Further Comments re Functions

Note: This website contains two other sections treating the topic of functions. They should be read in conjunction with the reading of this treatment. The links are: https://wp.me/P2R9Im-aC and https://wp.me/P2R9Im-bx

The name of this website is Functional Macroeconomic Dynamics. The objective economic process is constituted by interdependent velocitous and accelerative functionings.

We wish to explain this process, rather than merely compile account balances or report statistics. We are headed towards explanatory theory.

The terms function and functioning will be used to denote the constituent economic activities of the process. The functions-activities will be implicitly defined by the functional relations in which they stand with one another. Thus, the whole explanatory structure of the current, concrete, purely dynamic process is a purely relational structure.

Our explanatory terms will be the velocities of these interdependent functions. The overall economic functioning is constituted by component velocitous functionings. These velocities – so much or so many every so often – have the status of explanatory conjugates, just as do the terms such as mass, charge, space, and time in Newtonian mechanics and in electromagnetics.

A function-activity, especially as conducted by human beings, is characterized by

  • a velocity of so much per period
  • an immanent, purely relational intelligibility
  • a vital macroeconomic purpose with regard to continuity or full use of the economic resources
  • human agency effecting the activity, with the human individual’s purpose – which may be in conflict with normative macroeconomic puropose – having its basis in some combination of the psychological and the intellectual. This human agency may be called the efficient cause in contrast to the normative formal cause, which is the intelligibility of the process prior to or apart from human adaptation or maladaptation.
  • a normativity, or ideal velocitous enactment in its interdependence with other functionings
  • a relation of conditional upon or conditional to, interdependence and interaction with, other functions
  • a differentiating, intelligible, functional form defined by its relation to other intelligible functional forms.
  • a degree of velocitous enactment or implementation relative to its ideal velocitous enactment in the economic process. Human agency may non-normatively cause excesses or deficiencies, booms or slumps, stagflation or crises.

The noun “function” is often interchangeable with the verbal noun “functioning.” Thus an activity may be called interchangeably a function or a functioning as one chooses for purposes of clarity of meaning, emphasis, or rhythm.

In one’s use of the term function, one may focus on the aspect of macroeconomic purpose and say, for example, in regard to the functional activity of saving money,

“The function of savings is to underpin the whole redistributional area of the economy. Savings are sums of money that as yet have not been spent. They are built up by savings banks, insurance companies, pension funds. They are mobilized by the agencies that float new issues of stocks and bonds, that grant credit to firms, that extend credit to consumers.” [CWL 15, 70]

However, the reader must understand that that, in addition to savings’ purposive aspect, “to underpin”, the savings function must be regarded formally as a current, dynamic activity. Thus, in the excerpt above we could deliberately employ the verbal noun and rewrite in part, “In any instance of the always-current dynamic process, the functioning of current savings is the current underpinning of the whole redistributional area of the economy.”[1][2] That is, the monetary SOURCE called functional savings is formally identified with its monetary correlative of USE for lending and investing.

The term function is, of course, also used in mathematics in two respects. In calculus we write y = mx and we call the entire equation a function. Second, we also say, “y is a function of x”. The mathematical function is a formula specifying the precise relation of y to x, meaning it specifies the correspondence of the independent variable x with the dependent variable y. One variable may be explicitly defined by the other, as y is explicitly defined in terms of x; or the variables may implicitly define each other as in x2 + y2 = 20.

In functional macroeconomics, as in calculus, the notion of correspondence is foundational.

In the course of further development of mathematics it turned out to be necessary and useful to give a more general and abstract content to the concept of function. The essence of the concept is not the dependence of quantities, by which one usually understands numbers that can be compared in a ‘less than or greater than’ relationship, but the fact of correspondence itself, on the basis of which certain objects are regarded as being assigned to certain other objects. The concept of a function is reduced to set-theoretical definitions. [VNR Encyclopedia, 1977, 107]

The relation of correspondence between one explanatory term and another defines the terms; the terms are defined by their functional correspondences and the relations of correspondence define the terms. They implicitly and mutually define each other. A distinct functioning has an analytic relation to other functionings; in fact this precise analytic relation implicitly defines the function. This is to say that our functions are not defined nominally or descriptively, but rather analytically.

