The Einsteinian Context: Curvature and Relativity

Albert Einstein, Steven Weinberg, Lillian Lieber, Douglas Giancoli, Raymond A. Serway, Bernard Lonergan, Philip McShane, Peter Burley,

.1. Introductory

Graduate students seeking a thesis topic may expand this treatment of the Einsteinian context of Functional Macroeconomic Dynamics. It should be of special interest to those having a strong background in theoretical physics and, thus, able to appreciate the analogies from physics. “Similars are similarly understood.” (CWL 3, 288/313)

Philip McShane alerted us to the resemblances between Lonergan’s context of general macroeconomics and Einstein’s context of general relativity.

(Part Two entitled Fragments) belongs almost entirely in what I call the Einsteinian context of Part Three, in contrast to the Newtonian achievement of Part One; … [CWL 21, Index, 325]

A new science has emerged. Lonergan has elevated conventional macrostatics to a macrodynamics explaining economic accelerations.

Einstein used the idea of curvature and the technique of tensor calculus to replace the idea of the force of gravity, and, to advance from special relativity to a unified general relativity, Lonergan revealed two circuits and their curvatures in the expansion of the economic process; and he used the fact of accelerational curvature of production and the existence of correlated monetary flows as the basis of complete explanation of the double-circuited, pure cycle of expansion. A new and higher explanatory systematics of macroeconomic dynamics was achieved, and has now emerged to replace academe’s single-circuited macrostatics. Government and private-sector macroeconomists take note.

A science emerges when thinking in a given field moves to the level of system. … The system really emerged with Newton. He laid down a set of basic, definitions, and axioms, and proceeded to demonstrate and conclude from general principles and laws that had been established empirically by his predecessors. Mechanics became a science in the full sense at that point where it became an organized system. … there is a point in the history of any science when it comes of age, when it has a determinate systematic structure to which corresponds a determinate field.

The main analytic apparatus is now complete. The two acceleration systems have been defined: a circulatory system consisting of two connected circuits that are accelerated by an external redistribution function; a quantity system of two parts in which one part is the long-term accelerator of the other. In each of these acceleration systems … an inner logic or ground in the nature of things indicates the normative or pure cycle of the quantity process. Finally, indices of price increments serve as markers of the divergence between the two systems. [CWL 21, 134]

This would be the beginnings of a new economics of measurable flows, one that would yield norms [McShane, 2017, Preface v]

Further Contents:

.2. Analytic distinctions and the principle of concomitance
.3. The general form of acceleration in Functional Macroeconomic Dynamics
.4. Two key relativistic equations
- .4a. Einstein’s field equations
- .4b. Lonergan’s field equations
.5. Einstein’s relativity principle and principle of equivalence
.6. Unifying dualities by tensor analysis of spatio-temporal curvature (Einstein) and vector analysis of pretio-quantital curvature (Lonergan)
.7. Curvature of space in physics and curvature in the structure of expansion in economics
.8. Concomitance of the curvatures in production and in distribution of incomes in the hierarchical brackets of income
.9. Miscellaneous Addenda

.2. Analytic distinctions and the principle of concomitance

While formulating special relativity, Einstein was puzzled that there were two equivalent ways of explaining both mass and acceleration – by the gravitational force or by inertial resistance. He pondered this “principle of equivalence” over several years – 1906-1913. Finally, by his general relativity of 1913, the equivalence of gravitation and inertia were conjoined in a single unified general theory.

Schumpeter puzzled over the relation of capital-spending to booms and slumps.

By the end of the period (1870-1914) most workers agreed – or tacitly took for granted – that the fundamental fact about cyclical fluctuations was the characteristic fluctuation in the production of plant and equipment. …(W)e seem to behold nothing but disagreement and antagonistic effort. … The contradiction is only apparent however. Agreement on the list of features, even if it had been complete, does not spell agreement as to their relations to one another, and it is the interpretation of these relations and not the list per se which individuates an analytic scheme or business cycle ‘theory.’ … it leaves the decisive question of interpretation wide open. (Schumpeter, Joseph, History of Economic Analysis (New York: Oxford University Press, 1954) 1125 (Lonergan’s reference) (CWL 15, 8)

By precise analytic distinctions, Lonergan first divided the always-current, purely-dynamic process of production-and-sale by analysis of correspondences. He divided the process into analytically-distinct correspondences of products under process with their level of end use. (see CWL 15, 23-28)

Current-determinate-point to current-determinate-point,
Current-determinate-point to future-indeterminate-line,
Current-determinate-point to future-indeterminate-surface,
Current-determinate-point to future-indeterminate-volume,
Current-determinate-point to higher correspondences

Plantations cotton

Cotton gins clean cotton

Spindles cotton thread

Looms cotton cloth

Sewing machines cotton dresses

Extraction machinery coal; iron ore

Blast furnaces pig iron

Steel mills steel

Machine tools gins, spindles, looms, sewing machines (CWL 15, 22)

For simplicity and clarity of exposition of principles, laws and their formulation, he distinguished, then formulated the correspondences. Point-to-point (“basic”) products exit the process to constitute a standard of living; point-to-line (“surplus”) producer goods remain in the process for acceleration of the point-to-point process Each product type would have its own distinct circuit of payments correlated with production and sale.. And, by the abstract correlations of measured data and the principle of concomitance, and by the technique of implicit definition, he related the quantities and timings of these distinct productive subprocesses and their normatively-correlated monetary circulations in a single, unified, explanatory theory. His Functional Macroeconomic Dynamics is a field theory of the functional relations of explanatory conjugates among themselves, i.e. the theory is an explanatory field theory.

… again, as to the notion of cause, Newton conceived of his forces as efficient causes, and the modern mechanics drops the notion of force; it gets along perfectly well without it. It thinks in terms of a field theory, the set of relationships between n objects. The field theory is a set of intelligible relations linking what is implicitly defined by the relations themselves; it is a set of relational forms. The form of any element is known through its relations to all other elements. What is a mass? A mass is anything that satisfies the fundamental equations that regard masses. Consequently, when you add a new fundamental equation about mass, as Einstein did when he equated mass with energy, you get a new idea of mass. Field theory is a matter of the immanent intelligibility of the object. (CWL 10, 154)

Einstein conjoined two equivalent explanations of his special relativity into the single explanation of general relativity by using tensors of rank two; Lonergan first divided the process analytically, then linked the subprocesses by tensors of rank one.

Both Einstein and Lonergan achieved unitary theories characterizable by their curvatures.

There exists, then, a point-to-point correspondence between bushels of wheat and loaves of bread, between head of cattle and pounds of meat, between bales of cotton and cotton dresses, between tons of steel and motorcars. In each case the elements in the standard of living are algebraic functions of the first degree with respect to elements in the productive process….(there is) an inexorable law of limitation. [CWL 15, 23]

… The analysis that insists on the indeterminacy is the analysis that insists on the present fact: estimates and expectations are proofs of the present indeterminacy and attempts to get round it; and, to come to the main point, an analysis based on such estimates and expectations can never arrive at a criticism of them; it would move in a vicious circle. It is to avoid that circle that we have divided the process in terms of indeterminate point-to-line and point-to-surface and higher correspondences. [CWL 15, 28]

Again, Philip McShane alerted us to the resemblances between Lonergan’s context of general macroeconomics and Einstein’s context of general relativity.

Einstein’s unified context is grounded in the theoretical unification of gravitation and inertia. Lonergan’s unified theoretical context is grounded in a) the principles of correlation and concomitance, b) Hilbert’s technique of implicit definition, and c) the axiomatic precise analytic distinction of current-determinate-point-to-current-determinate-point processes and current-determinate-point-to-future-indeterminate-line processes, whose products are loosely and “colloquially” – rather than “scientifically” – spoken of as consumer goods and producer goods.

Lonergan’s correlations and principle of concomitance, ground and conjoin all explanatory aspects of his macroeconomic dynamics.

Concomitance is, I would claim, the key word in Lonergan’s economic thinking. [Philip McShane, [Fusion 1, page 4 ftnt 10]

Lonergan’s principle of concomitance underlies a) a balance of pretio-quantital supply and demand within and between circuits of monetary flows, and b) the constraints and accommodations with one another of the flows in analytically-distinct circuits; i.e. Outlays-Incomes and Expenditures-Receipts within a circuit, and c) price and quantity in a pretio-quantitality. The normative constraints and accommodations are properly achieved by properly adjusted distributions of Income-for-Expenditure in accord with the requirements imposed by the particular phases of the pure cycle of expansion of the productive process.

.3. The general form of acceleration in Functional Macroeconomic Dynamics

The full field equations of Lonegan’s general theory are based upon his

lagged technical accelerator – connecting levels of the process.

k_n [f’_n(t-a)-B_n] = f”_n-1(t) – A_n-1(CWL 15, 37)

principle of concomitance underlying and linking the abstract definitive correlations as explanatory conjugates
curvatures in the pure cycle of double-circuited expansion of the productive process
vectorial equatings of “macroeconomic costs” and “macroeconomic revenues,” (P’Q’) = (p’)(a’Q’) + (p”)(a”Q”), and
the central condition of equilibrium between monetary circuits:

G = c”O” – i’O’ = 0 (CWL 15, 49-51).

Lonergan’s principle of concomitance is exemplified by the equating of macroeconomic revenues (monetary demand) and macroeconomic costs (monetary supply) – not the accountant’s sense of costs:

(P’Q’) = (p’)(a’Q’) + (p”)(a”Q”) and

(P”Q”) = i”(O”)_Expansionary+ i”(O”)_{Repair & Maintenance}. (CWL 15, 158)

The concomitance of outlay and expenditure follows from the interaction of supply and demand. The concomitance of income with outlay and expenditure is identical with the adjustment of the rate of saving to the requirements of the (particular phases of the pure cycle of the) productive process. [CWL 15, 144]

.4. Two key relativistic equations

.4a. Einstein’s field equations

The full field equations of Einstein’s general theory are

G_ab = kT_ab

(ten equations connecting 20 quantities) (see d”Inverno, 1992, 143).

