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The discrete continuous choice of economic modeling or quantum economic chaos
Mathematical
Social Sciences
21 (1991) 261-286
261
North-Holland
THE DISCRETE CONTINUOUS CHOICE MODELING OR QUANTUM ECONOMIC Amos
GOLAN
Department
of Economics,
Communicated
by K.H.
A framework developed. structed
for
University of Haifa, Haija, Israel 31999
Kim
investigating
An Essential
economic
the
discrete/continuous
Unit (EU) is defined
in terms of the EU multiplied
that, in most cases, there is no definite an interesting blems,
OF ECONOMIC CHAOS
paradox
arises.
well-known
representation)
result is that in economic
properties
of deterministic
describing
discrete
economic
economic.
Possible
applications
Key words: Deterministic
chaos;
modeling
chaos.
variables
Two main results continuum)
quantum
modeling variable
are shown.
is
is con-
The first is
choice of modeling results in information
results in model fluctuations.
and pro-
The second and
chaotic
structures
emerge that exhibit all the
These chaotic
structures
are universal
and are proved
for quantum
of economic
to the discrete/continuous
A small EU (approaching
while a large EU (a discrete
most surprising
by an integer. answer
choice
such that each (decision)
economic chaos;
to be universal chaos
discrete
research
modeling;
to all models
to systems
other
than
are suggested. non-linear
dynamics.
1. Introduction The objective of this work is to investigate the conditions under which the economy should be modeled as a continuum or as discrete, that is, to develop a rule for the discrete/continuous choice of economic modeling - a crucial choice for economists. In nature, as well as in economics, most observables are discrete in the sense that they are based on small quantities indivisible beyond some critical minimum size. Energy levels, bits of information, and production levels are three examples of indivisible entities describing natural, social, and economic systems (e.g. Samuelson, 1973; Golan, 1988a). In economic systems we call these discrete quantities Essential economic Units (EUs). When one models an economy (system) with a large number of components (such as agents, goods, or firms), it is argued that individual components become weightless and that a continuous specification is correct (Aumann, 1964). Except for the mathematical convenience, this is the main argument in moving to the continuum. At the heart of this argument is the assumption that for each economy there exists some critical size of the EU (e.g. one bushel of wheat, a ton of apples, a table, a household, a firm) such that for any EU smaller than this critical size, the economy could be modeled as a continuum, while for all essential units larger than this critical 0165-4896/91/$03.50
G 1991-Elsevier
Science
Publishers
B.V. All rights
reserved
262
A. Golan / Quantum economic chaos
size the economy should be modeled as discrete. This work investigates theoretically and empirically whether such a critical size exists. It will be shown that, in most cases, such a critical size does not exist and that the answer to the above question is surprisingly non-trivial. It will be shown that each economy is governed by some characteristic in modeling
parameter, and when this parameter exceeds some critical level, chaos will emerge regardless of the size of the EU. However, when this
parameter is below its critical level, there is a critical EU size below which a continuum model is accurate. These results are proved to be universal to other systems (e.g. physical, traffic) as well. An exact definition of the EU is given in the next section. Then, a simple example to introduce the main results of the paper is given. This is followed by a description of the model and the experiment. The results are then discussed in detail, and a discussion of deterministic chaos and its relationship to the main results of this work is presented. A summary and discussion of the implications of results conclude the paper.
2. The Essential
Unit
The discrete continuous choice is best explored by defining each variable in terms of an EU multiplied by an integer. Thus, each observed quantity increases in multiples of its EU. To select the correct EU, one should choose the natural quantities of the economy modeled, that is, the basic units of the variables (decision variables) representing the economy. For example, production should be defined based on the smallest quantity possible of producing for a given technology. In that case, when the natural EU is especially small relative to the economy (e.g. one apple), it is convenient to define multiples of EUs that will translate the EUs into larger quantities (Golan, 1988a). A firm (the smallest firm capable of existing independently given some technology) is another EU one might use in analyzing an industry. The next part of this work deals with production EUs. These results are then generalized. Define x as an EU of production with production units (i.e. the smallest unit of production capable of existing independently) and n as an integer. Then, y, = nx is the production level, where yn depends on the integer n. An EU of input can be defined using a similar approach. A production function will be defined as nx= g(kf), wherefis an EU of input, k is an integer, and g( . ) is a well-defined function. A vector representation is possible by defining a vector of EUs with a corresponding vector of integers.
3. Example Following the above definitions, troduce the main results.
a short
mathematical
example
will help in-
A. Golan / Quantum economic chaos
263
Assume an industry consisting of three firms. Firm A produces 15 EUs, say 15 tons of tomatoes, firm B produces 20 tons, and firm C produces 25 tons. The production, yi (i = A, B, C), is constructed in terms of the EU multiplied by an integer. Specifically, let x be the EU (in units of tomatoes, say 1 ton of tomatoes) and nj the integer describing the production of firm i. Then, Y; = n,x,
i=A,B,C.
(1)
The question to be investigated is whether the size of the EU affects the basic properties of the model. It is sufficient, however, to prove that at least one property
Table 1 Average production ample). jumps N
vs. EU. Thirty
Case A: EU ranges of l/3.
between
Case C: EU ranges Case A EU
different
EUs are calculated
0 and 30 in jumps between
0 and 2 in jumps
EU
(the ex-
0 and 10 in
Case C Average
EU
prod.
mod.
industry
between
of l/15.
Case B Average
for the three-firm
of 1. Case B: EU ranges
Average prod.
1 2
1.000
20.000
0.333
20.000
0.066
2.000
19.334
0.666
19.778
0.133
20.000 19.556
3 4
3.000 4.000
19.000
20.000
0.200
18.667
1.000 1.333
19.556
0.266
20.000 19.910
5
5.000
20.000
1.666
20.000
0.333
19.778
6
6.000
18.000
2.000
0.400
19.866
7
7.000
16.334
2.333
19.333 18.666
0.466
19.755
8
8.000
16.000
2.666
18.666
0.533
19.733
9
9.000
15.000
3.000
19.000
0.600
19.800
10 11
10.000 I1 .ooo
16.667 14.667
3.333 3.666
18.889 18.333
0.666 0.733
19.777 19.800
12
12.000
16.000
4.000
18.666
0.800
19.733
13
13.000
13.000
4.333
17.333
0.866
19.644
14
14.000
14.000
4.666
18.666
0.933
19.600
15
15.000
15.000
5.000
20.000
1.000
20.000
16
16.000
10.667
5.333
16.000
1.066
19.555
17 18
17.000 18.000
Il.334 12.000
5.666 6.000
17.000 18.000
1.133 1.200
19.644 19.200
19
19.000
12.667
6.333
16.889
1.266
19.000
20
20.000
13.334
6.666
17.778
1.333
19.556
21
21 .ooo
7.000
7.000
16.333
1.400
19.133
22
22.000
7.334
7.333
17.111
1.466
19.556
23 24 25
23.000
7.667 8.000 8.334
7.666
15.333
1.533
19.442
8.000 8.333 8.666
1.600 1.666 1.733
19.200 20.000
0.000 0.000
16.000 16.666 14.444 15.000
1.800 1.866
19.200 19.289
1.933
18.689
2.000
19.333
26 27 28 29 30
24.000 25.000 26.000 27.000 28.000 29.000 30.000
0.000 0.000 0.000
9.000 9.333 9.666 10.000
15.555 16.111 16.666
19.067
264
A. Golan / Quantum economic chaos
of the model is affected by the size of the EU. The selected property is discussed below. Of the large number of properties representing the modeled industry, the first moment (average) of its production is fundamental. This property has been chosen for investigation in this example and in the rest of the present paper. It should be noted that any property (such as second moment, variance, etc.) would be affected by the
a 20*
. .**
.
i
I o! 0
10
20 EU
(tons)
.
.
.
0.
l
.
. .
.
.
.
.
. .
. 14 0
2
6
4
10
6
I
1:
EU (tons)
Fig. 1. Example
of the effect of EU size on average (b) EU=
l/3,2/3
,...,
production
in a three-firm
10; (c) EU= l/15,2/15
,..., 2.
industry.
(a) EU = 1,2,30;
A. Golan / Quantum economic
EU size, but by the sufficiency be asked: Does the average that the EU of the modeled all of the firms.) Naturally, unchanged,
condition,
chaos
265
it is not necessary
production of the industry industry must be identical
to show it. It can then
change with the EU? (Note at each moment of time for
if one increases the EU size arbitrarily and the integers, {n,}, remain the average production must increase as well. Therefore, it is necessary
to keep the total production fixed. More precisely, for each EU size, the total production is forced to remain not larger than the total production observed in the industry (60 units). In more economic terms, one might think of a fixed level of demand or of limited resources such that, on average, the industry’s production is constrained to 60 units. Starting with the smallest EU of this industry, say x = 1, the average production is y=$
c n;x=$ i
c n;=+x1
x(15+20+25)=20.
1
Next, the model is examined for some larger EU, say 2. For x= 2, firm A will produce only 14 units, since 7 is the largest possible integer for which A’s production will not exceed 15 (note, if nA= 8, then y,= 16). Similarly, one sees that nn= 10, and nc= 12. The average production of the industry is then y=fx2x(7+10+ Now,
12)= 19:.
for x= 3 the average
production
will be
y=$x3x(5+6+8)=19.
-_ _ C
. .
‘0
l **
.
. .
. .
.
.
. .
.
. 18.51 0
0.2
0.4
0.6
0.8
I.0
1.2
EU (tons) Fig. 1 - contd.
1.4
1.6
I.8
2-o
: ,2
266
A. Golan / Quantum economic
chaos
Increasing the EU by 1 up to, say 30, gives a series of numbers representing the average production of the industry for each EU size. This is shown in Table 1 and Fig. l(a). The average production is a continuous scale represented on the vertical axis of Fig. l(a), while the discrete EU size is on the horizontal axis. As the size of the EU increases (moving to the right in Fig. l(a)), one sees the change in the average production of the industry. The change, however, is not monotonic, and many critical jumps occur. Nonetheless, for EU sizes larger than 26 (26,27, . . . , a), the average production calculated by the model remains zero. This is because the EU size is larger than the total production of the largest firm. The following example clarifies this point. Consider a sample taken from an industry with a large number of firms. Each firm is characterized by its production level. If one chooses an EU that is larger than the production level of the largest firm, say EU of 11 for an industry with a production range of 4 to 10 units of production, there will be no firms in the sample. The sample is an empty set and is represented by the zero ‘line of points’ on the figure. This simple exercise shows that the wrong choice of EU size yields a model whose average differs from that of the economy it wishes to describe. A basic characteristic of this representation is now shown. The above exercise is repeated, but this time instead of changing the EU 30 equal times between 0 and 30, the EU size is changed 30 equal times between 0 and 10 (i.e. l/3, 213, 1, . . . , 10). Table 1 and Fig. l(b) show the average production that corresponds to each one of the 30 EUs. The same exercise is repeated for 30 equal jumps in the EU size between 0 and 2 (i.e. l/15, 2115, 3/15, . . . . 2). This is shown in Table 1 and Fig. l(c). From these figures one can see that there is self-simifarity between all the graphs in the sense that, in a general way, the same structures repeat themselves except for the line of zeros shown for EU of 26 and higher in Fig. l(a). In fact, no matter how fine one divides the EUs, a similar structure appears. Furthermore, it also shows that, as EU size increases, the fluctuations between the average values of the observed industry and those of the model increase. To conclude, this simple example shows that there is a special mathematical property of the above-defined variables that results in the behavior of the model presented here. This mathematical property is the non-linearity of the system described by this example. This non-linearity, in turn, adds to the special chaotic structures shown, and is discussed in detail after describing the different experiments done in this work. The next task is to explore the implications for economic modeling, specifically for the discrete continuous choice of modeling. This is done in the following section.
4. The experiment 4.1.
The model
The experiment
analyzed
in this section investigates
the discrete continuous
choice
A.
