The Formulation of Functional Interdependencies

The Diagram of Monetary Flows below is an image of interconnected flows; it is an image of interdependencies and conditionings.  It has a unified set of laws.

to know many laws, not as a mere congeries of distinct empirical generalizations, but in the network of interrelations of each to all the others, is to reach a system. [CWL 3, 76/99]

Among this website’s topics Topics in Functional Macroeconomics are four separate treatments which concentrate on the explanatory mathematical formulae, but present them in different orderings.  Our purpose in each of the four treatments is to wall off the reader from the verbiage and make him concentrate on and understand solely the relationships in the abstract formulations which explain the objective economic process.  The reader must check his psyche at the door.  No political leanings may be expressed.  No psychic preferences allowed.  Objections are restricted to proving the error in the insight expressed in the formula or in the coherence of the formulas with one another.

The deepest and most precise understanding is contained in concise and precise mathematical forms which are isomorphic with the patterns of the data of the dynamic process.  The four treatments are titled:

  1. Cluster of Key Relations: Seeing Production, Exchange, and Finance All in a Single View (URL)
  2. Key Equations Module (URL)
  3. Edifice of Formulae (URL)
  4. The Formulation of Functional Interdependencies (This section)

Equations are devoid of hot air and excess verbosity.  Equations offer the advantages of a.) stating concisely and precisely the nexus of relationships of implicitly-defined functionings to their conjugate functionings. A few pages of equations of a few general governing forms may encapsulate the contents of tens, or even hundreds, of pages of descriptive and postulational verbiage.  Properly read and struggled with, equations pressure understanding down into the brain.  In contrast, verbiage may get continuing nods of acceptance by the reader but leave the reader’s brain empty.

The totality of formulae below comprises a unified whole which constitutes a scientific, explanatory macrodynamics.  The next subsection (URL) will discuss The Achievement of Explanation.

We begin with a challenge or tease.  We assume that the reader has read CWL 15 and, thus, we test his understanding of the theory he has read.  Prescinding from international trade and government deficits or surpluses, we simply list the main equations in a unified and scientific macroeconomic dynamics. Our single comment is the reminder that the huge-little subscript krepresents a compensated application of a factor of production by a human.

ki= the rate of application of a factor of production[1]

qi = ΣΣqijk[2]           (1)

Σpijqij= PŸQ= PQ cosA[3]

kn[f’n(t-a)-Bn] = f”n-1(t) – An-1[4]             (2)

ΔM’ = (S’ – s’O’) = ΔT’ + ΔR’ + (O’ – R’)[5]        (3)

G = O[6]  (4)  (The condition of dynamic equilibrium)

P’Q’ = p’a’Q’ + p”a”Q”[7]   [8]        (5a)

J = a’ + a”R

dJ = da’ + a”(dR) + R(da”)[9]      (5b)

Π”Κ”Expansionary Pure Surplus = π”a”Κ”Expansionary Pure Surplus[10]       (6a)

Π”Κ”=ΣFi= vI” [11]

f  = v[I”/(I’+I”)] =  vw[12]

df = v(dw) + w(dv)     (6b)

dI’ = Σ((widni+ nidwi)yi[13]        (6c)

GDFF = P’Q’ + Π”Κ”=  p’a’Q’ + p”a”Q”R&M+ π”a”Κ”Expansionary Pure Surplus +π”α”Κ”R&M        (7)

A process is a motion or movement in time; it occurs over time; therefore, it must be understood in terms of its velocities and accelerations.

From one standpoint, our overall process is a threefold process – two operative processes and one financing process.  Within the two operative functionings there are three classes of products being supplied and demanded:

  1. consumer goods for a standard of living
  2. repair and replacement goods to maintain existing capacity and capability
  3. expansionary capital goods to widen and deepen the stock of capital and, so, to increase goods and services available in the future

The flows of these goods through and out of production are enabled and accompanied by payments of money.

basic terms are defined by their functional relations.  The maintaining of a standard of livingis attributed to a basic process(distinct process 1), an ongoing sequence of instances of so much every so often.  The maintenance and acceleration (positive or negative) of this basic process is brought about by a sequence of surplus stages, in which each lower stageis maintained and accelerated (distinct process 2) by the next higher.  Finally, transactions that do no more than transfer titles to ownership are concentrated in a redistributivefunction, whence may be derived changes(distinct process 3) in the stock of money dictated by the acceleration in the basic and surplus stages of the process. … So there is to be discerned a threefoldprocess 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, 53-54]

From another standpoint, our operative process consists of two interdependent velocitous and accelerative systems:

  1. a rectilinear quantity system consisting of an accelerated and an accelerator process, and
  2. a circulatory monetary system consisting of two monetary circuits connected by crossovers

in the long run, and especially in the very long run, such a correlation exists.  It is that surplus production is the accelerator of basic production.  In other words the correspondence between the two is not a point-to-point but a point-to-line correspondence; … Now such a correspondence, if it is to be expressed not in terms of expectations of the future but in terms of present fact, is a correspondence of accelerator to accelerated. … If the system is to move into a long-term expansion, this movement has to begin with a surplus quantity acceleration: surplus production has not merely to maintain or renew existing capital equipment but has to reach a level at which it turns out new units of production and maintains or renews a greater number of existing units; this gives the quantity surplus expansion. [CWL 21, 132]

The main analytic apparatus is now complete. The two acceleration systems have been defined: “a circulatory system consisting of two connected (monetary) circuits that are accelerated by an external redistribution function”; a quantity systemof two parts in which one part is the long-term accelerator of the other. … an inner logic or ground in the nature of things (is) indicated as 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]

Within the overall process there are functional interdependencies.  In general, the condition of the occurrence of B is the occurrence of A.  The continuity and the expansion of the entire economic process are conditioned by the satisfaction of the dependencies of functional flows by one another.

