V. Bazarov

ON “RECOVERY PROCESSES” IN GENERAL

The term “recovery process” was naturalized in our contemporary economic literature at the happy suggestion of V. G. Groman. He performed an unquestionable service in this matter, for without a clear conception of the nature of the recovery processes one can understand nothing of our present-day economy. Unfortunately, the nature of the recovery process as such, its inner pattern and logic, is precisely what most people discoursing and writing on this subject have thus far poorly grasped. On the strength of the gigantic coefficients of our present economic growth, some of them have been indulging in the most fantastic optimism while others, failing to perceive any solid base behind this growth, are becoming thoroughly ironical about the very idea of a recovery process, and go to the length of perversely denying wholly obvious facts.

Since it is not very useful to engage in polemics on these two lines determined by intuitive feelings, we trust the reader will forgive us if, with a certain pedantry, we force him to take a close look at the concept of the recovery process in its-so to speak-pure form, before turning to the concrete phenomena of concrete reality.

What, then, is the recovery process?

Any system of relations having a definite organizational structure, if some external forces have caused the deformation of that system, seeks to regain the state of equilibrium (static or dynamic), provided the deformation is not of such significant extent as to make the system fall apart.

What is the mechanism of processes of this sort?

Since the greater the deviation from equilibrium, the more strained are the inner powers of cohesion seeking to restore the deformed system to a state of equilibrium, it is obvious that the rate of speed of the “recovery process” must slacken^1. in proportion as the difference diminishes between the given state of the system and the state of its stable equilibrium. We re-emphasize, furthermore, that stable equilibrium can be not only static but dynamic, the characteristic of the latter as it applies to biotic and, specifically, societal processes being definite quantitative relationships between the parts of a normally functioning whole.

The principle just formulated is applicable to any system-mechanical, biological, or social-of organized relations. For example, a pendulum which has received a push oscillates, gradually shortening the span of its swings as it is affected by friction, until finally it comes to rest in a position of stable equilibrium. Subject to this same law of “fading oscillation” are the movements of a sounding string, the discharge of an electric battery, waves spreading out as the effect of a stone thrown into the water, “conjunctural” fluctuations of market supply and demand, and even the succession of political forms in transitional

Actually, phenomena of the type which interests us still appear to be far more common than is usually believed. As the statistical method has scored greater and greater gains in the exact sciences, the stringent laws of the old physics, chemistry, and mechanics have been proving to be no more than statistical averages, i.e., combinations which are not absolutely necessary but merely most likely, oscillating around and gravitating toward what are all manner of real processes which, in their quirky concrete diversity, can be accommodated in no hard-and-fast laws.

However, even though fading periodic oscillations are widespread in societal life, they relate in this sphere primarily to phenomena of a so-called “conjunctural,” and on the whole transitory, character. Nevertheless, the effects of conjunctural, seasonal, and other periodic or in general undulating oscillations are precisely what must be eliminated if the recovery processes which constitute the object of our investigation are to be brought out in their pure form. They themselves are not undulatory in character, are not among the periodic phenomena, and represent a particular instance of fading oscillation where the process gradually dies down, approaching a state of equilibrium, and in no case goes beyond the equilibrium line.

For a more graphic explanation of the matter, let us first analyze a fictitious example. Let us suppose that a certain country which manufactures a particular mass-consumption article (cotton fabrics, for example) from imported raw material and is unable, owing to climatic conditions, to develop domestic production of that raw material, has been subjected by its neighbors to a protracted blockade involving the particular raw material that it needs. Obviously, while the blockade is on, our hypothetical country will be compelled to replace the output which new circumstances make impossible by manufacturing some surrogate of domestic derivation. Let us assume that when a number of years have gone by the blockade is in an instant terminated and the country reverts from the surrogate to the original product. At what rate, it may be asked, will the output of this product develop if the country’s productive capabilities are unlimited?

