Untangling Complex Systems, page 34
If you want to know more about models of business cycles, you can visit the website of the Institute
for New Economic Thinking. On this webpage (https://www.ineteconomics.org/), there is a link to “The History of Economic Thought Website.” In the “Essays and Surveys,” there is a contribution dedicated completely to “Business Cycle Theory” (http://www.hetwebsite.net/het/essays/cycle/
cyclecont.htm).
6.9 EXERCISES
6.1. Goodwin’s model into practice: solve numerically the system of two differential equations
[6.17] and [6.18] for the conditions listed in Table 6.1.
You may calculate the time evolution of the two variables μ (employment rate) and ν (wage
share) in the time interval [0 250] and for the initial conditions μ( t = 0) = 0.4 and ν( t = 0) = 0.6.
6.2. The multiplier and accelerator model into practice: solve the nonhomogeneous second
order linear difference equation [6.27] for the values of parameters c , ,
0 I 0 c, κ, and the
initial conditions listed in Table 6.2. To find the solutions, read carefully Box 6.1 of this
chapter. These data have been extracted from Samuelson (1939).
The Emergence of Temporal Order in the Economy
161
TABLE 6.1
Values of the Capital-Output Ratio ( κ), the Labor Productivity ( λ), the
Rate of Workers Supply ( n), the Slope of the Wage Rate ( β) and the
Intercept of the Wage Rate ( α) for Four Different Cases, Labeled as
A, B, C, and D, Respectively
Case
κ
θ
n
β
α
(A)
0.5
0.04
0.003
0.52
2.54
(B)
0.5
0.4
0.003
0.52
2.54
(C)
1
0.4
0.003
0.52
2.54
(D)
0.5
0.4
0.003
1
2.54
TABLE 6.2
Values of the Autonomous Consumption ( c0), the Autonomous
Investment ( I0), the Marginal Propensity to Consume ( c), the Capital-
output Ratio ( κ), the Two Initial Conditions of the Output Y ( Y( t = 0) and Y( t = 1))
Case
c0 + I0
c
κ
Y( t = 0)
Y( t = 1)
A
1
0.5
0
1.00
B
1
0.5
2
1.00
2.50
C
1
0.6
2
1.00
2.80
D
1
0.8
4
1.00
5.00
6.3. The data in the Table 6.3 refers to the percent annual GDP growth for the two strongest economies in the world: that of the USA and that of China (data extracted from the
World Bank website). By using the Fourier Transform (see Appendix C), determine and
compare the most important frequencies in the two GDP trends (Table 6.3).
6.10 SOLUTIONS TO THE EXERCISES
6.1. To solve numerically the equations [6.17] and [6.18] you may use MATLAB. The script file
should look like the following: [ t, y] = ode45(‘Goodwin’,[0 250], [0.4 0.6]);
The outputs are reported in Figures 6.9 and 6.10.
When the labor productivity increases ten times as from condition (A) to (B), the ampli-
tude of the oscillations grows, as well. In particular, the excursion of employment rate μ
raises. If now, at high labor productivity, the capital-output ratio doubles (condition [C]),
the period of the oscillations becomes longer. Finally, if the capital-output ratio is low, but
the slope β of the wage rate doubles, the amplitude of the oscillations shrinks significantly.
6.2. The complete solution of equation [6.27] is the sum of two contributions: Ytot = Ys + Yh. The term Y
t
t
s is obtained by applying equation [6.28]. The general form for Yh is M r
1 1 + M 2 r 2 ,
where r and are the two roots of the characteristic equation [6.29], whereas
and
1
r 2
M 1
M are arbitrary constants to be defined after knowing the initial conditions.