In words, on the left side of the equation, the distribution of (gravitational) matter determines the geometric curvature of spacetime symbolized in the Einstein curvature tensor, G_ab; and on the right, the energy-momentun tensor, T_ab, expresses the motion of matter along the geodesics of the curvature of spacetime. k is Einstein’s coupling constant in relativistic units. (in non-relativistic units: k = 8πG/c⁴ ) (d’Inverno, 1992, 143)

John A. Wheeler (Princeton) commented colloquially about Einstein’s general explanatory scheme,

“Matter tells space how to curve, and space tells matter how to move.” [John A. Wheeler as quoted in Lieber, 2008, 350]

and we would say colloquially about Lonergan’s general explanatory scheme,

The curvatures in the potential of the distribution of invention, skills, and finite resources tells kinetic pretio-quantital production how to implement the basic and surplus expansion; and the normative implementation of the two normative curvatures by double-circuited production tells the potential how it should kinetically and coordinatedly move.

A collaboration with the mathematician Marcel Grossman led Einstein by 1913 to the view that the gravitational field must be identified with the 10 components of the metric tensor of Riemannian space-time geometry. … the Principle of Equivalence is incorporated into this formalism through the requirement that the physical equations be invariant under the general coordinate transformations, (Weinberg, 1971, 19) … the appropriate mathematical technique for implementing the Principle of Equivalence is tensor analysis, … . (Weinberg, 1971, 67)

It was Grossmann who emphasized the importance of a non-Euclidean geometry called Riemannian geometry (also elliptic geometry) to Einstein, which was a necessary step in the development of Einstein’s general theory of relativity. Abraham Pais‘s book^[7] on Einstein suggests that Grossmann mentored Einstein in tensor theory as well. Grossmann introduced Einstein to the absolute differential calculus, started by Elwin Bruno Christoffel ^[8] and fully developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita.^[9] Grossmann facilitated Einstein’s unique synthesis of mathematical and theoretical physics in what is still today considered the most elegant and powerful theory of gravity: the general theory of relativity. The collaboration of Einstein and Grossmann led to a ground-breaking paper: “Outline of a Generalized Theory of Relativity and of a Theory of Gravitation”, which was published in 1913 and was one of the two fundamental papers which established Einstein’s theory of gravity. (Wikipedia, under “Marcel Grossman”)

… the laws of special relativity apply in the frame of each observer. But relative to one frame the laws of special relativity do not describe the phenomena in another frame in a different position in a gravitational field – say at a different distance or direction from the center of the earth. The difficult part of the general theory of relativity is weaving together all the local descriptions into a global description that works everywhere. [Serway 1986, 904-5]

With the help of tensor calculus, Einstein was able to accomplish just this task (of “weaving together”) by including time as a dimension and including curvature. [Serway 1986, 905]

The Einstein curvature tensor, G_ab, equals k times the energy momentum tensor T_ab. (where k is the coupling constant in relativistic units).

the shortest distance between two points is called a geodesic On a sphere, a geodesic is an arc of a “great circle (an arc in a plane passing through the center of the sphere) such as the Earth’s equator and the Earth’s longitudinal lines. [Giancoli 2005, 928]

G_ab= kT_ab

(ten equations connecting twenty quantities) [d’inverno, 1992, 143]

In general relativity, the Einstein curvature tensor, G_ab, equals k times the energy momentum tensor T_ab. (where (G_ab = R_ab– 1/2g_abR, where g_ab is the metric geodesic, R_ab is the Ricci tensor, R is the Ricci scalar, [d’Inverno 192, 83-87] and k is the coupling constant in relativistic units.)

where

- the 4×4 tensor G_ab represents the curvature of spacetime, as determined by the distribution of matter,
- the 4×4 tensor T_abstands for the energy-momentum tensor, and
- the constant k is determined by the correspondence principle, since the equation must reduce to Poisson’s equation (4.5) in the appropriate limit … this is given in non-relativistic units by k = 8πG/c⁴(10.44) … in relativistic units, in which we take both c = 1 and G = 1, and then the coupling constant is k = 8π (10.45). [d’inverno, 1992, 143]

The distribution of matter gives space a Riemannian curvature; and curved spacetime “tells” matter how to move.

We keep in mind that, given the curvature resulting from the distribution of matter, g_ab is the metric geodesic or shortest path between A and B, So, under the heading of “Chapter 13, The Structure of the Field Equations”, d’Inverno reads Einstein’s implicit field equations from right to left, left to right, and back and forth. A “metric” is a covariant tensor of rank 2 used to define distances and lengths of vectors; and the infinitesimal distance (or interval in relativity) is defined by

ds² = g_ab(x) dx^a2dx^b2

See (d’Inverno, 1992, 81), and Lonergan’s (CWL 3, 147 and 140-72 titled “Space and Time”)

We quote d’Inverno reading right to left, left to right, and back and forth:

Before attempting to solve the field equations we shall consider some of their important physical and mathematical properties … . Again, the full field equations (in relativistic units) are

G_ab= kT_ab

1. The field equations are differential equations for determining the metric tensor g_abfrom a given energy-momentum tensor T_ab. Here we are reading the equations from right to left. … one specifies a matter distribution and then solves the equations to ascertain the resulting geometry.
2. The field equations are equations from which the energy-momentum tensor can be read off corresponding to a given metric tensor g_ab. Here we are reading the equations from left to right.
3. The field equations consist of ten equations connecting twenty quantities, namely, the ten components of g_aband the ten components of T_ab. Hence, from this point of view, the field equations are to be viewed as constraints on the simultaneous choice of g_aband T_ab. This approach is used when one can partly specify the geometry and the energy-momentum tensor from physical considerations and then the equations are used to try and determine both quantities completely. [d’Inverno, 1992, 169]

.4b. Lonergan’s field equations

Again, we would comment colloquially about Lonergan’s general explanatory scheme,

The full field equations of Lonegan’s general theory are implicit in Lonergan’s

.a. lagged technical accelerator

k_n [f’_n(t-a)-B_n] = f”_n-1(t) – A_n-1(see CWL 15, 37)

connecting two production processes, each having its own velocitous and accelerative monetary circulation of Outlays-Incomes-Expenditures-Receipts.

.b. field equations of CWL 15, 50-54; represented in the Diagram of Rates of Flow

.c. forms of the cycle of expansion graphed in the figures listed below: (graphs are appended at the end of this topic)

Figure 24-1 Rate of Change of dQ”/Q” for dQ” = constant and dQ”/Q” = constant
Figure 24-2 Growth of Rate of Surplus Production (Q”) for dQ” = constant and dQ”/Q” = constant
Figure 24-3 Rate of change of dQ”/Q” over a pure cycle
Figure 24-4 Growth of Rate of Surplus Production (Q”) over a Pure Cycle
Figure 24-5 Rate of Change of dQ’/Q’ over a Pure Cycle
Figure 24-6 Growth of Rate of Basic Production (Q’) over a pure cycle
Figure 24-7 Rate of Change of dQ’/Q’ and dQ”/Q” over a Pure Cycle
Figure 27-1 Rate of Change of v, w, and f during a Pure Cycle, Ideal Maximum f

.d. vectorial (tensor of rank one) equatings of “macroeconomic costs” and “macroeconomic revenues,”

(P’Q’) = (p’)(a’Q’) + (p”)(a”Q”), and

(P”Q”) = i”(O”)_Expansionary+ i”(O”)_{Repair & Maintenance}. (CWL 15, 158)

and,

.e. central condition of dynamic equilibrium between monetary circuits as adjusted by crossover flows:

G = c”O” – i’O’ = 0 (CWL 15, 49-51).

The key principle of concomitance, suffused in the names of correlation and interdependence throughout Lonergan’s theory, is exemplified by the equating of the vectorial dot-products of macroeconomic revenues and macroeconomic costs:

(P’Q’) = (p’)(a’Q’) + (p”)(a”Q”) and

(P”Q”) = i”(O”)_Expansionary+ i”(O”)_{Repair & Maintenance}. (see CWL 15, 158)

In discussions of interpretation, the meaning and significance of the mathematical equals sign may be expressed in several linguistic symbolizations and phrasings:

=
is in functional correspondence with
is identified as
is mutually, implicitly defined as
tells and is told by
determines and is determined by
constrains and is constrained by
accommodates and is accommodated by
conditions and is conditioned by
is isomorphic with
is equivalent to
is one with
is theoretically concomitant with

Colloquially and analogously with respect to the intelligibility of the two curvatures in the particular phase of the pure cycle of expansion (static phase, or proportionate phase, or surplus phase, or basic phase), the possible normative curvatures are determined by the potentials of invention, skills, and finite resources. The differently-timed curvatures of the velocities of the point-to-point process succeeded by point-to-line (“basic” and “surplus”) production, (p’)(a’Q’ + (p”)(a”Q”), tell velocitous Expenditures on the left, (P’Q’) and (P”Q”), how to curve, and those Expenditures, by correlative concomitance with macroeconomic costs, tell those costs on the right (p’)(a’Q’ + (p”)(a”Q”), how to move in a curved coordination with Revenues. Similarly for equations of surplus production.

Conversely, the curvatures of point-to-point and point-to-line production costs on the right, by the potentials of invention, skills , and finite resources, tell point-to-point and point-to-line revenues, by correlative concomitance, how to curve.

Colloquially and briefly: Curvatures “instruct”, according to the principle of concomitance and the technique of implicit definition, both Expenditures and Outlays; and Revenues and Outlays, according to the principle of concomitance and the technique of implicit definition, instruct, constrain, and accommodate one another.

Thus is the objective economic process explained on the basis of current curvature of invention, the principle of concomitance, the technique of implicit definition, two circuits, and the condition of equilibrium. And, thus, one can understand how a cycle of expansion should be normatively equilibrated and implemented, rather than ignorantly converted into a boom followed by a slump.

