Untangling Complex Systems, page 33
λ . [6.12]
Goodwin assumed that the wage rate g can be approximated by the following linear relation:
w
gw = −α + βµ. [6.13]
It is known as Phillips curve where α and β are positive constants. Phillips straight line tells that wage rate rises as employment increases. From the definition of λ, we infer that gλ = θ. Therefore, g
( α βµ)
v = −
+
−θ. [6.14]
Merging equations [6.11] and [6.14], we obtain a system of two differential non-linear equations
describing how the employment rate μ and wage share v change mutually over time:
dµ
µ
= (1− v) − (θ + n)µ [6.15]
dt
κ
dv
= (−α + βµ) v −θ v [6.16]
dt
The system of equations [6.15] and [6.16] is formally equivalent to the Lotka-Volterra equations
describing the predator-prey dynamics in a natural ecosystem. This is evident if we arrange [6.15]
and [6.16] in the following form:
dµ 1
v
=
−θ −
n µ µ [6.17]
dt
−
κ
κ
dv
= βµ v − (α +θ ) v [6.18]
dt
The Emergence of Temporal Order in the Economy
155
The system of equations [6.17] and [6.18] has two fixed points, namely the trivial fixed point at the
origin and
(α +θ )
v
1 (θ
)
ss = −
+ n κ and µ ss =
. [6.19]
β
The Jacobian evaluated at the non-trivial fixed point is
(α +θ )
0
−
J =
βκ . [6.20]
β (1− (θ + n)κ )
0
The trace of J is null, and its determinant is positive. The eigenvalues are imaginary numbers.
The values of the two variables, μ, and v, oscillate: when μ is very high, v starts to grow, but when v increases, μ drops (see Figure 6.6a). μ and v trace closed orbits (see Figure 6.6b). In
[6.17 and 6.18], the employment rate μ serves as the prey while the wage rate v acts as the
predator. Equation [6.17] implies that in the absence of wage rate, the employment rate grows
exponentially at rate 1
( κ −θ − n) because labor does not cost. On the other hand, equation
[6.18] suggests us that in the absence of employment rate, the wage rate decreases exponen-
tially at the rate ( α + θ).
Goodwin’s theoretical model has attracted much interest since its publication. It is appeal-
ing because it is simple. It has been found an adequate model at the qualitative level. Proofs
underlying Goodwin’s model have been collected (Harvie 2000; Molina and Medina 2010). They
encourage its theoretical development to more complex models of business cycles in the capitalist
economies.
TRY EXERCISE 6.1
µ
ν
ν
ν.
µ.
(a)
Time
(b)
µ
FIGURE 6.6 Oscillations of the “prey-predators” variables μ (dashed gray trace) and v (continuous black trace) in (a), and a closed orbit traced by them in (a). In (b),
µ and v represent the null-clines, and their intersec-
tion is the steady-state solution. The closed orbit is outlined counterclockwise.
156
Untangling Complex Systems
6.5.2 The mulTiPlier and acceleraTor model
The American economist Paul Samuelson (1939) developed another famous macroeconomic model
to interpret the business cycles. It is based on the concepts of multiplier and accelerator.
The theory of multiplier describes how an increase in the level of investment I raises income or
output Y:
It I
Y
t−1
t − Yt−1 =
−
(
, [6.21]
1− c)
where c is the marginal propensity to consume. 8 The increase in the income induces growth in investment through the accelerator:
I
κ(
−1 )
t =
Yt − Yt . [6.22]
In [6.22], κ is the capital-output ratio ( K/Y). Thus, the multiplier implies that investment increases output, whereas the accelerator implies that growth in output induces increases in investment. There
is a reciprocal positive feedback action between income and investments. It is analogous to the posi-
tive feedback relation between science and technology.9
A national income at time t, Y , can be expressed as the sum of two terms, assuming away govern-
t
ment and foreign sector:
Yt = Ct + It, [6.23]
where:
C is consumption expenditure
t
I is private investment.
t
The consumption depends on past income Y and autonomous consumption c :
t− 1
0
Ct = c 0 + cYt−1. [6.24]
Investment is taken to be a function of the change in consumption and the autonomous invest-
ment I :
0
I
0
κ (
−1 )
t = I +
Ct − Ct . [6.25]
Inserting equations [6.24] and [6.25] in [6.23], we obtain the definition [6.26] of how changes in
income are dependent on the values of marginal propensity to consume ( c ) and the capital-output
0
ratio κ:
Y
0
−1
0
κ ( −1
−2 )
t = c + cYt
+ I + c Yt − Yt . [6.26]
8 The marginal propensity to consume c is the derivative of the consumption function C with respect to the income Y: c = ( dC dY
/
).