First of all, it is clear that this transition cannot be instantaneous. However poor and unprofitable may have been the surrogate employed during the blockade, its long-time use has created certain habits which, despite the obvious preferability of the genuine product, take a certain time, albeit very brief, to disappear, clearing the way for the new contrivances. Thus at no finite distance from the level of equilibrium can the rate of the recovery process be infinitely great; however abruptly the line which graphically delineates the recovery process may ascend to the level of equilibrium, it must have a smooth appearance. It may not have acute angles, discontinuities, points of reversion, or isolated points. Nor may it have points of flection, i.e., there cannot be moments when, before it has attained equilibrium, the recovery process of its own accord, without in any way being affected by outside forces,^2. temporarily pauses and then resumes with great intensity. Thus in our example demand for the resumed output will obviously grow at its maximum rate at the very beginning; then this rate will taper off, for two reasons: first, because as a larger and larger part of the public needing the new product has its need satisfied, the unsatisfied part of the consumers will diminish quantitatively; second, because the first to gratify their wants will be those consumers who are most alert and quickest to find their bearings in the new setting, while the people who emerge as customers from then on will be those who are slower to change their habits. In other words, the growth of the process is attended by falling quality as well as numbers of the elements which it has not yet embraced. Not only are there fewer and fewer people remaining who have not been turned to account, but each person left contributes with less and less intensity to the development of the process. Because of this fact, the further the point the recovery process has reached is from the level of equilibrium, the speedier that process is, whereas as it draws nearer that level its rate steadily declines, becoming infinitely slow at an infinitely short distance from the equilibrium level. When an insignificantly small number of unsatisfied customers remains, the recovery process as such is practically over. After that the development of production is attributable to the rise in population and to its well-being, or to the lowering of prices on output, resulting in fuller gratification of needs. These are all processes of normal organic development and have nothing to do with recovery processes, which are subject to altogether different laws and, by their rate, determine the level of dynamic equilibrium sought by the recovery processes in particular elements of the societal whole, elements which are for some reason retarded in their growth and have therefore proved to be at their minimum.

Needless to say, the only way to tell whether we are confronted with a recovery process in the precise meaning of the term or a phenomenon of a different order is through a concrete analysis of available empirical data. Before the formula of the recovery process is actually applied and any practical conclusions drawn on the strength of it, it must be proved that in each given instance all the essential prerequisites for this are on hand.

But as long as we describe the recovery process solely in terms of qualitative features, the practical results of applying this concept will be rather slim. Admittedly, we shall be able to orient ourselves better in the available material on our socialized economy, gain better insight into the structure of its separate sectors or elements, clear up some seeming paradoxes, and remove some misunderstandings and prejudices. This is by no means a trifle, but it is still not enough for the practical man, and not sufficient for the theorist either. The main task of practice, and accordingly of theory as its primary tool, is not classification and qualitative description of what exists but prevision of the future, quantitative appreciation of developmental trends. To transform quality into quantity, logic into mathematics (and this is, in the final analysis, the problem of so-called exact cognition), there is no necessity in our case for recourse to any arbitrary hypotheses. It is enough to employ mathematical symbols to designate the elementary characteristics which we have just searched out in the pattern of the recovery process. Comparison of the mathematical designations thus obtained will yield the formula that expresses the general law of fee recovery process, making it possible to proceed from qualitative description to quantitative analysis.

For convenience of inquiry, we shall operate not with the absolute proportions attained through the recovery process by a given moment in time, but with the distance it still has to travel before arriving at the level of equilibrium.

