The Two Components of Concrete Relations

One cannot help but think that Bernard Lonergan had functional macroeconomic dynamics clearly in mind as he treated the intelligibility of world process in CWL 3, Insight: …, which is very much an implementation of the act of understanding of mathematicians and natural scientists.  In his understanding of mathematics, the natural sciences, and the science of macroeconomics in particular, he grasped that the explanation of the dynamic concrete process is expressed by a mathematical conjunction of component abstract primary relativities with component concrete secondary determinations from the non-systematic manifold.  And these secondary determinations, such as particular prices and quantities, are to be interpreted in the light of the significant, abstract, explanatory variables rather than in the obscurity of the IS-LM, and AD-AS models.

Lonergan agreed with Schumpeter on the importance of systematic or analytic framework in order to explain, rather than merely record or describe, the aggregate phenomena of macroeconomics; … and he agreed that the economist had to know what are the significant variables in the light of which price changes are to be interpreted.  According to Lonergan, standard economic theory had successfully achieved none of these desiderata. [CWL 15, Editors’ Introduction liii]

Lonergan thought clearly about a) distinctions between description and explanation in science, b) the requirement for complete explanation in the form of a field theory in which terms are related to one another, c) the required concomitance of outlays and expenditures, and of incomes with outlays and expenditures d) primary and fundamental abstract analytic relations such as those of identity, proportion,  point-to-point correspondence, point-to-line correspondence, algebraic first-degree functions, and first order and second order functions e) functional correlations such as  basic expenditures with monetary costs, and expansionary expenditures with net savings plus expansionary borrowing.

His understanding of empirical method, with its six canons, and his application of scientific method to the phenomena of the concrete dynamic economic process enabled him to advance beyond external-interventions Keynesianism and psychological, ceteris-non-paribus neo-classicalism to a radically-different, purely relational, scientific, macroeconomic field theory.

The heretofore untested applicability of the pure scientific spirit of empirical method and of mathematics to macroeconomics now presents a golden opportunity to candidates for advanced degrees in economics to write influential theses and peer-reviewed articles in order to follow Lonergan in the advancement of macroeconomics from a pseudo-science and mythmaking to the level of explanatory science.

General laws contain a primary relativity and are applied to the concrete “only through the addition of further determinations, and such further determinations pertain to a non-systematic manifold. … it is not enough to think about the general law; one has to add further determinations that are contingent from the very fact that they have to be obtained from a non-systematic manifold.” [CWL 3, 492/516]

The secondary determinations are  constitutive neither of the relation nor of its reality as relation but simply of the differentiation of concrete relations; [CWL 3, 495]

The explanatory primary relativities of the dynamic economic process are the abstract relations of interdependent, explanatory, velocitous flows and their changes to one another.  The flows are congruent with the proprietary network of production and exchange.  They are flows of prices times quantities.   And, while prices from a non-systematic manifold will have an isolated primary relativity of proportion to other prices, they are to be understood and interpreted in scientific macroeconomics in the light of the primary relationships of the significant, explanatory variables of which they are pretio-quantital components; i.e. in light of significant, interdependent, mutually conditioning, mutually defining explanatory flows.

Lonergan’s functional field theory of economics is purely relational and fully explanatory.  It distinguishes between description and explanation, and it explains the economic process in the formalisms of terms related to one another rather than to ourselves.  It is scientific.

A distinction has been drawn between description and explanation.  Description deals with things as related to us. Explanation deals with the same things as related among themselves.  … description and explanation envisage things in fundamentally different manners.  The relations of things among themselves are, in general, a different field from the relations of things to us. … The scientist selects the relations of things to us that lead more directly to knowledge of the relations of things among themselves.  Ordinary description is free from this ulterior preoccupation. [CWL 3, 291-92/316-17]

Lonergan identified primary economic activity as the rate of application by a human of a factor of production; i.e.

qi = Σqijk  [CWL 15, 30]

And, since the humans – whether employees or entrepreneurs – are compensated, each instance of qijk is mated with a price of

pijk

and, thus, we have the elemental flow of incomes, and indeed macroeconomic costs as we define them, associated with productive activities.

Then, by vector analysis, we can work our way to an adequate analytical index for any (cost or revenue) price P vector or quantity vector.

