THE MODEL

G.A. Fel’dman was a Russian economist who wrote an article ‘On the Theory of National IncomeGrowth’ which was published in The Planned Economy, the journal of the Soviet State PlanningCommission (GOSPLAN) in 1928. It is a theoretical model which is concerned with long-runplanning.1

AssumptionsThe Fel’dman model is built around the following assumptions :

1. It assumes constant prices in the economy.2. Capital is assumed to be the only limiting factor.3. There are no lags in the growth process.4. There is a closed economy.5. There are two sectors in the economy—the consumer goods sector and the capital goods

sector.6. Production is assumed to be independent of consumption,7. There is no government expenditure except on consumption and investment.

1. The Fel’dman Model has been translated and interpreted in English by Evcsy D Domar. Essays in theTheory of Economic Growth (1957). Essay IX. Our analysis is based io Prof. Domar’s interpretation.

C H A P T E RC H A P T E RC H A P T E RC H A P T E RC H A P T E R

TTTTThe Fhe Fhe Fhe Fhe Fel’el’el’el’el’dam Modeldam Modeldam Modeldam Modeldam Model

49

356 The Economics of Development and Planning

8. There are no bottlenecks in the economy.9. The supply of labour is unlimited.

Given these assumptions, Fel’dman based his model on the Marxian division of the total outputof an economy (W) into category 1 and category 2. The former relates to capital goods meant forboth producer goods and consumer goods, while the latter category relates to all consumergoods including raw materials for them. The production of each category is expressed as thesum of constant capital (C), variable capital (wages), V, and surplus value S. It can be shown as

W1 = C1 + V1 + S1

+ = + += + +

W C V S

W C V S2 2 2 2

“The division of the economy between the two categories is complete, in the sense that noexisting capital can be transferred from one to another. Thus the rate of investment is rigidlydetermined by the capital coefficient and the stock of capital in category 1. Similarly, the outputof consumer goods is determined by the stock of capital and the capital coefficient of category2. Hence the division of total output between consumption and investment at any given momentdepends on the relative productive capacities of the two categories. The division of totalinvestment (that is, of output of category 1) between the two categories is, however, completelyflexible. Indeed the fraction of total investment allocated to category 1 is the key variable to themodel.”In this two-sector model he demonstrated that ifγ = the fraction of total investment allocated to category 1;I = the annual rate of net investment allocated to the respective categories, so that I=I1 + I2;t = the time measured in years;V - the marginal capital coefficient for the whole economy;V1 and V2 = the marginal capital coefficients of the respective categories;C = the annual rate of output of consumer goods;Y = the annual net rate of output of the whole economy or national income;α = the average propensity to save;α’= the marginal propensity to save;I0 , C0 and Y0=the respective initial magnitudes of these variables (when t = 0); the annual rate ofnet investment allocated to category 1 is given by I1 = γI. And since only I1 increases the capacityof category 1, it is shown by

dl

dt

I

V

I

VI It= = =

1 11

γ γ[ ]Q

In time t, total investment will grow at an exponential rateI = eγγγγγ/vt . . . (1)

In other words, total investment will grow at a constant exponential rate of γ/ V1-Similarly, the annual rate of net investment allocated to category 2 is given by I2 = (1—γ)I. AndI2 being the source of increased capacity in category 2,

dC

dt

I

V Ve I ev t v t= = − =2

2 2

11 1

( )[ ]/ /γ γ γQ

The annual rate of output of consumer goods is given by

The Fel’dman Model 357

C CV

Ve v t= + −F

�����

−01

2

111

γγ

γ( )/...(2)

The elements which determine the national income and the growth rate of the economy aregiven by

Y = I + CBy substituting the values of I and C in the above equation

Y e CV

Vev t v t= + + −F

�����

−γ γγγ

/ /( )1 10

1

2

11 [From (1) and (2)]

Y e CV

Vev t v t= − + + + −F

�����

−γ γγγ

/ /( )1 11 11

101

2

= − + + + −F��

���

−( ) ( )/ /e CV

Vev t v tγ γγ

γ1 11 1

110

1

2

= + + −F��

���

+ −[ ] ( )/11

1 101

2

1CV

Ve v tγ

γγ

Assuming I0 = 1, the equation becomes

Y I CV

Ve v t= + + −F

�����

+�

�

�� −0 0

1

2

11 11

γγ

γ( )/

or Y YV

Ve v t= + −F

�����

+�

�

���

−01

2

11 11

γγ

γ( )/ [ ]QY I C0 0 0= +

The fundamental equation shows that C and Y each represent a sum of a constant and anexponential in t. Their rates of growth will differ from γ/V1. The values of C and Y will.begreater than the value of I. With the passage of time, the exponential eγγγγγ/v1t will dominate thescene and the rates of growth of C and Y will gradually approach γ/V1. But this may take quite

a long time, unless of course it so happens that CV

V01

2

1= −( )γγ in which case the constants

will vanish, and C and Y will grow at the rate of γ/V1 from the very beginning.”

