In the graphs of CWL 15, pages 121-25, it is easy to become disoriented by the symbols on the vertical axes and by the titles and annotations. In particular, one might tend mistakenly to view Q as a symbol for an absolute quantity or an accumulation rather than for a rate of flow of a quantity. Recall:
In Lonergan’s circulation analysis, the basic terms are rates – rates of productive activities and rates of payments. The objective of the analysis is to discover the underlying intelligible and dynamic (accelerative) network of functional, mutually conditioning, and interdependent relationships of these rates to one another. [CWL 15 26-27 ftnt 27]
Lonergan never used terms for magnitudes, only for rates and their accelerations (‘rates of rates’) in the Essay in Circulation Analysis. [CWL 15, 182]
But if the ultimate product qi is related by a double summation to the contributions of factors of production qijk, then the total flow of ultimate products Qi is also related by a double summation to the rates of the contributions of the factors of production Qijk, where both Qiand Qijk are instances of the form ‘so much or so many every so often.’ (CWL 15, 30)
In the graphs superscripts identify basic (‘) or surplus (“) elements. Absence of superscripts here indicates “in any case.”
Q stands for a velocity or rate of flow – a so much or so many per interval. The calculus operator of instantaneity would be d/dt. However, the editors use the simple upper-case Q to represent a flow of quantity with respect to time.
dQ stands for a change of velocity, i.e. an acceleration, which could be positive, zero, or negative; the calculus operator would be d2/dt2
dQ/Q stands for a change of velocity divided by an underlying (initial) velocity; i.e. it stands for the change of a rate compared to its base or initial rate. For example, in the case of an initial rate of .05 per interval and a change of that rate by .01 to .06 per interval, dQ/Q stands for .01/.05 = .20, i.e. a 20% increase in velocity
k, in the k -1 annotation, is a factor representing the amount of change of velocity; e.g. say, 1.05 – 1 = .05. This constant, k -1, would represent a constant rate of change in the underlying velocity, thus a constant rate of increase of the underlying rate; so, over n intervals the increase would be 1.05n; so in each of the first 3 intervals the increase factor applied to the initial rate would be 1.05, 1.052 or 1.1025, 1.053 or 1.1576. This form of increase is called a geometric increase.
r, in (1/r -1)Q1, stands for, say, (1/.95 -1)Q1 = (1.05263 – 1) times an initial rate, or, say, (1.05263 – 1) (.05) = 0.5263. This would be a “uniform acceleration” each and every interval.
Therefore, regarding Figures 24-1 and 24-2 wherein higher line corresponds to higher line and lower line to lower line:
- in Figure 24-1, a constant rate of change, k -1, of the velocity would amount to a geometric increase of the initial velocity; while a constant absolute increase of the initial velocity, (1/r -1)Q1, would amount to an increasing denominator and a constant numerator in dQ/Q, and, thus, a deceleration of the rate of increase. These affects are shown in figure 24-2.
- For Figure 24-3, with vertical axis labeled dQ”/Q”, and for Figure 24-4, with vertical axis labeled Q”, the growth rate of an underlying velocity corresponds to the increase of the rate of the underlying velocity.
(Editors’ footnote: CWL 15, ftnt 29, page30) In order to avoid confusion, … the following scheme has been adopted by the editors throughout the remainder of the text. ¶Rates of flows are generally designated by upper-case letters (primed or unprimed), and upper-case letters (primed or unprimed) always designate rates or flows, unless otherwise specifically noted. ¶Quantities are generally designated by lower-case letters (primed or unprimed), using the same letter as for rate or flow, where pertinent. At times lower-case letters are also used … to designate fractions, but these and other exceptions to the scheme will be specifically noted. … ¶Note that while this shift in notation is intended to minimize confusion, the importance of a distinct symbolism for rates or flows cannot be overemphasized.
Lonergan interprets the graphs:
Now, when the surplus stage of the process is effecting a long-term acceleration of surplus activity but as yet not affecting basic activity, one may expect successive values of Q” to increase in geometrical progression. This gives an initial period, in which the graph of dQ”/Q” is approximately a level straight line. Next, as the surplus expansion develops and devotes more and more of its activity to the long-term acceleration of the basic stage, one may expect no more than a uniform acceleration of the surplus stage. This gives a second period in which dQ”/Q” is curving downwards with successive values in decreasing geometrical progression. Thirdly, as the expansion approaches its maximum in the surplus stage, dQ” reverts to zero and Q” becomes constant. In this third period dQ”/Q” is again a level straight line but now coincidental with the x-axis; h” is zero, but h’Q” may be great for a notable period to effect a long-term acceleration of the basic stage which, however, gradually declines as replacement requirements begin to mount. [CWL 15, 121-24]
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]
Thus the titles of the graphs of velocities and accelerations have clear, unambiguous meanings:
- 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
And, thus, the Baseball Diagram, Figure 14-1, is entitled Diagram of Rates of Flow, and alternate titles suggesting velocities might be
- The Diagram of Operative Functional Flows of Products, Payments, and Financings
- The Diagram of Circulations
- The Diagram of Interdependent, Implicitly-Defining, Mutually-Conditioning Velocitous Functionings
In Functional Macroeconomic Dynamics, the Pure Cycle of Expansion stands in contrast and opposition to the trade cycle consisting of boom and systematically-necessary corrective slump. The pure cycle is an ideal and general conception. It is a normative model and framework wherein a) flows in a circular conditioning within a circuit keep pace, b) crossovers flows in a crossover conditioning between circuits balance, c) money is supplied to the supply functions by the issuing authorities in proper proportion to the expanded transactions, d) the basic expansion is implemented subsequent to the surplus expansion, and thus d) the full potentials of production and employment are achieved.
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, … 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 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)
For further interpretation of the four-phase cycle the reader should consult CWL 15, pp. 120-28, especially pp. 125-28 and footnote 165 on pp. 125-26.
Without urging the necessity of such a (particular) cycle, one may say that it is solidly grounded in a dynamic structure of the productive process; and one has only to think of the practical impossibility of calculating the acceleration ratios dQ’/Q’ and dQ”/Q” to smile at the suggestion that one should try to ‘smooth out the pure cycle.’ (CWL 15, 128)