In macroeconomics the precise analytical distinction between two types of functional correspondence with elements in the standard of living constitutes their precise, analytical functional relation to one another. The abstract, analytical, functional correspondences are

  1. current-determinate-point-to-current-determinate-point, and
  2. current-determinate-point-to-future-indeterminate-series.

An example of a point-to-point correspondence is the correspondence in the current period of a bale of cotton processed into a cotton shirt. An example of a point-to-series correspondence is the correspondence of a sewing machine manufactured in the current period to a series of cotton shirts to be produced in an indefinite number of future periods.

The current incorporation of the cotton in a cotton shirt vs. indeterminate future correspondence of the sewing machine with an indeterminate number of cotton shirts distinguishes different functional correspondences to elements in the standard of living and, thus, to each other. This precise, analytic distinction and relation between types of functionings is foundational and of systematic significance. This distinction virtually contains other relations that can be built upon it. The whole structure of foundational and superstructural relations will comprise a complete system of coherent relations which explain the process. The whole structure is purely relational. And the whole structure of productive correspondences and their monetary correlatives comprises a normative explanatory theory.

The identification of foundational and systematic relations in macroeconomics bears resemblance to the identification of foundational and systematic relations in Euclidean geometry and among space, time, mass, and charge in the equations of Newtonian mechanics and electromagnetics.

In the field of pure mathematics, a set of relational formulae do not stand for any physical content. The equation is simply an abstract relational form based on rules of operation on symbols. But in the application of mathematics to physics and functional macroeconomics, a relational formula will represent the real relation – (static with no time subscripts, or dynamic with time subscripts, velocities, and accelerations) – of explanatory concepts or empirical unities to other explanatory concepts or empirical unities; e.g. distance resulting from the combination of rate multiplied by time,

m (say 20) loaves of bread y yielded by the processing of each bushel of wheat x, as in

y = mx

or basic revenues correlated with basic costs, as in

[3]

And, in each particular case, the relation of one term to another is, in fact, the defining of one by the other.

Sometimes meanings of the word function overlap or shade into one another without necessarily conflicting. If there is no conflict, we may interpret a statement in two or more legitimate ways, but if there is conflict, we may declare in frustration that the statement is hopelessly ambiguous. Suppose we say, “Wheat-bread is functionally dependent on wheat for wheat flour.”: first, we may be making a simple statement of condition and composition: you can’t have wheat bread without wheat flour; second, we may be talking mainly mathematically and abstractly relating the rate of production of loaves of bread to the rate of use of bushels of wheat, thus referring to an actual formula based upon empirical data, e.g. y = mx; third, we may intend both meanings in a case where the material basis or limiting condition and the empirical quantitative relation are both designated in a somewhat sloppy and ambiguous manner, but without conflict, by the statement. “Wheat bread is functionally dependent on wheat.”[4]

An advanced exchange economy – in contrast to a primitive, hunter-gatherer economy – is a process of producing things to sell them by an exchange of products and money. The means of exchange is the dummy called money. The flows of products and their payments are unitary. They occur together. Prescinding for the moment from pointless quibbling about nuances of credit, you can’t have one without the other.

At the end of these lines of production are the finished products (which are exchanged for money; these exchanges constitute) the aggregate receipts of industry and commerce, and the aggregate expenditures of final buyers. … (The connection of expenditures and receipts) with production is immediate: they … are, so to speak, the immanent manifestation of the productive process as a process of value. … production is not a merely technical affair;… intrinsically it is an economic affair, … production for sale, production in view of and at every instant adapted to payments. [CWL 21, 114]

Thus the overall economic functioning is intrinsically a process of value and cannot be conducted without humanly-intended, functional flows of money.

Continuity or stability of the overall forward-functioning process is always at stake. As one reads the next two excerpts please refer to the Diagram of Rates of Flow.[5] Indeed, prior to the full excerpts, let us extract the end pieces from both to demonstrate our course and goal of explanation of well functioning and poorly functioning economies.

