# 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 (R2), Durbin-Watson statistic (D-W) and statistical variances.

Change Rates in Per Capita Birth Rates:

(1)     BRt = 0.48 + 0.373*LWt + 1.079*MRt – 0.928*CMt

(0.0061)        (0.0297)        (0.0314)

R2 = 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)     MRt = 0.82 + 0.638*GPt + 0.015*AFt-1 – 0.495*BRt-2

(0.0092)       (0.0000)         (0.0087)

R2 = 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)     LWt = -4.78 + 2.955*RAt + 1.705*HSt-1 + 1.042*BRt-1

(0.8509)        (0.1263)            (0.0333)

R2 = 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)     HSt = 1.55 – 0.343*LWt-2 – 0.341*BRt-2 +0.396*GPt-2

(0.0269)          (0.0147)         (0.0255)

R2 = 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)      BEt = 1.39 – 0.688*IAt + 0.164*IMt + 0.047*IMt-2

(0.0178)      (0.0006)      (0.0003)

R2 = 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)   RAt = 0.76 – 0.070*HSt-1 + 0.450*BEt-1 – 0.111*IAt-2

(0.0027)         (0.0030)        (0.0006)

R2 = 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)    IAt = -5.05 – 2.519*RAt + 8.450*URt

(0.7570)        (2.5359)

R2 = 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)    URt = 1.18 – 0.100*HSt – 0.059*CMt-1 + 0.003*AFt-1

(0.0018)      (0.0010)           (0.0000)

R2 = 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)    CMt = 1.89 – 1.624*RAt + 0.611*GPt + 0.250*GPt-1

(0.2610)       (0.0110)       (0.0105)

R2 = 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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