Golan
/ Quantum
economic
chaos
267
of modeling based on real-world data. Using a discrete definition of the variables, the average values of an economy are calculated for each EU size. Next, the theoretical economic model is developed. Using this model and figures from the Californian wheat industry, an empirical estimation is made. The estimation yields the economic and technological parameters representing this economy. Based on these parameters, the experiment is performed through a series of steps described in the next section. The experiment is then repeated for different theoretical parameters. The characteristics of the economy are modeled by using the maximum likelihood approach subject to constraints. This approach is also known as the maximum entropy (ME) formalism (Golan, 1988a). In this approach all possible economic states are equally probable, with the exception of those excluded by the available information (constraints). That is, one maximizes the missing information, which implies there is no white noise in the model. The ME formalism describes the most probable macroeconomic state characterizing an economy. Each event has a certain probability of occurring, and the sequence of events with the highest probability describes the most probable economic state. This is the best possible characterization of an economy subject to the available information. Although the theoretical results hold for other formalisms as well, the ME formalism was chosen because it is not a deterministic one; randomness is a basic property of the model and not just white noise. Furthermore, this framework is general in the sense that it is possible to show the conditions under which the resulting distribution collapses into any other well-known distribution (Golan, 1988b). Since this work concentrates primarily on the EU issue, the ME formalism is only described briefly. An extensive description of this approach is given in Levine and Tribus (1979), and its economic application and implications are given in Golan (1988a, 1988b). Based on limited available information, the ME approach determines the steadystate economic distribution in terms of the variables analyzed. In this paper, the ME formalism is used to model the size distribution of firms. For simplicity, the model is presented in terms of one output and one input, where the variables are defined as EUs multiplied by integers. Let the individual producer’s output be y,,,= nix, where n; is an integer related to producer i and x is the EU of production. Similarly, the resources of a producer are rk, = kJ, where k; is an integer and f is the EU of resources. For simplicity, the above definitions are written as y,, = nx and rk = kf. Next, the producers are grouped according to their level of production and resources, where the relationship between the production level and firm size (resources) is captured in the group specification (Golan, 1988a). Specifically, define q,,k as the number of producers having an output in the range nx to (n + 1)x and resources in the range kf to (k+ l)f, where n and k are not independent. It is then possible to construct the constraints of this economy (i.e. the available information) in terms of the discrete variables. Finally, the economic distribution (firm size distribution) can be estimated by maximizing (subject to the above constraints) the number of ways of partitioning all the firms into groups with similar characteristics
(the multinomial coefficient). The groups’ characteristics are represented by the integers. Since the EUs are necessarily identical within each system, the different integers represent different groups within a system. The above maximization process yields the estimated economic distribution that is characterized by some Lagrange parameters
(e.g. cr and /?) that correspond
1
n
constraints.
That is,
e-akf e-Dnx
N
qnk=
to the different
c
k
e-akf
’
e-anx
(2)
where N stands for the number of firms in the industry. This equation represents the steady-state distribution of firms according to their size, where size is defined over production and resources simultaneously. It is convenient to define the denominator as Q
c c e-akf e-Dnx,
=
n
k
where Q is a normalization factor that converts relative probabilities into absolute probabilities. The Q function is known as the partition function because it partitions the different elements of the model (e.g. type of producers). The probability of group
qnk is then Pnk
=
(Ink N
_
ePkf embnx s-2
.
This concludes the brief summary of the ME formalism. Lagrange parameters and other natural quantities that arise are discussed in detail in Levine and Tribus (1979), Reiss et (1988a). Having described the steady economic state, it is now the question: Does the EU size affect the average properties 4.2.
(4) Interpretation of the during this formalism al. (1986), and Golan possible to investigate of the model?
The experiment
The data set for the experiment is taken from Golan (1988a). It contains information on the resources and output of all Californian wheat farms in the years 1974, 1978, and 1982. Firm size is defined over both production (in units of bushels) and land (in units of acres). The different steps of the experiment are now discussed. In the first step the steady economic state distribution is calculated by solving the multinomial coefficient subject to the constraints. The calculated parameters for 1978 are (x=3.9199 and p= -4.163. Second, based on the above solution and parameters, it is possible (see proofs in Golan, 1988b) to infer the technological characteristics of the industry and to incorporate it directly into Q. It is found that the technology characterizing this industry is of the form nx=a(k#’ and, hence,
A.
The estimated
production
Golan
/ Quantum
parameters
economic
chaos
are (I= 1.62 and b = 0.815.
some EU size, say E, the total (and average) production and proved to be consistent with the empirical data.
269
Third,
choosing
of the industry is determined Fourth, taking the calculated
average production, the total production of this industry is calculated. Finally, constraining the industry’s total production to be no larger than the total production calculated for the case, x= E (the real-world observed level), the average production is calculated for each increasing EU size, i.e. for x= 2e, x= 3e, . . . . The reason for this type of normalization is now discussed. As was shown in the example, total production must be conserved when increasing EU size. Specifically, if n is an integer representing the average production of an industry with EU x, then y, =nx. Changing the EU size and keeping n unchanged effects y, in the same direction as x. It is therefore necessary to normalize the economy according to some criteria (e.g. total production, input level). This procedure is done as follows. Choose a small EU, say E, and calculate the average properties of this system, say average production and average vector of inputs. On the basis of these calculated averages, the total properties are then calculated. For some average production nx and N number of firms, the total production of the economy is NX nx. When calculating the economy’s average properties for the next EU size, 2&, it is important that the total number of properties in the economy remains the same, as in the case for X=E. That is, output and inputs are conserved between calculations of different EU size. This normalization framework enables one to investigate if a change of EU size changes the average calculated properties of an economy even though the aggregate properties are forced to remain constant. The objective of the experiment, then, is to calculate the properties (first moments) of the model for different EU sizes, while the total properties of the economy are held constant based on the observed data. This experiment was repeated for different empirical and theoretical values of the economic parameters (Q and p) and for different technologies. It was then repeated for different variables (e.g. production, inputs) and for different systems results are discussed in the following subsection. 4.3.
(not necessarily
economic).
The
Results
As in the previous example (Fig. l), a graphical analysis of the results is necessary and helpful in describing the outcome of the experiment. Figure 2 shows an economic system where the relationship between production with a>0 and b< 1. This figure is based on the and input is nx=a(/&, parameters of the 1978 Californian wheat industry, where 20000 different production EU sizes are calculated and each calculation is a point on this figure. Figure 2(a) shows all the 20000 calculated averages, where the vertical axis represents the continuous scale of average production, and the horizontal axis represents the 20000 different production EUs. From Fig. 2(a) one can observe the following properties: (1) as EU size increases, the fluctuations around the real observed average produc-
270
A. Golan / Quantum economic chaos
2
0
4
6
8
10
EU (log(x))
9.65
5 ‘Z v
9.60 9.55
4 g 9.50 gl 9.45
c
‘..‘.
.,....
‘.
.;.:,“.
L$ 9.40
,.
‘..
b
9.35
0.10
0.15
0.20
0.25
0.30
EU (log(x))
Fig. 2. Effect
of EU size on average
for the case of concave the 20000
technology. points;
production. Parameters
(b) an enlarged
Average
production
used: a=3.9199, version
vs. size of the EU of production
p=-4.162,
of the upper
left-hand
a= 1.62, b=0.815. corner
(a) All
of (a).
tion increase; and (2) there is a large number of critical points where a change of E in EU size results in a large discontinuous change in the average production level. It is also obvious from this figure that the average production of the modeled system becomes zero for all EU sizes that are bigger than the real-world observed average. It then follows that the integer associated with any EU larger than the observed average production (about 9.7 in Fig. 2) is necessarily zero. Going from right to left in Fig. 2(a) while investigating each ‘line of points’ also reveals the following property. In Fig. 2(a) the second line of points (from the right) is associated with the integer 1. The third line of points is associated with the integer 2. The next line is associated with the integer 3, and so on. Does this pattern continue all the way to EU = E, the furthermost point on the left-hand side? The answer is revealed in Fig. 2(b). This figure is just an enlarged version of the upper left-hand corner of Fig. 2(a). Here, it is obvious that this corner is qualitatively different than the right-hand side. What is the reason for this, and how can one explain it? To answer these questions it is enlightening to perform a short exercise. The exercise is similar in notion to the example shown in Fig. 1 (or I(a)-l(c)) where each
A. Golan / Quantum
figure represented the same distance between the smallest graphs depicted a continuous the same kind of magnification 20000 different calculations
economic
chaos
271
number of calculations (30 different EUs) but the EU (a) and the largest decreased. In other words, the magnification of the left-hand side. In this exercise, is performed but on the wheat farmer’s data. The of each EU size are repeated for smaller and smaller
EU differences. For example, 20000 calculations are done for the range 0 to 10 EU, so each EU size increases by (l/2000), i.e. the smallest EU is l/2000, the second EU is 2/2000, . . . , 10. Next, there are 20000 EUs between 0 and 2, so the smallest EU is l/10000, the second EU size is 2/10000 and so on. Next, there are 20000 different EUs for the sizes 0 to l/2 such that the smallest EU is l/40000, the second is 2/40000, and so on. The surprising results show that no matter how fine the distance between the smallest and largest EU, the left-hand side of the figure always looks the same. Of course the magnitude of the fluctuations becomes smaller, but the qualitative structure of the fluctuations remains the same for each magnification of the left-hand corner of the figure; the same basic structure keeps repeating itself. The figures generated by the exercises exhibit self-similarity. If one defines the distance between the smallest and largest EU as d and the number of different EUs calculated as N, then for each N as d + E > 0 there is self-similarity among the different graphical representations related to each d. Each figure corresponding to a different d will look almost exactly the same, the only difference appearing in the first figure (largest d) where there are points affiliated with an integer of zero (the EU size is larger than the average property). An immediate property evidenced by this observation is that no matter what the EU size, there will always be some fluctuations in modeling. In other words, the probability that the model will represent the economy it hopes to model diminishes with the EU size. The next question is whether this property is due to the specific data set analyzed or whether it is a universal property for all different economies. Specifically, is there some critical parameter associated with each economy that affects the above average calculations (the structures in the figures)? The answer is that such a parameter exists. For the economic production model presented here, it has been found that the critical parameter is the productivity level where productivity is defined as the set of parameters describing the production function such as a in y = a log(x) or a and b in y=axb. For each productivity level lower than some critical level, the discussed structures do not appear. That is, the average production, as calculated in the experiment, behaves differently than in Fig. 2. Above this critical level, however, the structures are universal for a/I different levels of the economic parameters (e.g. (Yand p) and for all possible technologies (production functions). The experiment is repeated for different economic parameters (e.g. the Californian wheat market in 1974 and 1982) and for different theoretical economies where the production functions are concave, linear, convex, or of many different mathematical forms [e.g. nx=alog(kf)]. The fascinating result is that the above structures repeat themselves throughout the experiment. It has been stated that below some critical level of the critical parameter these
212
A. Golan / Quantum economic
chaos
universal structures do not appear. In this case, a continuous line emerges, and the calculated average production is now a continuous function of the EU size. (Note that to prevent confusion the word ‘size’ is associated with EU, while the words ‘level’ and ‘value’ are associated with the parameters.) But even this continuous line
-
5 ‘Z u
.
5-
a
...“‘..
‘.
,:’
.:’
4(....._
9 ‘a
3-
i?
2-
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l-
‘......
,:’
,.:
,:’
,:’
. . .. . . . . . . .._..................................
O-
10
0
6
4
2
0
EU (log(x))
: H ,a
6-
m
4-
9
th I 9 <
,......._....................
-.., . ....’ .:’
,....._( .. ‘.
,,,..‘.
...”
. ...’
,...”
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,...‘.
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,._.”
.... . ,._..
b
..’
.._,,..”
2-
0 -_ 0
4
2
6
8
lo
6
8
10
EU ( log (x1)
0
2
4 EU (log(x))
Fig. 3. Effect of EU size on average production for different levels of the critical parameter (productivity) or, the Road to Chaos. The production function used is nx=a log(kf) and the parameters used are
a=3.9199
and p=-4.163.