9 Another action that exerts a positive feedback effect is the tax cut. In fact, a tax cut promotes investments by the producers and outgoings by the consumers.
The Emergence of Temporal Order in the Economy
157
Equation [6.26] can be rearranged as a nonhomogeneous second order linear difference equation
(see Box 6.1 in this chapter):
Y
(1 κ ) −1 κ −2 ( 0 0)
t − c
+ Yt + c Yt = c + I . [6.27]
The particular “steady-state” solution can be easily determined by fixing Yt = Yt−1 = Yt−2 = Yss: c 0 I
Y
0
ss =
+ . [6.28]
1− c
The solution of the homogeneous equation is of the type Y
t
t
h = M r
1 1 + M 2 r 2 , where M and
are
1
M 2
arbitrary constants (that can be determined by fixing particular initial conditions, Y and ), and
0
Y 1
r and are the two roots of the characteristic equation
1
r 2
r 2 − c 1
( +κ ) r + cκ = 0. [6.29]
The overall solution is Ytot = Yss + Yh. If we want to know the dynamics of the system, it is necessary to calculate the roots of equation [6.29]:
c( +κ ) ± c 2
2
1
1
( +κ ) − 4 cκ
r 1,2 =
. [6.30]
2
We may distinguish five kinds of solutions that correspond to five types of dynamical scenarios (see
Figure 6.7). To outline the five possible solutions, it is important to consider the sign of the discriminant ∆ = ( c 2 ( +κ )2
1
− 4 cκ ) and the conditions for stable roots (see Box 6.1 in this chapter), which are
BOX 6.1 SOLUTION OF A NONHOMOGENEOUS SECOND
ORDER LINEAR DIFFERENCE EQUATION
A nonhomogeneous second order linear difference equation may take the form
yt + ayt−1 + byt−2 = k, where k is a constant for all t. Its overall solution is the sum of two contributions: ytot = ys + yh. The first contribution, y , is obtained by setting yt = yt−1 = y s
t −2.
It derives that ys = k / (1+ a + b). The second term y represents the solution of the homoge-h
neous equation: y
t
t
t + ayt −1 + byt −2 = 0. It has the general form yh = M r
1 1 + M 2 r 2 , where M and
1
M are arbitrary constants to be defined, and and are the two roots of the character-
2
r 1
r 2
istic equation r 2 + ar + b = 0. The two roots are given by r
2
(
)
1,2 = − a ±
a − 4 b /2. We dis-
tinguish three cases. First, when a 2 > 4 b. The characteristic equation has two distinct real
roots, and the general solution is y
t
t
2
h = M r
1 1 + M 2 r 2 . Second, when a = 4 b. The character-
istic equation has a single root, r = − a/2, and the solution is y
t
h = ( M 1 + M t
2 ) r . Third, when
a 2 < 4 b. The characteristic equation has roots that are complex numbers, and the solution is
y
t
t
1
cos(θ )
h = M r
t + M 2 r sin(θ t). The overall solution ytot = ys + yh is stable if and only if the modulus of each root of the characteristic equation is less than 1. If the characteristic equation has real roots, then the modulus of each root is its absolute value. Therefore, for stabil-
ity, we need the absolute values of each root to be less than 1, or (− a + a 2 − 4 b )/2 < 1 and (− a − a 2 − 4 b )/2 >1. If the characteristic equation has complex roots, then the modulus of each root is b. For stability, we need b < 1. It can be demonstrated (Elaydi 2005) that in terms
of the coefficients appearing into the second order linear difference equation, the solution is
stable if and only if a < 1+ b and b < 1.
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Untangling Complex Systems
1.0
(1)
(2)
0.8
0.6
c
(4)
(3)
0.4
0.2
(5)
0.0 0
1
2
3
4
5
6
κ
FIGURE 6.7 The output of Samuelson’s model. The c vs. κ plot is partitioned in five regions by the discriminant Δ of equation [6.30] represented by the continuous black curve, and the stability conditions for the roots
of the characteristic equation [6.29], represented by the dashed gray straight line ( c = )
1 and the dotted gray
curve ( c = 1/κ ).
1
( +κ ) c < 1(+κ c) and κ c <1. The discriminant can be rearranged in the following form c = 4 /(1+ )2
κ
κ ,
and it is plotted as a continuous black curve in Figure 6.7. The conditions for stability reduce to the relations c < 1 and c < 1/κ . The functions c = 1 and c = 1/κ are plotted as dashed gray straight line and dotted gray curve, respectively (see Figure 6.7). Merging the three functions, the graph of marginal propensity to consume ( c) versus the capital-output ratio ( κ) is partitioned into five regions.