P = √P= √Σpi2      (CWL 15, 74)

Q = √Q= √Σqi2      (CWL 15, 74)

Thus, for example, employing the familiar dot product, the total monetary flow called basic final expenditures in the interval j would be

Σp’ijq’ij =  PjŸQj  =  P’jQ’jcos A  (CWL 15, 74)

Lonergan discovered a set of general, primary explanatory relationships among significant interdependent variables applicable currently and continuously and, thus, in every instance of the evolution of the economic process. And by the application of the secondary boundary conditions of time lags, prices, quantities, labor supply, money supply and technical coefficients, one can reach the particular law of the concrete, overall, economic functioning.

The first set of primary relativities is listed  below. The first equation is the lagged technical accelerator of the productive process with which monetary circulations are correlated.  The rest are monetary relativities.  One of the monetary relations in this set provides the condition of equilibrium universally applicable throughout the temporal process.  And note that, though the relations are symbolized in algebraic functions,  the symbols represent interdependent rates of velocitous flows, i.e. so much or so many every so often, d/dt and Δ/Δt, as diagrammed in the double-circuited, credit centered, Rates of Flow.

kn[f’n(t-a)-Bn] = f”n-1(t) – An-1    (CWL 15, 37)

  • R’ = E’     (CWL 15, 54)
  • R” = E”      (CWL 15, 54)
  • I’ = O’ +M’      (CWL 15, 54)
  • I” = O” +M”     (CWL 15, 54)
  • G = c”O” –i’O’   (CWL 15, 54)
  • G = c”O” –i’O’ = 0     the condition of dynamic equilibrium   (CWL 16, 50)
  • M’ = (S’ – s’O’) + (D’ – s’I’) + G   (CWL 15, 54)
  • M” = (S” – s”O”) + (D” – s”I”) – G   (CWL 15, 54)
  • (S’-s’O’) = ΔT’ + (O’ – R’) + ΔR’  (CWL 16, 67)
  • (S”- s”O”) = ΔT” + (O” – R”) + ΔR”  (CWL 16, 67) 
Diagram of Rates of Flow 2

Diagram of Rates of Flow

 Those familiar with elementary statics and dynamics will appreciate the shift in thinking involved in passing from (static) equilibrium analysis … to an analysis where attention is focused on second-order differential equations, on d2θ/dt2, d2x/dt2, d2y/dt2, on the primary relativities of a range of related forces, central, friction, whatever.  Particular secondary boundary conditions in Functional Macroeconomic Dynamics, past and future pricings and quantities, are relatively insignificant for the analysis of the primary relativity immanent in, and applicable to, every instance of the process.  What is significant is the Leibnitz-Newtonian shift of context. [McShane, 1980, 127]

A second set of primary relativities dealing with how the process plays out over time is provided in CWL 15, sections 26-28 which treat the cycles of

  • basic income,
  • pure surplus income,
  • the aggregate basic price-spread.

These sections explain both the normative equilibria and the violative disequilibria in the phases of the evolutionary process as it works out to the tune of the lagged technical accelerator.

kn[f’n(t-a)-Bn] = f”n-1(t) – An-1  (CWL 15, 37),

per the possibilities of a) the Table of Possibilities on CWL 15, 114, and b) Figure 24-7, p. 125.

 

The differentials of the expansionary cycle:

  • dI’ = Σ(widni+ nidwi+dnidwi)yi       [CWL 15, 134]
  • df = vdw + wdv       [CWL 15, 148-49]
  • P’Q’ = p’a’Q’ + p”a”Q’      [CWL 15, 156-58]
  • P’/p’ = a’ + a”(p”Q”)/(p’Q’)      [CWL 15, 156-58], or
  • J = a’ + a”R       [CWL 15, 156-58]
  • d(P’/p’) = dJ = da’ + a”dR + Rda”       [CWL 15, 158]

The pattern of a pure cycle would would have an exigence for a normative balance of crossovers between circuits.  And, as exemplified by Burley’s and Csapo’s Characteristic Equations and Their Root Solutions, the pure cycle would exhibit a) normative relative intensities of productive activities, and b) normative relative pricing.  And, per the lagged technical accelerator, the pure cycle would implement the magnitude of coefficient k, and the timing indicated by t, t-a, t-b, etc, in that lagged technical accelerator.  Neither too fast nor too slow, neither excessive nor deficient, and dependent upon the state of technology, culture, and institutions.

