# HERBERT SIMON, PAUL THAGARD, PAT LANGLEY AND OTHERS ON DISCOVERY SYSTEMS

## APPENDIX I- Page 12

**EQUATIONS OF THE THEORY**

Each equation of the theory is displayed below together with its
coefficient of determination (R^{2}), Durbin-Watson
statistic (D-W) and statistical variances.

**Change Rates in Per Capita Birth Rates:**

(1)**
BR**_{t} = 0.48 + 0.373***LW**_{t} + 1.079***MR**_{t}
– 0.928***CM**_{t}

(0.0061) (0.0297) (0.0314)

R^{2}
= 0.8688
D-W = 2.4321

The
change rates of the crude birth rates (**BR**) increase with
increases in the change rates of the per capita rates of
conformity with criminal law (**LW**) in the same four-year
period, with increases in the change rates of the per capita
marriage rates (**MR**) in the same four-year period, and
with declines in the change rates of per capita voluntary
exposure rates to mass communications media (**CM**) in the
same period.

**
Change Rates in Per Capita Marriage Rates:**

(2)**
MR**_{t} = 0.82 + 0.638***GP**_{t} + 0.015***AF**_{t-1}
– 0.495***BR**_{t-2}

**
**(0.0092)
(0.0000)
(0.0087)_{}

**
**R^{2}
= 0.9582
D-W = 2.0527

The change rates
of marriage rates (**MR**) increase with increases
in the change rates of real per capita income (**GP**)
in the same period, with increases in change rates in per capita
armed forces active duty personnel (**AF**), four
to eight years earlier, and with declines in change rates of the
crude birth rates (**BR**) eight to twelve years
earlier. The average age of first marriage during the
fifty-year sample is twenty-one years (Commerce, P. 19). Thus
this equation relates the peaks of the marriage rates to the
troughs of the earlier birth rates instead of relating the peaks
of the marriage rates and the peaks of the still earlier birth
rates, to maximize the degrees of freedom. The time lag between
the marriage rates growth and mobilization change rates (**AF**)
is due to wartime postponements of marriage.

**
Change Rates in Per Capita Criminal-Law Compliance Rates:**

(3)**
LW**_{t} = -4.78 + 2.955***RA**_{t} +
1.705***HS**_{t-1} + 1.042***BR**_{t-1}

**
**
(0.8509) (0.1263)
(0.0333)

R^{2}
= 0.9165
D-W = 1.5671

The
growth rates of the per capita rates of compliance with criminal
laws proscribing homicide (**LW**) increase with increases of
the change rates of religious affiliation rates in the same
period (**RA**), with increases in the change rates of high
school graduation rates of seventeen-year olds in the previous
period (**HS**), and with increases in change rates in birth
rates (**BR**) in the prior period. This equation
reveals the institutional reinforcement between the civic value
orientation and those of the religious and educational
institutions. The positive relation between compliance with
criminal law and birth rates suggest Ryder’s comment that
nothing makes a younger generation settle down faster than a
younger one coming up (1965).

**
Change Rates in High School Graduation Percentage Rates:**

(4)**
HS**_{t} = 1.55 – 0.343***LW**_{t-2} –
0.341***BR**_{t-2} +0.396***GP**_{t-2}

(0.0269) (0.0147) (0.0255)

R^{2}
= 0.9519
D-W = 2.2653

The
change rates of the percent of seventeen-year-olds who graduate
from high school (**HS**) decrease with increases in change
rates of the per capita rates of compliance with criminal laws
proscribing homicide (**LW**), with increases in change rates
in birth rates (**BR**), and increase with increases of the
change rates of real per capita income (**GP**). All
these operate with a time lag of eight to twelve years.
These lengthy time lags suggest that the effects on high-school
age students are mediated by the socializing efforts of adults
such as the school authorities and/or parents. Aberle
reports that the socializing function of parents implies a
prospective attitude toward their children, and that the
children’s futures as envisioned by the parents will be
influenced by the parents’ experiences, as these are affected by
conditions prevailing in the adult world at the time of their
socializing efforts (1963, p. 405). Thus the equation
identifies compliance rates with criminal law as influential
conditions in the adult world.

**
Change Rates in Business Formation Per Capita Rates:**

(5) **BE**_{t}
= 1.39 – 0.688***IA**_{t} + 0.164***IM**_{t}
+ 0.047***IM**_{t-2}

(0.0178) (0.0006) (0.0003)

R^{2}
= 0.9669
D-W = 2.8134

Change
rates of the per capita rates of net new business formation (**BE**)
increases with decreases in the change rates of per capita
patent applications (**IA**), and increase with increases in the
growth rates of per capita immigration (**IM**) with lags
from zero to twelve years.