The analysis gives new meanings to old terms.

Lonergan was seeking the explanatory intelligibility underlying the ever-fluctuating rhythms of economic functioning.  To that end he worked out a set of terms and relations that ‘implicitly defined’ that intelligible pattern.  When all was said and done the relations, and the terms they implicitly defined, were markedly different from either the terms of ordinary business parlance or the terms of neoclassical and Keynesian economic theory. [CWL 15, 179]

We wish to grasp the scientificness of the analysis.  Let us escape from entrapment in corporate accounting and let us achieve the insight relating the concepts of “macroeconomic costs” and “macroeconomic profits” and “consumer-goods revenues” in the Revenue-Cost Identity Theorem.  This theorem states that the aggregate flows of consumer-goods purchase money are equated with, identified, and defined as “macroeconomic costs;” i.e. aggregate consumer-goods sales revenues are defined implicitly as aggregate functional costs.  In the following excerpt – arguably one of the most important passages in CWL 15 – these aggregate functional macroeconomic costs are defined, and, by the principle of the excluded middle, their functional complement, “aggregate and functional pure surplus income,” or “profits,” is implicitly defined.

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; for (our meaning of costs) would include among costs the standard of living of those who receive dividends but would not include the element of pure surplus in the salaries of managers; worse it would not include replacement costs, nor the part of maintenance that is purchased at the surplus final market………..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 (from total outlay), 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]

Note a sampling of differences between corporate accounting and functional macroeconomic dynamics.

 

Classification Accounting Macroeconomic Dynamics
Wages & Salaries Costs “Costs” if for consumption; but “pure surplus income” if used for investment
Interest Costs “Costs” if for consumption; but “pure surplus income” if used for investment
Dividends/Retained Earnings Not costs/not costs “Costs” if for consumption; but “pure surplus income” if used for investment

Now, consider the equation equating aggregate and functional velocitous basic incomes-expenditures (P’Q’) and aggregate and functional velocitous outlays-costs (p’a’Q’ andp”a”Q”Repair and Maintenance).

P’Q’ = p’a’Q’ + p”a”Q”Repair and Maintenance[14]

By this formulation a) outlays functioning as incomes-to-be-expended are mathematicized, b) the sum of two monetary cost velocities is identified as and, so, equated to a functional revenue velocity c) therefore, the terms are implicitly defined by their functional identity to one another, d) therefore, since science is constituted by explanation in the form of terms related to one another, the equation constitutes a fundamental relationship of explanatory macroeconomic science, and e) since the terms represent flows or velocities, the equation representing flows in a system requiring a dynamic equilibrium. We repeat

P’Q’ = p’a’Q’ + p”a”Q”Repair and Maintenance[15]

Our terms represent the magnitudes, per a designated time period of functional money flows in the basic circuit.  By our implicit definition of costs and expenditures, expenditures for purchase are identified as macroeconomic costs.  They are intrinsically related to one another. Thus monetary functionings become purely relational technical terms in an equation, which a) is a mutual defining of costs and expenditures in the manner of two points defining a line and a line defining two points, and b) specifies the law of the basic circuit’s monetary circulation within the system of Gross Domestic Functional Flows.

The purely relational functional interrelations define the interdependent functionings; in this case functional purchase-revenues are identified as the functional macroeconomic cost flows effecting purchase revenues in the period. The meaning of the terms is not fixed by the accounting convention calculating profit and loss nor by our everyday perception.  The initial meanings of the terms, based upon our everyday experience of working and purchasing have now been revised into technical terms in an explanatory equation.  We have discovered and postulated an economic law; and we have cast the functional relations of concrete economic phenomena in explanatory mathematical form; just as the physicists Galileo, Newton, Hamilton and Einstein applied mathematics to explain the particular phenomena which they were investigating.

Newton:

F = ma, or F = m d2x/dt2

F = gm1m2/d2

Hamilton:

I = St2t1 Ldt

δI = δSt2t1Ldt = 0[16]

The flowing of money is concrete, but the concept, “monetary function,” is an abstraction.  On the side of the knowing subject, our concept is the content of an act of insight into abstract relationships among data, rather than a word representing the sensation of an act of sensing.  On the side of the known object, a monetary functioning is an abstraction as are mass, heat, electrical intensity and magnetic intensity.  Our velocitous monetary functionings are now explanatory terms in an equation; and their meaning is now defined by the equation.  You cannot kick the abstract concept, functioning, nor can you smell it, nor can you hear it, nor does it press upon you in any sense.  Yet, just as abstract concepts in thermodynamic such as pressure, volume, temperature are explanatory terms which interconnect in a tight mathematical formalism the sensible experiences of hot, cold, pushing, expanding, etc., so as to explain the data of the phenomena of gases, so the abstract functionings implicitly defined in our P’Q’equation interconnect and explain the monetary phenomena of the economic process.

We begin from the experiences of our everyday work and enjoyment; we describe these experiences in categories of their relations to us; then we draw diagrams of the relations of dependence among these descriptive categories; then we revise the terms and relations of our concrete experiences into abstract terms which might somehow be related to one another and implicitly define one another; then, by an insight, we come to a grasp of the coherent relations among the abstract terms in our diagram; finally we grasp the abstract intelligibility of the whole in a set of terms and relations coherent with one another.