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]

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 21, 111]

Now, in full:

… In figure 14-1 the reader will notice five circles representing the monetary functions. … basic terms are defined by their functional relations. The maintaining of a standard of living is attributed to a basic process (distinct process 1), an ongoing sequence of instances of so much every so often. The maintenance and acceleration (distinct process 2, positive or negative) 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 to ownership are concentrated in a redistributive function, whence may be derived changes (distinct process 3) in the stock of money dictated by the acceleration (positive or negative) in the basic and surplus stages of the process. … 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]

On such a methodological model (i.e. implicit definition according to functional relation)…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 (in the hierarchical productive order) 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 21, 111]

In the Diagram of Rates of Flow, there are circular motions of money identifying and identified with a circuit. Normatively, compensation by entrepreneurs to employees must become incomes to employees; incomes must become purchase-expenditures; expenditures must become receipts by entrepreneurs; receipts must again become compensations in the next period to keep the process going. Continuity is at stake.

In addition to a circular flow of money, there is a crossover flow as mutually dependent circuits trade with one another.

Under the general heading of function-activity, we speak of a supply function and a demand function. In both of these designations there are functional flows of products and correlated functional flows of money. The supply function is constituted by the productive applications of factors of production by human persons and of the monetary payments of outlays to these human persons which become the incomes of these human persons. The demand function is constituted by the productive activity of vending and the monetary activity of paying purchase money.

Distinct production functionings are functionally related to other distinct production functionings. For example, production of consumer goods in a current-determinate-point-to-current-determinate-point correspondence with elements in the standard of living is dependent upon the production of its capital goods which are in a current determinate-point-to future-indeterminate-series correspondence with elements in the standard of living. Capital goods accelerate the production of an indefinite series of consumer goods. There is relation and congruence. Monetary functionings – i.e. payments made and received – are functionally correlated and functionally congruent with the distinct production functionings which they enable. Thus, in parallel with the interdependencies in production, distinct monetary functionings are interdependent with other monetary functionings.

Lonergan stated succinctly how to relate productive and monetary activities:

These differences and correlations (of a hierarchical, advanced exchange economy) have now to be projected into their monetary correlates to set up classes of payments. Thus a restrictive supposition is introduced into the argument. The productive process is now envisaged as occurring in an exchange economy. It will be supposed to be an economy of notable size, complexity, and development, with property, exchange, prices, supply and demand, money. [CWL 15, 39]

As we view the monetary flows of the current, purely dynamic process, there are the two categories of supply and demand applicable to three circuits of producing and vending – basic, ordinary surplus, and pure surplus

We specify twelve aspects of payments. We may arrange these twelve functional aspects first by division into a) basic, b) ordinary surplus, and c) pure surplus; and each of the six possible transactions has two aspects: a payout by one participant and a receipt by the other participant. Thus, we have twelve aspects of monetary activities in the operative circuits of the economic process.

We may set up trees in different orders, for example:

2 supply or demand functionings x 3 circuits: [a) basic, b) ordinary surplus, and c) pure surplus] x 2 aspects = 12

3 circuits x 2 supply or demand functionings x 2 aspects = 12

3 circuits x 2 aspects x 2 supply or demand functioning = 12

We give one arrangement of twelve aspects of payments (money flows) as follows: first by division into 3 circuits: [a) basic, b) ordinary surplus, and c) pure surplus], second, by aspects of transactions as outflow by one and inflow to another, and third, by the functioning of either supply or demand in that particular circuit

Four Basic-Circuit Monetary Categories or aspects

  1. Basic-circuit initial outlays for productive services (a ,monetary aspect of the basic supply function)
  2. Basic-circuit compensatory receipts – outlays becoming any-purpose incomes (a monetary aspect of the basic supply function)
  3. Basic-circuit expenditures to purchase basic materials-and-services (a monetary aspect of the basic demand function)
  4. Basic-circuit monetary receipts purchasing (a monetary aspect of the basic demand function)

