Chapter No.50

INTRODUCTION

In October 1952, Mahalanobis developed a single-sector model based on the variables ofnational income and investment. It was further developed into a two sector model in 1953where the entire net output of the economy was supposed to be produced in only two sectors—the investment goods sector and the consumer goods sector. Next he developed the famousfour-sector model in 1955. We will discuss his two-sector and four-sector models.

TWO-SECTOR MODEL

It was Mahalanobis’s two-sector1 model which became the basis for his formulation of the four-sector model for the Second Five-Year Plan. The Mahalanobis two-sector model was based on thefollowing assumptions:

(a) It is related to a closed economy where there is no foreign trade.(b) The economy consists of two sectors: the consumer goods sector and the capital goods

1. P.C. Mahalanobis, “Some Observations on the Process of Growth of National Income.” Sankhya,September 1953. This model turned out to be almost identical with the Fel’dman model.

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 Mahalanobis Modelhe Mahalanobis Modelhe Mahalanobis Modelhe Mahalanobis Modelhe Mahalanobis Model

50

sector. There is no intermediate sector. The industries producing intermediate goods are groupedtogether with the consumer goods and the capital goods which they help to produce.

(c) There is total non-shiftability of capital equipment once installed in any of the sectors.But products of the capital goods sector can be used as inputs in the two sectors.

(d) There is full capacity production in the consumer goods sector as well as in the capitalgoods sector.

(e) Investment is determined by the supply of capital goods.(f) There are no changes in prices.

Given these assumptions, Mahalanobis divides the economy into two parts: λk , the proportionof net investment used in the capital goods sector and λc, the proportion of net investment usedin the consumer goods sector:

λk + λC = 1 . . . (1)Further, net investment (I) can be divided into two parts at any point of time (t): one, λkIt toincrease the productive capacity of the capital goods sector and λC IC of the consumer goodssector. In this way

It = λkIt + λCIt . . (2)Taking βk and βC as the output-capital ratios of the capital goods sector and the consumergoods sector respectively and β as the total productivity coefficient, the latter can be shown as

β β λ β λλ λ

= ++

k k c c

k c

But λK + λC = 1 β = βk λk + βC λC ...(3)

The income identity equation for the entire economy isYt = It + Ct ...(4)

Now, when national income changes, investment and consumption also change. The change ininvestment depends upon previous year’s investment (It–1) and so does consumption on previ-ous years consumption (Ct_1 ). So the increase in investment in period t, is ∆It=It— It–1, andincrease in consumption is ∆Ct=Ct–Ct–1. As a matter of fact, the increase in the two sectors isrelated to the linking up of productive capacity of investment and the output-capital ratio.First, the investment growth path is determined by the productive capacity of investment in thecapital goods sector (λk Ik) and its output-capital ratio (βk). So that

It – It–1 = λk βk It–1It = It–1 + λk βk It–1

or It = (1+ λk βk) It–1 ...(5)Putting different value for t (t = 1, 2, 3,....) the solutions of equation (5) are

I1 = (1 + λk βk ) I0I2 = (1 + λk βk ) I1 = (1 + λk βk ) (1 + λk λk ) I0 = (1 + λk βk)2 I0 [ Q I1 = (1+λk βk) I0]

In the same manner by putting the value of t in equation (5), we getIt = I0 (1 + λk βk )

t

It = I0 = I0 (1 + λk βk )t – I0

It = I0 = I0(1 + λk βk )t –1 ...(6)

Similarly, by putting the value of t (t = l, 2,3,. . .) in the consumption growth path ∆Ct=Ct—Ct–1 =λcβc It–1, we get

362 The Economics of Development and Planning

C1 – C0 = λCβCI0C2 – C1 = λCβCI1

and, finally Ct – C0 = λCβC (I0+I1+I2+....+It)By substituting the values of I1, I2...It in equation (6) and its related equations, the above equa-tion can be solved as

Ct – C0 = λC βC [I0 + (1 + λk βk) I0 + (1+ λk βk)2 I0 +...+ (1+λk βk )t I0 ]

= λC βC I0[1 + (1 + λk βk) + (1+ λk βk)2 +...+ (1+λk βk )t ]*

or C C It c ck k

t

k k

− =+ −L

��

�

��

0 01 1

λ βλ βλ β

( )....(7)

Now, the growth path of income for the whole economy on the basis of equation (4) is∆Yt = ∆It, + ∆Ct

or Yt–Y0 = (It-I0 ) + (Ct–C0)By substituting the values of equations (6) and (7) in the above equation, we get

Y Y I It k kt

c ck k

t

k k

− = + − ++ −L

��

�

��

0 0 01 11 1

( [ ) ]( )