The act of understanding these functional interdependencies is followed by the test of verification. Is the understanding correct?  We test our hypothetical formulation in an heuristic analysis of time-series data of actual production and sale to determine if the hypothetical formulation does in fact account for the functioning of the economy in explanatory fashion.

The abstract concepts of the velocities of interdependent functional monetary flows exist within understanding – like the abstract concepts of the velocities of momentum and kinetic energy in theoretical mechanics; the concepts themselves are not representations of sensation.  Indeed, right from the start we possess some rudimentary notion of these explanatory conjugates of economics, for we all notice and speak, albeit only notionally, of the “functioning” or “malfunctioning” of the economy.[17]  Obscurely we know what we are talking about.  But we are still at the stage of “notion,” and we have not achieved the explanatory insight, much less the concepts and their terms yielded by the insight, which would enable us to state the complicated systematics of the “functioning.”  But then we push forward under the guide of an heuristic dynamic from vague notion to the actual clear understanding of the functional dynamic relations.  Finally, we test to verify that the terms and relations are in fact explanatory.

To pause for an aside exploiting certain conventional terms: As in applied science, the efficient cause of an economic functioning is the people performing the activities which constitute the function.  The final cause of an economicing function is the purpose or goal of the activity.  The exemplary cause is the normative rhythm to be followed in exploiting the potential of maximum possibility.  But as in theoretical or pure science, the formal cause of the entire system is the set of terms and relations which explain how things interconnect and the entire system actually works; i.e. the formal cause is the immanent intelligibility of the dynamic system as purely dynamic.

Scientific macroeconomics is not corporate accounting, National Income Accounting, or statistical economics.  One must not be trapped by indoctrination in Course 101.  The doing of explanatory macroeconomics is different from doing descriptive accounting and different from compiling the statistics of descriptive categories (with the categories often determined by the convenience and ease of gathering whatever statistical data is available).  Any analysis conducted with the objective of explaining an overall functioning exchange economy must seek its immanent intelligibility; a) it must be in terms of its inner, interdependent functionings, b) it must employ implicit definitions of terms by their functional interrelations to each other, and c) it must formulate first-order and second-order differential and difference equations of the primary relativity that generally govern their variations vis a vis each other as the economy progresses through its stages of growth (or decline).  The goal in macroeconomic science is to reach a set of tightly-knit terms and relations which explain the time-series data.

Conventionally, macroeconomic data is sorted or classified in terms of accounting unities or classifications easily gathered from commercial, industrial and banking data.  The GDP classifications of the Department of Commerce’s Bureau of Economic Research are aggregates of the Income-Statement accounts of domestic firms and governments in given periods.  The Flow of Funds classifications of the Federal Reserve Bank are aggregates of the changes in the Balance-Sheet positions among firms, households, banks, and government in the same periods.  Accounting categories are aggregated and presented as National Income and Product Accounts (NIPA) and Flow of Funds in a given period.

These accounting classifications are in fact abstract concepts, but on a lower level than is needed; e.g. the concepts of labor costs and pizza revenues are abstract, but lower-level, universal terms.  They are abstractions of  “things as related to us”; they are not abstract terms implicitly defined by their relations to each other; they are not explanatory.  They are just two of the many abstract terms of everyday, commonsense accounting; as are terms such as wages, interest, heat and power, legal fees, accounts payable, net plant and equipment, with which the government and shareholders are comfortable.  They are non-explanatory classifications.  They do not explain the functional, mutually-determinate operations of the functioning economy.  They do not answer the question, “Why?”.  They are related to each other only as descriptive unities convenient to the summation called the bottom line.

They may also be related to each other as ratios of one to another, or, as percentages of the whole, or even as rates or velocities per period.  They may also be compared in magnitude from period to period.  However, reports of the period’s magnitudes are not explanations of the process, any more than a report of electrical usage constitutes the schematic of an electric supply grid.  Reports of isolated magnitudes are not constituents of a comprehensive theory of macroeconomic functioning which, first, captures the economy’s nature, second, yields the economy’s implicit criteria for proper functioning, and, third, defines proper functionings or malfunctionings theoretically as conformity to or divergences from these implicit criteria.

Again, per our subtitle, we are interested in the mathematical formulation of functional interdependencies. Elsewhere we achieved a measure of familiarity with the foundational and systematic distinctions between

  • current-determinate-point-to-current-determinate-point vs. current-determinate-point-to-indeterminate-future-series relation of factors of production to elements exiting into a standard of living.[18]
  • The correspondence of accelerator to accelerated process, and accelerator quantities to accelerated quantities in the rectilinear productive process, and
  • The circulatory monetary system of two operative circuits connected by crossovers and furnished with transactions money by the money-creation system

So, let’s get to the formulation of functional interdependencies.  We shall review the following formulations:

The ultimate product or service

The lagged technical accelerator

The financing requirement

The condition of equilibrium

The identity of basic macroeconomic revenues and macroeconomic costs

The basic price-spread ratio

The differential of the aggregate basic price-spread ratio

The identity of expansionary investment and pure surplus income

The pure-surplus-income ratio

The differential of the pure surplus-income ratio

Gross Domestic Product sublated by Gross Domestic Flows

The ultimate product or service: Let us represent the composition of some ultimate product or service qas follows:

qi = ΣΣqijk[19]         (1)

where k represents a rate of application of a factor of production; e.g. so much management per period, so much capital per period, so much aluminum per period, so much labor per period, etc.; and where represents a unit of enterprise performing its specialty and adding value to the product or service.