**
Change Rates in Religious Affiliation Per Capita Rates:**

(6) **RA**_{t}
= 0.76 – 0.070***HS**_{t-1} + 0.450***BE**_{t-1}
– 0.111***IA**_{t-2}

(0.0027) (0.0030) (0.0006)

R^{2} = 0.9861
D-W = 1.8646

The
change rates of the per capita rates of religious affiliation (**RA**)
increase with decreases in the change rates of the percent of
seventeen-year-olds who graduate from high school (**HS**) in
the preceding period, increase with increases in the growth
rates of per capita net new business formation (**BE**) in
the preceding period, and increase with decreases in the change
rates of per capita applications for inventions (**IA**) two
periods earlier. The negative algebraic signs show the
conflicts of education and technology with religion and the
reinforcement between the religion and business.

**
Change Rates in Technological Innovation Per Capita Rates:**

(7)
**IA**_{t}
= -5.05 – 2.519***RA**_{t} + 8.450***UR**_{t}

_{
(0.7570) }(2.5359)

R^{2}
= 0.8697
D-W = 2.5601

The
change rates of the per capita rates of technological innovation
(**IA**) increase with decreases in the change rates of the
per capita rates of religious affiliation (**RA**) in the
same period, and increase with increases in the change rates of
per capita rates of urbanization (**UR**) in the same period.

**
Change Rates in Urbanization Percentage Rates:**

(8) **UR**_{t}
= 1.18 – 0.100***HS**_{t} – 0.059***CM**_{t-1}
+ 0.003***AF**_{t-1}

(0.0018) (0.0010) (0.0000)

R^{2} = 0.9831
D-W = 1.3162

The
change rates in the percent of the population not living on
farms (**UR**), i.e., the rate of urbanization, increase with
decreases in the growth rates of the percent of
seventeen-year-olds who graduate from high school (**HS**) in
the same four-year period, increase with decreases in the growth
rates of per capita exposure to mass media communications (**CM**)
in the prior four-year period, and increase with increases in
the growth rates of per capita memberships (**AF**) in the
prior four-year period.

**
Change Rates in Mass Communication Per Capita Rates:**

(9) **CM**_{t}
= 1.89 – 1.624***RA**_{t} + 0.611***GP**_{t}
+ 0.250***GP**_{t-1}

(0.2610) (0.0110) (0.0105)

R^{2}
= 0.9555
D-W = 2.6126

The change rates of per capita exposure to mass media
communications (**CM**) increase with decreases in growth
rates of the per capita rates of religious affiliation (**RA**)
in the same period, and increase with increases in the growth
rates of per capita real incomes (**GP**) in the current and
prior periods.

**
STATIC ANALYSIS**

In
quantitative functionalism the term “equilibrium” means a
solution of a model such that the values of each variable remain
unchanged for successive periods of iteration. This is
displayed by making all time subscripts current (**t**=0
for all) and then solving the equation system. Since the values
of this model’s variables are index numbers of change ratios of
per capita rates, the equilibrium solution is one of constant
change ratios of the per capita rates for all variables, and
they may be positive, zero or negative. The classical consensus
equilibrium is represented by constant per capita rates that are
near the maximum for all the institutional variables.

However examination of
the mathematical equilibrium solution of the model reveals that
a static or zero-growth solution for all the institutional
variables in the statistically estimated empirical model cannot
exist, and therefore that the classical functionalist consensus
equilibrium does not exist for the U.S. national macrosociety.
Some institutional variables must increase in order for others
to maintain a zero growth rate at any per capita level. Thus if
the former institutional variables are forced to represent zero
change, as when the maximum consensus per capita rate is
encountered as its upper limit, then the latter must decline
away from the per capita maximum or consensus equilibrium. This
condition is illustrated by equation (3) where an increasing per
capita rate of religious affiliation (**RA**) is
necessary to produce stabile constant per capita rates of
compliance with criminal law prohibiting homicide (**LW**).

Furthermore since the algebraic signs for some of the coefficients relating the institutional variables are negative, were a static equilibrium to exist, it might better be described as what Moore called a “tension-management” equilibrium rather than Parsonsian consensus equilibrium (1963, p. 10, 70). In summary: in classical functionalist terms the American national macrosociety is what Parsons called “malintegrated”.

**
DYNAMIC ANALYSIS**

In quantitative functionalism the term “dynamic” refers to the macrosociety’s adjustment and stability characteristics as exhibited by successive iterations of the macrosociometric model. This meaning of dynamics is not unrelated to that found in classical functionalism, since changes in per capita rates are changes in measures of consensus about institutional-group values and reflect the effects of socialization and social control. However, the problem addressed by the model is not the problem of explaining the operation of the social-psychological mechanisms of socialization and social control, and the macrosociological theory does not implement a social-psychological reductionist agenda. Rather the relevant problem is the macrosociological problem of tracing the pattern through time of the interinstitutional adjustment dynamics of the macrosociety. To this end three types of simulation analyses are made, in which the upper and lower limits of the per capita rates are ignored, and the values of the variables are allowed to be unrealistic to display their adjustment patterns.