λ β λ βλ βλ β

= + − +L

��

�

��I k k

t c c

k k0 1 1 1[ ) ]λ β

λ βλ β

= + −+L

��

�

��I k k

t k k c c

k k0 1 1[ ) ]λ β

λ β λ βλ β

Supposing I0 = α0Y0 and substituting it in the above equation, we have

= − = + −+L

��

�

��Y Y Yt k k

t k k c c

k k0 0 0 1 1α λ β

λ β λ βλ β

[ ) ]

or Y Y Yt k kt k k c c

k k

= + −+L

��

�

�� +α λ β

λ β λ βλ β0 0 01 1[ ) ]

or Y Ytk k c c

k kk k

t= ++

+ −L

��

�

��0 01 1 1[ {( ) }α

λ β λ βλ β

λ β ....(8)

where, Yt = gross domestic national income in year t;α0 = the rate of investment in the base year;λk = the share of net investment used in the capital goods sector;

*These are G.P. series, when they are solved, the result is the next equation. Its solution is not required.

The Mahalanobis Model 363

λc = 1—λk = the share of net investment going to the consumer goods sector;βk = incremental output-capital ratio in the capital goods sector;

βC = incremental output-capital ratio in the consumer goods sector.

The interpretative value of this model is that total investment in the economy consists of twoparts: one part λk is used to increase the production of capital goods, and the other part λc toincrease the production of consumer goods. Thus, the total investment is λk + λc = 1. The ratio

λ β λ βλ β

k k c c

k k

+

of the equation is the overall capital coefficient. Assuming βk and βc to be given, the growth rateof income will depend upon α0 and λk. Further assuming α0 (the rate of investment in the baseyear) to be constant, the growth rate of income depends upon the policy instrument λk.Given that βc>βk, it implies that the larger the percentage investment in consumer goodsindustries, the larger will be the income generated. The expression (l+λA λk)t of the equationshows, however, that after a critical range of time, the larger the investment in capital goodsindustries, the larger will be the income generated. In the beginning, a high value of λk increasesthe magnitude (l + λβk )t, and lowers the over-all capital coefficient

λ β λ βλ β

k k c c

k k

+

But as time passes, a higher value of λk would lead to higher growth rate of income in the longrun.

λ βλ β λ β

λk k

k k c ck+

=

If βc=βk , then the reciprocal of the overall capital coefficient = marginal rate of saving. Thisleads us to an important policy implication of the model that for a higher rate of investment(λk), the marginal rate of saving must also be higher. A higher rate of investment on capitalgoods in the short-run would make available a smaller volume of output for consumption, butin the long-run, it would lead to a higher growth rate of consumption.

RELATION OF THE MAHALANOBIS TWO-SECTOR MODEL WITH THE DOMAR MODEL

Mahalanobis derived his two-sector model from the Domar model. Therefore, both the modelshave a close relation. First, we present the Domar model in terms of the parameters of theMahalanobis model.The equilibrium equation for determining investment in the Domar model is

I = αYwhere, I is investment, α is the saving-income ratio and Y is the national income.The growth of investment in period t is

∆It = α∆Yt . . .(1)By taking investment in the initial period I0 = α0Y0 ...(2)

Dividing (1) by (2),∆ ∆I

I

Y

Yt t

0 0 0

= αα

.


or∆ ∆Y

Y

I

It t

0

0

0

= αα

.

orY Y

Y

I I

It t t− = −0

0

0

0

αα

.

[ Q ∆Yt= Yt – Y0 and ∆It = It – I0 ]

or

orY Y

Yt

t

− = + −0

0

0 1 1αα

αβ[( ) ]

Q

I

It t

0

1= +L

��

�

��( )αβ βand is output capital ratio

or Yt – Y0 = αα

0 Y0 [(1 + αβ)t – 1]

or Yt = αα

0 Y0 [(1 + αβ)t – 1] +Y0

or Yt = Y0 1 1 10+ + −L

��

�

��

αα

αβ{( ) }t....(3)

On the other hand, the final equation of the two-sector Mahalanobis model is

Y Ytk k c c

k kk k

t= ++

+ −L

��

�

��0 01 1 1α λ β λ β

λ βλ β( )� � ...(4)

There are certain similarities between the two models. First, the last expressions of the twoequations (3) and (4) are similar, i.e., (l + β)t and (1+λkβk )

t, since Domar’s αβ is Mahalanobis’sλkλk . Second, both use the concept of time lag. Finally, the policy conclusions of both are thesame, investment can be increased by raising the marginal savings rate.Despite these similarities, there is some difference between the two models. The Domar modelis a single-sector model, while the Mahalanobis model is a two-sector model. Mahalanobisdivides the economy into the capital goods sector and the consumer goods sector. On the otherhand, Domar treats the whole economy as one sector.