That little subscript k, and its monetary correlate compensation, are of huge importance in coming to understand functional macroeconomics.

Thus, a rate – repeat, “rate” – of application of a factor of production is the velocity of the application of this factor.  It is a so much per period.  And conjoined to the rates of the applications of factors are the rates of payments to human employees (including owners) and to rentiers. The two aspects of these initial payments are outlays by the unit of enterprise and incomes to the recipients. Outlays are the simultaneous obverse of incomes.

Thus the productive process is constituted at the front end by flows of both human work and monetary compensation, and at the back end, by flows of composite products and monetary expenditures to purchase these products.  It’s fairly simple, really.

One may look at the total flow of component factors in a period, where all flows have identical time subscripts, or one may look at the total flow of completed, composite, products out of “being-under-process” into no-longer-under-process use, where many contributions of kby joccurred previously.  Both cases may be represented by

Qi= ΣΣQijk[20]

where, again, both Qi and Qijk are rates, i.e.  instances of the form ‘so much or so many every so often.’

The basic stage of the process is, in its pure form, an aggregate of RATESof labor, of managerial activity, of the use of capital equipment for the sake of the goods and services that enter the standard of living. Let us say that some ultimate product … [CWL 15, 29-31]

… Since the form of the relation between them is a double summation, the emergent standard of living and the basic stage of the process are not identical aggregates of rates.  On the other hand, precisely because the relation is a double summation, they are equivalent aggregates of rates.  However, this statement requires three qualifications. cf. [CWL 15, 29-31]

Each factor has an associated compensation-“price” and final sale “price” because, as the product is built up from the potentialities of nature into a finished product, payments are made to humans applying the factors of production; and as the composite product is sold the vendors receive payments from human purchasers.

Giving each distinct final product qian axis in Cartesian coordinates, we may derive a sales-quantity vector index, the vector Q. Similarly, we may derive a production-quantity vector index (aQ).

By the same mathematics, we may derive a selling-price index and a cost-price index p.

And we may calculate the scalar dot products, PŸQ and pŸ(aQ).

Σpijqij = PŸ= PQ cosA[21]

The same mathematical form would be used whether we were calculating a cost price or a selling price

The lagged technical accelerator: The increase in machines, techniques, and skills precedes the acceleration of other production made possible by and effected by the use of these generic capital goods.  The rhythm constituted by increased capital-goods velocities followed by accelerated basic velocities is given general formulation in the lagged technical accelerator equation,

kn[f’n(t-a)-Bn] = f”n-1(t) – An-1[22]   

where is the application of factors of production, prime and double prime represent velocity and acceleration respectively, is a multiplier, t-a and trepresent earlier and later intervals of time, B represents the rate of production on level effecting merely replacements and maintenance on the next lower level, and A represents merely short-term acceleration (due to just taking up slack rather than new capital equipment) of the rate of production on the n-1 level.

Now, the process is always the current process, and the mathematical formulation of its primary relativity must always be in terms applicable to any instance, i.e. to any particular boundary conditions.  One does not seek to understand planetary motion by constructing a million theories from a million separate measurements of locations, velocities, or accelerations; rather one seeks a single general formulation – such as a general formulation of elliptical motion – applicable to any instance of measured secondary determinations or boundary conditions such as location and velocity or acceleration.

So, the lagged technical accelerator states a general relationship of any current rate of capital-goods production to a future amount of production on the next lower level.  The formula simply formulates the general relationship called an acceleration lag.  It is simply a general expression of a correspondence of rates in the purely dynamic, economic process.  It is an expression not in terms of expectations of the future but in terms of an always general present fact; it is a general expression of a correspondence of accelerator to accelerated.  It simply states a condition of acceleration.

in the long run, and especially in the very long run, such a correlation exists.  It is that surplus production is the accelerator of basic production.  In other words the correspondence between the two is not a point-to-point but a point-to-line correspondence; … Now such a correspondence, if it is to be expressed not in terms of expectations of the future but in terms of present fact, is a correspondence of accelerator to accelerated. … If the system is to move into a long-term expansion, this movement has to begin with a surplus quantity acceleration: surplus production has not merely to maintain or renew existing capital equipment but has to reach a level at which it turns out new units of production and maintains or renews a  greater number of existing units; this gives the quantity surplus expansion. [CWL 21, 132]

The non-Euclideans moved geometry back to premises more remote than Euclid’s axioms, they developed methods of their own quite unlike Euclid’s, and though they did not impugn Euclid’s theorems, neither were they very interested in them; casually and incidentally they turn them up as particular cases in an enlarged and radically different field. … Einstein went beyond Newton by employing the new geometries to make time an independent variable; and as Newton transformed the formulation and interpretation of Kepler’s laws, so Einstein transforms the Newtonian laws of motion. … It is, we believe, a scientific generalization of the old political economy and of modern economics that will yield the new political economy which we need. … Plainly the way out is through a more general field. [CWL 21, 6-7]

Paraphrasing:

Lonergan moved macroeconomics back to premises more remote than Walrasian statics, microeconomic price theory, neoclassical macroeconomics and Keynesian macroeconomics, he developed explanatory formulae quite unlike others’, and though he did not impugn them, neither was he very interested in them; casually and incidentally combinations of prices and quantities turn up as particular cases in an enlarged and radically different field. … Lonergan employed a new dynamics to make aggregate, mutually-defining, velocitous functionings the basic interdependent variables; and as Newton transformed the formulation and interpretation of Kepler’s laws, so Lonergan transforms the neoclassical and Keynesian laws of motion. … He achieved a scientific generalization of the old political economy and of modern economics that yields the new political economy which we need. … Plainly the way to settle disputes about economic change is through a sublating, more general, dynamics of functionings.