Now each phase in an exchange economy will have its exchange equilibrium, but the equilibria of the different phases differ radically from one another. … By this cyclic variation within the exchange equilibria there is effected the ‘curvature of the exchange equations.’ (CWL 21, 51-52)

Insights supply the concepts of explanatory relations.

The difficult part of the general theory of relativity and the general theory of macroeconomics is weaving together (by insight) all the (previously) isolated descriptions and insights of other struggling analysts into a single, unitary, completely-explanatory set of coherent insights that explains and is relevant in any instance. [paraphrasing Serway 1986, 904-5]

As the insights yielding the concepts and their relations comprising the intelligibility of the planar circle, so insight yields the unitary intelligibility of the velocitous and accelerative macroeconomic system. All the concepts tumble out together.

Let us say, then, that for every basic insight there is a circle of terms and relations, such that the terms fix the relations, the relations fix the terms, and the insight fixes both. If one grasps the necessary and sufficient conditions for the perfect roundness of this imagined plane curve, then one grasps not only the circle but also the point, the line, the circumference, the radii, the plane, and equality. All the concepts tumble out together, because all are needed to express adequately a single insight. All are coherent, for coherence basically means that all hang together from a single insight. [CWL 3, 12/36]

The equivalence of two ways of explaining mechanical motion (in terms of gravity or of inertia), which Einstein puzzled over and sought to unify and to mathematically formalize by his general relativistic G_ab = kT_ab, is, in Lonergan’s “coherent set of macroeconomic motions” implicitly and unitarily existent in the very beginning. In Lonergan’s correlations among measured data, concomitance of these abstract correlations, and implicit definition of these correlations as explanatory conjugates constitute Lonergan’s Explanatory Functional Macroeconomic Dynamics. Instead of fruitlessly attempting to analyze and explain the current process a) in the form of momentary macrostatic intersections, (IS-LM, and AS-AD) or b) in the form of a single circuit of households and firms, or c) in the form of non-explanatory, commonsense-accounting, terms of corporate and National-Income tallying– as investment analysts, academic economists, and econometricians are wont to do – Macroeconomic Field Theory expresses the immanent intelligibility of the wholly-connected dynamic economic process as a single unified field-theoretic system of interdependent, interconnected, mutually defining velocities: (P’Q’) = (p’)(a’Q’) + (p”)(a”Q”) and (P”Q”) = i”(”O”)_Expansionary+ i”(O”)_R&M. (CWL 15, 158) (See Explanation by Gross Domestic Functional Flows). To understand any single flow is to understand its functional connections in the unity of all flows. The economic process is a single organism comprised of several conceptually-distinct but interdependent flows.

Also see “An Einstein Relativistic Context … “

Lonergan, of course, uses Galilean time and the curvature of the production of quantities per Galilean interval; e.g. the curvature of geometric progressions or of logistical growth. For instruction as to abstract principles and laws of the concrete process, he provides one among many possible models. The editors of CWL 15 are careful to point out in a footnote that

Lonergan has introduced certain assumptions about rates of growth and acceleration. These assumptions constitute something of a ‘model’ of a pure cycle. That model is a four-phase cycle, where the successive phases are characterized by the parameters

(1) dQ’/Q’ = dQ”/Q” = 0; (@) dQ’/Q’ = 0, dQ”/Q” = k-1; (3) dQ’Q’ = k-1, dQ” = (1/r–1)Q₁”; (4) dQ’ = (1/r–1)Q₁’, dQ” = 0

followed by a repetition of phase 1. (See Figures 24-3 through 24-7, and Figure 27-1.) Such a model represents only one possible instance of a pure cycle; there are a manifold of other possible pure cycles, and a still larger set of possible aberrant ‘trade’ cycles. Lonergan’s use of this model at this point in the text merely illustrates his more general principles, and is not central to his argument. In later sections of this essay, where other graphs have been supplied to illustrate Lonergan’s points (for example, in §27, ‘The Cycle of Pure Surplus Income’), an effort has been made to stay as close as possible to the assumptions Lonergan introduced here. (CWL 15, 126, ftnt 165)

The general theories of both relativity and macroeconomics have “explanatory valence,” i.e. power of explanation. Lonergan has “weaved together” previously-isolated, non-explanatory insights into a complete, unified explanation, employing a) abstract correlations among measured data, b) implicit definition immanent in the correlations constituting functional relations, c) the compelling principle and fact of concomitance implicit in functional correlations, and d) precise analytic foundational terms, i.e. distinctions between point-to-point productive functions and point-to-line productive functions, each of the two having its own monetary circuit connected to and equilibrated with the other by balanced crossovers, i’O’ and c”O”. The condition of equilibrium is

G = c”O” – i’O’ =0 (CWL 15, 49-54)

Einstein’s general theory of relativity, by tensor calculus, united the Riemannian geometry of space and the physics of gravitational acceleration into a single, unitary theory – 10 equations connecting 20 quantities. (4×4 matrices, having 16 components, of which there are 6 identical pairs)

The general theory of macroeconomic relativity, by a) precise analytic differentials, b) abstract correlation, c) the principle of concomitance, d) Hilbert’s technique of implicit definition, and e) the vector calculus, united the structure of two productive processes (basic and surplus) and the structure of the two monetary circulations into the general systematics of a single unitary system. Curving Expenditures and Costs mutually define, constrain, and accommodate one another across the equals sign.

P’Q’ = (p’a’Q’ + p”a”Q”) (CWL 15, 158),” and similarly,

P”Q” = (i”O”_Expansionary) + (i”O”_{Repair & Maintenance})

implying that

a) the aggregate basic price-spread ratio

P’/p’ = a’ + a”p”Q”/p’Q’ (see CWL 15, 156-59)

will expand and contract with the curving accelerations and decelerations of production; i.e. in the monetary circuits, the ratio of surplus-to-total Outlays and basic-to-total Outlays will shift accordingly in the respective phases of expansion

O”/(O’ + O”), O’/(O’ + O”), and thus I”/(I’ + I”), I”/(I’ + I”) , and

b) the pure surplus-income ratio

f = v w, i.e. f = vp”Q”/p’Q’ (CWL 15, 148)

will rise and fall as the surplus expansion accelerates and decelerates, giving way to the basic expansion, during the pure cycle of expansion.

Einstein used the principle of equivalence of two previously separate explanations as the basis of the single explanation of general relativity. Lonergan used the principle of concomitance and precise analytic distinctions between two previously non-distinguished circulations as the basis of the single explanation of the expansionary economic process. Key elements in Functional Macroeconomic Dynamics:

abstract correlation yielding explanatory conjugates
principle of concomitance of explanatory conjugates
deliberate use of the technique of implicit definition,
unitary basis of precise analytic terms (point-to-point, point-to-line, point-to volume, and higher) and their relations comprising the structure of the production process
discovery of two (or more) circuits
rigorous deduction from that primal unitary basis of an explanatory superstructure and of economic theorems constituting the unitary immanent intelligibility of the whole macroeconomic system
projection of classes of production onto classes of payments, correlated with one another as to magnitudes and frequencies.

Lonergan provided a closely knit frame of reference explaining the systematics of Outlays-Incomes-Expenditures-Receipts of two connected circuits of monetary circulation. His explanatory frame of reference is a double-circuited, credit-centered system of velocitous and accelerated flows.

Insight yields “form” and, thus, the “formal causality” of the dynamic economic process.

Ought there not to be introduced a technical term to denote this type of intelligibility? … The intelligibility that is neither final nor material nor instrumental nor efficient causality is, of course, formal causality…What we have called the intelligibility immanent in sensible data and residing in the relations of things to one another might be named more briefly formal causality … [CWL 3, 78/101-102]

The principle of unity is intelligibility. The unity in Lonergan’s macroeconomics is recognized in its theory of relations by a sweeping insight into the unified intelligibility of the whole process and in practice by the satisfaction of the bullet-list above and the central condition of equilibrium between monetary circuits:

G = c”O” – i’O’ = 0 (CWL 15, 49-51)

Just as Einstein, by his theory of General Relativity equated, united, and formulated two previously undistinguishable frames of reference in a unity of a curved geometry on the left and the associated energy-momentum on the right, so Lonergan, by his Unitary Functional Macroeconomic Dynamics, equated, united, and formulated the interconnections of two circuits in order to explain the wholly-interconnected, threefold economic process of a) operative basic production and exchange, b) operative surplus production and exchange, and c) finance of all production and exchange through the release of money through the channels from the redistributive function.

Note in the following excerpt the phrases “Lonergan is alone” and “prior to and more fundamental”:

Lonergan is alone in using this difference in economic activities to specify the significant variables in his dynamic analysis… no one else considers the functional distinctions between different kinds of productive rhythms prior to, and more fundamental than, wealth, value, supply and demand, price levels and patterns, capital and labor, interest and profits, wages, and so forth….only Lonergan analyzes booms and slumps in terms of how their (explanatory) velocities, accelerations, and decelerations are or are not equilibrated in relation to the events, movements, and changes in two distinct monetary circuits of production and exchange as considered both in themselves (with circulatory, sequential dependence) and in relation to each other by means of crossover payments (crossover interdependence). [CWL 15, Editors’ Introduction, lxii]

Lonergan pointed out that this (functional) differentiation of economic activities … is discussed by traditional economists such as S. M. Longfield (1802-1884), John Rae (1796-1872), Nassau Senior (1790-1864), Eugen von Bohm-Bawerk (1851-1914), and in the heavily disputed “Ricardo effect.” But Lonergan credits Piero Sraffa (1898-1983) as having clarified it most thoroughly in his famous essay, “Production of Commodities by Means of Commodities” (1960). Yet even Sraffa does not use his sophisticated explanation of the “Ricardo effect” and the “roundabout” or “concertina”-like phenomena associated with it in the way Lonergan does. [CWL 15, Editors’ Introduction, lxii]

The double-circuited economic process is understood as a threefold process.