Lonergan’s internal primary relations include point-to-point and point-to-line productive relations, monetary circular and crossover conditions; productive and monetary velocities (d/dt); accelerations (d2/dt2); correlations of productive structure with monetary circulations.  And the primary relations would be applied to secondary determinations from the concrete non-systematic manifold, which include determinate price and quantity vectors, determinate time lags, and determinate state of technology symbolized by k.

Accordingly, while we must grant that the shift from description to explanation involves a shift from external to internal relations, still we also contend that the internal relations constitute no more than the component of primary relativity and, since in concrete relations there is also a component of contingent secondary determinations, external relations also survive a definitive explanatory account of our universe. [CWL 3, 494/ 518]

“Since in concrete relations there is also a component of contingent secondary determinations, external relations also survive a definitive explanatory account of our universe.”

There is always a “size” of price and a “size” of quantity.  One particular concrete size suggests comparison to other concrete sizes.  And the primary relations of one price to another and one quantity to another would be relations of proportion, while the secondary determinations from the non-systematic manifold would be a numerical ratio such as 1:1, 2:1; 3.5:1, 8:1, etc..

… it is necessary to distinguish in concrete relations between two components, namely, a primary relativity and other secondary determinations.  Thus, if it is true that the size of A is just twice the size of B, then the primary relativity is a proportion and the secondary determinations are the numerical ratio, twice, and the two observable sizes.  Now ‘size’ is a descriptive notion that may be defined as an aspect of things standing in certain relations to our senses, and so it vanishes from an explanatory account of reality.  Again, the numerical ratio, twice, specifies the proportion between A and B, but it does so only at a given time under given conditions; moreover, this ratio may change, and the change will occur in accord with probabilities; but while probabilities will explain why objects like A and B every so often have sizes in the ratio of two to one, they will not explain why A and B are in fact in that relation here and now; and so the numerical ratio, twice, is a non-systematic element in the relation.  However, if we ask what a proportion is, we necessarily introduce the abstract notion of quantity and we make the discovery that quantities and proportions are terms and relations such that the terms fix the relations and the relations fix the terms.  For the notion of quantity is not to be confused with a sensitive or imaginative apprehension of size; a quantity is anything that can serve as a term in a numerical ratio; and, inversely, a proportion, in the present context, is a numerically definable ratio between two quantities. [CWL 3, 491/515]

The distinction of primary relativity and its secondary “coincidental” and absolute determinations separates the systematic and the non-systematic.

The point, then, to our distinction between the primary relativity of a relation and its secondary determinations is that it separates the systematic and the non-systematic.  If A and B are things of determinate kinds, then they must be quantitative; and if they are quantitative, there must be some proportion between their quantities.  But just what that proportion will be at any given time, will depend on the manifold of factors that form the non-systematic pattern of a diverging series of conditions, and so there is within the limits of human science no ultimate and fully determinate explanation of why A happens to be just twice B at a given moment. [CWL 3, 491/515]

Distinguish the primary, abstract, purely relative relation from the non-systematic secondary determinations to which it can be applied.

There is a further point to our distinction.  As it separates the systematic from the non-systematic, so also it separates the relative from its absolute determinations. All that is relative in the notion, twice, is also found in the notion, proportion; the difference between them is that ‘twice’ is a proportion specified by a pair of quantities such as one and two, or two and four, etc.; and such pairs of quantities, simply as pairs of quantities, prescind from the relations of one to the other. [CWL 3, 492/515-16]

Again,

General laws contain a primary relativity and are applied to the concrete “only through the addition of further determinations, and such further determinations pertain to a non-systematic manifold. … it is not enough to think about the general law; one has to add further determinations that are contingent from the very fact that they have to be obtained from a non-systematic manifold. [CWL 3, 492/516]

In Functional Macroeconomic Dynamics two simple, yet fundamental and explanatory relations of proportion are the Aggregate Basic Price-Spread Ratio and its differential:

  • P’/p’ = a’ + a”R     (CWL 15, 156-60)
  • dJ = da’ + a”dR + Rda”     (CWL 15, 156-60)

This variable ratio of the indexes of selling price to cost price has vast implications for understanding and interpreting the economic process.  The variable ratio, as variable, changes.  That primary relation of  proportion is, in this case defined by the collocation of significant explanatory variables of acceleration, surplus productive outlays and basic productive outlays.  Variables implicitly define variables.  The discovery of that central, purely-relational, key intelligibility of the economic process is last in the analysis.  The gem hidden in the process is discovered and becomes the capstone of the process.