The financing requirement:

… to bridge the gap between the initiation of his turnover and its completion, (the entrepreneur) has to purchase materials, pay wages, and it is only when he has sold his product that the monetary return of his outlay begins.  To bridge this gap he must have circulating capital. [CWL 21, 59]

The supposition that circuit acceleration to some extent postulates increments in the quantities of money in the circulation accounts for both mercantilism and for the substitution of more elegant techniques in place of mercantilism.  Further, it points to excess transfers to supply, to (S’ – s’O’) and (S” – s”O”), as the mode in which increments of money enter the circuits…the function of monetary circulating capital is to bridge the gap between payments made and payments received; as goods and services are in process, the unit of enterprise makes payments to earlier units in the production series and to its own factors; only when goods and services are sold are payments received.  Now this gap increases with increments in turnover magnitude. [CWL 15, 61-62]

The supplying of money for the basic process is given by

ΔM’ = (S’ – s’O’) = ΔT’ + ΔR’ + (O’ – R’)[23]

The substituting of double superscripts for single superscripts would give the supplying of money for the surplus process:

ΔM” = (S” – s”O”) = ΔT” + ΔR” + (O” – R”)

This financing requirement is presented in CWL 15, 65-68 and in the Fragments Section of CWL 21, 163-174. Elsewhere on this website, we have treated this subject of financing under the title The Series, the Stack, and The Economic-Work Theorem of Money, in which we show the financing required by a) the series of contributions of producers from the potentialities of nature to a completed product, and b) the stackof producers when a product is under production simultaneously by all producers in the rectilinear series.  The spreadsheets of that section work well as a visual image of the credit money associated with the production of a single product.

The condition of equilibrium:  In order that one monetary circuit may not drain the other, the crossovers between the two circuits must balance.

G = O[24]

That is the condition of dynamic equilibrium

If this condition of equilibrium is not achieved normatively through the crossovers i”O” and c’O’ of the operative circuits, it can be achieved non-normatively by taxations and redistributions flowing undesirably through the redistributive function.

We would add a second condition of equilibrium.  For example, a fishing village making its own nets (capital items), saves and invests each period but makes more nets than it can ever use.  The excess of nets results in less than full use of all nets and finally they are replaced by even better combinations of nets and motors.  The old nets get set aside and then are burned to make space for something else.  During the original excess capital expansion, the crossovers balanced, but the excess production more and more cannibalized itself.  The flow of pure surplus income should have declined, but it lasted longer. The industry became plagued by excess capacity.  The production and consumption of basic consumer goods (fish) during the excess expansion should have been greater.  There should have been less investment and greater consumption.  The crossovers balanced, but at too great a magnitude.

It is critical for continuity and dynamic equilibrium that entrepreneurs estimate carefully the normative capital requirements of the process so that, not only will crossovers balance, but the magnitude of the crossovers is precisely correct.  We quote (CWL 21, 55):

It has been shown that a condition of continuity is defined by the cancellation of the income crossover.  By equation (9)

G’DI’ = G”DI”

Where G’ is the precise and correct fraction of basic outlay directed to real maintenance required by the process, rather than to excess surplus production or in insufficient maintenance.

By transposing the terms, we find the ratio of basic income to pure surplus income as the reciprocal of the ratio of the surplus crossover fraction to the basic crossover fraction.

DI’/DI” = G”/G’

So that on the assumption of continuity, G”/G’equals the Normative Proportion in the example of CWL 21, p. 55. …

The Crossover Ratio

As the table makes clear, a variation in G’ is much more significant that a variation in G”.  … Inversely, when G” is 90% and G’ is really 10% but estimated to be 20% by over-zealous depreciation charges and by depressed wages, then a normative proportion of 9 is given a monetary distribution corresponding to a proportion of 4.5.  The result is an overproduction or an insufficient purchasing power or a maldistribution (or whatever it is safe to call it, for superficial economists fancy the thing cannot exist) that generously slices off about half of existing economic activity.  We say ‘about half’ for the proportion 4.5 is a relative term: secondary activity may increase, and then the proportion is four-and-a-half times something greater than what it was nine times greater; on the other hand, as eventually will be the case, secondary activity may decrease, and then the proportion becomes 4.5 times something smaller than before [CWL 21 55 (for full explanation of “about half”)]

The process of correction is addressed on this website in The Theory of Booms and Slumps (URL) and in CWL 15, pp 588 ff.

The Basic-Circuit Identity Equation; basic macroeconomic expenditures equal “macroeconomic costs”: The implicit defining of flow-rates of macroeconomic, basic-circuit expenditures and costs, as previously described is represented by the following formulation:

P’Q’ = p’a’Q’ + p”a”Q”Repair and Maintenance[25]   [26]

Lonergan’s Basic-Circuit Identity Equation interrelates

  • prices (P’, p’, p”,),
  • integrated factor of production velocities (Q’, Q”Repair and Maintenance, a’Q’, and a”Q”
  • repair and Maintenance)
  • crossover flows (c”O”-i’O’) between interdependent circuits, and
  • intra-period supply-vs.-demand quantity differences (a’Q”=q’, a”Q”R&M=q”)

all in one neat explanatory equation.