2
162
Untangling Complex Systems
TABLE 6.3
Data of the Percent Annual GDP Growth for the USA and
China in the Period Included between 1961 and 2014
Year
GDP Growth for the USA
GDP Growth for China
1961
2,3
−27,3
1962
6,1
−5,6
1963
4,4
10,2
1964
5,8
18,3
1965
6,4
17
1966
6,5
10,7
1967
2,5
−5,7
1968
4,8
−4,1
1969
3,1
16,9
1970
3,2
19,4
1971
3,3
7
1972
5,3
3,8
1973
5,6
7,9
1974
−0,5
2,3
1975
−0,2
8,7
1976
5,4
−1,6
1977
4,6
7,6
1978
5,6
11,9
1979
3,2
7,6
1980
−0,2
7,8
1981
2,6
5,2
1982
−1,9
9
1983
4,6
10,8
1984
7,3
15,2
1985
4,2
13,6
1986
3,5
8,9
1987
3,5
11,7
1988
4,2
11,3
1989
3,7
4,2
1990
1,9
3,9
1991
−0,1
9,3
1992
3,6
14,3
1993
2,7
13,9
1994
4
13,1
1995
2,7
11
1996
3,8
9,9
1997
4,5
9,2
1998
4,4
7,9
1999
4,7
7,6
2000
4,1
8,4
2001
1
8,3
2002
1,8
9,1
2003
2,8
10
2004
3,8
10,1
2005
3,3
11,4
( Continued )
The Emergence of Temporal Order in the Economy
163
TABLE 6.3 ( Continued )
Data of the Percent Annual GDP Growth for the USA and
China in the Period Included between 1961 and 2014
Year
GDP Growth for the USA
GDP Growth for China
2006
2,7
12,7
2007
1,8
14,2
2008
−0,3
9,6
2009
−2,8
9,2
2010
2,5
10,6
2011
1,6
9,5
2012
2,2
7,8
2013
1,5
7,7
2014
2,4
7,3
Source: World Bank website: (http://www.worldbank.org/)
25
(A)
20
15
1050
200
210
220
230
240
250
30
25
(B)
20
15
105
ν 0 200
210
220
230
240
250
μ and 25
(C)
20
15
1050
200
210
220
230
240
250
14
12
(D)
1086420
200
210
220
230
240
250
Time (years)
FIGURE 6.9 Trends of employment rate μ (dashed gray traces) and wage share ν (continuous black trace) for the four conditions A, B, C, and D listed in Table 6.1.
164
Untangling Complex Systems
(C)
(A)
7
(B)
(C)
6
(D)
(B)
5
(A)
4
ν
(D)
3
2
1
0
0
5
10
15
20
25
μ
FIGURE 6.10 Limit cycles in the ν-μ phase space for the four conditions A, B, C, and D, listed in Table 6.1
of the exercise.
Case (A): the marginal propensity to consume c = ( dC dY ) = .
0 5 and the capital-output ratio
is κ = ( K Y ) = 0. The accelerator does not work. We are in the region (1) of Figure 6.7.
The general solution is Y
t
= 2 − (0 5
. ) , because Ys = 2, and the roots are r 1 = 0 5
. , r 2 = 0.
With a constant level of governmental expenditure through time, the national income
approaches asymptotically the value Ys = 2 (see graph (A) in Figure 6.11).
Case (B): the marginal propensity to consume c = ( dC dY ) = .
0 5 and κ = 2. Both the
multiplier and the accelerator are in action. Ys = 2. The roots of the characteristic
equation are r 1,2 = 3 4 ± i 7 4. The general solution of the homogeneous part of the
equation [6.27] is Y
t
t
t
t
1 3 4
7 4
2 (3 4
7 4)
h = c (
+ i
) + c
− i
= c 1 (α + iβ ) + c 2 (α − iβ ) .
In polar coordinates α = r co θ
s , β = r sinθ , r = α 2 + β 2 , θ =
−
tan 1 (β α).
It derives that Y
t
(
θ −
θ)
1 ( cosθ
sinθ )
h = c
r
+ ir
+ c r
ir
t
2
cos
sin
. By using De Moivre’s
theorem
[ r( θ + i
t
θ )] = rt
cos
sin
(cos( tθ )+ i sin( tθ )), the solution becomes
Y
t
h = r [ M 1 cos ( tθ ) + M 2sin ( tθ )], with M 1 = c 1 + c 2 and M 2 = i( c 1 − c 2 ). In our case, the general solution is Y
t
tot = 2 + ( )
1 [ M 1 cos(41. t
41 ) + M 2sin(41. t
41 )]. From the initial con-
ditions, we can infer the values of the two coefficients, M 1 = 1
− , M 2 = 1 8
. 9.
