G. A . Fel’dman

THE ANALYTICAL METHOD OF CONSTRUCTING PERSPECTIVE PLANS

1. ECONOMIC BASIS OF THE ANALYTICAL METHOD OF PLANNING

...The equations that follow from our schema were derived and discussed in our article published in Planovoe khoziaistvo, 11 and 12, 1928.¹

For continuity with that article and for those who cannot make use of it, we shall present the equations, after some short explanations.² To simplify the system of symbols we shall denote the growth rate of a function [first derivative of a function] by a single prime sign, the growth rate of a growth rate [the second derivative of a function] by a double prime sign. Thus the rate of growth of D will be denoted by D’, the rate of growth of D’ by D”.

We define a growth rate as the ratio of the increment of any variable per unit of time to the growing variable itself. In mathematical language it is the ratio of the first derivative to the function; in other words, it is the derivative of the logarithm of the function. One may consider it a most interesting property of growth rates that in many respects they resemble logarithms. Thus, for example, the growth rate of the product of two functions is equal to the sum of the growth rates of the factors. On a semilogarithmic scale, a function that increases or decreases at a constant rate is represented by a straight line.

The level of national income and of each of its parts clearly depends upon the size of the corresponding capitals (D_p depends on K_p and D_u depends on K_u) and on the productivity of labor.

We shall express this dependence by the following equations:

We define the coefficients S, S_p, and S_u as the coefficients of the effectiveness of utilization of the corresponding capital; e, e_p, and e_u as the productivity of labor (output per worker); n, n_p, and n_u as the corresponding number of workers.

The nature of the coefficients of the effectiveness of utilization of capital and of labor productivity are defined by the following expressions:

To simplify the mathematical analysis we assume that the processes of production, consumption, and accumulation are continuous. The equation which describes the distribution of new capital will then have the following form:

where A_mu and A_mp are amortization due to obsolescence.

From the division itself is determined the part D_p − α_p which goes to sector u to satisfy the direct needs of that portion of workers, white-collar employees, and others engaged in u in one capacity or another. When we calculate in constant prices, p ⋅ D_p³ determines that part of D_u produced for the account of the value of net output of p, at the expense of its accumulation. The share of u in productive accumulation is then expressed by the formula:

α_u ⋅ D_u = D_u − α_p ⋅ D_p.

Essentially, these formulas exhaust what our schema can yield directly.

However, the following formulas which are derived mathematically from the above fundamental formulas and which establish the mutual relationships of the growth rates of the elements of u and p are also interesting:

where A_m = a_m ⋅ K, etc.

Finally we introduce the most general formulas of the rate of growth of national income as a whole

and with A_m = 0

In analyzing the class relationships of the capitalist economy one must keep in mind that D_p = D_pv + D_pm, i.e., that consumption equals consumption of the proletariat + consumption of the bourgeoisie. The total surplus value is:

D_m = D_pm + D_u.

The rate of surplus value is then expressed by D_m/D_pv and the rate of profit by D_m/K.

The connection between these two Marxian categories is expressed by the formula:

but

In Table 1 we show in concrete figures how the rate of surplus value must increase with a decline of the effectiveness of capital, depending on the level of the annual rate of profit. We also introduce a calculation of the organic composition of capital [organicheskogo stroeniia kapitala] (OSK).

One can easily derive the dependence of the rate of surplus value D_m/D_v on the organic composition of capital, given the turnover period of variable capital (n) and the effectiveness of utilization of capital (S).

The organic composition of capital is expressed as follows:

Table 1

On the basis of these formulas we have calculated the organic composition of capital, having arbitrarily chosen n = 2.

2. APPLICATION OF THE ANALYTICAL METHOD TO ILLUSTRATIVE VARIANTS OF A PERSPECTIVE PLAN

The means at our disposal did not make it possible to make precise and thoroughly complete calculations, which are necessary for responsible concrete planning. Our calculations are an illustration of our theoretical presentation and they come more or less close to real planning.

