAnalysis.pdf)
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If the firms pay out their profits completely, the tendency for F vanishes (the firm does not
accumulate money). This gives the following profit
Π = C + I − wL − iB
The investment I will of course not directly be paid out in the form of money to the firm-holders.
Instead, the investment will be paid to other firms that produce capital goods (production
facilities). So, much of the investment will eventually be paid out in the form of wage (capital
goods can be viewed as “coagulated work”). Therefore, a part of wL compensates much of the
investment I.
We already see a problem here. If the assets B increase monotonically, the debt of the firms
(-B) will also increase. To avoid a negative profit due to increasing interest payment, the firm
has to raise its production and sales.
However, firm-holders could avoid further investments because of the increasing interest
payments. In such a case saved money will be increasingly extracted from the monetary
circuit. This leads to a decrease of income (stemming from the investments) and increasing
unemployment that may cause an economic crisis. To circumvent this, the state can borrow
saved money in the form of a bond and the borrowed money can be feeded back into the
monetary circuit by public spending. This kind of politics was suggested by the economist J.M.
Keynes and is called Keynesianism. However, the increasing debt and the accompanying
interest payments of the government also lead to problems in the long term (debt crisis).
Therefore, Keynesianism also does not make the economic system really sustainable.
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So far, the model includes monetary exchanges without a rule to evaluate them. To determine
these exchanges more relations have to be introduced. First, the economy produces goods
and its amount per time unit can be measured by the so-called production function:
Y = Y ( K , L)
Y describes the production of consumer and capital goods. Furthermore, depreciating capital K
(production facilities) must be recovered. Therefore, we have the relation
Y=
dK
+ δK + C
dt
where δ is the depreciation rate of capital goods. The households spend a part of their income
on consumption and another part will be saved. The saving leads (in this model) to
investments, that is, the firms use this money to extend and recover their production facilities.
dK
S=
+ δK → Y = S + C
dt
With the saving ratio s (the fraction of saved money) we obtain
dK
S = sY =
+ δK
dt
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Therefore, the economy will be governed by the following differential equation:
dK
= sY ( K , L ) − δK
dt
This is the so-called neoclassical Solow-Swan model.
The so-called Cobb-Douglas production function is widely used and is given by
Y = AK α L1−α
,
where A denotes the level of technology. α is a constant with 0<α <1. This function allows the
substitution of labour L by capital K and vice versa. However, an optimal ratio K/L exists
depending on the costs for capital (interest and depreciation) and labour (wage).
Prescribing the labour L, the level of technology A and the saving ratio s as exogenous
constants we obtain the following dynamical system equation
dK
= sAK α L1−α − δK
dt
In more elaborate models L, A and s are endogenous variables and will be predicted.
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This equation has two equilibria:
K e1 = 0, K e 2 = (sA )
1
1−α
L
The first trivial equilibrium is unstable. This instability triggers economic growth. The second
equilibrium is stable. One can see this by looking at the following graph
sY
δK
Ke1=0
Ke2
K→
When sY>δK, economic growth takes place and this appears for Ke1<K<Ke2. Therefore, the
system changes monotonically towards the state Ke2 away from state Ke1.
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This nontrivial equilibrium state is, however, in disagreement with the monetary cycle. Due to
equation III we have (assuming that all saved money will be invested)
t
t
dK
B (t ) − B (t 0 ) = ∫ Sdt' = ∫
+ δKdt'
dt
t0
t0
Therefore, the assets of the households (debt of the firms) will increase further although the
development of capital K has reached equilibrium. On the other hand the increasing debt leads
to increasing capital costs and diminishing profits that eventually causes a collapse of the
economic system. The only way to circumvent this is to abolish the interest or to introduce
never-ending technological progress that will increase A.
In the former case the investor will not receive a return
for their investment. Then, it is likely that the investor will retain their money and this also
provokes a collapse of the economy. The cost for
Real Capital K
depreciation could be hided in the price for the goods.
Financial Capital B
Then however, a steady state must be accompanied
with zero savings. This appear unlikely as rich people
can hardly spend their complete income on consumption.
Technological progress together with inventing new
products for consumption remains as the last
possibility. Then, the economy can never reach
Time t→
equilibrium. This so-called growth imperative clearly Figure: Divergence of real and financial capital
disagrees with sustainable development.
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3.6 World-2 model
World-2 is a global ecological-economic model that makes a prediction of population growth,
economic growth, food production, non-renewable resources and pollution.
Figure: Schematic diagram of Forrester’s World-2 model (from Forrester 1971)
CM-136
World-2 has five variables:
N
R
Population
non-renewable resources
P
X
Pollution
Agricultural fraction of capital
K
Capital
These are predicted by the following equations
dN
= N [c B BS ( S ) BF ( F ) B N ( N ) BP ( P ) − c D DS ( S ) DF ( F ) D N ( N ) DP ( P )]
dt
dK
= c K NK S ( S ) − δK
dt
dP
P
K
= N PK   −
dt
 N  τ P (P )
N0
dR
=−
NR S ( S )
dt
R0
dX X F ( F , S ) − X
=
dt
τX
where S denotes the material standard of living and F the food ratio
CM-137
The auxiliary variables material standard of living S and food ratio F are defined by
S=
K
1− X
S R ( R)
N
1− X 0
,
 K
F = FN ( N ) FP ( P ) FX  X 
 N
The material standard of living S increases with capital per person and amount of nonrenewable resources but it decreases with the agricultural fraction of capital. The food ratio F
decreases with population and pollution (FN and FP are monotonically decreasing functions) but
increases with the amount of agricultural capital per person.
All functions occurring in this system are tabulated.
The population N, the capital K, the pollution P have been normalized with dimensional values
N0, K0 and P0 so that their values in 1970 are near to one. The non-renewable resources have
been normalized with the value R0 occurring in the year 1900.
The time is measured in years. The following constants are used in World-2
N0=3.6x109,
cB=0.04,
K0=3.6x109,
cD=0.025,
P0=3.6x109,
cK=0.05,
R0=900x109,
δ=0.025,
X0=0.3,
τX=15
To understand World-2 better it is useful to consider isolated model parts. We begin with the
macroeconomic core of the model.
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Economic part of World-2
dK
= c K NK S ( S ) − δK
dt
For simplicity KS is approximated by a linear function of S, namely
K S ( S ) = 0.85 S + 0.15
The latter is indeed nearly true in a suitable range of S. Then, the economic equation becomes
dK
1− X