Having, on a horizontal axis (see figure) moving to the right from any point arbitrarily taken as the start of the process, marked off segments proportional to the time which has elapsed in the period of observation (the letters t_l t₂, t₃,...t designate the number of days, weeks, months, or any other units of time), we shall on ordinatês drawn at each such point mark off segments proportional to the distance which at the moment of observation separates the process under study from the equilibrium level for which it is aiming. The level of equilibrium thus coincides on our drawing with axis t. The distances we shall designate by the letters X₀, X₁, X₂...X , depending on the moments in time to which they refer. The course of the restoration process is represented by curve X. The characteristics of the restoration process, with which we are already familiar, are expressed geometrically in the fact that curve X descends smoothly to the level of equilibrium, has no points of flection, acute angles, or discontinuities, and its convexity is directed steadily downward, i.e., as it approaches the level the intensity of its fall diminishes; for an infinitely long time it approaches axis t at an infinitely short distance without, however, ever intersecting it. In other words, axis tis an asymptote of tne curve.

In reality every recovery process starts at a definite moment in time, at a finite distance from the level of equilibrium. But theoretically there is nothing to prevent us from assuming that line X, symbolizing a process, extends infinitely far to the left, while retaining all the characteristics enumerated above. Sincé at a finite distance from the level of equilibrium the speed of the recovery process is inevitably finite, the ordinate of X can attain an infinitely great magnitude only at an infinitely great distance from point 0, and that means that curve X has no second asymptote perpendicular to axis t.

If, to continue, at various moments in time we measure not the actual distances separating the process from equilibrium, but the rate or speed at which these distances change, we shall find that the curve of the rate possesses the same characteristics as the curve of the process itself with one exception: if we take the distance from the level of equilibrium as a positive magnitude, then the rate of its change is a negative magnitude, for we have in this case the rate at which the distances diminish, not grow, with the passage of time. The graphic expression of this will be that the rate curve, which we shall designate with the letter X’, will have to be traced not above but below line t. For the rest, the rate curve must be characterized by all the features identified above for the curve of distances, for deviation from even one of them would with logical inevitability cause the description which we have established for curve X to be in one way or another distorted. Thus, for example, if curve X’ had a point of flection, i.e., straightened out for ãn instant in order thereafter to have its convexity turned in the other direction, it would mean that at a certain moment the distance begins to diminish not at a slackening but at a constant speed, in defiance of the basic pattern of the recovery process; if X’ did not have axis t as its asymptote, process X would not draw to an end either as it approached line t, but would seek to go beyond it, and line t would not be the level of equilibrium; if curve X’ had a second asymptote, perpendicular to axis t, at a finite distance from point 0, the speed of the recovery process would take on infinitely great magnitude at a finite distance from equilibrium, etc.

Since curve X’ possesses all the characteristics of the recovery process and is distinguished from curve X only by its location below axis t, if we invert the sketch we can rightfully take line X’ as our basic curve. From this it follows in turn that a curve X” should exist, depicting the speed with which X’ changes, the rate of the change of rate, so to speak, and having exactly the same relationship to X’ as X’has to X; it is opposite curve X in its sign (i.e., as does X it lies above line t), but is in all other respects analogous to X and X’. Deriving in exactly the same way from the existence of curve X”, clearly, is the existence of a curve X’”, expressing the rate at which X” changes, and so on ad infinitum.

Mathematicians term the rate of a process, measured at a given moment in time, its first time derivative, the rate of the change of rate the second derivative, etc. Our symbols X’, X’, and X’” consequently signify the first t derivative of X, the second derivative of X, and so forth.

The patterns deduced above may be briefly expressed as follows. X, the curve of distances, has at all points a numberless series of derivatives of consecutively higher order; the first derivate of X all the way from t = - ∞ to t = + ∞, is negative, the second X is positive, the third X”’ negative, etc.; finally, if t = - ∞. the X = - X’ = + X”= - X’”...= + ∞; if t = + ∞, then X = - X’ = + X” = - X’” = ... (Xⁿ) = 0.

By what law, then, can the magnitude of X change, if all the conditions named above must be met?

...We shall not here adduce the proofs^3. but shall simply point out that functions of one type and one type only meet all the requirements we have set; these are model functions which have the following form: X = ca^-kt , or x = c/a^kt, where a, c, and k are some constants and X and t variables (in our case distance and time).