(The Diagram’s) basic terms are (implicitly) defined by their functional relations. The maintaining of a standard of living (distinct process 1) is attributed to a basic process, an ongoing sequence of instances of so much every so often. The maintenance and acceleration (distinct process 2) of this basic process is brought about by a sequence of surplus stages, in which each lower stage is maintained and accelerated by the next higher. Finally, transactions that do no more than transfer titles (distinct process 3) to ownership are concentrated in a redistributive function, whence may be derived changes in the stock of money dictated by the acceleration (positive or negative) in the basic and surplus stages of the process. … So there is to be discerned a threefold process in which a basic stage is maintained and accelerated by a series of surplus stages, while the needed additions to or subtractions from the stock of money in these processes is derived from the redistributive area. … it will be possible to distinguish stable and unstable combinations and sequences of rates in the three main areas and so gain some insight into the long-standing recurrence of crises in the modern expanding economy. [CWL 15, 53-4 and 177]

If one observes Descartes’ admonitions,

In the midst of the vast and profound stirring of human minds, which we name the Renaissance, Descartes was convinced that too many people felt it beneath them to direct their efforts to apparently trifling problems. Again and again, in his Regulae ad directionem ingenii, he reverts to this theme. Intellectual mastery of mathematics, of the departments of science, of philosophy, is the fruit of a slow and steady accumulation of little insights. Great problems are solved by being broken down into little problems. The strokes of genius are but the outcome of a continuous habit of inquiry that grasps clearly and distinctly all that is involved in the simple things that anyone can understand. ¶ I thought it well to begin by recalling this conviction of a famous mathematician and philosopher, for our first task will be to attain familiarity with what is meant by insight, and the only way to achieve this end is, it seems, to attend very closely to a series of instances all of which are rather remarkable for their banality. (CWL 3, 3)

both Einstein’s full field equations and Lonergan’s equations will be apparently-simple, but they are implicitly and implicatively dense. In partial analogy and resemblance to d”Inverno’s reading general relativity’s equations right to left, left to right, and back and forth across the equals sign, in normative macroeconomics, macroeconomic Expenditures on the left are concomitant with, implicitly define, constrain, and accommodate macroeconomic Costs on the right, and vice versa. These macroeconomic costs are not at all an accountant’s sense of costs. Prescinding from any improper and ill-timed money supply effected by the executive and legislative branches, in Lonergan’s implicit equation, P’Q’ = p’a’Q’ + p”a”Q” [CWL 15,156-62], the terms are implicitly defined by the functional relations in which they stand with one another.

Reading from right to left: “Macroeconomic costs” limit or constrain macroeconomic Expenditures-Revenues. Reading from left to right: Macroeconomic Expenditures are correlative to both adequate productivity and an adequate expending of a level of aggregate basic incomes. The left and the right simultaneously tell each other how they must act. They imply one another and yoke one another together in an implicit equation. They implicitly define one another.

P’Q’ = (p’a’Q’ + p”a”Q”) (CWL 15, 158)

where,

p’a’Q’ = c’O’ and p”a”Q” = c”O” in the Diagram of Rates of Flow.

There is a sense in which one may speak of the fraction of basic outlay that moves to basic income as the “costs” of basic production. It is true that that sense is not at all an accountant’s sense of costs; … But however remote from the accountant’s meaning of the term “costs,” it remains that there is an aggregate and functional sense in which the fraction… is an index of costs. For the greater the fraction that basic income is of total income, I’/(I’ + I”), the less the remainder which constitutes the aggregate possibility of profit. But what limits profit may be termed costs. Hence we propose ….to speak of c’O’ and c”O” as costs of production, having warned the reader that the costs in question are aggregate and functional costs…. (CWL 15 156-57)

In contrast to the intelligibility of physical mechanics, the intelligibility of macroeconomics involves several twos:

two (or more) monetary circuits: point-to-point (“basic”) and point-to-line (“surplus”)
two types of transactions within a circuit: making and selling
two orders: productive and monetary
two uses of surplus goods: expansion and repair and maintenance
two destinations of outlays to a) incomes within their own circuit, and b) through the crossovers to incomes in the other circuit.

.5. Einstein’s relativity principle and the principle of equivalence:

Einstein faced the theoretical problems of a) two equivalent explanations of mechanical phenomena of mass and acceleration, and b) the constancy of the speed of light in an inertial frame of reference, resulting in 4-dimensional curved spacetime. The puzzle of the principle of equivalence in special relativity became Einstein’s motive for, and the basis of, general relativity. Two theories of explanation had to be combined into a single theory.

Conversely, we might think we are being pulled by gravity when in fact we are undergoing an “inertial” acceleration having nothing to do with gravity. [Giancoli 2005, 926]

Einstein tackled the problem of accelerating reference frames in his general theory of relativity and in it also developed a theory of gravity. [Giancoli 2005, 926]

The principle of equivalence is related to the concept that there are two types of mass. Newton’s second law, F = ma, uses inertial mass. We might say that inertial mass represents “resistance” to any type of force. The second type of mass is gravitational mass. When one body attracts another by the gravitational force (Newton’s law of universal gravitation, F = Gm₁m₂/r², Chapter 5) the strength and force … [Giancoli 2005, 926]

… no experiment – not even of high precision –has been able to discern any measurable difference between inertial mass and gravitational mass. This is another way to state the equivalence principle: gravitational mass is equivalent to inertial mass. [Giancoli 2005, 927]

To accomplish (the democratic result of the General Theory) Einstein introduced the PRINCIPLE OF EQUIVALENCE, by which the idea of a FORCE OF GRAVITY was replaced by the idea of the CURVATURE OF SPACE. [Lieber, 2008, 98-99]

Einstein’s two postulates in special relativity are;

First postulate (the relativity principle): The laws of physics have the same form in all inertial reference frames. – footnote The first postulate can also be stated as: There is no experiment you can do in an inertial reference frame to tell if you are at rest or moving uniformly at constant velocity. (Giancoli 2005, 731)

Second postulate (constancy of the speed of light): Light propagates through empty space with a definite speed c independent of the speed or the source or observer. (Giancoli 2005, 731)

That first postulate is to say that

An upwardly accelerating reference frame is equivalent to a downward gravitational field. [Giancoli, 2005, 926-27]

One can find other wordings and qualifications of the relativity principle and the principle of equivalence, the latter becoming Einstein’s motive for, and the basis of, general relativity:

Einstein’s general relativity is grounded in the principle of equivalence, i.e. “the equivalence for explanation of Gravitation and Inertia. The idea of a FORCE OF GRAVITY was replaced by the idea of the CURVATURE OF SPACE.” [Lieber, 2008, 98-99]

In the special theory of relativity, Einstein concluded that there is no way for an observer to determine whether a given frame of reference is at rest or is moving at constant velocity in a straight line. Thus the laws of physics must be the same in different inertial reference frames [Giancoli, 2005, 926-27]

The equivalence principle also means that the two different ways in which we define mass are equivalent. One way to define the mass of a body is by its inertia properties – its resistance to acceleration (through the equation F = ma). The other way is by the body’s gravitational attraction for other bodies (through Newton’s formula for gravitation). [Serway 1986, 904]

In the vicinity of a gravitating body, however, the coordinates of spacetime are distorted. Within that curved spacetime, light and all other objects move without acceleration in straight lines – more properly called geodesics – but Euclid’s geometry does not apply to measurements in curved spacetime. [Serway 1986, 905]

… we formulate the equivalence principle as the statement that at every space-time point in an arbitrary gravitational field it is possible to choose a “locally inertial coordinate system” such that, within a sufficiently small region of the point in question, the laws of nature take the same form as in unaccelerated Cartesian coordinate systems in the absence of gravitation. (Weinberg, 1971, 68]

In my view, it is much more useful to regard general relativity above all as a theory of gravitation, properties summarized by Einstein’s Principle of the equivalence of Gravitation and Inertia. For this reason, I have tried throughout this book to delay the introduction of geometrical objects, such as the metric, the affine connection, and the curvature, until the use of these objects could be motivated by considerations of physics. (Weinberg, 1971, 3)

This SINGLE new idea (that time was not absolute but rather relative to an observer) was SUFFICIENT to accomplish the task undertaken in the Special Theory. We shall now see that again by the addition of ONLY ONE idea, called “THE PRINCIPLE OF EQUIVALENCE,” Einstein made possible the GENERAL Theory. [Lieber, 2008, 97]

To accomplish (the democratic result of the General Theory) Einstein introduced the PRINCIPLE OF EQUIVALENCE, by which the idea of a FORCE OF GRAVITY was replaced by the idea of the CURVATURE OF SPACE. The study of this curvature required the machinery of the TENSOR CALCULUS by means of which the CURVATURE TENSOR was derived. This led immediately to the NEW LAW OF GRAVITATION which was tested by THREE CRUCIAL PHENOMENA and found to work beautifully. [Lieber, 2008, 300]

It remained to construct a relativistic theory of gravitation. A crucial step toward this goal was taken in 1907, when Einstein introduced the Principle of Equivalence of Gravitation and Inertia, and used it to calculate the red shift of light in a gravitational field. … A collaboration with the mathematician Grossman led Einstein by 1913 to , view that the gravitational field must be identified with the 10 components of the metric tensor of Riemannian space-time geometry. As discussed in Chapters 4 and 5, the Principle of Equivalence is incorporated into this formalism through the requirement that the physical equations be invariant under the general coordinate transformations, not just Lorentz transformations, though I do not know to what extent this “General Principle of Relativity” took on in Einstein’s mind a life of its own, apart from the Principle of Equivalence. (Weinberg, 1971, 19)

… the appropriate mathematical technique for implementing the Principle of Equivalence is tensor analysis, and only after we complete the introduction to tensor analysis in the next chapter will we be able to make use of the full content of this principle. (Weinberg, 1971, 67)

.6. Unifying dualities by tensor analysis of spatio-temporal curvature (Einstein) and vector analysis of pretio-quantital curvature (Lonergan)

… the laws of special relativity apply in the frame of each observer. But relative to one frame the laws of special relativity do not describe the phenomena in another frame in a different position in a gravitational field – say at a different distance or direction from the center of the earth. [Serway 1986, 904-5]

Einstein tackled the problem of accelerating reference frames in his general theory of relativity and in it also developed a theory of gravity. [Giancoli 2005, 926]

Again, Einstein used tensor analysis expressed in his succinct summation notation,

… the appropriate mathematical technique for implementing the Principle of Equivalence is tensor analysis, and only after we complete the introduction to tensor analysis in the next chapter will we be able to make use of the full content of this principle. (Weinberg, 1971, 67)

Recall Wheeler’s colloquial statement:, “Matter tells space how to curve, and space tells matter how to move.” [John A. Wheeler as quoted in Lieber, 2008, 350]

Einstein’s general relativity is not merely a reformulation of Newton’s gravitational theory. It is a new theory.