We have first made calculations of the coefficients of effectiveness in the basic sectors of the economy, both according to data for the period 1925/26 to date and according to materials of the Five-Year Plan (Table 2).

The calculated coefficients show great stability and confirm our view regarding the possibility of using them for the analytical method of planning.

On the other hand our attention is forcefully brought to the notion of the necessity of a more thorough analysis of their change over time. Of particular interest is the indicated time path of the coefficients of effectiveness in industry, which shows growth for the current period and decline for the period of the Five-Year Plan.

Table 2

This decline occurs, in the main, because of a rapid relative growth of fixed capital. This is confirmed by the following series, which shows the ratio of net output to fixed capital in industry.

Assume that we aim to raise the rate of growth of D, by the end of the Five-Year Plan, from 7.5 per cent in 1927/28 to 17 per cent, as is approximately the case in the adopted Five-Year Plan. Considering that under our conditions depreciation due to obsolescence is practically zero, we then obtain the following data for D_u/D. We shall perform the calculation for two variants-for S = .45 and .75—in order to show the significance of increasing the effectiveness of utilization of capital.

We shall make use of the formula:

where D₀ = 1 (national income at the beginning of the Five-Year Plan) = .45 (see Table 2).

Assume now that S gradually increases from .45 to .75: S = .45, .51, .57, .63, .69, .79 (see Table 3).

Table 3

Then, in order to determine D_u/D one must apply the formula:

.

We can see from Tables 2 and 3 how differently one may realize the tasks depending on how we shall make use of productive capital. In the second case the consumption of the population is maintained all the time on a very high level even though it grows somewhat more slowly. In the first two years there is not only no need to accelerate accumulation, but, on the contrary, a part of the capital appears to be superfluous for the realization of the plan and can be sold abroad.

On the other hand it is obvious that it would be possible to achieve in the second case considerably higher growth rates of national income if we were to adopt the same relation D_u/D as we had in the first case.

These examples illustrate with what facility one can select, with the use of the analytical method, any type and rate of development of the economy. The only difficulty, but one that is easily overcome, consists in determining the coefficient of effectiveness. It would be necessary to do this for its separate components: (1) the intensity of labor, (2) the duration of the utilization of equipment, (3) the technical relation between net output and the volume of the required capital.

The above rough outline presented to the attention of the reader did not take into consideration the necessity of re-examining, from the viewpoint of readability, the utilization of capital in the two basic sectors p and u which follows from the relationships obtained in this way. If we were to deal with a more rigid and determined development of S_p and S_u, we should have to consider the equations:

If the magnitude of S_p or S_u turns out to be mandatory or if it is adopted in one way or another, then all the other magnitudes will be determined from the two equations presented, given S and K, and this will determine the whole structure of capital and of production as a whole.

Of course, one would have to leave abstraction and deal with concrete things and all planning would have to follow the method of successive approximations or else the analytical-synthetic method.

We have attempted to reveal, by means of the categories which we have established, the nature and the dynamics of our Five-Year Plan and we have outlined two variants: one that is least satisfactory from the viewpoint of utilization of capital but which carries on, in general and as a whole, those lines of development which took place until last year; and a second one, a high intensity variant.

The results of our calculations are contained in Table 4.

We shall present some clarification concerning this table.

The sum of national income (D) and the sum of total capital (K) were obtained directly from Gosplan data. They are expressed in 1925/26 prices. The increment of capital equal to the productively accumulated part of national income (D_u) is obtained from data concerning the growth of capital stock. The consumed part of national income (D_p) is the difference D − D_u = D_p.

The effectiveness [of utilization] of all capital is determined by the equation

The growth rates D', D'_u, D'_p, and K' as well as the ratios D_u/D and D_u/D_p can be calculated directly from the data obtained earlier and we have determined them for each October 1 as average interpolated magnitudes.