Four Ordinary-Surplus-Circuit Functionings

  1. Ordinary-surplus -circuit initial outlays for productive services (a ,monetary aspect of the ordinary surplus supply function)
  2. Ordinary-surplus -circuit compensatory receipts – outlays becoming any-purpose incomes (a monetary aspect of the ordinary surplus supply function)
  3. Ordinary-surplus -circuit expenditures to purchase basic materials-and-services (a monetary aspect of the ordinary surplus demand function)
  4. Ordinary-surplus -circuit monetary receipts purchasing (a monetary aspect of the ordinary surplus demand function)

Four Pure-Surplus-Circuit Functionings

  1. Pure-surplus-circuit initial outlays for productive services (a ,monetary aspect of the pure-surplus supply function)
  2. Pure-surplus -circuit compensatory receipts – outlays becoming any-purpose incomes (a monetary aspect of the pure-surplus supply function)
  3. Pure-surplus -circuit expenditures to purchase basic materials-and-services (a monetary aspect of the pure-surplus demand function)
  4. Pure-surplus -circuit monetary receipts purchasing (a monetary aspect of the pure-surplus demand function)

Using another point of view, we focus first on productive activities. We arrange the current process as 3 circuits, each producing-supplying and vending into demand, with each of these 6 producing-and selling activities having a pair of monetary aspects: ouflow from one participant being inflow to another participant. Thus, we have

Six Productive Functionings:

  1. Basic-circuit production of materials-and-services (basic productive function as supply)
  2. Basic-circuit vending of materials-and-services (basic productive function as demand)
  3. Ordinary-Surplus-circuit supply of materials-and-services (ordinary-surplus productive function as supply)
  4. Ordinary-Surplus-circuit vending of materials-and-services (ordinary-surplus productive function as demand)
  5. Pure-Surplus-circuit supply of materials-and-services (pure-surplus productive function as supply)
  6. Pure-Surplus-circuit vending of materials-and-services (pure-surplus productive function as demand)

Six Correlated Transaction Aspects

  1. Basic-circuit monetary compensation – outlays as any-purpose incomes (basic monetary function for supply)
  2. Basic-circuit monetary purchase – expenditures as any-purpose receipts (basic monetary function for demand)
  3. Ordinary-Surplus-circuit monetary compensation – outlays as any-purpose incomes (ordinary-surplus monetary function for supply)
  4. Ordinary-Surplus-circuit monetary purchase – expenditures as any-purpose receipts (ordinary-surplus monetary function for demand)
  5. Pure-Surplus-circuit monetary compensation – outlays as any-purpose incomes (pure-surplus monetary function for supply)
  6. Pure-Surplus-circuit monetary purchase – expenditures as any-purpose receipts (pure-surplus monetary function for demand)

A functional activity, whether in the productive order or the monetary order, will be effected to some degree of its normative level in an intelligible system of normative relations. Human agency may be adaptive or maladaptive. That is, complementary to the classical laws of normative economic functioning, there are the probabilities of maladaptation due to difficulties of measurement, ignorance, or malice.

 

 

[1] Similarly we find the following use of the word function to indicate purpose: “The purpose or function of monetary circulating capital is to bridge the gap between payments made and payments received.” But to stick closer to the notion of current, dynamic activity, we may rewrite and say, “The actual functioning of monetary circulating capital is the actual bridging of the gap between payments made and payments received.”

 

[2] Also, this use of the participle ending in “ing” suggests a practical continuousness of the activity in the aggregate in a large economy of millions of transactions, which in turn would permit an analysis, at least for pedagogical purposes, in terms of differentials rather than in terms of differences.

 

[3] CWL 15, 156-59; Here we have entered d/dt to indicate the change in cumulative revenues and costs. The text normally omits the d/dt since it repeatedly states that we are dealing in the first place with velocities as basic rates.

[4] Also, analytically we regard a loaf of bread as an empirical unity composed of components of factors of production such as wheat, labor, management, oven usage, and heat. To the hungry it is simply bread; to the economist is an integral of compensated factors of production.

[5] CWL 15, 55