Thus, in Einstein’s theory we do not speak of the “force” of gravity acting on bodies. Instead we say that bodies and light rays move as they do because space-time is curved. A body at rest or moving slowly near the great mass of Fig. 33-18 would follow a geodesic (the equivalent of a straight line in plane geometry) toward that body. [Giancoli 2005, 929]

In general relativity, then, the concept of a (Newtonian) gravitational field is replaced by a curved spacetime within which all things move without acceleration, locally always obeying the laws of special relativity. It is really a beautiful concept. It is not merely a reformulation of Newton’s gravitational theory. … Any change in the gravitational field, say, by a redistribution of matter or the collapse of a great mass, must result in a reshaping of spacetime. It does not reshape instantaneously, however, rather a disturbance in spacetime moves outward with the speed of light as a gravitational wave. [Serway 1986, 905]

This SINGLE new idea that time was not absolute but rather relative to an observer was SUFFICIENT to accomplish the task undertaken in the Special Theory. We shall now see that again by the addition of ONLY ONE idea, called “THE PRINCIPLE OF EQUIVALENCE,” Einstein made possible the GENERAL Theory. [Lieber, 2008, 97]

To accomplish (the democratic result of the General Theory) Einstein introduced the PRINCIPLE OF EQUIVALENCE, by which the idea of a FORCE OF GRAVITY was replaced by the idea of the CURVATURE OF SPACE. The study of this curvature required the machinery of the TENSOR CALCULUS by means of which the CURVATURE TENSOR was derived. This led immediately to the NEW LAW OF GRAVITATION which was tested by THREE CRUCIAL PHENOMENA and found to work beautifully. [Lieber, 2008, 98-99]

Below, we insert a single paraphrase of (Serway 1986, 905) relevant to macroeconomics, wherein basic pretio-quantital macroeconomic Expenditures, P’Q’, are conjoined to pretio-quantital macroeconomic Costs, p’a’Q’ + p”a”Q”. Note that human invention and its change of economic potential will – like a redistribution of matter or the collapse of a great mass reshaping spacetime – result in the reshaping of the potential curvature of the pretio-quantitality of both basic and surplus production and sale. And ever more invention will serve to increase potential energy, and thus, total energy. Macroeconomics is not subject to a law of conservation of energy.

Invention of new and better methods, constrained by the scarcity of resources, will result in the reshaping of the potential curvature of the pretio-quantital process for kinetic implementation in both basic and surplus circuits. It does not reshape the actual implementation instantaneously, however, rather a disturbance in the potential of pretio-quantitality by invention is actually propagated through the kinetic implementation in both the surplus and basic circuits and concomitantly – by the identity of Outlays and Incomes and Incomes and Expenditures – through the income brackets and their propensities to save and borrow for investment and to spend for a standard of living. [paraphrase of Serway 1986, 905]

Again, we would say colloquially about Lonergan’s general explanatory scheme,

Decisions regarding spending, saving, and borrowing are made by efficient-causal, more-or-less-educated, human recipients of income. Ignorance of normative macroeconomic theory accompanied by egoistic bias, group-tribal bias, and general bias, which prefers unreliable common sense over reliable theory, play a role. (Consult CWL 3, 218-42/244-67) (also see Commonsense Economics vs. Scientific Economics.

A condition of circuit acceleration was seen in Section 15 to include the concomitant keeping in step of basic outlay, basic income, and basic expenditure, and on the other hand, the concomitant keeping in step of surplus outlay, surplus income, and surplus expenditure. Any of these rates may begin to vary independently (rather than in normative concomitance with) the others, and adjustment of the others may lag. But any systematic divergence brings automatic correctives to work. The concomitance of outlay and expenditure follows from the interaction of supply and demand. The concomitance of income with outlay and expenditure is identical with the adjustment (through monetary crossovers) of the rate of saving to the requirements of the productive process. [CWL 15, 144]

Thus, Outlays and Expenditures “flank” Incomes in the Diagram of Rates of Flow. And the flows of Incomes in both the basic and surplus circuits are properly coordinated by proper flows through the crossovers and proper credit as the double-circuited process expands. All con-junctions of interdependencies are connected proximately or remotely with all other con-junctions of interdependencies. There are no gaps or discontinuities in the Diagram of Rates of Flow or in the velocitous and accelerative system it represents..

In the field theory of macroeconomic dynamics, then, the concept of the Walrasian momentary, non-dynamic, intersection devoid of subscripts representing time, is replaced by an implicitly-formulated, dynamic pretio-quantitality within which all correlatives should move concomitantly in their normative proximate and remote interdependence, always obeying the coherent set of laws of the normative macroeconomic field theory. It is really a beautiful concept. It is not merely a reformulation of Walras’s macrostatics. It is a new idea and a new dynamical theory … Any independent non-normative divergence of flows in the macroeconomic field from the normative flows of the normative pure cycle of expansion, by, say, a) a careless and ineffective manipulation of interest rates), or b) a careless flooding or draining of the money supply by the Fed or Treasury, c) imbalance of crossovers, or a drain of one circuit by the other, d) government deficit or surplus, or e) non-concomitant variation of normative implicitly-defined prices and quantities, must result in the reshaping of the newly-distorted curves of the pretio-quantitality of P’Q’ = p’a’Q’ + p”a”Q” through the hierarchical brackets of incomes for purchase. It does not reshape instantaneously; rather there results a torturing or a stress and strain in the intelligible normative pretio-quantitality of the pure cycle of expansion through a) the linear supply chain, b) the circulations in the two circuits, and c) the hierarchical brackets of income, to an eventual new ratio of I’/(I’ + I”) proper to each phase of the newly distorted cycle of expansion. And, in the brackets of incomes considered as purchasing power, there are winners and losers. The relatively more vulnerable get swindled; the relatively less vulnerable can win big [paraphrase of Serway 1986, 905]

Inflation tortures the flows and unjustly creates winners at the expense of losers. Ideally, money would be constant in exchange value,

… the dummy (money) must be constant in exchange value, so that equal quantities continue to exchange, in the general case, for equal quantities of goods and services. The alternative to constant value in the dummy is the alternative of inflation and deflation. Of these famous twins, inflation swindles those with cash to enrich those with property or debts, while deflation swindles those with property or debts to enrich those with cash; in addition to the swindle each of these twins has his own way of torturing the dynamic flows; deflation gives producers a steady stream of losses; inflation yields a steady stream of gains to give production a drug-like stimulus. [CWL 21, 37-38]

The scientific analysis of the dynamics of the objective economic process should be functional and should reveal the field-theoretic functional relations represented in the Diagram of Rates of Flow.

The whole structure of Functional Macroeconomic Dynamics is purely relational (relativistic). A macroeconomic functioning is not a mere compilation, tallying, or aggregation of particular income-statement categories, such as wages or interest expense. A macroeconomic functioning is implicitly defined by its functional relation to other functionings. The whole structure is unitary and purely relational.

“Lonergan’s analysis is concrete but heuristic. It focuses on functional relations intrinsic to the productive process to reach eventually a general theory of dynamic equilibria and disequilibria.” [McShane 1980, 117]

I have insisted on focusing on the central issue: the need of a functional analysis of the productive process and its correlated monetary flow. [McShane 1980, 200]

The division is not a matter of social relations or of property or of the properties of things: it is a functional analysis. … The aim of the analysis is to reveal the possibilities of the productive process as a dynamic system. One moves forward to that revelation in so far as one appreciates the different ways in which basic and surplus stages may relate. [McShane 1980, 119-20]

The analysis is functional and leads us to define five monetary functions which reveal a set of circulations of money. [McShane 1980, 121]

Now whatever the difficulties of measurement, the functional distinction is undeniably valid. [McShane 1980, 121]

… the diagram is an aid to separating and understanding functions. The circles are not places, nor are they, say, groups of capitalists, workers, bankers, exporters. … The diagram represents the functional journeys. [McShane 2017, 79]

you begin to glimpse the necessity and the plausibility of the functional analysis for the understanding and guiding of the globe’s economy. [McShane, 2017, 81]

We stick with our simple illustrations … to get you used to thinking in terms of these functional distinctions. [McShane, 2017, 85]

The algebra and vector calculus of Macroeconomic Field Theory can be symbolized traditionally or, as in Burley’s linear von Neumann models, using matrix and tensor of rank-1 form. So, in Macroeconomic Field Theory, 1) the abstract correlatives in measured concrete flows implicitly determine and define the abstract explanatory conjugates, 2) P’Q’ implicitly defines, is correlative to, and satisfies the principle of concomitance in its equality with (p’a’Q’ + p”a”Q”), and 3) in vector, dot-product form.

P’Q’ = (p’)(a’Q’) + (p”)(a”Q”) (CWL 15, 158)

and assuming vectors, thus,

P’Q’ = p’a’Q’ + p”a”Q”, and

P”Q” = i”O”_Expansionary+ i”O”_R&M

and the condition of equilibrium equals zero.

G = c”O” – i’O’ = 0 (CWL 15, 49-50.