Upon closer examination of the dynamics of D_u/D, D', D'_u, and D'_p for the period of the Five-Year Plan (in the first variant) we observe the absence of symmetry in the distribution of national income. Particularly striking is the decline of the rate of growth of consumption as of October 1, 1929, and a substantial tendency to oscillation of D_p for the following years.

Table 4

The growth rate structure of national income (D_u/D_p) rises according to the plan from 1928 to 1929 from 23.8 per cent to 43.5 per cent in a jump; later on the dynamics slows down somewhat and we have the following figures: 55.9 per cent, 67.0 per cent, 72.0 per cent.

The second variant fully guarantees the construction of socialism in the indicated period....

It would be necessary to calculate a whole series of variants, changing only within the limits of the possible the coefficients of utilization of productive capital and the portion of accumulation in national income. Each variant does not require more than two to three days of work of a single qualified statistician. The chief difficulty consists in the determination of the limits for the coefficient of utilization of capital, but even this difficulty is considerably smaller than that of determining how much labor productivity will rise in the future. An outline of these limits would require the utilization of experience coefficients from the practice of the technologically most advanced countries.

In order to give greater reliability to all calculations it would be necessary to organize a more thorough study of the tendencies of the development of the coefficients of effectiveness S, S_u, and S_p in their whole structural complexity.

Having thus calculated a whole series of variants of the possible development of the productive forces and of production, one could decide on a selection of one of them depending on particular political premises.

In the present calculation we have not touched upon either the question of wages, or the question of labor productivity, or the question of the growth of the labor force. We have not done so for the following reasons: (1) given the growth of consumption that we have projected, the question of the wages fund reduces itself to the question of the class distribution of national income, which does not yield itself to mathematical elaboration. In any case, the projected growth of consumption exceeds many times anything that has ever taken place in any country and it represents in any case a full opportunity for unparalleled satisfaction of the growing needs of popular masses; (2) from a theoretical viewpoint these questions Represent nothing new on the plane on which we have undertaken our investigations. They have been clarified by us in a previous statement.

We believe that we have elucidated with sufficient thoroughness, and that we have sufficiently justified, the possibilities presented by the analytical method of planning the economy suggested by us. In our opinion it would be quite inadmissible to ignore them and save a few thousand rubles by not using them, considering that they can reveal with exhaustive thoroughness the entire process of our material development in an abstract form and give us a clear, logical foundation for all our plans, calculated for many years to come, whose fulfillment will cost hundreds of billions of rubles. Without such a scale of our economic plans, theoretical errors could cost us hundreds and hundreds of millions of rubles.

1. G. A. Fel’dman, “K teorii tempov narodnogo dokhoda.” Pianovoe khoziaistvo, nos. 11 and 12, 1928. The first part of this article is included in Part I of this book (Macro-economic Models); the second part is in Part IIB (Economic Growth: Pace and Efficiency).

2. Here is a list of the symbols used in this article by Fel’dman. The symbols are not always identical with those used in “K teorii tempov narodnogo dokhoda.”—Ed.

u = producers’ goods sector

p = consumers ’ goods sector

K = total capital (gross fixed assets and circulating capital)

D = net output

α_u, α_p = fractions of net output destined for accumulation

D_u = accumulation

D_pv = consumption of workers

D_pm = consumption of bourgeoisie

D_p = D_pv + D_pm = total consumption

D_m = D_pm + D_u = surplus product

e, e_p, e_u = output per man

n, n_p, n_u = number of workers

A_mu, A_mp = amortization due to obsolescence

OSK = organic composition of capital

effectiveness of utilization of capital

' = rate of growth of a function (e.g., D', K', etc. )

Certain errors in the mathematical formulas in the original text have been corrected.—Ed.

3. This symbol “p” presumably stands for price index deflator, the reciprocal of the price index number.—Ed.

4. OSK is here defined as the ratio of constant capital to variable capital —Ed.

“Analiticheskii metod postroeniia perspectivnykh planov,” Planovoe khoziaistvo. no. 12, 1929, pp. 95-127.