Region (1) embeds positive real roots that are stable solutions. Therefore, in (1), the income, or
the GDP, moves from the original steady-state upward or downward at a decreasing rate and finally
reaches a new steady-state. In region (2), we encounter positive real roots of c and κ that cause the system to diverge and explode from the original steady state, at increasing rate. In regions (3), (4),
and (5) that are below the function c = 4 /(1+ )2
κ
κ (having a maximum at c = 1 and κ = 1), the roots
are complex numbers. The points in (3) represent stable solutions. Therefore, change in investment
or consumption will give rise to damped oscillations in income or GDP, until the original state is
restored. The solutions in (4) are combinations of c and κ values that are relatively high and cor-
respond to such values of multiplier and accelerator that bring about explosive cycles, that is, the
oscillations of income or GDP with greater and greater amplitude. Finally, when the values of c
and κ lie over curve labeled as (5), they generate oscillations of constant amplitude in income. It is
evident that only a precise constellation of c and κ values yield constant cycles. All the other com-
binations bring to either complete stability or complete instability, whether monotonic or oscillat-
ing. Thus, the multiplier-accelerator model is incomplete as a theory of permanent business cycles,
although it has been a great advance in understanding the dynamics of macroeconomy.
TRY EXERCISE 6.2
6.5.3 oTher models
Other models to elucidate business cycles have been proposed. For example, the real business cycles
theory. It supposes that business cycles are always triggered by an exogenous cause, such as new
revolutionary technology, geopolitical event, war, natural disaster, and so on (Plosser 1989), and
The Emergence of Temporal Order in the Economy
159
they are responses to the changes in real markets’ conditions. Another theory is that of credit and
debt cycles, which attributes business cycles to the dynamics of over-borrowing by businesses dur-
ing the periods of economic booms, followed by the inevitable economic slowdown that brings to a
debt crisis and a recession (Fisher 1933). There is also the political cycles theory (Nordhaus 1975)
that attributes business cycles to political administrators. For example, soon after an election, new
politicians impose austerity to reduce inflation, which increases prices of goods and services. The
inflation reflects a reduction in the purchasing power per unit of money. The electors soon take the
bitter medicine against inflation. Hopefully, they have time to forget its savor before the new election,
a few years later. In fact, around one year before the new vote, politicians, who like to be re-elected,
boost the economy by reducing taxes, increasing public investments, and persuading the central
banks to maintain the interest rates low. The economic activity of many capitalistic democracies has
an electoral rhythm.
6.5.4 The real business cycles
The real business cycles are not perfectly regular. But they are not perfectly random, either. The
economic time series data, whether they refer to the GDP, or unemployment, or inflation, wiggle
with many characteristic frequencies (an example is the GDP change data shown in Figure 6.8a,
whose Fourier Transform is in b).
18
9
0
GDP change −9
−18
(a) 1700
1750
1800
1850
1900
1950
2000
Years
1
Amplitude
0 0.0
0.1
0.2
0.3
0.4
0.5
(b)
Frequency (year−1)
FIGURE 6.8 The trend of the GDP change for Sweden from 1700 up to 2000 in (a) and its Fourier Transform
in (b). The source of the GDP data is the Economics Web Institute at the link http://www.economicswebinsti-
tute.org/ecdata.htm.
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Untangling Complex Systems
Nowadays, scientists are aware that the not-quite-regular, not-quite-random economic time
series are emergent phenomena typical of complex adaptive systems (Beinhocker 2007). They
have three causes: two endogenous and one exogenous. The endogenous are, first, the behavior of
the players in the economic system; second, the structure of the economic system, like the mar-
ket, the production chains, the supply chains, and so on. The third cause is any exogenous input
into the economic system, such as the technological changes. All these factors put together are
responsible for the not-quite-regular and not-quite-random ups and downs observed in the real
business cycles.
TRY EXERCISE 6.3
6.6 KEY QUESTIONS
• Why are ecology and economy related disciplines?
• What is the “economic problem” and how is it possible to solve it?
• Which are the factors that allow to produce goods and provide services?
• What distinguishes a linear and a circular economy?
• Explain the Law of Supply and Demand.
• How is it possible to measure the GDP of a nation?
• Which are the causes of business cycles?
• Describe the Goodwin’s predator-prey model.
• Describe the multiplier and accelerator model.
6.7 KEY WORDS
Economy and political economy; Ecological Economics; Micro- and Macro-economic systems;
Producers and Consumers; Economic environment; Aggregate demand and supply; Employment
rate; Wage share; Marginal propensity to consume; Capital-output ratio.
6.8 HINTS FOR FURTHER READING