… the immediate task is to state as accurately as possible the general character of the unknowns to which, we hope, investigators will endeavor to approximate. [In other words, can one define two numbers, say P and Q, such that P varies with a set of numbers p₁, p₂, p₃, … and Q varies with another set of numberts q₁, q₂, q₃, …? ¶ A universally valid answer to this question may be had when P and Q are not mere numbers but vectors in an n-dimensional configuration space, where n is the number of objects offered in exchange, so that for every difference in objects there is a distinct dimension. ¶ Let P and Q be the vectors …(CWL 15, 72-73)

In CWL 15, on pages 70-75:

P²= Σp_i²

Q²= Σq_i²

P’_j²= Σp’_ij²

Q’_j²= Σq’_ij²

Σp’_ijq_ij = P’_j Q’_j = P’_jQ’_j cos A’

Then, in CWL 15, pages 107-709:

Z = Σ p_iq_i[for expenditures][CWL 15, 07-109]

Z + DZ = Σ(p_iq_i + p_idq_i + q_idp_i +dp_idq_i) [for expenditures] [CWL 15, 107-109]

Therefore, Z = Σ(p_i)(aq_i)[for costs]

Z + DZ = Σ(p_iaq_i + p_idaq_i +aq_idp_i +dp_idaq_i) [for costs]

Z = Σ p_iq_i= P.Q = PQ cosA [CWL 15, 107-109]

(P + dP)² = Σ(p_i + dp_i)²

(Q + dQ)² = Σ(q_i + dq_i)²

Z + DZ = (P+dP)(Q + dQ) cos(A + dA) [CWL 15, 107-109]

Therefore, Σ(p_iq_i + p_idq_i + q_idp_i +dp_idq_i) = (P+dP)(Q + dQ) cos(A + dA) [CWL 15, 107-109]

DZ = PQ[(dP/P + dQ/Q + dPdQ/PQ) cos (A + dA) – 2 sin(dA/2) sin(A + dA/2)] (21) [CWL 15, 107-109]

(Also, see ) [ VNR Encyclopedia, 1977] p. 234

Within each of the three price-times-quantity products of P’Q’, and p’a’Q, + p”a”Q” in the equation which conjoins monetary demand and monetary supply, a) some units of enterprise will combine higher prices with lower quantities; others might combine higher quantities with lower market-clearing prices. And, P’Q’, and p’a’Q, + p”a”Q” , so to speak, contend relativistically with one another across the equals sign, and b) price and quantity contend relativistically with one another within the parentheses.

Now every unit of enterprise involves a turnover magnitude and a turnover frequency. The statement would be merely a truism if it meant no more than that the rates of payment received and made by the unit of enterprise involved quantities and velocities of money. But the statement is not a truism, for it involves a correlation between the quantities and velocities of rates of payment and the quantities and velocities of goods and services. (CWL 15, 57)

The existence of this correlation may be seen readily enough. … There are, then, alternatives between quantity and velocity in both rates of payment and rates of production. But the quantity alternative in the rates of payment is conjoined with the quantity alternative in the rate of production, and the frequency alternative in the rate of payment is conjoined with the frequency alternative in the rate of production. The two cases of quantity-velocity are not only parallel but also correlated. [CWL 15, 57]

Lonergan’s Functional Macroeconomic Dynamics is the purely relativitistic explanation of the dynamic economic process by the functional relations of the implicitly-defined, explanatory conjugates among themselves – i.e. by the functional relations in which they stand with one another, not as determined (per the Establishment textbook macroeconomics) by unexplained or mistakenly-conceived, so-called external shocks of supply or demand. (See The IS-LM, AD-AS Models and the Phillips Curve Correlation)

As the Diagram of Rates of Flow represents, Outlays and Expenditures are simply and in principle concomitant; they are linked or “yoked” in a particular normative circulation – aided by incidental credit to fill brief time gaps and long-term credit to finance the requirements of the pure cycle of expansion – in the macroeconomic field. Just as, in Hilbert’s geometry constituted by implicit definitions, where two points determine a straight line, and vice versa, so, in Macroeconomic Field Theory, Outlays constituting Incomes define and explain Expenditures constituting Receipts, and vice versa. These Outlays, Incomes, Expenditures, Receipts constitute the abstract explanatory conjugates in a pure field theory; they are mutual to and implicit in one another. They are mutually conditioning and mutually defining by their correlation to, and concomitance with, one another. Thus these abstract, explanatory conjugates tumble out together in the brilliant light of the sweeping, comprehensive insight that grasps all the interrelations and interdependencies which completely explain the entire unitary process.

Again,

A condition of circuit acceleration was seen in Section 15 to include the concomitant keeping in step of basic outlay, basic income, and basic expenditure, and on the other hand, the concomitant keeping in step of surplus outlay, surplus income, and surplus expenditure. Any of these rates may begin to vary independently of – rather than in normative concomitance with – the others, and adjustment of the others may lag. But any systematic divergence brings automatic correctives to work. The concomitance of outlay and expenditure follows from the interaction of supply and demand. The concomitance of income with outlay and expenditure is identical with the adjustment of the rate of saving to the requirements of the productive process. [CWL 15, 144]

Functional Macroeconomic Dynamics provides a closely knit frame of reference.

On this (methodological) model (i.e. concomitance by abstract correlation and implicit definition) circulation analysis raises a large superstructure of terms and theorems upon a summary classification and a few brief analyses of typical phenomena. Classes of payments quickly become rates of payment standing in the mutual conditioning of a circulation; to this mutual and, so to speak, internal conditioning there is added the external conditioning that arises out of transfers of money from one circulation to another; in turn this twofold conditioning in the monetary order is correlated with the conditioning constituted by productive rhythms of goods and services………There results a closely knit frame of reference that can envisage any total movement of an economy as a function of variations in rates of payment, and that can define the conditions of desirable movements as well as deduce the causes of breakdowns…CWL 15, 18

A new science has emerged. Lonergan has elevated conventional macrostatics to a macrodynamics explaining economic accelerations. Walrsian macrostatics has been reconfigured and animated. As Newton focused on second-order accelerations, F = ma, and F = Gm₁m₂/d², so Lonergan also focused on the changes of velocity (i. e. accelerations) in the pure cycle of expansion.

Taking into account past and (expected) future values does not constitute the creative key transition to dynamics. Those familiar with elementary statics and dynamics (in physical mechanics) will appreciate the shift in thinking involved in passing from equilibrium analysis (of for example a suspended weight, a steel bridge, or a static intersection of curves)…to an analysis where attention is focused on second-order differential equations, on d²θ/dt², d²x/dt², d²y/dt², on a range of related forces, central, friction, whatever. Particular boundary conditions, “past and future values” are relatively insignificant for the analysis. What is significant is the Leibnitz-Newtonian shift of context. [McShane, 1980, 127]

… A science emerges when thinking in a given field moves to the level of system. Prior to Euclid there were many geometrical theorems that had been established. The most notable example is Pythagoras’ theorem on the hypotenuse of the right-angled triangle, which occurs at the end of Book 1 of Euclid’s elements. Euclid’s achievement was to bring together all these scattered theorems by setting up a unitary basis that would handle all of them and a great number of others as well. … similarly, mechanics became a system with Newton. Prior to Newton, Galileo’s law of the free fall and Kepler’s three laws of planetary motion were known. But these were isolated laws. Galileo’s prescription was that the system was to be a geometry; so there was something functioning as a system. But the system really emerged with Newton. This is what gave Newton his tremendous influence upon the enlightenment. He laid down a set of basic, definitions, and axioms, and proceeded to demonstrate and conclude from general principles and laws that had been established empirically by his predecessors. Mechanics became a science in the full sense at that point where it became an organized system. … again, a great deal of chemistry was known prior to Mendeleev. But his discovery of the periodic table selected a set of basic chemical elements and selected them in such a way that further additions could be made to the basic elements. Since that time chemistry has been one single organized subject with a basic set of elements accounting for incredibly vast numbers of compounds. In other words, there is a point in the history of any science when it comes of age, when it has a determinate systematic structure to which corresponds a determinate field. [Method, 241-42]

.7. Curvature of space in physics and curvature in the structure of expansion in economics

Steven Weinberg states:

The attentive reader may have noticed a certain resemblance between the Principle of Equivalence and the axiom which Gauss took as the basis of non-Euclidean geometry. The Principle of Equivalence says that at any point in space-time we may erect a locally inertial coordinate system in which matter satisfies the laws of special relativity. We saw in Chapter 1 that Gauss assumed that at any point on a curved surface we may erect a locally Cartesian coordinate system in which distances obey the law of Pythagoras. (Weinberg, 1971, 68-9)

Alike, Lonergan’s vectorial treatment of prices and quantities in CWL 15, Section 17, entitled Measuring Change in the Productive Process, 70 ff., assumes that price and quantity changes obey the law of Pythagoras.

ds² = Σg_ijdx_idx_j, [i, j=1, 2 … n]

Also, for simplicity of instruction and foundational understanding, the possible curvatures in Functional Macroeconomic Dynamics may be exemplified in the form of a geometrical progression, or of a uniform progression, or of a flat-zero progression, or in the form of a logistical expansion – as the particular set of invention, skills and resources currently warrants.

A logistic differential equation (the logistic growth curve or, colloquially, the S-curve), would be such as

dP/dt = kP(1- P/K)

where P is the population, k is a constant of proportion, and K is an absolute “carrying capacity.” Initially the curve accelerates exponentially, then decelerates exponentially until it tapers toward a flatness where dP/dt approaches zero.