Each of the eight implicitly-defined terms of the above equation could be partially differentiated with respect to every other term in a multitudinous set of relations of interdependence,  However, we can let the pricing system decide who performs what task for what reward, and we can concentrate our attention on the three functional aggregates, (P’Q’) and (p’a’Q’) and (p”a”Q”), which can be easily handled.

The basic price spread ratio: Dividing equation (5a) by p’Q’, letting  stand for the basic price-spread ratio P’/p’, and letting stand for p”Q”/p’Q’, the ratio of surplus costs to basic costs of the products being sold, we have:

P’/p’ = J = a’ + (a”){(p”Q”)/(p’Q’)} = a’ + a”R

i.e.     J = a’ + a”R

Lonergan’s J Equation, giving the basic price-spread ratio, implicitly encompasses the functional relationships of the rates of monetary functionings of the basic circuit to one another and by its differential, explains how and why price spreads expand and contract.

The differential of the aggregate basic price-spread ratio: Assuming continuity and differentiating, we get an equation specifying the dependency of the rate of change of the basic price-spread ratio upon a) the acceleration factors a’ and a”, and b) the ratio of surplus-to-basic activity p”Q”/p’Q’:

dJ = da’ + a”(dR) + R(da”)[27]    (5b)

The usefulness of the Basic-Circuit Identity Equation, the J Equation, and the differential of the J Equation cannot be overemphasized.  They provide an explanatory framework.  They explain how a key section of the economy actually functions.  They point to a clear distinction between the functional concepts of basic and pure surplus.  No textbooks do this.

The Pure-Surplus Identity Equation; the identity of expansionary investment and pure surplus income: The implicit defining of flow rates of “pure” surplus investment expenditures and “pure” capital outlays is represented by the following formulation.

Π”Κ”Expansionary Pure Surplus = π”a”Κ”Expansionary Pure Surplus[28]      

where Π” is the pure surplus selling price index, π” is the “profit price” index and Κ is the quantity sold.

The pure-surplus-income ratio: And, purely in its incomes aspect, letting w= I”/( I’+I”) or the ratio of all functional surplus income to the sum of both all functional surplus income and functional basic income, and letting vrepresent the fraction of functional surplus income associated with expansionary capital equipment (rather than repair and maintenance of existing capital), the formula for the pure-surplus-income ratio is

f = v[I”/(I’+I”)]

f = vw[29]

And, we have for total, expansionary, pure, surplus income and investment:

Π”Κ” = ΣF= vI” [30]  (and ΣF= Π”Κ” = π”α”Κ”)

The pure surplus income ratio is ultimately implicit in the process; it is a relation of terms implicitly defined by their functional relations.  I’ and I” are implicitly defined by their correspondent relations to one another.  is simply a fraction representing the pure-surplus part-to-the-whole of functional surplus income; and the fractions vand (1-v)simply apportion the complementary  functional pieces of total functional surplus monetary demand I”→E”→Π”Κ”Repair and Maintenance Π”Κ”Pure Surplus .

The differential of the pure surplus-income ratio: Further,

f  = v[I”/(I’+I”)] =  vw[31]

df = v(dw) + w(dv)     (6b)

Over the course of a long-term expansion, the pure surplus income correlated with expansionary spending, may move – d(ΣFi= vI”) – from zero to a large amount before declining back to zero.  The pure surplus income ratio, therefore, moves from zero, upward, then back down to zero.  Lonergan provides a graph based on the inner logic of its normative course.[32]

Pure surplus income is called by at least five different but equivalent names depending somewhat on the context, or on the aspect of it which we observe, or on the point from which we view:

  • pure surplus income, (as the correlative of pure surplus expenditure)
  • net aggregate savings, (from the view of its not being spent on consumption)
  • the social dividend, (as a functional social responsibility to be spent wisely)
  • the monetary correlate of capital expansion. (as the projection of quantities onto their payments)
  • macroeconomic profit (π”α”Κ”)as distinguished from macroeconomic costs (p’a’Q’) and (p”a”Q”), and as a nodding concession to society’s conventional psychological need to keep using the term profit)

We repeat the two differentials:

dJ = da’ + a”(dR) + R(da”)

df = v(dw) + w(dv)

With his background in mathematics, physics, and scientific method, Lonergan understood the difference between a general specification of the primary relativity of a dynamic macroeconomic system and the probabilistic coincidental boundary values (secondary determinations) of particular prices and quantities to which the general specification is applicable.  Mere reports of prices and price changes or quantities and quantity changes do not explain the economy.  They are, in and of themselves, merely isolated bits of data rather than explanatorily related data.

Lonergan (emphasized) that prices and their changes are not explanatory but accountants’ entities. (He) insists that the mechanism of the pricing system does not furnish economists with distinctions among the significant variables, any more than Galileo Galilei’s discrete measurements of distances and times at the Tower of Pisa of themselves provided the law of the acceleration of falling bodies.  In short, the lack of ultimacy that Lonergan ascribes to prices and price theory can scarcely be overemphasized. [CWL 15 Editors’ Introduction, xlvi-xlvi]

With his own background in mathematics, physics, and scientific method, McShane addresses the subject of the general specification of dynamics.