(a) a=3;
(b) a=4;
(c) a=4.5;
(d) a=4.6;
(e) a=5;
(f) a=7.
A. Golan / Quantum economic chaos
becomes
erratic
up to a certain
for some EU greater critical
than a critical
273
size of the EU. This means that
size of the EU both continuous
and discrete
models
prove
to be completely consistent with each other and both are correct in the sense that the averages of the model are the same as the averages of the modeled economy. Above the critical EU size, however, the model starts to behave erratically in that the quantities
0
derived
from
the model
2
cease to reflect
4
the real modeled
6
0
system.
10
EU (log(x))
IO -
5
,:...
c
8
Y D 2
6-
... ‘.. ..‘.‘.“..‘.‘. ‘. .. .’
.’
.
...‘.
e
,...”
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,a
0
2
4
6
8
10
EU (log(x))
0-l
0
2
4
6 EU (log (xl)
Fig. 3 - conrd.
8
10
274
Figure
A. Golan / Quantum
3 shows the behavior
of the critical
economic
chaos
parameter.
The
This figure can be titled
Road to Chaos. At this point a closer examination
of the Road to Chaos is in order.
On the basis
a
0
5
10
15
EU ( log scale
10 EU, (log
20
1
scale)
10 EU (log scale)
Fig. 4. Effect of EU size on average input. Average input vs. size of the EU of input. The production function used is nx = a log(kf). (a) a = 13; (b) a = 15; (c) a = 20; (d)-(f) the enlarged versions of the lefthand
corners
of (a)-(c).
A. Golan / Quantum economic
of the 1978 economic
parameters
calculated
215
chaos
for the wheat industry,
different
levels
of a one-parameter theoretical production function, nx = a log(kf ), are investigated. The experiment was repeated for different levels of the parameter a. Figures 3(a)-3(f) show the structural change of the model as productivity increases. The series of figures start with a = 3 where average production is equal to the theoretical one, and continues
1391 , 0
16.2
-
16.1
-
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-
up to some critical EU beyond
,
1
which the calculated
average pro-
,
,
\,
2
3
4
3
4
EU (log
scale)
EU (log
2 scale)
EU (log
2 scale)
& 1592 $ 15.8 Q 15.7 15.6
0
1
Fig. 4 - contd.
216
A. Golan / Quantum economic chaos
duction is no longer compatible with the real average value. The next figure in the series (Fig. 3(b)) is for a=4. Here a continuous line also exists for the small EUs, but beyond the critical EU size the structure becomes more erratic. Next, for a = 4.5 in Fig. 3(c), some discontinuous points emerge. Figure 3(d), with a = 4.6, begins to exhibit the chaotic structure of Fig. 2. With a= 5, the structure becomes more chaotic, but not quite as chaotic as the final form (Fig. 2). Figure 3(f), with a=7, is identical qualitatively to Fig. 2, where a complete chaotic structure of the model emerges. This example shows the sensitivity of the modeled economy to the level of the critical parameter. A small change in the parameter (e.g. 0.4 between Figs. 3(d) and 3(e)) changes the model enormously. The main result, however, is that for any larger productivity level the structure will remain as in Fig. 3(f). Furthermore, any other change in the model, such as increasing a or /3 or even changing the production structure to be convex, will result in no qualitative change. It should be noted that an increase in the productivity level a results in an increase of the average level of production. This can be seen in Figs. 3(a)-3(f) and is expected in such a modeling; it does not have any qualitative effect on the experiment. One should also note that for production technologies characterized by more than just one parameter, say nx = a(kf)b, an increase of all parameters, say a and b, simultaneously shifts the structure (represented by the figures) much faster than a change in just one of the parameters. But as before, when the model reaches its final chaotic structure, this structure will not change for any additional increase of any of the parameters. It has been shown that chaotic structures emerge in economic modeling when one chooses the wrong EU size for the model. Beyond some level of the critical set of parameters (e.g. productivity) governing an economy, the model exhibits chaotic structure (Fig. 2) for all levels of economic parameters (e.g. (Yand p) and for all different forms of technology (e.g. concave, linear, convex). It was also shown that there is self-similarity in the model in the sense that for each level of magnification of the left-hand side of the graph (i.e. reducing the initial EU size) the same structures emerge over and over again and will disappear only for an EU size of zero. The next set of questions concerns the universality of the above results. Two such questions are considered. The first is whether the above results are universal for the different variables of the model. That is, do changes in the EU size of variables other than productivity (say input level) affect the average properties of the system? The second question is whether these results are universal to systems other than economic ones. The experiments needed to investigate these questions, and the related answers, are presented in the following section. Variables other than production are examined first. Figure 4 shows the change in the model’s structure as productivity increases, where the average input (land) is calculated based on the EU size of the inputs. For each size of the EU, in units of acres of land, and for a given value of productivity, the average input used by the industry is calculated. Twenty thousand points are calculated and, as in the previous exercise, it is shown (Figs. 3(a)-3(f)) that as the productivity value increases, the model becomes more and more chaotic up to the critical level at which the two struc-
A. Golan / Quantum economic chaos
271
tures become qualitatively identical (Figs. 4(c) and 3(f)). Nevertheless, it is shown that for values lower than the critical level of productivity (the critical parameter), the two models behave somewhat differently. This difference is, however, not a basic one. In both cases the fluctuations, together with the number of discontinuous points, increase with the productivity level. Furthermore, the qualitative result of a Road to Chaos is identical in both cases. Figures 4(d)-4(f) are enlarged versions of Figs. 4(a)-4(c) for an EU size between zero and four. This exercise (Fig. 4) has shown the universality of the result to a variable other than production within the system. In fact, this result is universal for all variables within the system. Each variable investigated is necessarily defined as a discrete one. All economic systems represented by such variables will result in chaotic structures similar to these shown in Fig. 2. This is due to the purely mathematical property of the discrete variables as is shown in the example introduced in the beginning of this work and in Fig. 1. The critical set of parameters is, however, different for each class of economies modeled. This argument proves that this result is universal to all variables. The universality of the result to systems other than economic ones will now be investigated. The system investigated is a single-lane traffic system, which is modeled identically to many physical systems (Reiss et al., 1986). The data used are from Table 2 of Edie et al. (1964), where the measurements are made in the Holland Tunnel in New York for 23377 cars. The experiment was to measure the average velocity of the above traffic system for the different sizes of the velocity quantum (EU). A similar experiment was conducted by Reiss et al. (1986) but, as will be discussed shortly, their results are somewhat different from the results introduced in this paper. The traffic system’s parameters are calculated following Reiss et al. (1986). The relationship between velocity, u, and the distance between two cars, I, is measured according to u =nu~(kw/c), where u is a quantum (EU) of velocity in ft/sec, w is a quantum of distance in ft, n and k are integers, 2 is the drivers’ sensitivity parameter in ft/sec, and c is the observed characteristic distance between the centers of two cars when the two are motionless. Figure 5 shows the average velocity of this traffic system for 20000 different quanta of velocity and for a given level of the drivers’ sensitivity parameter, A. As is expected by now, Fig. 5 is qualitatively identical to the previous figures. Figure 5(a) shows the whole quanta range, while Figs. 5(b) and 5(c) show smaller ranges, where it is possible to see the exact structures that emerge during the calculations. Figure 5(d) shows the same calculations as before, but for a higher level of the parameter A. Once again, when the critical parameter (A in this case) reaches a certain level, a fully chaotic structure emerges, which is identical to the chaotic economic structures shown. Increasing 1 further does not change the structure of Fig. 5(d). This chaotic structure remains for all values of A larger than the critical one. The difference between this experiment and the one done by Reiss et al. (1986) is that they used the real observed A, which is very low (compared with the critical level, A*), and therefore the structure in their figure is qualitatively identical to Fig. 3(a). That is, average velocity is equal to the observed one up to some
278
A. Golan / Quantum economic chaos
critical quantum (EU) size, beyond which the calculated average velocity is no longer compatible with the observed average value. This result enabled them to conclude that below some critical value of the velocity quantum (EU), a discrete or a continuous modeling of the above traffic system is completely identical. However, for any traffic system with A> A*, their result is incorrect. The question whether A > A* exists or not is interesting, but not significant for the results introduced here. This concludes results.
the basic experiments
needed
to show the great universality
of these
This experiment has shown the universality of the results to a system other than an economic one. In fact, this result is universal for many systems. Defined in terms of at least one discrete variable, where the discrete variable is in turn defined as an integer multiplied by an EU, each model that represents a system (economic, physical, chemical social, etc.) exhibits the characteristics represented here and shown in the figures. This argument is based on the mathematical property shown in the initial example (Fig. 1) and explored in the different experiments.
&
lo-
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20
30 Quantum
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size (EU) on average
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6
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velocity
are shown based on the parameters
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10
W/s)
vs. the quantum
calculated
of velocity
by Reiss et al. (1986).
of the left-hand corner of (a); (d) a higher level of the critical structure as in the previous figures, representing an economic system.
A.
Golan
It is clear by now that chaotic
/ Quantum
structures
sion of this special type of chaos,
economic
279
chaos
do emerge in economic
its comparison
modeling.
with deterministic
Discus-
chaos such as
is shown by Baumol and Benhabib (1989), and the implications to economic modeling, are now in place. However, before engaging in this discussion, the initial objective concerning the discrete continuous choice in economic modeling should finally be answered. This is done in the next section.
5. The discrete continuous
paradox
Is there some critical EU size for each variable in an economy below which an economy could be modeled as a continuum rather than as a discrete one? This was the question that introduced this work. There are two answers to this question, one for an economy where the critical parameter is below its critical level and another for the economy where the critical parameter is above the critical level. The former case is simpler and is discussed first. In this instance the critical parameter (e.g. productivity, A) of the economy (system) is below its critical value. In fact, this critical parameter should be much
I
0.0
0.2
0.4
0.6
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10
20
30 Quantum
08
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(ftls)
40 (ftls)
Fig. 5 - contd.
50
60
280
A. Golan / Quantum economic chaos
below its critical value so that the structure generated by an experiment similar to the one suggested here will be qualitatively similar to Figs. 3(a)-3(c). When this qualification is met, an economic model may be continuous or discrete for any EU size smaller than some critical size (e.g. about 1.75 in Fig. 3(a)); both models are accurate. For any EU larger than the critical size, a continuous modeling of the system will not be accurate. How would one choose the basic EU of the economy modeled? The answer is that it depends on the economy, its size, and its basic units (an apple, a farm, an industry). The economist must choose the EU that is believed to be the basic unit, as defined earlier, representing this economy. For example, if 1 is (in units of bushels) the basic production EU for an economy such as the one represented by Fig. 3(a), a continuous modeling will be correct. This point will be discussed further in the next section. The second answer to the discrete continuous choice question involves those cases where the critical parameter governing the economy (system) is beyond its critical value (e.g. a> 7 in Fig. 3(f)). The results introduced previously (Figs. 2, 3(f), 5) show that as the EU increases, the fluctuations around the mean always increase up to some EU size at which the average properties all become zero. This zero is associated with the case where the EU becomes greater than the average property investigated. These results also show that there is no critical EU size below which the economy should be modeled as a continuum and above which the economy (system) should be represented by a discrete model. Furthermore, the results show that even a discrete modeling is not always sufficient. As EU-,O, the fluctuations become smaller and smaller, but still exist. Assuming the economist is willing to settle for the ‘slight but never zero fluctuations’ associated with a small EU (those on the left-hand side of the figures), she or he must now decide whether that small EU contains the correct amount of information for the construction of a meaningful model. An example drawn from another field may help explain the importance of this decision. Consider the literary critic who studies a text syllable by syllable rather than word by word, sentence by sentence, or chapter by chapter. The critic risks examining and manipulating thousands of incoherent bits of information (syllables) that add nothing to comprehension. Ironically, too many pieces of information are processed without results precisely because each piece of information is too small. An economist runs the same risk when deciding whether an economy should be modeled in terms of EUs of 1 bushel (or 1 dollar), or in terms of firms (or $lOOO), where the EU is the smallest firm capable of existing independently in this economy for a given level of technology. It might appear at first that a model based on small EUs is always preferred, but then an information problem might arise. By using too small an EU, the economist might enter unnecessary (‘too much’) information into the model. Entering unnecessary information is not only costly in terms of collecting the data, but is also costly in terms of analyzing the data. The information-processing mechanism is affected by the fact that the information bits are too small. In terms of the partition function, L?, it means that the partition among the basic elements
A. Golan / Quantum
economic
chaos
281
of the model is incorrect. This would happen, for example, if one models an industry with a large number of firms and selects an EU of 1 machine or a level of production too this a waste of inefficient and On the other In this case the
small to exist independently in any one of these firms. Not only is resources, but it also makes the information-processing mechanism inaccurate. hand, assume that one chooses a relatively large EU, say one firm. information level could be the correct one, but at this level the fluc-
tuations are already enormous (see x= 5 in Fig. 2) and there is a large number of discontinuous points. Even if the discrete variable is defined correctly, the result is an inaccurate model. To summarize: the choice presented here is between a small EU, which entails an information problem, or a large EU, which results in model fluctuations. There is no unique rule for this choice; there is no golden rule for the discrete continuous choice question. This is the paradox that arises in the experiments presented here. So, what is one to do? The decision about EU size cannot be made arbitrarily, not even for social systems. However, in modeling social and economic systems, one can control the information bits (or the EU) up to a point and then gather the required information based on the desired EU size. That is, the partition between the elements of the system should be consistent with nature and with the system (economy) modeled.