The main analytic apparatus is now complete. The two acceleration systems have been defined: a circulatory system consisting of two connected circuits that are accelerated by an external redistribution function; a quantity system of two parts in which one part is the long-term accelerator of the other. In each of these acceleration systems … an inner logic or ground in the nature of things indicates the normative or pure cycle of the quantity process. Finally, indices of price increments serve as markers of the divergence between the two systems. [CWL 21, 134]

We quote again two excerpts from Giancoli and two excerpts from Lieber, each of the four to be paraphrased immediately:

First quote:

Thus, in Einstein’s theory we do not speak of the “force” of gravity acting on bodies. Instead we say that bodies and light rays move as they do because space-time is curved. A body at rest or moving slowly near the great mass of Fig. 33-18 would follow a geodesic (the equivalent of a straight line in plane geometry) toward that body. [Giancoli 2005, 929]

Paraphrasing:

Thus, in Lonergan’s theory we do not speak unscientifically of the external “forceful” shocks of absolute price or absolute quantity acting on the process. Instead we say that production moves as it does because the pretio-quantital process consists of two circuits having normative curvatures defined by the potential of invention, skills, finite resources, and proper financing as to magnitudes and frequencies. The magnitudes and frequencies of turnovers are correlated with the magnitudes and frequencies of correlated proportionate payments. The motion of the process would follow a geodesic in the normative curvature of the process. [paraphrasing Giancoli 2005, 929]

Second quote:

In general relativity, then, the concept of a (Newtonian) gravitational field is replaced by a curved spacetime within which all things move without acceleration, locally always obeying the laws of special relativity. It is really a beautiful concept. It is not merely a reformulation of Newton’s gravitational theory. … Any change in the gravitational field, say, by a redistribution of matter or the collapse of a great mass, must result in a reshaping of spacetime. It does not reshape instantaneously, however, rather a disturbance in spacetime moves outward with the speed of light as a gravitational wave. [Serway 1986, 905]

Paraphrasing:

In general relativistic macroeconomics, then, the concepts of an absolute price, an absolute quantity, a plane geometry and a Galilean time are replaced by a curved pretio-quantitality according to which the process would normatively progress. It is really a beautiful concept. It is not merely a reformulation of a Gross-Domestic-Product straight-line trend. Any change in the normative macroeconomic curvature by a redistribution of new invention should result in a change in the curvature of the normative implementation of the new curved pretio-qantitality. It does not reshape instantaneously, however, rather a disturbance in the pretio-quantital potential is propagated forward kinetically through the value-adding supply chain and the hierarchical brackets of income as a double-circuited pure cycle of expansion. [paraphrasing Serway 1986, 905]

Third quote:

Paraphrasing:

These new ideas that a) two (or more) circuits are analytically distinct and have their own circulations, b)neither price nor quantity are external absolutes (as in IS-LM and AD-AS models), but rather relative to one another within a two-curved pretio-quantitality, c) by the principle of concomitance, Outlays are concomitant with Expenditures; and Incomes in the two circuits are adjusted according to the curvatures of the two interacting circuits, and d) abstract correlations determine explanatory conjugates yoked together in explanatory relations. We now see that by the addition of only one idea, precise analytical distinction of correspondences of levels of production and sale, Lonergan made possible a new science called Functional Macroeconomic Dynamics or Macroeconomic Field theory. [paraphrasing Lieber, 2008, 97]

Fourth quote:

To accomplish (the democratic result of the General Theory) Einstein introduced the PRINCIPLE OF EQUIVALENCE, by which the idea of a FORCE OF GRAVITY was replaced by the idea of the CURVATURE OF SPACE. The study of this curvature required the machinery of the vector CALCULUS by means of which the CURVATURE of velocitous and accelerative pretio-quantital production. was derived. This led immediately to the NEW LAW OF GRAVITATION which was tested by THREE CRUCIAL PHENOMENA and found to work beautifully. [Lieber, 2008, 300]

Paraphrasing:

To accomplish the General Theory of Macroeconomics, Lonergan introduced two analytically distinct (non-Euclidean) correspondences of production and end-use, each having its own monetary circulation and curved pretio-quantital geometry. These distinctions and scientific formulations can now replace the academy’s faulty notions of external, unexplained shocks of price and/or quantity by the idea of the curvatures of double-circuited expansion. The study of these curvatures requires the machinery of the vector calculus by means of which the curvature can be derived. This can lead immediately to the new science of the field of macroeconomics which can be learned, tested, and implemented in routines by staff with a background in physical theory at the Bureau of Economic Analysis, the Federal Reserve Bank, the National Bureau of economic Research, the Congressional Budget Office, the Wharton School and many banks and investment-research entities. [Lieber, 2008, 98-99]

Also, see CWL 15, footnote 165, pp. 125-26 re assumptions in Lonergan’s model.

Insert Figures 24-2, 24-6, 24-7, and 14-1.

So far attention has been directed to the latter parts of the graphs of dQ”/Q” and dQ’/Q’. It has been said that when the surplus stage devotes all its energies to self-acceleration, then Q” will be increasing in geometrical progression and dQ”/Q” will be a level straight line. When this period of gestation is coming to an end, the acceleration of Q” tends to become uniform, and then gradually to decrease to zero; when it is uniform, dQ”/Q” is decreasing, and when it is zero, dQ”/Q” is zero. [CWL 15, 125-26]

Now, when the acceleration of Q” is uniform, the long-term potential of the surplus stage is increasing, and so the surplus stage is devoting more and more of its efforts to the long-term acceleration of the basic stage; then Q’ will be increasing at an increasing rate, and time series of its values may stand in a geometrical progression to make dQ’/Q’ a level straight line. When, however, Q” becomes constant, the acceleration of Q’ becomes uniform, and then dQ’/Q’ will curve downwards; and as replacement requirements begin to mount, this downward curve is accentuated until dQ’ reverts to zero. Thus, both dQ”/Q” and dQ’/dQ’ are described as initially straight level lines that eventually curve downwards till the acceleration ratios become zero. One may well ask for an account of the movement of the acceleration ratios from their initial zeros to the level straight lines. ¶ There are two factors in such a movement: short-term acceleration and the period of generalization of a long-term acceleration. … [CWL 15, 126]

Walras’s macrostatic graphs miss the dynamics of the phases of the productive rhythms in a pure cycle of expansion.

Leon Walras developed the conception of the markets as exchange equilibria. Concentrate all markets into a single hall. Place entrepreneurs behind a central counter. Let all agents of supply offer their services, and the same individuals, as purchasers, state their demands. Then the function of the entrepreneur is to find the equilibrium between these demands and potential supply. … The conception is exact, but it is not complete. It follows from the idea of exchange, but it does not take into account the phases of the productive rhythms. … … Now each phase in an exchange economy will have its exchange equilibrium, but the equilibria of the different phases differ radically from one another. … By this cyclic variation within the exchange equilibria there is effected the ‘curvature of the exchange equations.’ … In the capitalist pase, the secondary rhythms are widening and deepening themselves. … Hence in the capitalist phase the surplus ratio S[1] is increasing. Surplus activity, surplus expenditure, and net surplus income are becoming greater and greater. To make a large profit is, in the general case, not a matter of brilliant enterprise. It is inevitable. …In the materialist phase, the secondary rhythms are widening and deepening the primary rhythms. … S is some proper fraction, but it is decreasing. No matter how intelligent and efficient traders may be, S cannot but be decreasing; for with surplus expenditure decreasing, net surplus income cannot but follow suit. … In the static phase. S is zero. The industrial structure is not becoming bigger. In the aggregate, there is no surplus income. When the whole motive force of economic activity is based on the anticipation of profits, this variation in net surplus income will be projected resonantly throughout the whole economic field. The increasing surplus ratio S of the capitalist phase heralds the bright dawn of a boom. The decreasing surplus ratio of the materialist phase overclouds the heavens and foretells a hurricane. The zero surplus ratio of the static phase is the most incomprehensible of mysteries, for what can be done when there is no net surplus income? CWL 21, 51-52

If we can say in doing physics that it is the gravitational mass that causes the curvature in mechanics, can we then say in doing macroeconomics that, by an inner logic or ground in the nature of things, if the initial stage of a pure cycle of economic expansion follows by its inner logic a curved convex path, increasing geometrically or logistically, then pretio-quantitality is curved per the very nature and inner logic of the process? If a light beam can follow a curved path, … then perhaps we can say that space itself is curved and that it is the gravitational mass that causes the curvature. Indeed, the curvature of space – or rather of four-dimensional space-time – is a basic aspect of Einstein’s General Relativity. And, indeed, the curvature of quantity and price in the pretio-quantitality in two circuits is a basic aspect of Lonergan’s relativistic macroeconomics. [see Giancoli 2005, 927]

Quoting Giancoli, 2005 re the curvature of space:

One way would be to measure the sum of the angles of a triangle. If the surface is a plane, the sum of the angles is 180°, as we learn in geometry. But if the space is curved, and a sufficiently large triangle is constructed, the sum of the angles will not be 180°. … the shortest distance between two points is called a geodesic On a sphere, a geodesic is an arc of a “great circle (an arc in a plane passing through the center of the sphere) such as the Earth’s equator and the Earth’s longitudinal lines. [Giancoli 2005, 928]

Another way to test the curvature of space is to measure the radius r and circumference C of a large circle. On a pleane surface, C = 2πr. But on a two-dimensional spherical surface, C is less than , as can be sen in Fig. 33-16. The proportionality between C and r is less than 2π. Such a surface is said to have positive curvature. On the saddle-like surface of Fig. 33-17, the circumference of a circle is greater than 2πr, and the sum of the angles of a triangle is less that 180°. Such a surface is said to have negative curvature. [Giancoli 2005, 928]

Thus, in Einstein’s theory we do not speak of the “force” of gravity acting on bodies. Instead we say that bodies and light rays move as they do because space-time is curved. A body at rest or moving slowly near the great mass of Fig. 33-18 would follow a geodesic (the equivalent of a straight line in plane geometry) toward that body. [Giancoli 2005, 929]

And, to satisfy the reader’s growing curiosity, in non-Euclidean geometry, (re space as infinite or closed, depending on curvature, the reader can check out (Giancoli 225, 928-29).

If the universe had a positive curvature, the universe would be closed or finite in volume. This would not mean that the stars or galaxies extended out to a certain boundary, beyond which there is empty space. There is no boundary or edge in such a universe. If a particle were to move in a straight line in a particular direction, it would eventually return to the starting point – perhaps eons of time later. On the other hand, if the curvature of space was zero or negative, the universe would be open. It could just go on forever. An open universe could be infinite, … [Giancoli 2005, 928]

.8. Concomitance of the curvatures a) in production, and b) in distribution of incomes in the hierarchical brackets of income

The form of the actual cycle of economic expansion has not the rigidity of physical mechanics. Humans are the efficient cause of the actual configuration of the process; but they do not understand the process. They commonly suffer from individual-egoistic bias, group-tribal bias, and general bias – ignorantly favoring flawed common sense, which disregards theoretical norms. Any of the rates of flow effected by human agents may vary independently of the others. One recent divergence was the flooding of the system with inflationary free money rather than the amount of money needed for payments to be correlated in magnitudes and frequencies with the magnitudes and frequencies of the production-for-sale process.