One might be reminded here of a parallel in hydrodynamics: if what is at issue is a general specification of the dynamics of free water waves, a premature introduction of general boundary conditions or worse, specific channel conditions, botches the analytic possibilities….the Robinson-Eatwell analysis is hampered … by their building the economic priora quoad nos of profits, wages, prices, etc., into explanation, when in fact the priora quoad nos[33] are last in analysis: they require explanation. [McShane 1980, 124][34]

Taking into account past and expected future values (particular boundary values in macroeconomics) does not constitute the creative key transition to dynamics.  Those familiar with elementary statics and dynamics (in mechanics) will appreciate the shift in thinking involved in passing from (static) equilibrium analysis…to an analysis where attention is focused on second-order differential equations, on

d2e/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.[35]

Prices and quantities are not explicatoria; rather they are explananda.  They don’t explain.  They require expoanation.  They are to be explained.  Their combination as functional velocities are implicitly defined by The Basic-Circuit Identity Equation and The Pure-Surplus Identity Equation.

P’Q’ = p’a’Q’ + p”a”Q”[36]   [37]

Π”Κ”Expansionary Pure Surplus = π”a”Κ ”Expansionary Pure Surplus[38]      

The definite integrals of the two differential equations – dJ = and df = – give the actual course of basic price spreads and pure surplus-income ratios over the course of the expansion.

Lonergan provides a careful heuristic analysis of the behavior of both and in terms of dJ and df through the phases of the normative pure cycle and through its possible distortions.

We can solve the first-order differential equations governing the behavior of and f; then a.) with the aid of the coefficients for a’, a”, and functions for the course of Q’, and Q”+Κ”in their cyclical movements, get the particular and definite algebraic law of the magnitude of the basic price spread ratio J, and b.) with the aid of functions for and win their cyclical movements, get the course of the magnitude of the pure surplus income ΣFand the pure surplus income ratio f.

The dJ and df formulae apply universally for all boundary values.  They apply in any instance.  The dJ function, with its interacting factors (a’, a”, p”Q”/p’Q’), may be applied heuristically over the course of a pure cycle to show how the basic price spread ratio expands and contracts over the phases of the cycle.  Additionally, it can be shown how, as the factors increase and decrease, misinterpretation of the increases and decreases tend to be manifested in optimism and pessimism, which in turn lead to activities which deviate from the systematic norms and distort the economy by torturing the flows of products and payments.  These distortions such as overinvestment and a stifled basic expansion, as distortions of the economic process, impose hardships on people.

In similar fashion, the df equation, df = vdw + wdv, with its four interacting factors, may be applied heuristically over the course of a pure cycle to show how the pure surplus income ratio expands systematically from zero, reaches a maximum, then contracts systematically back to zero in a dynamically-stable static phase.

Gross Domestic Product sublated by Gross Domestic Functional Flows: Finally, to explain Gross Domestic Activities we combine equations (4a) and (5a) into formula (6).  The Gross Domestic Functional Flows is a unity of four distinct functional flows, each of which can be viewed in the perspective of its differential to explain how the economy works.

GDFF = P’Q’Basic+ Π”Κ”Surplus  = p’a’Q’ + p”a”Q”R&M + π”a”Κ”Expansionary +  π”a”Κ”R&M

One formula is worth a ton of prose.  Let us close by listing one symbol for compensated factors of production plus fundamental key equations for the reader’s review.

ki= the rate of application of a factor of production[39]

qi = ΣΣqijk[40]         (1)

Σpijqij = PŸ= PQ cosA[41]

kn[f’n(t-a)-Bn] = f”n-1(t) – An-1[42]           (2)

ΔM’ = (S’ – s’O’) = ΔT’ + ΔR’ + (O’ – R’)[43]      (3)

G = O[44](4)

The condition of dynamic equilibrium

P’Q’ = p’a’Q’ + p”a”Q”[45]   [46]     (5a)

J = a’ + a”R

dJ = da’ + a”(dR) + R(da”)[47]    (5b)

Π”Κ”Expansionary Pure Surplus = π”a”Κ”Expansionary Pure Surplus[48]       (6a)

Π”Κ” = ΣF= vI” [49]

f  = v[I”/(I’+I”)] =  vw[50]

df = v(dw) + w(dv)     (6b)

dI’ = Σ((widni+ nidwi)yi[51]        (6c)

GDFF = P’Q’ + Π”Κ”=  p’a’Q’ + p”a”Q”R&M+ π”a”Κ”Expansionary Pure Surplus +π”α”Κ”R&M        (7)

[1]CWL 15, 30

[2]CWL 15, 30

[3]CWL 15, 74

[4]CWL 15, 37

[5]CWL 15, 67“Transfers to or from supply, (S’ – s’O’), tend to equal the sum of the increments of aggregate turnover magnitudes in final payments (ΔR’)and transitional payments (ΔT’).  Of these, two, the increment in transitional payments will be the larger, since for each sale at the final market there  commonly is a sale at a number of transitional markets.”  CWL 15, 67

The history of the development of money points to a preponderant role of increasing turnover magnitude in circuit accelerations

[6]CWL 15, 50

[7]CWL 15, 158

[8]Basic and surplus cost indices p’ and p” respectively are defined implicitlyby equations. They are not defined by their relation to us as we get and spend every day.  Their macroeconomic functional definitions cannot be found in Webster. One must consult the equations. Their definitions are arrived at as follows:[8]

c’O’ = p’a’Q’

c”O” = p”a”Q”

 

p’ = c’O’/a’Q’

p” = c”O”/a”Q”

 

 

[9]For a reminder that we are dealing in dynamics or evolution over time, we may insert the time denominator

 

(x)  dJ= da’ + a”dR + Rda

dt     dt       dt         dt

 

However, the same period denominator is common to all terms and, for interpretation of the Rule of Change of the Basic Price Spread Ratio, a period denominator is unnecessary.