COMPARISON WITH THE DOMAR MODEL

In the Domar model, the average propensity to save (α) is equal to the marginal propensity tosave (α’), i.e., α = α’. But in the Fel’dman model α = α’. To compare the Domar model with theFel’dman model, it is necessary to rework their results without the assumption that α ≠ α’,treating α’ as a constant. But since α = α’, α has now become a variable. The rate of growth ofinvestment will now be α ’/v while that of income α /v (by disregarding the difference betweenα and s : s being the reciprocal of v). The expression α’/v is the ratio of marginal propensity to

358 The Economics of Development and Planning

save to the overall capital coefficient. In Fel’dman’s model, however, we have obtained γ/V1 asthe growth rate of investment where γ is the fraction of investment allocated to category 1 andV1 is the capital coefficient of this category only. In the special case when V=V1, we obtain α = γ,that is Fel’dman’s fraction of investment allocated to category 1 and the marginal propensity tosave become, identical. If V1>V2, than γ>α’. When V1 = V2, Fel’dman’s γ and Domar’s α ’ areclosely related. “But it is merely a reflection of the fact that if a certain fraction of increment innational income (α’) is to be devoted to investment, a corresponding fraction of investment (γ)must be allocated to capital goods industries to make the production of this increment ininvestment possible. In other words, in a growing economy some capital is used to make morecapital.”

IMPLICATIONS FOR ECONOMIC DEVELOPMENT

The Fel’dman model has important implications for economic development. Since V1= V2, theexpressions I, C and γ in the model are all inverse functions of Vl and V2. Fel’dman treated themagnitudes of his capital coefficients as variables for the purpose of economic development. Ifthe purpose of economic development is maximisation of investment or national income at apoint of time, or of their respective rates of growth, or of integrals overtime, γ should be set ashigh as possible. This is always true for investment and nearly always for income. The onlyexception being when V1 greatly exceeds V2 and even then for a short period of time. A high γdoes not imply, however, any reduction in consumption. With capital assets assumed to bepermanent, even γ = 1 would merely freeze consumption as its original level. If assets weresubject to wear, consumption would be slowly reduced by failure to replace them. Finally, atransfer of resources from consumption to investment industries would reduce consumptionstill further, though the latter possibility is excluded from the Fel’dman model.Summing up the Fel’dman model, Prof. Domar observes: “it contains an important element oftruth, a closed economy without well-developed metal, machinery and subsidiary industries(the complex of the so-called heavy industries) is unable to produce a sizable quantity of capitalgoods and thus to invest a high fraction of its income, however, high its potential saving propensitymay be. In Soviet economic thinking the former consideration has been predominant; in ourrecent literature the ability to save has been emphasized.” This was because he felt that morecould be achieved with greater utilization of capital than from its expansion as happened inRussia from 1924-25 through 1927-28. He, therefore, favoured a fall in the magnitudes of hiscapital coefficients.But his own analysis of data given in the optimal version of the First Five-Year Plan of Russiaindicated movement and variation in coefficients of specific industries, while the average forthe whole economy remained almost unchanged at 2.4 for every year for the period 1925-26 to1932-33. However, in the ‘two versions of his own long-run plan, he showed in the first analmost constant capital coefficient at 2.4 for the period 1926 to 1932 which gradually rose to 3.3in 1950; and in the second, a sharply declining coefficient to about 2.0 in 1930 which stabilisedat 1.4 over the period 1932-50. The purpose of working out such contrasting versions was toillustrate the effect of the variation in the size of the capital coefficient on the growth rate ofnational income.Treating the capital coefficient as given, the variable y (the fraction of total investment allocatedto category 1) can be varied as an instrument of planning. Since there is complete intra-categoryflexibility γ can vary between zero and one. But the choice of the optimum size of r will dependon the objective of economic development.

The Fel’dman Model 359

But the Fel’dman model does not determine the magnitude of capital coefficient because noattempt is made to relate it to any other variables, such as the desirability of the assets, thelength of the construction period, the supply of labour and of other factors like the magnitude,composition and the rate of growth of investment and the industrial structure.Moreover, it is difficult to distinguish clearly between consumer goods and capital industries,when a large number of industries are in the nature of intermediate goods industries whichhelp produce both consumer goods and capital goods. For instance, metals, coal, transportation,chemicals, petroleum, power, etc. are some of the industries whose goods and services are usedin both categories of Fel’dman. “Perhaps Fel’dman could claim that in the Russia of his day,practically all metals were used in category 1 only. But what would he say about the rest? Norwould it help to divide an industry (like coal or transportation) between two categories, becausethe respective proportions would by their very nature lack stability. Of course, any division ofan economy by industries, or even of output between consumption and investment, is difficultand arbitrary but it is clear that Fel’dman’s method creates special difficulties.”