** Type I Simulation:**
In the first type of simulation the model was iterated with all
of its exogenous variables and all of the initial lagged-values
assigned their index number equivalents to represent zero change
in their per capita rates. When the model is thus iterated, it
propagates a time path that oscillates with increasing amplitude
and a phasing of eight four-year periods, i.e., it generates an
explosively oscillating intergenerational cycle of between
twenty-eight and thirty-two years. This is due to the
exogenously fixed constant real per capita GNP, so that there is
no negative feedback to living standards (**GP**)
that would dampen such explosive decline and growth rates in
birth rates as occurred during the dire Great Depression years
and the affluent post-WWII “baby boom” years. Capturing this
feedback requires integrating this macrosociometric model with a
macroeconometric model.

Furthermore examination
of the structure of the model reveals that equations (1), (2)
and (3), which determine the growth rates of the birth (**BR**),
marriage (**MR**) and criminal-law compliance (**LW**)
rates, are interacting to capture an intergenerational cyclical
pattern in the national demographic profile. With historical
birth rates gyrating from 15.7 in 1933 to 21.7 in 1947 to 14.9
in 1972, the empirical model has captured a cycle in the
national demographic profile and shows its sociological
effects. Thus when a new generation born at the peak of a “baby
boom” is in their infancy, the simulation shows a coincident
peak in the per capita rate of religious affiliation reflecting
the practice of infant initiation. When they are in their teens,
it shows a peak in the crime rate. When they are in their late
twenties, it shows a peak in the marriage rates, and then the
birth rates come full circle for another demographic cycle.
Also when they are in their later twenties the simulation shows
a peak in new business formation.

Another simulation was
run with the criminal-law compliance change rates variable (**LW**)
set exogenously to its index equivalent of constant zero-growth
rate representing a continuing stable level of law-abiding
social order. And the real per capita income change rate
variable (**GP**) is set to its index equivalent of
an atypically high annual growth rate of twelve percent, as
occurred between the depths of the Great Depression in 1933 and
peak production and employment levels of World War II in 1945.
When the model is thus iterated, all of the institutional
variables and the birth rate variable quickly settle into a
stable moving equilibrium pattern of constant positive change
rates in the direction of consensus equilibrium. But as noted
in the static analysis above, the U.S. macrosociety cannot
achieve stable consensus equilibrium due to its institutional
malitegration.

**
Type II Simulation: **As with the term “dynamic”, so too with
the phrase “integrative mechanism”, its meaning in quantitative
functionalism is different from but related to its meaning in
classical functionalism. For a macrosocial negative feedback in
the model to be compatible with a classical functionalist
integrative mechanism, it must produce a tendency to stabilize
the rates of social change in constant positive growth paths for
all the institutional variables, and thus trend upward toward
macrosocial consensus equilibrium, even if such consensus is
unattainable.

In order to isolate and
make evident the interinstitutional integrative mechanisms, the
birth-rate equation is removed from the model in these
simulations, and the **BR** change rate is
exogenously set to its index equivalent of zero making the per
capita birth rate constant. As in the prior simulation all the
exogenous and initializing lagged-values are assigned their
index number equivalents representing zero change in their per
capita rates. When the model is thus iterated but with the real
per capita income change rate (**GP**) set to its
index number equivalent of a high annual growth rate of twelve
percent, then the model propagates a damped eight-year
oscillating time path that converges into constant positive
growth rates toward consensus equilibrium in the per capita
rates of all the institutional variables.

**The operative
integrative mechanism** is a dampening negative feedback
due to equations (3) and (4), which determine the change rate of
the compliance rate (**LW**) and the change rate of
the high-school completion rate (**HS**). The
model shows that an increase in social disorder as indicated by
rising rates of noncompliance with criminal law proscribing
homicide calls forth a delayed reaction by the socializing
educational institution, which in turn tends to restore order by
reinforcing compliance with criminal law. This negative
feedback produced by the educational institution (**HS**)
results in the positive growth paths toward consensus
equilibrium; it is a macrosocial integrative mechanism.

But these positive
growth rates of all the institutional per capita rates need not
necessarily result from the effective operation of this
negative-feedback mechanism. As it happens, if all the
exogenous variables are assigned index-number equivalents to
zero-growth values, including the per capita real income
variable (**GP**), then the resulting equilibrium
is one in which the change in the criminal-law compliance rates
(**LW**) is negative. That is because the
zero-growth rate of the per capita real income variable
represents a divisive social condition that Lester Thurow in
1980 called a “zero-sum society” with destabilizing effect.