= c K 0.85 KS R ( R )
+ 0.15 N  − δK
dt
1− X 0


The similarity to the Solow-Swan model is evident when we assume that labor L is
proportional to the population N. In this case the equation for capital takes formally the form
dK
= sY ( K , L, X , R ) − δK
dt
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Therefore, the production function now also depends upon the agricultural fraction X of capital
and the amount of natural resources R. X decrease and R increase the production. By
assuming constant X and R we obtain the following form for the production function
Y ( K , L ) = aK + bL
This is the so-called von Neumann production function which is characterized by a linear
increase of production with capital and labour. Therefore, the impact of substitution of labour
by capital is independent of the actual amount of capital. This has important consequences for
the solution. For as>δ no equilibrium exists and the solution exhibits unlimited exponential
growth. This is indeed the case in World-2 when no resources are depleted (R=R0).
δK
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Effect of non-renewable resources on economic growth in World-2
A termination of economic growth takes place when the equation for the non-renewable
resources is included. The function RS(S) is approximately given by RS(S)=S. In this case the
dynamical system takes the form (assuming X=X0):
dK
= c K [0.85 KS R ( R ) + 0.15 N ] − δK
dt
N
dR
= − 0 KS R ( R )
dt
R0
The function SR(R) can be written as (giving a perfect fit with the tabulated data):
S R ( R ) = −0.5 cos(π R ) + 0.5 for 0 ≤ R ≤ 1
By setting tendencies to zero we obtain the equilibrium
R = 0, K = 0.15c K N / δ
This means that in this state humans are maintaining some capital without using nonrenewable resources. This could be a state similar to the preindustrial era.
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Numerical solution
Starting values: K(1900)=0.1, R(1900)=1
The economic growth ceases at around 2025 and afterwards economic decay takes place.
Note that even in the year 2200 non-renewable resources are available and decay very slowly
due to the low amount of capital.
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Effect of population dynamics on economic growth in World-2
The population equation is rather complicated since birth and mortality rates depend upon all
model variables. Here, we neglect the impact of pollution P and agricultural fraction X of
capital. Then, the equation becomes
dN
= N [1.02cB BS ( S ) BF ( F ) BN ( N ) − 0.92cD DS ( S ) DF ( F ) DN ( N )]
dt
Furthermore, the food ratio is approximately given by the following expression


K
F ( K , N ) = 1.02 0.4 + 0.6 [2.4 − 1.5 arctan(1.35 N )]
N


The various functions in the population equation can be approximated as follows
(4 − F ) 2
BF ( F ) = 2 −
,
8
B S ( S ) = −0 .2 S + 1 .2 ,
B N ( N ) = − 0 . 1N + 1 . 1
,
1
F
DS ( S ) = 2.5 exp( −1.5S ) + 0.5
DF ( F ) =
D N ( N ) = 0 .2 N + 0 .8
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The birth rate decreases with food ratio, material standard of living and population. The
mortality rate only decreases with food ratio and material standard of living while it increases
with population. Neglecting the dependence on material standard of living and population
yields a Malthusian population dynamics.
This hints at the possibility of multiple equilibria. In the first equilibrium the population should be
large with many fasting people and high birth rates. In the second equilibrium the population
should be low with a high material standard of living and high food ratio. This could lead to a
demographic transition dynamics. However, the model does not reveal this in a convincing
way.
In the equilibrium the capital K as a function of R and N becomes:
Ke =
0.15c K N e
0.15c K N e
=
δ − 0.85c K S R ( R ) δ − 0.85c K [1 − cos(π R )] / 2
Obviously, no equilibrium is possible for
R≥
2δ 

arccos1 −
 = 0.556
π
 0.85c K 
1
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Figure: Population of the equilibrium states as a function of R
In the model humans can survive even if all non-renewable resources are exhausted. Then,
the number becomes 380 million peoples. Multiple equilibria arise only near R=0.55. For
R→0.556 the equilibrium population diverges dramatically.
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Simulations with resource extraction and population dynamics
Simplified World-2 model
(only prediction of N,K and R)
Full World-2 model (including pollution
and agricultural dynamics)
We see that the simplified model already simulates an evolution which is quite similar to that of
the full model. Note that the real world population rises much more dramatically after 1970.
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A better agreement with the real world population results when we increase the economic
growth parameter from cK=0.05 to cK=0.1 after 1970.
Simulation with increased capital growth
parameter
Comparison of “poverty” index (death rate,
probability to die with in 1 year) simulations.
In this scenario the capital attains a much larger value and the poverty decreases below the
minimum of the standard simulation. However, the poverty increases much faster in the
economic decay phase.
CM-147