FOUR-SECTOR MODEL

The Mahalanobis model is not a growth model in the real sense, rather it is an allocation model.Being associated with the Planning Commission, Mahalanobis knew that the maximum fundsavailable for net investment during the Second Five-Year Plan would be approximately Rs 5,600crores and the aim was to provide additional employment to 10-12 million people. To these, he


added a 5 per cent per annum increase in national income during the Plan period. He furtherestimated one-third of the total investment in investment goods industries, leaving two-thirdsfor investment in the remaining three sectors of the economy. He put all this data in a simplesimultaneous equation system given below and obtained the solution which became the basisof India’s Second Five-Year Plan. The Mahalanobis model takes a four-sector economyconsisting of:

(a) the investment goods sector (k);(b) the factory produced consumer goods sector (C1);(c) the small household produced (including agricultural products) consumer goods

sector (C2); and(d) services (health, education, etc.) producing sector (C3).

These subscripts k ,1, 2, and 3 are used respectively in the model for the industries producinginvestment goods, consumer goods (both factory and household), and services.For each of these four sectors a set of three parameters is introducedβ’s (beta), i.e., βk, β1, β2, β3 —the ratios of net income generated to investment or output-capitalratios.θ’s (theta), i.e., θk, θ1; θ2, θ3, — the net investment required per engaged person or capital-labourratios.λ’s (lambda), i.e. λk, λ1, λ2, γ3,— the proportion of investment allocated to each sector or alloca-tion ratios.Further, A stands for the total amount of investment to be made for the plan period of fiveyears, E for the total increase in income and N for the total increase in employment over theplan period.Given these parametric ratios (β’s, θ’s and λ’s) and the total amount to be invested (A), an esti-mate of total income (E) and employment (N) generated in the different sectors of the economyduring the plan-period can be had on the basis of the system of equations.The equations of the model are:

E = Ek+E1+E2+E3 ....(l)N = nk+n1+n2+n3 ....(2)A = λkA + λ1A +λ2A + λ3A ...(3)

Now the increase in employment (N) in each sector isnk=λkA/θk or nkθk = λkA ....(4)n1=λ1A/θl or n1θl = λ1A ....(5)n2=λ2A/θ2 or n2θ2= λ2A ....(6)n3=λ3A/θ3 or n3θ3 = λ3A ....(7)

Substituting the values of λkA, λ1A, λ2A and λ3A in equation (3), the total investment equationbecomes

A = nkθk+ n1θ1 + n2λ2 + n3θ3-Similarly, the increase in income (E) generated in each sector can be estimated as follows:

Ek=λkA.β k ...(8)E1=λ1A.β1 ...(9)E2=λ2A.β2 ...(10)E3=λ3A.β3 ...(11)

Also, E= nkθkβk+ n1θ1β1 + n2θ2β2 + n3θ3β3 [ QλkA = nkθk and so onfrom equations 4, 5, 6, and 7]

= Y0 [(1 + η)5–1] ...(12)


In the Mahalanobis model the above equation is the final one where, η (eta) is a given 5 per centannual growth rate of income, and Y0 the initial income per year, the E is derived by applyingη rate to Y0. In the system of equations given above, A, E and N are the boundary renditions.They are constants. But at the same time they are the target variables to be achieved during theplan-period. The β’s, θ’s and λ’s are the instrument variables.The β’s and θ’s are, however, the structural parameters, determined by technological conditionsand assumed to remain constant during the plan-period. The λ’s are the allocation parameterswhich are at the choice of the planner within certain limits.In the Mahalanobis model, the allocation parameter (ratio) λk for the investment goods sector isgiven and the remaining ratios for the other three sectors (λ1, λ2, λ3) are obtained as solutions ofthe set of simultaneous equations given above. For example, as Mahalanobis explains, “the rateof increase of income or the employment generated may be treated as variables to which desiredvalue may be assigned. The model would then enable us, with the help of numerical estimatesof the various parameters, to study how the allocation ratios λ’s that is, the proportions of totalinvestment going into the different sectors should be chosen so that the desired aim can berealized.”Prof. Mahalanobis gives the following numerical solution of his model, whereA (total investment) = Rs 5,600 croresη (percentage increase in national income) =5 per cent per annumN (total employment to be created) = 110 lakhs (11-million)λk proportion of investment in investment goods industries = 1/3 (or 0.33)

The sectoral values of λ’s, β’s and θ’s are taken as

Sectors Parametersλ β θ

Investment goods (k) λk=0.33 βk=0.20 θk = Rs. 20,000Factory consumer goods (C1) λ1=0.17 β1=0.35 θ1 = Rs. 8,750Small and household industries including agriculture (C2) λ2=0.21 β2=1.25 θ2 = Rs. 2,500Services (C3) λ3=0.29 β3=0.45 θ3 = Rs. 3,750