6. On chaos in modeling A most interesting result emerged during the previous experiments: chaotic structures appeared in economic modeling. Furthermore, those chaotic structures were shown to be universal to all economic variables and to systems other than economic. In fact, most systems represented by discrete variables exhibit the same qualitative chaotic structures. There is self-similarity within each system modeled (for different ranges of the EUs), self-similarity among variables, and self-similarity among different systems. The meaning of these structures and their relationship to deterministic chaos is discussed next. Chaos is a result of some non-linearity in a system, and can occur in systems that are deterministic in the sense that no randomness is introduced at the outset. It is well known that even for such deterministic systems, if some critical parameter governing the system reaches a critical level, the model may become unpredictable within a certain range, i.e. chaos will emerge (e.g. the parameter governing the Logistic Equation; May, 1976). Deterministic chaos, which is associated with the macro properties of a system, can be characterized by two main properties. (A simple explanation and examples of deterministic chaos are given in Schuster, 1988, and a detailed survey of chaos in general and in economics is given in Baumol and Benhabib, 1989.) The first is that there is some ‘ordered structure’ in chaos. The second is that for each system there will always be some critical parameter (or a set
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of parameters) governing the system such that as this critical parameter (e.g. productivity) increases, the model becomes more chaotic up to a level that is stable for all other changes. It reaches some strange attractor that keeps the structure similar for ranges of values greater than this well-defined critical parameter. One example of this Road to Chaos (see, for example, Schuster, 1988, for more examples of Roads to Chaos) is that as the critical parameter increases, the period of an orbit increases through a set of critical (bifurcation) points (May, 1976). The critical parameter increases up to some level beyond which (in most cases) there is a different regime. This regime is characterized by an attractor to other states of the system, which is known as a strange attractor. At the heart of an understanding of deterministic chaos and strange attractors is the question of how the asymptotic behavior of deterministic systems can exhibit unpredictability, owing to sensitive dependence upon initial conditions, and yet at the same time preserve a coherent global structure. The figures shown in the present work show the asymptotic behavior (structure) of a deterministic model due to changes of the initial conditions, i.e. the economy’s set of technological parameters. Some properties of the global structure (a partial representation of the strange attractor) and the order of unpredictability (i.e. the Road to Chaos) are also shown graphically. However, instead of looking at the number of equilibria corresponding to some critical parameter (or the equilibria trajectory over time), the chaos here is a measure of the fluctuations around some observed property of the system, such as the first moment. As the critical parameter governing the system (e.g. productivity) increases, the model becomes more chaotic up to a level that is stable for all other changes (e.g. Fig. 2). It reaches some strange attractor that keeps the structure similar for all values greater than this well-defined critical parameter. Before discussing the mathematics behind this special chaotic behavior (i.e. before showing where the non-linearity enters), there is one philosophical point that needs to be discussed. That point concerns the relationship between chaos in modeling, as is defined in the present work, and chaos in systems (economies), as is described in most chaos models in the literature. This is the subject of the following section. Up to this point the discussion has been about chaos in modeling (or in an economic model) as opposed to chaos within the economic system (e.g. Baumol and Benhabib, 1989). Is there a difference between the two? The answer is that there is chaos in the system as described by the model. Owing to the large number of variables, parameters, and interactions in each observed system, it is almost impossible to describe most systems accurately. Therefore, the introduction of simplifying assumptions is needed in building a model describing some system. This ‘new’, simple (modeled) system, however, might exhibit chaotic behavior (structures) due to some non-linearity in it. It then follows that chaos in a system, based on some well-defined model, is essentially chaos in the simplified system, the modeled one. Does the real-world observed system exhibit chaotic behavior (structures)? This question can be answered in two parallel ways, one theoretical and one empirical.
A. Golan / Quantum
The theoretical
economic
answer is that since the modeled
283
chaos
system is basically
simpler
than the
real observed system, it is natural to expect that if chaos emerges in the simple model, it will emerge in the more complex one (e.g. Lorenz, 1963; May, 1976; Baumol and Benhabib, 1989). In seeking an empirical answer, one searches for chaos by investigating the behavior of data (e.g. Crutchfield and McNamara, 1987). However, this investigation is based on some mathematical (statistical) or graphical methods that are also subject to some underlying simplified structure (assumptions). One must then return to the first argument: if chaotic structures are observed in the data based on a simplified model, then it is natural to expect the basic structure of the data to be chaotic. Nevertheless, it is inaccurate to argue that a real-world complex system will always exhibit chaotic structure if the model describing it is chaotic. There might be a case where a number of non-linear relations cancel each other such that the modeled system behaves different from the model. In that case chaos emerges only in the model. The conclusion of the short argument presented here is that, in essence, the chaos found in modeling and the chaos in systems, economic or others, are basically the same. To summarize: the notion of chaos in modeling is introduced in the present work. Other papers, such as Lorenz’s (1963) famous work and all the references given before, show the existence of deterministic chaos in natural and economic systems. Following the above argument, one concludes that both types of models represent chaos in some simplified version of an observed system (economy). In searching for the explanation for the chaotic structures emerging here, one is essentially searching for non-linearity in the model. Going back to the example represented by Fig. 1 and Table 1, one sees immediately that the series of numbers representing the average production of the three firms’ industry is non-linear. For each fixed EU, one calculates an average production that cannot exceed its initial value. For example, when x = 1, the average is 20, but because production is defined in terms of integers, the average production will not be exactly 20, except for a limited number of cases. The model is therefore non-linear. (Table 1 shows the corresponding averages for x = 1,2, . . . , 30 as well as the averages corresponding to two different ranges of EUs.) This concludes the search for non-linearity in this discrete system. It should be noted, however, that the simple example introduced at the beginning of this work does not produce all the results presented in this work: the critical parameter governing a system, the universality of the results, and the Road to Chaos. For this, a model of some real economy is best. To summarize: at the heart of an understanding of deterministic chaos and strange attractors is the question of how the asymptotic behavior of deterministic systems can exhibit unpredictability due to sensitive dependence upon initial conditions, and yet at the same time preserve a coherent global structure. The figures shown in this work show the asymptotic behavior (structure) of a deterministic model due to changes of the initial conditions, i.e. the economy’s set of technological parameters. The global structure (strange attractor) and the order of unpredictability (i.e. the Road to Chaos) are also shown.
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A. Golan / Quantum economic chaos
One more issue is left undiscussed in this section; it concerns a type of chaos that might be qualitatively different from the well-known deterministic chaos, the chaos dealing with the macro level of a system. It is the chaos related to the micro level of a system, which is known as quantum chaos, or in the terminology of the present work, EU chaos. The fascinating search for quantum chaos is now being conducted simultaneously in many different disciplines (e.g. O’Connor and Heller, 1988; Casati, 1985). One way of trying to characterize this type of chaos is by looking at the quantum (micro) behavior of a system that exhibits deterministic chaos at the macro level. A second way is to investigate the conditions under which the quantum (EU) behavior exhibits some kind of chaotic, but not purely stochastic, behavior. Both these lines of research are necessary for the understanding of quantum chaos. The model described here would help shed some light on the second way of approaching the question. This would be done by investigating the left-hand side of the figures represented here, where the EU becomes smaller and smaller but never vanishes. However, this is beyond the scope of the present paper and will be done in a separate study.
7. Implications
of results and summary
Two main results depend upon one another and are shown here. First, there is no critical EU size below which the economy should be modeled as a continuum, and, furthermore, even a discrete representation of an economy is not always sufficient. Second, chaos in modeling emerges for some critical level of a set of parameters governing an economy. This result is universal to all variables and to systems other than economic. Furthermore, the chaotic structures emerging are selfsimilar for the different cases. The first result implies that models based on discrete observed quantities need some special adjustments. This adjustment, call it rescafing (renormalization), is a procedure aimed at decreasing the fluctuations that arise from modeling large EUs or large bits of information. This resealing is done such that, at the limit, all the points (see figures) collapse into one line at the level of the observed average property. A general resealing of the model is possible due to the universality of the results and due to the fact that each modeled system is self-similar for the different EU sizes. Figure 2 illustrated the ordered structure of the fluctuations. In general, it is expected that as EU increases, n will decrease. However, n is an integer and can change only in discrete values (units). The observed straight lines on the right-hand side of the graphs are due to this property. As the EU size decreases (moving left on the figures), the straight lines made by each collection of points on the graph no longer appear as linear. They have a completely different structure, but that structure is common to each experiment. It is suggested here that this difference is an ‘optical illusion’ in the eyes of the observed owing to the relative small size of the
A. Golan / Quanium
EU at this range.
That is, concentrating
economic
chaos
on the analysis
285
of the figure at this range
only and decreasing the distance between sequential EUs (EU + 0), will yield the same self-similar structure. The right-hand side will consist of straight lines, while the left-hand side will look as before. This is to say, the left-hand side of each figure will always look the same, no matter how small the EU is. One more property is observed in the experiment. The right-hand line of points is at the value of zero average production and will remain zero for any larger EU since, simply, the EU is larger than the average property calculated. The second line of points from the right has a slope of 1 (y = nx = lx for all x at this range). The next line has a slope of 2, the next one has a slope of 3, and so on. From right to left the slope of each additional ‘line’ increases by one. As was discussed before, this linear property is due to the discrete specification of the model and makes it possible to rescale the properties derived from the model such that they will collapse, at the limit, into their observed level. However, this renormalization procedure is yet unsolved and not the subject of the present work, which aimed to investigate the conditions under which an economist can model an economic system as a continuous one and under what conditions the same economy should be modeled as a discrete one, based on discrete defined variables. The second result, which is not independent of the first, implies that there is chaos in modeling. This chaos exhibits the same basic properties as deterministic chaos, but instead of looking at the equilibria structure one is looking at the structure of the model’s fluctuations. However, having defined the discrete variable of some economy as an EU multiplied by an integer, it is easy to show chaos in terms of equilibria. For example, consider the modeling of an industry with N identical firms, where the total output is measured by N multiplied by an EU, which is defined as the firm’s output. Under certain conditions this type of economy will yield deterministic chaos in the sense that the long-run equilibrium number of firms, N, will change chaotically. The present work, however, deals with the discrete continuous choice of modeling, and the exact development of the above example is the subject of a different paper. Finally, definition vestigating of further
it is suggested that the approach described in this work, including the of the variables and the treatment of the chaotic results, is helpful in inquantum economic (EU) chaos phenomena. This, however, is the subject work.
Acknowledgments I wish to thank Amotz Agnon, Robert Anderson, field, Elise Hardy Golan, and seminar participants Rehovot for many helpful comments. Special thanks with the computer analysis.