This case (B) gives rise to regular oscillations (see graph (B) in Figure 6.11). In fact,
it lies on the curve (5) of Figure 6.7.
Case (C): What happens when the marginal propensity to consume is slightly larger than in
case (B)? c = ( dC dY ) = .
0 6, and the capital-output ratio is κ = ( K Y ) = 2. Ys = 2 5
. . The
roots of the characteristic equation are r 1,2 = 0 9
. 0 ± i 0 62
. . In polar coordinates, the gen-
eral solution becomes Y
t
2 5
.
1 0
. 9 1cos(34 5
. 6 )
2 sin (34.56 )
tot =
+ (
) M
t + M
t . From the
initial conditions, we can infer the values of the two coefficients, M 1 = 1
− 5
. , M 2 = 2 6
. 5.
Case (C) that lies in the region (4) of Figure 6.7 originates explosive oscillations of
income (see graph (C) in Figure 6.11).
Case (D): What happens when both the marginal propensity to consume and the
capital-output ratio are high? We have c = ( dC dY ) = .
0 8 and κ = ( K Y ) = 4. Ys = 5.
The Emergence of Temporal Order in the Economy
165
2.0
4
1.8
Y
1.6
tot
Ytot 2
1.4
1.2
0
1.0
0
2
4
6
8
10 12 14
0
2
4
6
8
10 12 14
(A)
t (years)
(B)
t (years)
12
10
8
10000
6
Y
4
tot
Ytot
2
5000
0
−2
−4
0
0
2
4
6
8
10 12 14
0
2
4
6
8
(C)
t (years)
(D)
t (years)
FIGURE 6.11 Outputs of the multiplier and accelerator model in the four situations, labeled as A, B, C,
and D, corresponding to the four conditions A, B, C, and D listed in Table 6.2.
The roots of the characteristic equation are r 1 = 2 9
. 0 and r 1 = .
1 10. The general solution is
Y
t
t
5
1 (2.90)
2 (1.10)
tot =
+ M
+ M
. The values of the coefficients M 1 and M 2 are determined
from the initial conditions. The final expression is Y
t
t
tot = 5 + 2 45
. (2.90) − 6.45(1.10) .
The national income explodes, literally, without oscillations if the population is inclined
to spend and if the capital-output ratio is high. A κ = 4 means that four units of capi-
tal are required to produce one unit of output. In general, the lower the capital-output
ratio, the higher the productive capacity of capital, and the lower will be the investment
according to the accelerator.
6.3. Figure 6.12 shows the trend of the percent annual GDP growth for the USA and China.
In the last twenty years or so, the percent annual GDP growth of China is impressive.
Nevertheless, the gap of GDP per capita between the two countries is still huge according
to the data in the World Bank website (http://data.worldbank.org/). For example, in 2015, the GDP per capita was 55,836.80 US$ in the USA and seven times smaller in China
(7,924.70 US$).
The two trends of percent annual GDP growth appear different. However, if we look
at their Fourier spectra (see Figure 6.13), we find that, in both, the main frequency is
0.148 years−1. It corresponds to almost seven years. This result could suggest that the two
economies are somehow linked or experience analogous cycles.
The French economist Clement Juglar (1862) was one of the first to develop a theory
of business cycles. He identified cycles of seven to eleven years’ long that were attributed
to credit oscillations. Other recurrent cycles have been, then, proposed or discovered. For
instance, the shortest is the Kitchin (1923) cycle of three to five years, generated by the time
lags in information movements affecting the making of commercial firms. Another kind of
cycle is the Kuznets swing having a period of 15–25 years attributed to demographic pro-
cesses, in particular with immigrant inflows and the changes in construction intensity they
cause (Kuznets 1930). Finally, the longest type of cycle, ranging between 45 and 60 years,
166
Untangling Complex Systems
25
20
15
10
5
0
−5
−10
USA
annual GDP growth
China
% −15
−20
−25
−30
1960
1970
1980
1990
2000
2010
Years
FIGURE 6.12 Percent annual GDP growth for the USA and China in the period included between 1961
and 2014.
Frequency
Frequency
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
0.4
0.5
−500
300
−1000
200
−
Phase 1500
Phase 100
1.5
04
USA
China
1.0