(x)  dJ = da’ + a”dR + Rda”

 

[10]CWL 15, 150

[11]CWL 15, 146-48

[12]CWL 3, 147-49

[13]The increment per interval of basic income to adjust the rate of savings to the pure cycle of the process; CWL 15, 133-34

[14]CWL 15, 158

[15] where prime and double prime refer to the basic and surplus circuit respectively, Prepresents a selling price index, Q’a quantity index, p a cost index, and athe ratio of quantities produced to quantities sold.

CWL 15, 158

[16]Source ?

[17]In another context, Lonergan states: Man affirms the divine, and obscurely he knows what he means.  As best he can he expresses his meaning, but his resources for expression are unequal to the task. CWL 3, 681/

[18]CWL 15, 23-28

[19]CWL 15, 30

[20]CWL 15, 30

[21]CWL 15, 74

[22]CWL 15, 37

[23]CWL 15, 67“Transfers to or from supply, (S’ – s’O’), tend to equal the sum of the increments of aggregate turnover magnitudes in final payments (ΔR’)and transitional payments (ΔT’).  Of these, two, the increment in transitional payments will be the larger, since for each sale at the final market there  commonly is a sale at a number of transitional markets.”  CWL 15, 67

The history of the development of money points to a preponderant role of increasing turnover magnitude in circuit accelerations

[24]CWL 15, 50

[25]CWL 15, 158

[26]Basic and surplus cost indices p’ and p” respectively are defined implicitlyby equations. They are not defined by their relation to us in our getting and spending every day.  Their macroeconomic functional definitions cannot be found in Webster. One must consult the equations. Their definitions are arrived at as follows:[26]

c’O’ = p’a’Q’

c”O” = p”a”Q”

 

p’ = c’O’/a’Q’

p” = c”O”/a”Q”

 

 

[27]For a reminder that we are dealing in dynamics or evolution over time, we may insert the time denominator

 

(x)  dJ= da’ + a”dR + Rda

dt     dt       dt         dt

 

However, the same period denominator is common to all terms and, for interpretation of the Rule of Change of the Basic Price Spread Ratio, a period denominator is unnecessary.

(x)  dJ = da’ + a”dR + Rda”

 

[28]CWL 15, 150

[29]CWL 3, 147-49

[30]CWL 15, 146-48

[31]CWL 3, 147-49

[32]After considering this, we can change the magnitudes of a’. a”, and R in an example of the basic price spread ratio to represent that ratio in a growing economy.

[33]The priora quoad nos– first for us – are the things which we notice first because they are related to our sensitive selves, e.g. hot and cold, fast, slow.  The priora quoad se– first among themselves – are the things or terms which are related to each other, e.g. pressure, volume, temperature, space, time, mass, etc.

 

[34]More fully, the quote is: One might be reminded here of a parallel in hydrodynamics: if what is at issue is a general specification of the dynamics of free water waves, a premature introduction of general boundary conditions or worse, specific channel conditions, botches the analytic possibilities….the Robinson-Eatwell analysis is hampered, not only by an absence of paradigmatic heuristic thinking in a field whose principles involve ends, but also by their building the economic priora quoad nosof profits, wages, prices, etc., into explanation, when in fact the priora quoad nosare last in analysis: they require explanation. McShane, Philip (1980) Lonergan’s Challenge to the University and the Economy, (Washington, D.C.: University Press of America) P. 124[34]

 

[35]McShane, 1980, 127

[36]CWL 15, 158

[37]Basic and surplus cost indices p’ and p” respectively are defined implicitlyby equations. They are not defined by their relation to us as we get and spend every day.  Their macroeconomic functional definitions cannot be found in Webster. One must consult the equations. Their definitions are arrived at as follows:[37]

c’O’ = p’a’Q’

c”O” = p”a”Q”

 

p’ = c’O’/a’Q’

p” = c”O”/a”Q”

 

 

[38]CWL 15, 150

[39]CWL 15, 30

[40]CWL 15, 30

[41]CWL 15, 74

[42]CWL 15, 37

[43]CWL 15, 67“Transfers to or from supply, (S’ – s’O’), tend to equal the sum of the increments of aggregate turnover magnitudes in final payments (ΔR’)and transitional payments (ΔT’).  Of these, two, the increment in transitional payments will be the larger, since for each sale at the final market there  commonly is a sale at a number of transitional markets.”  CWL 15, 67

The history of the development of money points to a preponderant role of increasing turnover magnitude in circuit accelerations

[44]CWL 15, 50

[45]CWL 15, 158

[46]Basic and surplus cost indices p’ and p” respectively are defined implicitlyby equations. They are not defined by their relation to us as we get and spend every day.  Their macroeconomic functional definitions cannot be found in Webster. One must consult the equations. Their definitions are arrived at as follows:[46]

c’O’ = p’a’Q’

c”O” = p”a”Q”

 

p’ = c’O’/a’Q’

p” = c”O”/a”Q”

 

 

[47]For a reminder that we are dealing in dynamics or evolution over time, we may insert the time denominator

 

(x)  dJ= da’ + a”dR + Rda

dt     dt       dt         dt

 

However, the same period denominator is common to all terms and, for interpretation of the Rule of Change of the Basic Price Spread Ratio, a period denominator is unnecessary.

(x)  dJ = da’ + a”dR + Rda”

 

[48]CWL 15, 150

[49]CWL 15, 146-48

[50]CWL 3, 147-49

[51]The increment per interval of basic income to adjust the rate of savings to the pure cycle of the process; CWL 15, 133-34