George Judge, James Crutchat the Hebrew University at to Gary Casterline for helping
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economic
chaos
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THE DISCRETE CONTINUOUS CHOICE MODELING OR QUANTUM ECONOMIC Amos
GOLAN
Department
of Economics,
Communicated
by K.H.
A framework developed. structed
for
University of Haifa, Haija, Israel 31999
Kim
investigating
An Essential
economic
the
discrete/continuous
Unit (EU) is defined
in terms of the EU multiplied
that, in most cases, there is no definite an interesting blems,
OF ECONOMIC CHAOS
paradox
arises.
well-known
representation)
result is that in economic
properties
of deterministic
describing
discrete
economic
economic.
Possible
applications
Key words: Deterministic
chaos;
modeling
chaos.
variables
Two main results continuum)
quantum
modeling variable
are shown.
is
is con-
The first is
choice of modeling results in information
results in model fluctuations.
and pro-
The second and
chaotic
structures
emerge that exhibit all the
These chaotic
structures
are universal
and are proved
for quantum
of economic
to the discrete/continuous
A small EU (approaching
while a large EU (a discrete
most surprising
by an integer. answer
choice
such that each (decision)
economic chaos;
to be universal chaos
discrete
research
modeling;
to all models
to systems
other
than
are suggested. non-linear
dynamics.
1. Introduction The objective of this work is to investigate the conditions under which the economy should be modeled as a continuum or as discrete, that is, to develop a rule for the discrete/continuous choice of economic modeling - a crucial choice for economists. In nature, as well as in economics, most observables are discrete in the sense that they are based on small quantities indivisible beyond some critical minimum size. Energy levels, bits of information, and production levels are three examples of indivisible entities describing natural, social, and economic systems (e.g. Samuelson, 1973; Golan, 1988a). In economic systems we call these discrete quantities Essential economic Units (EUs). When one models an economy (system) with a large number of components (such as agents, goods, or firms), it is argued that individual components become weightless and that a continuous specification is correct (Aumann, 1964). Except for the mathematical convenience, this is the main argument in moving to the continuum. At the heart of this argument is the assumption that for each economy there exists some critical size of the EU (e.g. one bushel of wheat, a ton of apples, a table, a household, a firm) such that for any EU smaller than this critical size, the economy could be modeled as a continuum, while for all essential units larger than this critical 0165-4896/91/$03.50
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Science
Publishers
B.V. All rights
reserved
262
A. Golan / Quantum economic chaos
size the economy should be modeled as discrete. This work investigates theoretically and empirically whether such a critical size exists. It will be shown that, in most cases, such a critical size does not exist and that the answer to the above question is surprisingly non-trivial. It will be shown that each economy is governed by some characteristic in modeling
parameter, and when this parameter exceeds some critical level, chaos will emerge regardless of the size of the EU. However, when this
parameter is below its critical level, there is a critical EU size below which a continuum model is accurate. These results are proved to be universal to other systems (e.g. physical, traffic) as well. An exact definition of the EU is given in the next section. Then, a simple example to introduce the main results of the paper is given. This is followed by a description of the model and the experiment. The results are then discussed in detail, and a discussion of deterministic chaos and its relationship to the main results of this work is presented. A summary and discussion of the implications of results conclude the paper.
2. The Essential
Unit
The discrete continuous choice is best explored by defining each variable in terms of an EU multiplied by an integer. Thus, each observed quantity increases in multiples of its EU. To select the correct EU, one should choose the natural quantities of the economy modeled, that is, the basic units of the variables (decision variables) representing the economy. For example, production should be defined based on the smallest quantity possible of producing for a given technology. In that case, when the natural EU is especially small relative to the economy (e.g. one apple), it is convenient to define multiples of EUs that will translate the EUs into larger quantities (Golan, 1988a). A firm (the smallest firm capable of existing independently given some technology) is another EU one might use in analyzing an industry. The next part of this work deals with production EUs. These results are then generalized. Define x as an EU of production with production units (i.e. the smallest unit of production capable of existing independently) and n as an integer. Then, y, = nx is the production level, where yn depends on the integer n. An EU of input can be defined using a similar approach. A production function will be defined as nx= g(kf), wherefis an EU of input, k is an integer, and g( . ) is a well-defined function. A vector representation is possible by defining a vector of EUs with a corresponding vector of integers.
3. Example Following the above definitions, troduce the main results.
a short
mathematical
example
will help in-
A. Golan / Quantum economic chaos
263
Assume an industry consisting of three firms. Firm A produces 15 EUs, say 15 tons of tomatoes, firm B produces 20 tons, and firm C produces 25 tons. The production, yi (i = A, B, C), is constructed in terms of the EU multiplied by an integer. Specifically, let x be the EU (in units of tomatoes, say 1 ton of tomatoes) and nj the integer describing the production of firm i. Then, Y; = n,x,
i=A,B,C.
(1)
The question to be investigated is whether the size of the EU affects the basic properties of the model. It is sufficient, however, to prove that at least one property
Table 1 Average production ample). jumps N
vs. EU. Thirty
Case A: EU ranges of l/3.
between
Case C: EU ranges Case A EU
different
EUs are calculated
0 and 30 in jumps between
0 and 2 in jumps
EU
(the ex-
0 and 10 in
Case C Average
EU
prod.
mod.
industry
between
of l/15.
Case B Average
for the three-firm
of 1. Case B: EU ranges
Average prod.
1 2
1.000
20.000
0.333
20.000
0.066
2.000
19.334
0.666
19.778
0.133
20.000 19.556
3 4
3.000 4.000
19.000
20.000
0.200
18.667
1.000 1.333
19.556
0.266
20.000 19.910
5
5.000
20.000
1.666
20.000
0.333
19.778
6
6.000
18.000
2.000
0.400
19.866
7
7.000
16.334
2.333
19.333 18.666
0.466
19.755
8
8.000
16.000
2.666
18.666
0.533
19.733
9
9.000
15.000
3.000
19.000
0.600
19.800
10 11
10.000 I1 .ooo
16.667 14.667
3.333 3.666
18.889 18.333
0.666 0.733
19.777 19.800
12
12.000
16.000
4.000
18.666
0.800
19.733
13
13.000
13.000
4.333
17.333
0.866
19.644
14
14.000
14.000
4.666
18.666
0.933
19.600
15
15.000
15.000
5.000
20.000
1.000
20.000
16
16.000
10.667
5.333
16.000
1.066
19.555
17 18
17.000 18.000
Il.334 12.000
5.666 6.000
17.000 18.000
1.133 1.200
19.644 19.200
19
19.000
12.667
6.333
16.889
1.266
19.000
20
20.000
13.334
6.666
17.778
1.333
19.556
21
21 .ooo
7.000
7.000
16.333
1.400
19.133
22
22.000
7.334
7.333
17.111
1.466
19.556
23 24 25
23.000
7.667 8.000 8.334
7.666
15.333
1.533
19.442
8.000 8.333 8.666
1.600 1.666 1.733
19.200 20.000
0.000 0.000
16.000 16.666 14.444 15.000
1.800 1.866
19.200 19.289
1.933
18.689
2.000
19.333
26 27 28 29 30
24.000 25.000 26.000 27.000 28.000 29.000 30.000
0.000 0.000 0.000
9.000 9.333 9.666 10.000
15.555 16.111 16.666
19.067
264
A. Golan / Quantum economic chaos
of the model is affected by the size of the EU. The selected property is discussed below. Of the large number of properties representing the modeled industry, the first moment (average) of its production is fundamental. This property has been chosen for investigation in this example and in the rest of the present paper. It should be noted that any property (such as second moment, variance, etc.) would be affected by the
a 20*
. .**
.
i
I o! 0
10
20 EU
(tons)
.
.
.
0.
l
.
. .
.
.
.
.
. .
. 14 0
2
6
4
10
6
I
1:
EU (tons)
Fig. 1. Example
of the effect of EU size on average (b) EU=
l/3,2/3
,...,
production
in a three-firm
10; (c) EU= l/15,2/15
,..., 2.
industry.
(a) EU = 1,2,30;
A. Golan / Quantum economic
EU size, but by the sufficiency be asked: Does the average that the EU of the modeled all of the firms.) Naturally, unchanged,
condition,
chaos
265
it is not necessary
production of the industry industry must be identical
to show it. It can then
change with the EU? (Note at each moment of time for
if one increases the EU size arbitrarily and the integers, {n,}, remain the average production must increase as well. Therefore, it is necessary
to keep the total production fixed. More precisely, for each EU size, the total production is forced to remain not larger than the total production observed in the industry (60 units). In more economic terms, one might think of a fixed level of demand or of limited resources such that, on average, the industry’s production is constrained to 60 units. Starting with the smallest EU of this industry, say x = 1, the average production is y=$
c n;x=$ i
c n;=+x1
x(15+20+25)=20.
1
Next, the model is examined for some larger EU, say 2. For x= 2, firm A will produce only 14 units, since 7 is the largest possible integer for which A’s production will not exceed 15 (note, if nA= 8, then y,= 16). Similarly, one sees that nn= 10, and nc= 12. The average production of the industry is then y=fx2x(7+10+ Now,
12)= 19:.
for x= 3 the average
production
will be
y=$x3x(5+6+8)=19.
-_ _ C
. .
‘0
l **
.
. .
. .
.
.
. .
.
. 18.51 0
0.2
0.4
0.6
0.8
I.0
1.2
EU (tons) Fig. 1 - contd.
1.4
1.6
I.8
2-o
: ,2
266
A. Golan / Quantum economic
chaos
Increasing the EU by 1 up to, say 30, gives a series of numbers representing the average production of the industry for each EU size. This is shown in Table 1 and Fig. l(a). The average production is a continuous scale represented on the vertical axis of Fig. l(a), while the discrete EU size is on the horizontal axis. As the size of the EU increases (moving to the right in Fig. l(a)), one sees the change in the average production of the industry. The change, however, is not monotonic, and many critical jumps occur. Nonetheless, for EU sizes larger than 26 (26,27, . . . , a), the average production calculated by the model remains zero. This is because the EU size is larger than the total production of the largest firm. The following example clarifies this point. Consider a sample taken from an industry with a large number of firms. Each firm is characterized by its production level. If one chooses an EU that is larger than the production level of the largest firm, say EU of 11 for an industry with a production range of 4 to 10 units of production, there will be no firms in the sample. The sample is an empty set and is represented by the zero ‘line of points’ on the figure. This simple exercise shows that the wrong choice of EU size yields a model whose average differs from that of the economy it wishes to describe. A basic characteristic of this representation is now shown. The above exercise is repeated, but this time instead of changing the EU 30 equal times between 0 and 30, the EU size is changed 30 equal times between 0 and 10 (i.e. l/3, 213, 1, . . . , 10). Table 1 and Fig. l(b) show the average production that corresponds to each one of the 30 EUs. The same exercise is repeated for 30 equal jumps in the EU size between 0 and 2 (i.e. l/15, 2115, 3/15, . . . . 2). This is shown in Table 1 and Fig. l(c). From these figures one can see that there is self-simifarity between all the graphs in the sense that, in a general way, the same structures repeat themselves except for the line of zeros shown for EU of 26 and higher in Fig. l(a). In fact, no matter how fine one divides the EUs, a similar structure appears. Furthermore, it also shows that, as EU size increases, the fluctuations between the average values of the observed industry and those of the model increase. To conclude, this simple example shows that there is a special mathematical property of the above-defined variables that results in the behavior of the model presented here. This mathematical property is the non-linearity of the system described by this example. This non-linearity, in turn, adds to the special chaotic structures shown, and is discussed in detail after describing the different experiments done in this work. The next task is to explore the implications for economic modeling, specifically for the discrete continuous choice of modeling. This is done in the following section.
4. The experiment 4.1.
The model
The experiment
analyzed
in this section investigates
the discrete continuous
choice
A.