**
Type III Simulation: **The third type of simulations examines
the stability characteristics of the growth equilibrium by
disturbing it with shocks. In the shock simulations the
magnitude of the shock is unrealistically large and the upper
and lower boundaries of the per capita rates are ignored, in
order to display the dynamic properties of the model. The
results are thus intentionally eccentric to exhibit adjustment
patterns.

Some sociologists such
as Ogburn have cited technological invention as an initiating
cause of social change. Thus a simulation was made in which the
growth rate of the per capita rate of patent applications for
inventions (**IA**) was increased from zero growth
to one hundred percent growth for only one iteration. This
one-time shock is an improbable permanent doubling of the per
capita rate of inventions. When the model is initially
iterated, the per capita rate of technological invention is kept
constant at the index-number equivalent of zero-growth rate for
fifteen iterations, i.e., sixty years, so that the model can
adjust and settle into a long-term constant change-rate
equilibrium solution. Then in the sixteenth iteration the
shock, the onetime permanent doubling of the per capita rate, is
made to occur. The result is a damped oscillation, a shock wave
that propagates through the social system with a phasing of four
four-year periods generating a sixteen-year cycle and having a
small amplitude that nearly disappears after two cycles to
return to the initial per capita change-rate equilibrium levels
for all of the institutional variables. This is suggestive of a
Schumpeterian economic-development cycle scenario of the
economy’s reaction to technological innovations, save for the
noteworthy fact that the real GNP variable has been exogenously
held constant, and thus can receive no reinforcing positive
feedback raising the economy to a higher equilibrium level
through a consequent shift in the macroeconomy’s aggregate
production function.

Similar simulations
using the other variables as shocks yielded comparable results.
But a very different outcome occurs when the shock is a
permanent doubling of the change rate of the per capita urban
residence rate (**UR**). As in the other shock
simulations, the model is initially iterated with the
index-number equivalent of zero change for fifteen iterations,
i.e., sixty years, before the one-time doubling of the urban per
capita rate is made.

The constant proportion
of urban population during the initial fifteen iterations
produces accelerating positive change rates of the all the
institutional per capita rates but the educational institutional
variable (**HS**), which exhibits accelerating
decline. The permanent agrarian share of the population makes
the other institutional variables accelerate in the direction of
consensus equilibrium with no cyclical reversals, because the
educational institution’s negative feedback is ineffective.
This phase of the simulation scenario suggests the
traditionalism of an agrarian society having a low valuation for
education and a tendency toward high macrosocial integration.

But when the one-time
doubling of the growth rate of the urban residents’ share of the
population is made to produce a sudden permanent doubling of
their share of the macrosociety in the second phase, the
opposite outcome happens. The sudden surge into cities that the
shock represents sends the variable representing civil order (**LW**)
together with all the other institutional variables except the
educational variable into accelerating decline. The negative
feedback from the educational variable’s positive change rate is
overwhelmed and cannot effectively function as an integrative
mechanism to reverse the accelerating negative change rates of
the other institutional variables. In other words the model
describes a **macrosociety disintegrating** toward
the Hobbesian chaos that Parsons says institutions exist to
preclude. Such is the lot of a failed and collapsing society.

**
SUMMARY OF FINDINGS **

The static and dynamic analyses with the quantitative functionalist theory of macrosocial change yield four findings about the American national society based on the fifty years of history following World War I:

1. Static mathematical equilibrium analysis shows that the interinstitutional cultural configuration of value orientations is malintegrated, such that macrosocial consensus equilibrium theorized by classical functionalists does not exist for the American national society.

2. Dynamic simulation reveals that fluctuations in the growth rate of the birth rate exhibit an intergenerational demographic life cycle which is explosively oscillating in the absence of a negative feedback reducing the level of per capita real income measured by per capita real GNP.

3. If the birth rate is
exogenously made constant, the national society exhibits
movement toward macrosocial consensus, when per capita real
income grows at the historically high rate of twelve percent
annually. This movement is due to **an interinstitutional
cultural configuration that constitutes an integrative mechanism
consisting of a negative feedback reaction** to criminal
social disorder operating through the socializing functions of
the universal public educational institution.

4. Finally a static
urban/rural share of the national population suggests a
traditionalist agrarian society with all of the institutional
variables except education exhibiting ** growth
toward consensus macrosocial equilibrium**. But a very
large and sudden inundation of population from the nation’s
hinterlands into the cities sends the institutional variables
into accelerated

*decline*producing disintegration of the institutional order and apocalyptic social disorganization.**
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