Indeed, if the relationship between X and t is expressed by the equation just written down, then if t equals negative infinity, X equals positive infinity; if t equals positive infinity, X equals zero; all through this the curve descends smoothly to axis t, its convexity being turned toward it all along; the function has a countless number of derivatives which exactly duplicate it and differ from it only in the constant coefficient, the equivalent ratios X’/X, X”/X’, etc., furthermore, having a minus sign. Neither algebraic nor trigonometric functions meet all the conditions of our problem, nor do model functions of any other type.

There thus derives from the very pattern of the recovery process, from the combination of the most general features qualitatively distinguishing processes of this sort from all others, the law which quantitatively defines the course of these processes.

This law states: X = ca^-kt or, employing symbols more convenient for computation purposes, X = X.e*” . In this last formula X_o is the distance of the process, measured directly, from the level of equilibrium at the moment which we agreed to take as the start (the zero point on the drawing) for measuring off our observations; e is the base of the natural logarithm; k the coefficient, different for the various processes and describing their intensity; X the distance from the level of equilibrium at the moment of observation; and, finally, t the time which has elapsed from the start to the moment of observation.

As a specimen of a recovery process we cited above the example of a country which revives a branch of production after it has been blighted by an extended blockade. The vast majority of present-day recovery processes in Soviet Russia bear the same general character but are, needless to say, immeasurably more involved and complex than our systematically oversimplified examples.

We have assumed that the productive capabilities inthe given field were limitless and that the potential demand to be completely satisfied (the level of equilibrium) was precisely known in advance. Actually, rather narrow limits, defined by the productivity of this or that branch when the available means of production have the maximum possible work load, have been set for our productive capabilities within the bounds of a pure recovery process. Given the present state of information on the capital stock of our industry, this limit, which determines the level of equilibrium from the technical side, may be indicated-to be more accurate, guessed-only very roughly. On the other hand, the potential extent of demand backed by ability to pay in this or that field (the second element determining the level of equilibrium), after the profound socio-political upheaval through which the country has passed, appears highly speculative. Finally, we are not dealing with a temporary retardation of the economic process in one field, but with a severe deformation of the whole of the national economy, a deformation which has grossly disrupted all the relationships of the parts. And if, furthermore, the elements which have proved to be at their minimum go through the recovery process under a “double strain,” so to speak, pulling themselves up to the general level of equilibrium and at the same time experiencing special acceleration by virtue of their lag behind the recovery process as a whole, on the other hand-and perhaps to an even greater degree-the presence of the particular “minimums,” which appear one moment here and the next there, hampers the general course of the recovery of our national economy. The picture is rather complicated and involved and calls for close and detailed analysis....

APPENDIX

The following considerations lead to the conclusion that the characteristics of the recovery process can be expressed only by an equation of the type: X = X_o e^-kt.

(1) The ratios X’/X, X”/X’, etc., should have finite value all through the recovery process from X = ∞ to X = 0, including the limits themselves. Indeed, if X’/X turned into infinity with X equalling 0 it would mean that X’ retains finite dimensions while X becomes smaller than any given magnitude, i.e., that the axis of abscissas is not an asymptote for X’-and this conflicts with the conception of the recovery process. Likewise the ratio X’/X cannot turn into infinity with X equalling ”, for that assumes that as X grows, X’ attains infinitely great values, while X itself (and consequently t as well) still retains finite dimensions, i.e., that besides the asymptote X= 0, curve X’has still another asymptote, namely the ordinate, the equation of which is t = -a, where a is a finite magnitude; this once again conflicts with our definition of the concept of the recovery process.

(2) It follows directly from point (1) that if X = 0 the ratios X’/X, X”/X; etc. gravitate toward one and the same finite limit C, and if X = ∞, to the limits C₁, C₂, C₃, etc. (C₁, C₂, C₃...also being finite magnitudes).