Golan
/ Quantum
economic
chaos
267
of modeling based on real-world data. Using a discrete definition of the variables, the average values of an economy are calculated for each EU size. Next, the theoretical economic model is developed. Using this model and figures from the Californian wheat industry, an empirical estimation is made. The estimation yields the economic and technological parameters representing this economy. Based on these parameters, the experiment is performed through a series of steps described in the next section. The experiment is then repeated for different theoretical parameters. The characteristics of the economy are modeled by using the maximum likelihood approach subject to constraints. This approach is also known as the maximum entropy (ME) formalism (Golan, 1988a). In this approach all possible economic states are equally probable, with the exception of those excluded by the available information (constraints). That is, one maximizes the missing information, which implies there is no white noise in the model. The ME formalism describes the most probable macroeconomic state characterizing an economy. Each event has a certain probability of occurring, and the sequence of events with the highest probability describes the most probable economic state. This is the best possible characterization of an economy subject to the available information. Although the theoretical results hold for other formalisms as well, the ME formalism was chosen because it is not a deterministic one; randomness is a basic property of the model and not just white noise. Furthermore, this framework is general in the sense that it is possible to show the conditions under which the resulting distribution collapses into any other well-known distribution (Golan, 1988b). Since this work concentrates primarily on the EU issue, the ME formalism is only described briefly. An extensive description of this approach is given in Levine and Tribus (1979), and its economic application and implications are given in Golan (1988a, 1988b). Based on limited available information, the ME approach determines the steadystate economic distribution in terms of the variables analyzed. In this paper, the ME formalism is used to model the size distribution of firms. For simplicity, the model is presented in terms of one output and one input, where the variables are defined as EUs multiplied by integers. Let the individual producer’s output be y,,,= nix, where n; is an integer related to producer i and x is the EU of production. Similarly, the resources of a producer are rk, = kJ, where k; is an integer and f is the EU of resources. For simplicity, the above definitions are written as y,, = nx and rk = kf. Next, the producers are grouped according to their level of production and resources, where the relationship between the production level and firm size (resources) is captured in the group specification (Golan, 1988a). Specifically, define q,,k as the number of producers having an output in the range nx to (n + 1)x and resources in the range kf to (k+ l)f, where n and k are not independent. It is then possible to construct the constraints of this economy (i.e. the available information) in terms of the discrete variables. Finally, the economic distribution (firm size distribution) can be estimated by maximizing (subject to the above constraints) the number of ways of partitioning all the firms into groups with similar characteristics
(the multinomial coefficient). The groups’ characteristics are represented by the integers. Since the EUs are necessarily identical within each system, the different integers represent different groups within a system. The above maximization process yields the estimated economic distribution that is characterized by some Lagrange parameters
(e.g. cr and /?) that correspond
1
n
constraints.
That is,
e-akf e-Dnx
N
qnk=
to the different
c
k
e-akf
’
e-anx
(2)
where N stands for the number of firms in the industry. This equation represents the steady-state distribution of firms according to their size, where size is defined over production and resources simultaneously. It is convenient to define the denominator as Q
c c e-akf e-Dnx,
=
n
k
where Q is a normalization factor that converts relative probabilities into absolute probabilities. The Q function is known as the partition function because it partitions the different elements of the model (e.g. type of producers). The probability of group
qnk is then Pnk
=
(Ink N
_
ePkf embnx s-2
.
This concludes the brief summary of the ME formalism. Lagrange parameters and other natural quantities that arise are discussed in detail in Levine and Tribus (1979), Reiss et (1988a). Having described the steady economic state, it is now the question: Does the EU size affect the average properties 4.2.
(4) Interpretation of the during this formalism al. (1986), and Golan possible to investigate of the model?
The experiment
The data set for the experiment is taken from Golan (1988a). It contains information on the resources and output of all Californian wheat farms in the years 1974, 1978, and 1982. Firm size is defined over both production (in units of bushels) and land (in units of acres). The different steps of the experiment are now discussed. In the first step the steady economic state distribution is calculated by solving the multinomial coefficient subject to the constraints. The calculated parameters for 1978 are (x=3.9199 and p= -4.163. Second, based on the above solution and parameters, it is possible (see proofs in Golan, 1988b) to infer the technological characteristics of the industry and to incorporate it directly into Q. It is found that the technology characterizing this industry is of the form nx=a(k#’ and, hence,
A.
The estimated
production
Golan
/ Quantum
parameters
economic
chaos
are (I= 1.62 and b = 0.815.
some EU size, say E, the total (and average) production and proved to be consistent with the empirical data.
269
Third,
choosing
of the industry is determined Fourth, taking the calculated
average production, the total production of this industry is calculated. Finally, constraining the industry’s total production to be no larger than the total production calculated for the case, x= E (the real-world observed level), the average production is calculated for each increasing EU size, i.e. for x= 2e, x= 3e, . . . . The reason for this type of normalization is now discussed. As was shown in the example, total production must be conserved when increasing EU size. Specifically, if n is an integer representing the average production of an industry with EU x, then y, =nx. Changing the EU size and keeping n unchanged effects y, in the same direction as x. It is therefore necessary to normalize the economy according to some criteria (e.g. total production, input level). This procedure is done as follows. Choose a small EU, say E, and calculate the average properties of this system, say average production and average vector of inputs. On the basis of these calculated averages, the total properties are then calculated. For some average production nx and N number of firms, the total production of the economy is NX nx. When calculating the economy’s average properties for the next EU size, 2&, it is important that the total number of properties in the economy remains the same, as in the case for X=E. That is, output and inputs are conserved between calculations of different EU size. This normalization framework enables one to investigate if a change of EU size changes the average calculated properties of an economy even though the aggregate properties are forced to remain constant. The objective of the experiment, then, is to calculate the properties (first moments) of the model for different EU sizes, while the total properties of the economy are held constant based on the observed data. This experiment was repeated for different empirical and theoretical values of the economic parameters (Q and p) and for different technologies. It was then repeated for different variables (e.g. production, inputs) and for different systems results are discussed in the following subsection. 4.3.
(not necessarily
economic).
The
Results
As in the previous example (Fig. l), a graphical analysis of the results is necessary and helpful in describing the outcome of the experiment. Figure 2 shows an economic system where the relationship between production with a>0 and b< 1. This figure is based on the and input is nx=a(/&, parameters of the 1978 Californian wheat industry, where 20000 different production EU sizes are calculated and each calculation is a point on this figure. Figure 2(a) shows all the 20000 calculated averages, where the vertical axis represents the continuous scale of average production, and the horizontal axis represents the 20000 different production EUs. From Fig. 2(a) one can observe the following properties: (1) as EU size increases, the fluctuations around the real observed average produc-
270
A. Golan / Quantum economic chaos
2
0
4
6
8
10
EU (log(x))
9.65
5 ‘Z v
9.60 9.55
4 g 9.50 gl 9.45
c
‘..‘.
.,....
‘.
.;.:,“.
L$ 9.40
,.
‘..
b
9.35
0.10
0.15
0.20
0.25
0.30
EU (log(x))
Fig. 2. Effect
of EU size on average
for the case of concave the 20000
technology. points;
production. Parameters
(b) an enlarged
Average
production
used: a=3.9199, version
vs. size of the EU of production
p=-4.162,
of the upper
left-hand
a= 1.62, b=0.815. corner
(a) All
of (a).
tion increase; and (2) there is a large number of critical points where a change of E in EU size results in a large discontinuous change in the average production level. It is also obvious from this figure that the average production of the modeled system becomes zero for all EU sizes that are bigger than the real-world observed average. It then follows that the integer associated with any EU larger than the observed average production (about 9.7 in Fig. 2) is necessarily zero. Going from right to left in Fig. 2(a) while investigating each ‘line of points’ also reveals the following property. In Fig. 2(a) the second line of points (from the right) is associated with the integer 1. The third line of points is associated with the integer 2. The next line is associated with the integer 3, and so on. Does this pattern continue all the way to EU = E, the furthermost point on the left-hand side? The answer is revealed in Fig. 2(b). This figure is just an enlarged version of the upper left-hand corner of Fig. 2(a). Here, it is obvious that this corner is qualitatively different than the right-hand side. What is the reason for this, and how can one explain it? To answer these questions it is enlightening to perform a short exercise. The exercise is similar in notion to the example shown in Fig. 1 (or I(a)-l(c)) where each
A. Golan / Quantum
figure represented the same distance between the smallest graphs depicted a continuous the same kind of magnification 20000 different calculations
economic
chaos
271
number of calculations (30 different EUs) but the EU (a) and the largest decreased. In other words, the magnification of the left-hand side. In this exercise, is performed but on the wheat farmer’s data. The of each EU size are repeated for smaller and smaller
EU differences. For example, 20000 calculations are done for the range 0 to 10 EU, so each EU size increases by (l/2000), i.e. the smallest EU is l/2000, the second EU is 2/2000, . . . , 10. Next, there are 20000 EUs between 0 and 2, so the smallest EU is l/10000, the second EU size is 2/10000 and so on. Next, there are 20000 different EUs for the sizes 0 to l/2 such that the smallest EU is l/40000, the second is 2/40000, and so on. The surprising results show that no matter how fine the distance between the smallest and largest EU, the left-hand side of the figure always looks the same. Of course the magnitude of the fluctuations becomes smaller, but the qualitative structure of the fluctuations remains the same for each magnification of the left-hand corner of the figure; the same basic structure keeps repeating itself. The figures generated by the exercises exhibit self-similarity. If one defines the distance between the smallest and largest EU as d and the number of different EUs calculated as N, then for each N as d + E > 0 there is self-similarity among the different graphical representations related to each d. Each figure corresponding to a different d will look almost exactly the same, the only difference appearing in the first figure (largest d) where there are points affiliated with an integer of zero (the EU size is larger than the average property). An immediate property evidenced by this observation is that no matter what the EU size, there will always be some fluctuations in modeling. In other words, the probability that the model will represent the economy it hopes to model diminishes with the EU size. The next question is whether this property is due to the specific data set analyzed or whether it is a universal property for all different economies. Specifically, is there some critical parameter associated with each economy that affects the above average calculations (the structures in the figures)? The answer is that such a parameter exists. For the economic production model presented here, it has been found that the critical parameter is the productivity level where productivity is defined as the set of parameters describing the production function such as a in y = a log(x) or a and b in y=axb. For each productivity level lower than some critical level, the discussed structures do not appear. That is, the average production, as calculated in the experiment, behaves differently than in Fig. 2. Above this critical level, however, the structures are universal for a/I different levels of the economic parameters (e.g. (Yand p) and for all possible technologies (production functions). The experiment is repeated for different economic parameters (e.g. the Californian wheat market in 1974 and 1982) and for different theoretical economies where the production functions are concave, linear, convex, or of many different mathematical forms [e.g. nx=alog(kf)]. The fascinating result is that the above structures repeat themselves throughout the experiment. It has been stated that below some critical level of the critical parameter these
212
A. Golan / Quantum economic
chaos
universal structures do not appear. In this case, a continuous line emerges, and the calculated average production is now a continuous function of the EU size. (Note that to prevent confusion the word ‘size’ is associated with EU, while the words ‘level’ and ‘value’ are associated with the parameters.) But even this continuous line
-
5 ‘Z u
.
5-
a
...“‘..
‘.
,:’
.:’
4(....._
9 ‘a
3-
i?
2-
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l-
‘......
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,.:
,:’
,:’
. . .. . . . . . . .._..................................
O-
10
0
6
4
2
0
EU (log(x))
: H ,a
6-
m
4-
9
th I 9 <
,......._....................
-.., . ....’ .:’
,....._( .. ‘.
,,,..‘.
...”
. ...’
,...”
,...”
,...‘.
_..._ ,_.‘.
,._.”
.... . ,._..
b
..’
.._,,..”
2-
0 -_ 0
4
2
6
8
lo
6
8
10
EU ( log (x1)
0
2
4 EU (log(x))
Fig. 3. Effect of EU size on average production for different levels of the critical parameter (productivity) or, the Road to Chaos. The production function used is nx=a log(kf) and the parameters used are
a=3.9199
and p=-4.163.