A condition of circuit acceleration was seen in Section 15 to include the keeping in step of basic outlay, basic income, and basic expenditure, and on the other hand, the keeping in step of surplus outlay, surplus income, and surplus expenditure. Any of these rates may begin to vary independently of the others, and adjustment of the others may lag. But any systematic divergence brings automatic correctives to work. The concomitance of outlay and expenditure follows from the interaction of supply and demand. The concomitance of income with outlay and expenditure is identical with the adjustment of the rate of saving to the requirements of the productive process. [CWL 15, 144]

In Macroeconomic Field Theory, expenditures and incomes are “woven together” by Lonergan’s use of vector calculus (and later by Burley’s linear von Neumann models; see Bibliography).

P’Q’ = (p’)(a’Q’) + (p”)(a”Q”) (CWL 15, 158)

P”Q” = (p”)(a”Q”)_Expansionary + (p”)(a”Q”)_{Repair and Maintenance}

And assuming vectors but in simplified notation, thus,

P’Q’ = p’a’Q’ + p”a”Q”

P”Q” = i”O”_Expansionary+ i”O”_R&M

where the vector dot-product P’Q’ stands for basic Expenditures, and the sum of vector products (p’)(a’Q’) + (p”)(a”Q”) stands for the basic-plus-surplus Outlays which become total basic Incomes (macroeconomic “costs”) and, thus, limit so-called ”surplus profit for investment”.

The violation of proportionality of concomitant flows of products and money is treated in The Road Up is The Road Down; the Mechanism of Rising and Falling Prices; and in Facing Facts, and in The Ideal of Constant Value of The Currency vs. The Fact of Inflation.

So, what is meant by the so-called “consumer,” when financial-talk-show commentators say, “The consumer is ‘resilient’?” Does the commentator know that the current part of the curvature of economic expansion may be greater than or less than a constant zero? Is the commentator describing metaphorically a race-track hunch or does the commentator know the marginal propensity to consume in the hierarchy of income brackets and the present state of the bracket incomes and balance sheets? In analyzing and discussing the macroeconomic state of affairs, the consumer must be conceived by a macroeconomic scientist in the aggregate abstractly, theoretically, and coldly as, first, the duality–identity of outlay-receiver and income-expender; second, the aggregate duality-identity of level of productivity and level of compensation; and, third, the aggregate duality-identity level of final products being sold and level of final prices being paid. Difficult though it may be to say, the macroeconomic scientist must be a scientist of interconnected flows and leave it to a culture of intelligence and values to ensure desserts and equity. In the science of macroeconomics, one should not talk of consumers apart from their identity as receivers of Incomes for their productive services, perhaps qualified by consideration of level of their net borrowing, (D’ – s’I’). Macroeconomic science prescinds from human psychology and deals with the abstract correlations determining concomitances, implicit definitions, continuity, and equilibrium.

The whole structure of Functional Macroeconomic Dynamics is purely relational. A macroeconomic functioning is not a compilation or aggregation of particular income statement categories, such as wages or interest expense. A macroeconomic functioning is implicitly defined by its functional relation to other functionings. The whole structure is unitary and purely relational. “Lonergan’s analysis is concrete but heuristic. It focuses on functional relations intrinsic to the productive process to reach eventually a general theory of dynamic equilibria and disequilibria.” [McShane 1980, 117]

.9. Miscellaneous addenda for consideration:

xxxx

See “An Einstein Relativistic Context … “

Also see The Einsteinian Context

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Just as, in defining a circle, one does not understand the equality of radii without understanding the implications for the diagonal, π, circumference, area, etc, so in Functional Macroeconomic Dynamics one cannot understand anything explanatory about one particular flow in the unified set of circulations without grasping the unified explanation of the whole unitary circulatory process. All the concepts yielded by the insight into the dynamic interconnections and interdependencies tumble out together.

… one may begin at any function to move in either direction. One may begin anywhere because the total movement is circular. One may move in either direction, for one may ask where the money goes or where it is coming from. … [CWL 15, 165]

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There are classical laws and statistical laws; and there are canons of empirical method. (CWL 3, 70-103/93-126)

… there is a canon of complete explanation. Everything is to be explained. … … … The canon of complete explanation demands that the scientific world, which expresses the relation of things to one another, be constructed completely. It is not the world of common sense. (CWL 10, 144)

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On this (methodological) model (i.e. concomitance by abstract correlation and implicit definition) circulation analysis raises a large superstructure of terms and theorems upon a summary classification and a few brief analyses of typical phenomena. Classes of payments quickly become rates of payment standing in the mutual conditioning of a circulation; to this mutual and, so to speak, internal conditioning there is added the external conditioning that arises out of transfers of money from one circulation to another; in turn this twofold conditioning in the monetary order is correlated with the conditioning constituted by productive rhythms of goods and services………There results a closely knit frame of reference that can envisage any total movement of an economy as a function of variations in rates of payment, and that can define the conditions of desirable movements as well as deduce the causes of breakdowns…CWL 15, 18

P’Q’ = (p’a’Q’) + (p”a”Q”) (CWL 15, 158)

Δ(P’Q’) = Δ(p’a’Q’) + Δ(p”a”Q”).

xxxx

A nation earns the productive contribution it effects. A nation earns what it produces.

q_i = ΣΣq_ijk, and Q_i = ΣΣQ_ijk. (CWL 15, 30)

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Concomitance implies abstract correlation, mutual definition, mutual reciprocal conditioning and keeping in step. However, while the normative theory of the immanent intelligibility or formal cause prescinds from human psychology, it remains that humans – as knowledgable or ignorant, cooperative or malicious – are the ultimate efficient cause of the economic process. Human desires and fears have not the rigidity of electric, magnetic, and electromagnetic phenomena. It would be good if academics understood how the economic process actually works, and then, educated the public about how the economic process actually works

xxx

… if someone has a scientific, dynamic heuristic and is doing macroeconomics and you open his book, what do you find? You find just mathematical equations. He is solving problems, and what is it? It is more mathematics. Why do you say he is doing macroeconomics? He seems to be doing mathematics all the time. It is because there are regions of mathematics that are isomorphic with macrodynamic reality. There is the same relational structure between a given mathematical theory or system as there is between macroeconomic functionings that can be observed. This is another case, a big case, of isomorphism: on the one hand, mathematical expressions, and on the other hand, interdependent, mutually defining macroeconomic functionings. There is the same relational structure. But in the mathematical case, the relational structure links symbolic expressions, or mathematical concepts, with one another, while in the macroeconomics case what are related are concrete, interdependent, dynamic, macroeconomic functionings. ¶ So there is an isomorphism [Paraphrase of CWL 18, 32-33]

xxxx

DP’ and DP” simply indicate what might be described metaphorically as the inertia of the quantity process of goods and services in its response to acceleration initiated in the circulatory process of payments. Rapid increases or decreases in the circulatory process have not a proportionate effect in the quantity process but are in part absorbed by positive or negative price increments. Thus booms are notoriously inflationary and slumps deflationary. Hence DP” and DP” are best taken as indices of divergence between circulatory and quantity phases. (CWL 21, 134)

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Addendum re a speculative boom in the stock market affecting basic spending:

It is to be recalled that the account given of the cycle of the basic price-spread ratio supposes (D’ – s’I’) to be zero throughout. A speculative boom in the stock market which encourages basic spending may be represented by a positive (D’ – s’I’); there is an excess release of money from the Redistributive Function to the basic demand function. Alternatively, it may be represented by an upward revision of the fractions w_i of total current income going to basic demand, while the fact that the surplus final market suffers no contraction then results from the excess of the rate of new fixed investment over the rate of pure surplus income, so that D” is positive. In either case, a movement of this type with its basis in redistributional optimism will offset any tendency towards a contraction of the price spread and will reinforce any tendency of the price spread to expand. On the other hand, the subsequent stock market break intensifies the crisis of the circuits, removing the props that had swollen expansive tendencies, and leaving the system with a greater height from which to fall. (CWL 15 162 )

Xxxx

A new science has emerged. Lonergan has elevated conventional macrostatics to a macrodynamics explaining economic accelerations. Einstein used the idea of curvature and the use of tensor calculus to advance from special relativity to general relativity. Lonergan discovered curvatures in macroeconomic dynamics. Also, he used the principle of concomitance, precise analytic distinctions between point-to-point and point-to-higher-order productive relations, the technique of implicit definition, abstract correlations in measurable data, and the vector calculus, to explain the curved productive flows and curved pretio-quantital monetary flows in interdependent circuits. He used the fact of curvature as the basis of complete explanation of the nature of the double-circuited, pure cycle of economic expansion. A new and higher explanatory science of macroeconomic dynamics has emerged to replace academe’s descriptive macrostatics.

xxxx

A certain fraction, say S, may be being devoted to purchasing new capital goods and services for the widening and deepening of existing industry. Further, this rate of expenditure at the secondary market S.DE”, will not appear initially in the accounts of any individual or any firm under the heading of costs in the ordinary sense of that term; on the contrary, it is part of the flow of investment; it is the outlay of fresh capital for capital goods and services. ¶Thus we are led to posit the theorem of the surplus. … … … … (CWL 21, 49) (i.e. the theorem of point-to-line)

xxxx

Graphs listed in the text:

Diagrams (below) show how the always current process plays out in intrinsically cyclical fashion many times over the long term. These are easily located by their Figure numbers corresponding to section numbers in CWL 15.

In these first two diagrams, suppose that k = 1.05 and that r = .9524.