That shifting proportion of price to price virtually contains, or brings together in a single expression, the productive and monetary orders of the credit-centered, double-circuited, threefold, time-lagged, process of production and exchange.  It consists of an internal primary relativity for application to secondary determinations. It might be compared in significance to e = mc2of physics.  Its simplicity is offputting to pseudosophisticates.  The prices and quantities they accept as an absolute external given basis for explanation is revealed to be merely the happenstantial secondary determinations of the differentials of a deeper, more profound, purely-relative explanation.

… conjugate forms are defined implicitly by their explanatory and empirically verified relations to one another.  Still, such relations are general laws; they hold in any number of instances; they admit application to the concrete only through the addition of further determinations, and such further determinations pertain to a non-systematic manifold. There is, then, a primary relativity that is contained in the general law; it is inseparable from its base in the conjugate form which implicitly it defines;[1]and to reach the concrete relation that holds at a given place and time, it is not enough to think about the general law; one has to add further determinations[2]that are contingent from the very fact that they have to be obtained from a non-systematic manifold. [CWL 3, 492/516]

Paraphrasing: [CWL 3, 492/516]  … conjugate forms or terms are defined implicitly by their explanatory and empirically verified relations to one another.  …  Such relations (P’Q’ = p’a’Q’ + p”a”Q”Basic circuit R&M; J = a’ + a”R; dJ = da’ + a”dR + R’da”) are general laws; they hold in any number of instances; they admit application to the concrete only through the addition of further determinations (such as the indices and coefficients of actual price and quantity), (however) such further determinations pertain to a non-systematic manifold. (CWL 3, 948) There are, then, primary relativities.   To reach the concrete relation that holds at a given place and time, it is not enough to think about the general law; one has to add further concrete determinations (such as actual pricings and quantities in the non-systematic manifold) that are contingent from the very fact that they have to be obtained from a non-systematic manifold.

The general laws of Functional Macroeconomic Dynamics are purely relational.

Moreover, there follows a clarification of the problem of internal and external relations. Relations are said to be internal when the concept of relation is intrinsic to the concept of its base; they are external when the base remains essentially the same whether or not the relation accrues to it.  Thus, if ‘mass’ is conceived as a quantity of matter and matter is conceived as whatever satisfies the Kantian scheme of providing a filling for the empty form of time, then the law of inverse squares is external to the notion of mass.  On the other hand, if masses are conceived as implicitly defined by their relations to one another and the law of inverse squares is the most fundamental of those relations, then the law is an internal relation, for the denial of the law would involve a change of the concept of mass. [CWL 3, 493/517]

Paraphrasing: Moreover, there follows a clarification of the problem of internal and external relations.  Relations of price are said to be internal when the concept of relation is intrinsic to the concept of price; they are external when the price remains essentially the same whether or not the relation accrues to it.  Thus, if ‘price’ is conceived as an absolute unchanging given filling the empty category of accounting cost, then the law of the basic price-spread ratio is external to the notion of price.  On the other hand, if prices are conceived as implicitly defined by their relations of proportion to one another and the law of the basic price-spread ratio is the most fundamental of those relations, then the law of the basic price-spread ratio is internal to the notion of price, for the denial of the law would involve a change of the concept of price as a term to which relation is intrinsic.

Similarly, for the law of economic point-to-point and point-to-line quantities: If quantities are conceived as implicitly defined by their distinctly-different relations to one another grounded in their relations to quantities exiting the process, and if the law of the basic price-spread ratio P’/p’ and its differential dJ are so fundamental and scientifically significant as to gather together in a single expression the productive and monetary flows of the process, then the law of the basic price spread ratio containing an internal pricing relativity of quantities is the most fundamental of those relations, then the law of the basic price-spread ratio is internal to the notion of quantity, for the denial of the law would involve a change of the concept of quantity in macroeconomics.

By a projection of the distinctions among classes of products onto classes of payments, a composite flow constituted by quantities happening to have prices in units of purchasing density is a an abstract monetary density moving at a velocity and subject to a change called an acceleration.  The composite flow itself is primarily defined by its functional relation to other flows in the dynamic production process.  The composite flow’s functional relativity is its primary relativity, as represented by both the lagged technical accelerator and the other equations above..

kn[f’n(t-a)-Bn] = f”n-1(t) – An-1  (CWL 15, 37)

The original precise analytic distinctions between point-to-point and point-to-line are scientifically significant so as to virtually contain the primary relations of both the lagged technical accelerator and the basic price-spread ratio.