(a) a=3;
(b) a=4;
(c) a=4.5;
(d) a=4.6;
(e) a=5;
(f) a=7.
A. Golan / Quantum economic chaos
becomes
erratic
up to a certain
for some EU greater critical
than a critical
273
size of the EU. This means that
size of the EU both continuous
and discrete
models
prove
to be completely consistent with each other and both are correct in the sense that the averages of the model are the same as the averages of the modeled economy. Above the critical EU size, however, the model starts to behave erratically in that the quantities
0
derived
from
the model
2
cease to reflect
4
the real modeled
6
0
system.
10
EU (log(x))
IO -
5
,:...
c
8
Y D 2
6-
... ‘.. ..‘.‘.“..‘.‘. ‘. .. .’
.’
.
...‘.
e
,...”
..’
,.Y
,,..‘.
.,....”
,....’
,a
0
2
4
6
8
10
EU (log(x))
0-l
0
2
4
6 EU (log (xl)
Fig. 3 - conrd.
8
10
274
Figure
A. Golan / Quantum
3 shows the behavior
of the critical
economic
chaos
parameter.
The
This figure can be titled
Road to Chaos. At this point a closer examination
of the Road to Chaos is in order.
On the basis
a
0
5
10
15
EU ( log scale
10 EU, (log
20
1
scale)
10 EU (log scale)
Fig. 4. Effect of EU size on average input. Average input vs. size of the EU of input. The production function used is nx = a log(kf). (a) a = 13; (b) a = 15; (c) a = 20; (d)-(f) the enlarged versions of the lefthand
corners
of (a)-(c).
A. Golan / Quantum economic
of the 1978 economic
parameters
calculated
215
chaos
for the wheat industry,
different
levels
of a one-parameter theoretical production function, nx = a log(kf ), are investigated. The experiment was repeated for different levels of the parameter a. Figures 3(a)-3(f) show the structural change of the model as productivity increases. The series of figures start with a = 3 where average production is equal to the theoretical one, and continues
1391 , 0
16.2
-
16.1
-
: .C a 16.0
-
up to some critical EU beyond
,
1
which the calculated
average pro-
,
,
\,
2
3
4
3
4
EU (log
scale)
EU (log
2 scale)
EU (log
2 scale)
& 1592 $ 15.8 Q 15.7 15.6
0
1
Fig. 4 - contd.
216
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duction is no longer compatible with the real average value. The next figure in the series (Fig. 3(b)) is for a=4. Here a continuous line also exists for the small EUs, but beyond the critical EU size the structure becomes more erratic. Next, for a = 4.5 in Fig. 3(c), some discontinuous points emerge. Figure 3(d), with a = 4.6, begins to exhibit the chaotic structure of Fig. 2. With a= 5, the structure becomes more chaotic, but not quite as chaotic as the final form (Fig. 2). Figure 3(f), with a=7, is identical qualitatively to Fig. 2, where a complete chaotic structure of the model emerges. This example shows the sensitivity of the modeled economy to the level of the critical parameter. A small change in the parameter (e.g. 0.4 between Figs. 3(d) and 3(e)) changes the model enormously. The main result, however, is that for any larger productivity level the structure will remain as in Fig. 3(f). Furthermore, any other change in the model, such as increasing a or /3 or even changing the production structure to be convex, will result in no qualitative change. It should be noted that an increase in the productivity level a results in an increase of the average level of production. This can be seen in Figs. 3(a)-3(f) and is expected in such a modeling; it does not have any qualitative effect on the experiment. One should also note that for production technologies characterized by more than just one parameter, say nx = a(kf)b, an increase of all parameters, say a and b, simultaneously shifts the structure (represented by the figures) much faster than a change in just one of the parameters. But as before, when the model reaches its final chaotic structure, this structure will not change for any additional increase of any of the parameters. It has been shown that chaotic structures emerge in economic modeling when one chooses the wrong EU size for the model. Beyond some level of the critical set of parameters (e.g. productivity) governing an economy, the model exhibits chaotic structure (Fig. 2) for all levels of economic parameters (e.g. (Yand p) and for all different forms of technology (e.g. concave, linear, convex). It was also shown that there is self-similarity in the model in the sense that for each level of magnification of the left-hand side of the graph (i.e. reducing the initial EU size) the same structures emerge over and over again and will disappear only for an EU size of zero. The next set of questions concerns the universality of the above results. Two such questions are considered. The first is whether the above results are universal for the different variables of the model. That is, do changes in the EU size of variables other than productivity (say input level) affect the average properties of the system? The second question is whether these results are universal to systems other than economic ones. The experiments needed to investigate these questions, and the related answers, are presented in the following section. Variables other than production are examined first. Figure 4 shows the change in the model’s structure as productivity increases, where the average input (land) is calculated based on the EU size of the inputs. For each size of the EU, in units of acres of land, and for a given value of productivity, the average input used by the industry is calculated. Twenty thousand points are calculated and, as in the previous exercise, it is shown (Figs. 3(a)-3(f)) that as the productivity value increases, the model becomes more and more chaotic up to the critical level at which the two struc-
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tures become qualitatively identical (Figs. 4(c) and 3(f)). Nevertheless, it is shown that for values lower than the critical level of productivity (the critical parameter), the two models behave somewhat differently. This difference is, however, not a basic one. In both cases the fluctuations, together with the number of discontinuous points, increase with the productivity level. Furthermore, the qualitative result of a Road to Chaos is identical in both cases. Figures 4(d)-4(f) are enlarged versions of Figs. 4(a)-4(c) for an EU size between zero and four. This exercise (Fig. 4) has shown the universality of the result to a variable other than production within the system. In fact, this result is universal for all variables within the system. Each variable investigated is necessarily defined as a discrete one. All economic systems represented by such variables will result in chaotic structures similar to these shown in Fig. 2. This is due to the purely mathematical property of the discrete variables as is shown in the example introduced in the beginning of this work and in Fig. 1. The critical set of parameters is, however, different for each class of economies modeled. This argument proves that this result is universal to all variables. The universality of the result to systems other than economic ones will now be investigated. The system investigated is a single-lane traffic system, which is modeled identically to many physical systems (Reiss et al., 1986). The data used are from Table 2 of Edie et al. (1964), where the measurements are made in the Holland Tunnel in New York for 23377 cars. The experiment was to measure the average velocity of the above traffic system for the different sizes of the velocity quantum (EU). A similar experiment was conducted by Reiss et al. (1986) but, as will be discussed shortly, their results are somewhat different from the results introduced in this paper. The traffic system’s parameters are calculated following Reiss et al. (1986). The relationship between velocity, u, and the distance between two cars, I, is measured according to u =nu~(kw/c), where u is a quantum (EU) of velocity in ft/sec, w is a quantum of distance in ft, n and k are integers, 2 is the drivers’ sensitivity parameter in ft/sec, and c is the observed characteristic distance between the centers of two cars when the two are motionless. Figure 5 shows the average velocity of this traffic system for 20000 different quanta of velocity and for a given level of the drivers’ sensitivity parameter, A. As is expected by now, Fig. 5 is qualitatively identical to the previous figures. Figure 5(a) shows the whole quanta range, while Figs. 5(b) and 5(c) show smaller ranges, where it is possible to see the exact structures that emerge during the calculations. Figure 5(d) shows the same calculations as before, but for a higher level of the parameter A. Once again, when the critical parameter (A in this case) reaches a certain level, a fully chaotic structure emerges, which is identical to the chaotic economic structures shown. Increasing 1 further does not change the structure of Fig. 5(d). This chaotic structure remains for all values of A larger than the critical one. The difference between this experiment and the one done by Reiss et al. (1986) is that they used the real observed A, which is very low (compared with the critical level, A*), and therefore the structure in their figure is qualitatively identical to Fig. 3(a). That is, average velocity is equal to the observed one up to some
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critical quantum (EU) size, beyond which the calculated average velocity is no longer compatible with the observed average value. This result enabled them to conclude that below some critical value of the velocity quantum (EU), a discrete or a continuous modeling of the above traffic system is completely identical. However, for any traffic system with A> A*, their result is incorrect. The question whether A > A* exists or not is interesting, but not significant for the results introduced here. This concludes results.
the basic experiments
needed
to show the great universality
of these
This experiment has shown the universality of the results to a system other than an economic one. In fact, this result is universal for many systems. Defined in terms of at least one discrete variable, where the discrete variable is in turn defined as an integer multiplied by an EU, each model that represents a system (economic, physical, chemical social, etc.) exhibits the characteristics represented here and shown in the figures. This argument is based on the mathematical property shown in the initial example (Fig. 1) and explored in the different experiments.
&
lo-
2 0 -. 0
10
20
30 Quantum
40
50
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size (EU) on average
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6
6
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velocity
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(a) All 20000 points; (b), (c) magnification parameter /I reveals the same basic chaotic
10
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vs. the quantum
calculated
of velocity
by Reiss et al. (1986).
of the left-hand corner of (a); (d) a higher level of the critical structure as in the previous figures, representing an economic system.
A.
Golan
It is clear by now that chaotic
/ Quantum
structures
sion of this special type of chaos,
economic
279
chaos
do emerge in economic
its comparison
modeling.
with deterministic
Discus-
chaos such as
is shown by Baumol and Benhabib (1989), and the implications to economic modeling, are now in place. However, before engaging in this discussion, the initial objective concerning the discrete continuous choice in economic modeling should finally be answered. This is done in the next section.
5. The discrete continuous
paradox
Is there some critical EU size for each variable in an economy below which an economy could be modeled as a continuum rather than as a discrete one? This was the question that introduced this work. There are two answers to this question, one for an economy where the critical parameter is below its critical level and another for the economy where the critical parameter is above the critical level. The former case is simpler and is discussed first. In this instance the critical parameter (e.g. productivity, A) of the economy (system) is below its critical value. In fact, this critical parameter should be much
I
0.0
0.2
0.4
0.6
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10
20
30 Quantum
08
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(ftls)
40 (ftls)
Fig. 5 - contd.
50
60
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below its critical value so that the structure generated by an experiment similar to the one suggested here will be qualitatively similar to Figs. 3(a)-3(c). When this qualification is met, an economic model may be continuous or discrete for any EU size smaller than some critical size (e.g. about 1.75 in Fig. 3(a)); both models are accurate. For any EU larger than the critical size, a continuous modeling of the system will not be accurate. How would one choose the basic EU of the economy modeled? The answer is that it depends on the economy, its size, and its basic units (an apple, a farm, an industry). The economist must choose the EU that is believed to be the basic unit, as defined earlier, representing this economy. For example, if 1 is (in units of bushels) the basic production EU for an economy such as the one represented by Fig. 3(a), a continuous modeling will be correct. This point will be discussed further in the next section. The second answer to the discrete continuous choice question involves those cases where the critical parameter governing the economy (system) is beyond its critical value (e.g. a> 7 in Fig. 3(f)). The results introduced previously (Figs. 2, 3(f), 5) show that as the EU increases, the fluctuations around the mean always increase up to some EU size at which the average properties all become zero. This zero is associated with the case where the EU becomes greater than the average property investigated. These results also show that there is no critical EU size below which the economy should be modeled as a continuum and above which the economy (system) should be represented by a discrete model. Furthermore, the results show that even a discrete modeling is not always sufficient. As EU-,O, the fluctuations become smaller and smaller, but still exist. Assuming the economist is willing to settle for the ‘slight but never zero fluctuations’ associated with a small EU (those on the left-hand side of the figures), she or he must now decide whether that small EU contains the correct amount of information for the construction of a meaningful model. An example drawn from another field may help explain the importance of this decision. Consider the literary critic who studies a text syllable by syllable rather than word by word, sentence by sentence, or chapter by chapter. The critic risks examining and manipulating thousands of incoherent bits of information (syllables) that add nothing to comprehension. Ironically, too many pieces of information are processed without results precisely because each piece of information is too small. An economist runs the same risk when deciding whether an economy should be modeled in terms of EUs of 1 bushel (or 1 dollar), or in terms of firms (or $lOOO), where the EU is the smallest firm capable of existing independently in this economy for a given level of technology. It might appear at first that a model based on small EUs is always preferred, but then an information problem might arise. By using too small an EU, the economist might enter unnecessary (‘too much’) information into the model. Entering unnecessary information is not only costly in terms of collecting the data, but is also costly in terms of analyzing the data. The information-processing mechanism is affected by the fact that the information bits are too small. In terms of the partition function, L?, it means that the partition among the basic elements
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of the model is incorrect. This would happen, for example, if one models an industry with a large number of firms and selects an EU of 1 machine or a level of production too this a waste of inefficient and On the other In this case the
small to exist independently in any one of these firms. Not only is resources, but it also makes the information-processing mechanism inaccurate. hand, assume that one chooses a relatively large EU, say one firm. information level could be the correct one, but at this level the fluc-
tuations are already enormous (see x= 5 in Fig. 2) and there is a large number of discontinuous points. Even if the discrete variable is defined correctly, the result is an inaccurate model. To summarize: the choice presented here is between a small EU, which entails an information problem, or a large EU, which results in model fluctuations. There is no unique rule for this choice; there is no golden rule for the discrete continuous choice question. This is the paradox that arises in the experiments presented here. So, what is one to do? The decision about EU size cannot be made arbitrarily, not even for social systems. However, in modeling social and economic systems, one can control the information bits (or the EU) up to a point and then gather the required information based on the desired EU size. That is, the partition between the elements of the system should be consistent with nature and with the system (economy) modeled.