Let the looked-for equation of the recovery process be F(X₁t) = 0. Having solved it in terms of t and substituted the corresponding magnitudes in the expressions X’/X, X’’/X’, X’’’/X’’, etc., we can express these last in a function of X. Let us assume that X’/X = f(X). Then X”/X’ = f’(X)X + f(X) or X”/X’ = f’(X)X + X’/X. If X = 0 (taking into account that X”/X’ and X’/X are finite), we obtain X”/X’ = X’/X. In exactly the same way we discover that X’”/X” = X”/X’. And so, if X = 0, then X’/X = X”/X’ = ... = f(o) = C. On the other hand X”/X’ - X’/X = f’(X)X, or f’(X) = 1/X[(X”/X’) -(X’/X)]. Hence, if X = ∞, then f’(X) = o, and consequently f(X) = C,. Likewise if X = ∞, X”/X’ = C₂ etc.

(3) Thus, the function X’/X = Cif t = + ∞ (X = 0) and X’/X = C₁ if t = - ∞ (X = + ∞).

Let us first assume that X’/X is not a linear function of t (from the foregoing it follows that X’/X is a continuous function having a derivative at every point).

This being the case two hypotheses are in turn possible:

(1) Within the limits of a finite interval the function X’/X has several maximums and minimums, i.e., has an undulatory character. But as has already been indicated in the text, the recovery process is an essentially nonperiodic phenomenon alien to undulatory fluctuations. We still do not know whether or not the ratio X’/X can change, but we do already know that if it does change, this change, having once begun, mustunderthe influence solely of the inner forces of the recovery process swell or diminish smoothly. In no case must it pulsate, passing through a number of maximums and minimums and points of flection separating them. And even if, when engaging in a detailed factual study of this or that concrete recovery process, one should discover closely associated with the inner pattern of that process forces which give rise to undulatory fluctuations, even then it would be necessary at the first stage of the formal analysis to isolate them as conventional and conjunctural from the “secular” smooth level of the curve of the recovery process.

(2) The only assumptions which remain, therefore, are that curve X’/X has between points C and C₁ (a) one maximum, (b) one minimum, or (c) neither. But in view of the fact that C-C₁ is a finite magnitude, and the distance between C and C₁ along axis t is infinitely great, the curve in these three cases will at every point have an infinitely small curvature, i.e. will be expressed by a linear equation.

Thus the only hypothesis compatible with the concept of the recovery process is that X’/X = at - k (a minus sign with k because X’/X, as we know, is negative). In particular, C = a(+ ∞)-k and C₁ = a(- ∞) -k, i.e. a = o. The straight line X’/X is parallel to the axis of abscissas (the coefficient k remaining indeterminate: any straight line parallel to axis t satisfied the condition).

The looked-for equation of the recovery process, consequently, has the appearance: X’/X = -k; integrating it, we get: lg (X/X_o = -kt or X = X_oe^-kt.

Thus a more precise definition of the recovery process is that the speed of its X^1. at every given moment of time is proportional to the distance from the level of equilibrium, i.e. to the magnitude of X (and not to any function of X). Deriving from this, with logical necessity, are all the qualitative attributes of the recovery process which have been enumerated in the text, as well as the logarithmic law of its course.

“O ‘vostanovitel’nykh protsessakh’ voobshche i ob ‘emissionnykh voz-mozhnostiakh’ v chastnosti.” Ekonomicheskoe obozrenie, No. 1, 1925, pp. 11-29. [The full title of the article is “On ‘Recovery Processes’ in General and on the Possibilities of Currency Emission in Particular.” Bazarov’s discussion of currency emission has been omitted. -Ed.]

1. “Rate of speed” here signifies acceleration; with acceleration diminishing, speed itself may actually increase, provided the resistance of the environment is not great.

2. The reader must not forget that we are exploring the recovery process in its pure form, i.e., are abstracting ourselves from all changes in the external environment.

3. See appendix at the end of the article.