To recap, the determinate concrete prices and quantities in the non-systematic manifold of emergent probability are to be understood in the light of the significant explanatory variables of the primary relativities in which they are the concrete differentiations of the abstract relativistic variables.

One might be reminded here of a parallel in hydrodynamics: if what is at issue is a general specification of the dynamics of free water waves, a premature introduction of general boundary conditions or worse, specific channel conditions, botches the analytic possibilities….the Robinson-Eatwell analysis is hampered, not only by an absence of paradigmatic heuristic thinking in a field whose principles involve ends, but also by their building the economic priora quoad nos of profits, wages, prices, etc., into explanation, when in fact the priora quoad nos are last in analysis: they require explanation.[McShane 1980, 124]

Macroeconomics is a science composed of two components: abstract primary relativities and concrete secondary determinations.  There are primary relativities among the interdependent, explanatory pretio-quantital flows which we listed above; and there are primary relativities between prices themselves and quantities themselves such as 1:1, 2:1, 1.582:1, 8.3:1, etc. The prices and quantities themselves have primary relativities of proportion; but prices and quantities, as secondary determinations or differentials of their explanatory variables, are to be understood in the light of the explanation of the process by those significant variables.

Flows are comprised of so much pretio-quantital value per interval.

Our immediate task is to work out the correlations that exist between the velocity and accelerator rhythms of production and the corresponding rhythms of income and expenditure.  The set of such correlations constitutes the mechanical structure, a pattern of laws that stand to economic activity as the laws of mechanics to buildings and machines. [CWL 21, 43]

All such payments form a class by themselves.  They stand in a network that is congruent with the technical network of the productive process. …above all, their connection with production is immediate: they … are, so to speak, the immanent manifestation of the productive process as a process of value. [CWL 21, 114]

The whole structure is relational: one cannot conceive the terms without the relations nor the relations without the terms. Both terms and relations constitute a basic framework to be filled out, first, by the advance of the sciences and, secondly, by full information on concrete situations. [CWL 3, 492/516]

Further clarifications:

The reality of the real relation is in P prior to the change in Q; that reality is relative; it is the primary relativity inseparable from quantity; it involves everything quantitative in some relation of proportion to everything else that is quantitative; but it does not determine just what is the proportion between P and Q, or P and R, or P and S, etc.  To settle just what the proportion is in any case, one has to appeal to the secondary determinations, such as the size of P and the size of Q; and because the secondary determinations are found not only in P but also in Q, because variations in P and Q are not functionally related, the determinate proportion of P to Q can change without any change in P. [CWL 3, 495/519]

The primary relativity is inseparable from its base and for that reason all change is change in the base and and only incidentally and consequently change in the relativity. The secondary determinations are constitutive neither of the relation nor of its reality as relation but simply of the differentiation of concrete relations; and because the differentiation depends, not on the base alone but on the base and term together, it can vary without variation in the base [CWL 3, 495/519]

On the position, the real is being; it is whatever is to be grasped intelligently and affirmed reasonably.  Now within the limits of proportionate being, whatever is grasped intelligently is never a term without relations or a relation without terms.  To express an insight, one needs several terms and relations with the terms fixing the relation and the relations fixing the terms. To suppose that there are any terms without relations or any relations without terms is to suppose an oversight. Descriptive terms are no exception, for they express things as related to us. … On scientific terms we have been sufficiently abundant already.  But what cannot be affirmed, cannot be.  What cannot be conceived, cannot be affirmed.  But there is no intelligent conception of terms apart from relations or relations apart from terms, and so there is no possibility of their being apart. [CWL 3, 496/520]

The primary intelligibility of pricings and quantities consists in their being relativistic components of interdependent, explanatory, pretioquantital flows correlative to a flowing of goods and services in the process of production and exchange.

 

[1]That is, there is a general relation which is inseparable from the terms it relates; for the terms define the relations and the relations define the terms. The primary relativities are Productive point-to-point vs. point-to-line, and monetary P’Q’ = p’a’Q’ + p”a”Q” and Π”Κ” = π”α”Κ”.

[2]The further determinations are particular pricings-quantities combinations comprising the functional flows.