6. On chaos in modeling A most interesting result emerged during the previous experiments: chaotic structures appeared in economic modeling. Furthermore, those chaotic structures were shown to be universal to all economic variables and to systems other than economic. In fact, most systems represented by discrete variables exhibit the same qualitative chaotic structures. There is self-similarity within each system modeled (for different ranges of the EUs), self-similarity among variables, and self-similarity among different systems. The meaning of these structures and their relationship to deterministic chaos is discussed next. Chaos is a result of some non-linearity in a system, and can occur in systems that are deterministic in the sense that no randomness is introduced at the outset. It is well known that even for such deterministic systems, if some critical parameter governing the system reaches a critical level, the model may become unpredictable within a certain range, i.e. chaos will emerge (e.g. the parameter governing the Logistic Equation; May, 1976). Deterministic chaos, which is associated with the macro properties of a system, can be characterized by two main properties. (A simple explanation and examples of deterministic chaos are given in Schuster, 1988, and a detailed survey of chaos in general and in economics is given in Baumol and Benhabib, 1989.) The first is that there is some ‘ordered structure’ in chaos. The second is that for each system there will always be some critical parameter (or a set
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of parameters) governing the system such that as this critical parameter (e.g. productivity) increases, the model becomes more chaotic up to a level that is stable for all other changes. It reaches some strange attractor that keeps the structure similar for ranges of values greater than this well-defined critical parameter. One example of this Road to Chaos (see, for example, Schuster, 1988, for more examples of Roads to Chaos) is that as the critical parameter increases, the period of an orbit increases through a set of critical (bifurcation) points (May, 1976). The critical parameter increases up to some level beyond which (in most cases) there is a different regime. This regime is characterized by an attractor to other states of the system, which is known as a strange attractor. At the heart of an understanding of deterministic chaos and strange attractors is the question of how the asymptotic behavior of deterministic systems can exhibit unpredictability, owing to sensitive dependence upon initial conditions, and yet at the same time preserve a coherent global structure. The figures shown in the present work show the asymptotic behavior (structure) of a deterministic model due to changes of the initial conditions, i.e. the economy’s set of technological parameters. Some properties of the global structure (a partial representation of the strange attractor) and the order of unpredictability (i.e. the Road to Chaos) are also shown graphically. However, instead of looking at the number of equilibria corresponding to some critical parameter (or the equilibria trajectory over time), the chaos here is a measure of the fluctuations around some observed property of the system, such as the first moment. As the critical parameter governing the system (e.g. productivity) increases, the model becomes more chaotic up to a level that is stable for all other changes (e.g. Fig. 2). It reaches some strange attractor that keeps the structure similar for all values greater than this well-defined critical parameter. Before discussing the mathematics behind this special chaotic behavior (i.e. before showing where the non-linearity enters), there is one philosophical point that needs to be discussed. That point concerns the relationship between chaos in modeling, as is defined in the present work, and chaos in systems (economies), as is described in most chaos models in the literature. This is the subject of the following section. Up to this point the discussion has been about chaos in modeling (or in an economic model) as opposed to chaos within the economic system (e.g. Baumol and Benhabib, 1989). Is there a difference between the two? The answer is that there is chaos in the system as described by the model. Owing to the large number of variables, parameters, and interactions in each observed system, it is almost impossible to describe most systems accurately. Therefore, the introduction of simplifying assumptions is needed in building a model describing some system. This ‘new’, simple (modeled) system, however, might exhibit chaotic behavior (structures) due to some non-linearity in it. It then follows that chaos in a system, based on some well-defined model, is essentially chaos in the simplified system, the modeled one. Does the real-world observed system exhibit chaotic behavior (structures)? This question can be answered in two parallel ways, one theoretical and one empirical.
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The theoretical
economic
answer is that since the modeled
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chaos
system is basically
simpler
than the
real observed system, it is natural to expect that if chaos emerges in the simple model, it will emerge in the more complex one (e.g. Lorenz, 1963; May, 1976; Baumol and Benhabib, 1989). In seeking an empirical answer, one searches for chaos by investigating the behavior of data (e.g. Crutchfield and McNamara, 1987). However, this investigation is based on some mathematical (statistical) or graphical methods that are also subject to some underlying simplified structure (assumptions). One must then return to the first argument: if chaotic structures are observed in the data based on a simplified model, then it is natural to expect the basic structure of the data to be chaotic. Nevertheless, it is inaccurate to argue that a real-world complex system will always exhibit chaotic structure if the model describing it is chaotic. There might be a case where a number of non-linear relations cancel each other such that the modeled system behaves different from the model. In that case chaos emerges only in the model. The conclusion of the short argument presented here is that, in essence, the chaos found in modeling and the chaos in systems, economic or others, are basically the same. To summarize: the notion of chaos in modeling is introduced in the present work. Other papers, such as Lorenz’s (1963) famous work and all the references given before, show the existence of deterministic chaos in natural and economic systems. Following the above argument, one concludes that both types of models represent chaos in some simplified version of an observed system (economy). In searching for the explanation for the chaotic structures emerging here, one is essentially searching for non-linearity in the model. Going back to the example represented by Fig. 1 and Table 1, one sees immediately that the series of numbers representing the average production of the three firms’ industry is non-linear. For each fixed EU, one calculates an average production that cannot exceed its initial value. For example, when x = 1, the average is 20, but because production is defined in terms of integers, the average production will not be exactly 20, except for a limited number of cases. The model is therefore non-linear. (Table 1 shows the corresponding averages for x = 1,2, . . . , 30 as well as the averages corresponding to two different ranges of EUs.) This concludes the search for non-linearity in this discrete system. It should be noted, however, that the simple example introduced at the beginning of this work does not produce all the results presented in this work: the critical parameter governing a system, the universality of the results, and the Road to Chaos. For this, a model of some real economy is best. To summarize: at the heart of an understanding of deterministic chaos and strange attractors is the question of how the asymptotic behavior of deterministic systems can exhibit unpredictability due to sensitive dependence upon initial conditions, and yet at the same time preserve a coherent global structure. The figures shown in this work show the asymptotic behavior (structure) of a deterministic model due to changes of the initial conditions, i.e. the economy’s set of technological parameters. The global structure (strange attractor) and the order of unpredictability (i.e. the Road to Chaos) are also shown.
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One more issue is left undiscussed in this section; it concerns a type of chaos that might be qualitatively different from the well-known deterministic chaos, the chaos dealing with the macro level of a system. It is the chaos related to the micro level of a system, which is known as quantum chaos, or in the terminology of the present work, EU chaos. The fascinating search for quantum chaos is now being conducted simultaneously in many different disciplines (e.g. O’Connor and Heller, 1988; Casati, 1985). One way of trying to characterize this type of chaos is by looking at the quantum (micro) behavior of a system that exhibits deterministic chaos at the macro level. A second way is to investigate the conditions under which the quantum (EU) behavior exhibits some kind of chaotic, but not purely stochastic, behavior. Both these lines of research are necessary for the understanding of quantum chaos. The model described here would help shed some light on the second way of approaching the question. This would be done by investigating the left-hand side of the figures represented here, where the EU becomes smaller and smaller but never vanishes. However, this is beyond the scope of the present paper and will be done in a separate study.
7. Implications
of results and summary
Two main results depend upon one another and are shown here. First, there is no critical EU size below which the economy should be modeled as a continuum, and, furthermore, even a discrete representation of an economy is not always sufficient. Second, chaos in modeling emerges for some critical level of a set of parameters governing an economy. This result is universal to all variables and to systems other than economic. Furthermore, the chaotic structures emerging are selfsimilar for the different cases. The first result implies that models based on discrete observed quantities need some special adjustments. This adjustment, call it rescafing (renormalization), is a procedure aimed at decreasing the fluctuations that arise from modeling large EUs or large bits of information. This resealing is done such that, at the limit, all the points (see figures) collapse into one line at the level of the observed average property. A general resealing of the model is possible due to the universality of the results and due to the fact that each modeled system is self-similar for the different EU sizes. Figure 2 illustrated the ordered structure of the fluctuations. In general, it is expected that as EU increases, n will decrease. However, n is an integer and can change only in discrete values (units). The observed straight lines on the right-hand side of the graphs are due to this property. As the EU size decreases (moving left on the figures), the straight lines made by each collection of points on the graph no longer appear as linear. They have a completely different structure, but that structure is common to each experiment. It is suggested here that this difference is an ‘optical illusion’ in the eyes of the observed owing to the relative small size of the
A. Golan / Quanium
EU at this range.
That is, concentrating
economic
chaos
on the analysis
285
of the figure at this range
only and decreasing the distance between sequential EUs (EU + 0), will yield the same self-similar structure. The right-hand side will consist of straight lines, while the left-hand side will look as before. This is to say, the left-hand side of each figure will always look the same, no matter how small the EU is. One more property is observed in the experiment. The right-hand line of points is at the value of zero average production and will remain zero for any larger EU since, simply, the EU is larger than the average property calculated. The second line of points from the right has a slope of 1 (y = nx = lx for all x at this range). The next line has a slope of 2, the next one has a slope of 3, and so on. From right to left the slope of each additional ‘line’ increases by one. As was discussed before, this linear property is due to the discrete specification of the model and makes it possible to rescale the properties derived from the model such that they will collapse, at the limit, into their observed level. However, this renormalization procedure is yet unsolved and not the subject of the present work, which aimed to investigate the conditions under which an economist can model an economic system as a continuous one and under what conditions the same economy should be modeled as a discrete one, based on discrete defined variables. The second result, which is not independent of the first, implies that there is chaos in modeling. This chaos exhibits the same basic properties as deterministic chaos, but instead of looking at the equilibria structure one is looking at the structure of the model’s fluctuations. However, having defined the discrete variable of some economy as an EU multiplied by an integer, it is easy to show chaos in terms of equilibria. For example, consider the modeling of an industry with N identical firms, where the total output is measured by N multiplied by an EU, which is defined as the firm’s output. Under certain conditions this type of economy will yield deterministic chaos in the sense that the long-run equilibrium number of firms, N, will change chaotically. The present work, however, deals with the discrete continuous choice of modeling, and the exact development of the above example is the subject of a different paper. Finally, definition vestigating of further
it is suggested that the approach described in this work, including the of the variables and the treatment of the chaotic results, is helpful in inquantum economic (EU) chaos phenomena. This, however, is the subject work.
Acknowledgments I wish to thank Amotz Agnon, Robert Anderson, field, Elise Hardy Golan, and seminar participants Rehovot for many helpful comments. Special thanks with the computer analysis.
George Judge, James Crutchat the Hebrew University at to Gary Casterline for helping
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