On the basis of the given data, the amount of investment in sector k is λkA =

crores; the increase in income as a result of this investment comes to Ek= λkAβk = 185020

100× =

Rs 370 crores, while the increase in employment in sector k is of the order of nk = λkA/θk =

18501

200000 9× = . million (9 lakhs). Similarly, the allocation of increase in income, employ-

ment generate and investment for the other sectors during the planning period of 5 years inrounded figures, as calculated with the help of simultaneous equations are:


Increase in

Sectors Investment (A) Income (E) Employment (N)(Rs. crores) (Rs. crores) Million

k 1850 370 0.9 C1 980 340 1.1 C2 1180 1470 4.7 C3 1600 720 3.3

Total 5610 2900 10.0

We can sum up the Mahalanobis model thus:In a given time period, in order to achieve a certain growth rate for the economy, the totalinvestable amount has been divided in such a way that it leads to the required growth rate. Butsince the required growth rate is to be reasonably high, it can be achieved by expanding sectork and thereby producing larger quantities of investment goods. However, investment in sectork is bound to generate increased purchasing power and hence demand for consumer goodswhich require comparatively less capital but employ more labour. In this way a balance issought to be established between the investment goods sector and consumer goods sector.

A CRITICAL APPRAISAL

The above solution of the Mahalanobis model and its practical application to India in the formof the Second Five-Year Plan proves that it possesses great utility as an instrument of develop-ment planning. But it has its limitations and weaknesses.2

1. Fails to Solve any Definite Welfare Function. It is essentially an operational model. Asalready explained, it arrives at an optimal solution out of a multiplicity of solutions in relationto a preference or welfare function already prescribed. The numerical solution of the model,however, does not point towards any definite welfare function without which it is not possibleto arrive at an optimum allocation of resources.2. Arbitrary Value of λλλλλk. Mahalanobis assumes the value λk-=1/3 but he does not ascribe anycogent reason for this, and simply says that “it would not be possible to go beyond this valueunder present conditions.” He could very well choose any other value or any value for anyother allocation parameter with perhaps better results. The assumption of λ = 1/3 is, therefore,somewhat arbitrary and may not help the planners in arriving at correct solutions for the opti-mum allocation of investments of the different sectors of the economy.3. Technique not Applicable to Open Economy. Moreover, the use of λ technique suggeststhat investment is a single homogeneous fund which is utilised for a single type of investmentgoods. Since investment goods are of heterogeneous type, this requires the use of an invest-ment matrix. The λ technique can be applied as long as constant relative prices are assumed. Itcannot, therefore, be applied to a model of open economy where the system is not homoge-neous.

2. Alak Ghosh, New Horizons in Planning, pp. 41-42; see also S. Tsuru, “Some Theoretical Doubts on thePlan-Frame”, Economic Weekly, January 1957 Annual Number; and A. Mitra, ‘A Note on the MahalanobisModel’, Economic Weekly, March 16, 1957; P.N. Mathur. ‘A Note on Planning in India’, IEJ, Nov. 4, 1957.


4. Supply of Agricultural Produce not Infinitely Elastic. The Mahalanobis model is based onthe supposition that the supply of agricultural produce is infinitely elastic. This is untenable forthe supply of agricultural produce because it has failed to meet the increased demand for foodand raw materials ever since the beginning of the Second Five-Year Plan.5. Supply of Labour also not Infinitely Elastic. It also assumes an infinitely elastic supply oflabour which does not seem to be correct even though an underdeveloped country like India isfaced with the serious problem of unemployment and under-employment. What is requiredfor productive structure is not simple labour but skilled and trained labour and management.6. Production Technique not constant. Like Harrod, Mahalanobis assumes the techniques ofproduction to be constant during the Plan period. In fact, technological change is bound tooccur during the process of development. Thus his model does not seem to take us very far.7. Arbitrary Values for Structural Parameters. The values assigned to the structural parameters(the β’s and θ’s) are also arbitrary. In fact, it is extremely difficult to have a correct estimate-values of β’s and θ’s in an underdeveloped country which completely lacks in sufficient reliabledata. Moreover, the assumption of independence between capital-output ratios and capital-labour ratios is not realistic. These parameters may change in the process of development.8. Silent over Investment in a Mixed Economy. Further, the Mahalanobis model fails to guidethe planners in deciding the share of investments in the private and public sectors. It is silentwith regard to important problem of development planning in a democratic country with amixed economy.9. Ignores Factor Prices. Another important defect of this model is that Mahalanobis ignoresthe pattern of factor prices while fixing targets on the basis of his model.10. Closed Model. This model is confined to a closed economy. Mahalanobis assumed “thatthere will be no imports or exports of investment goods.” Thus he ignored the impact of foreigntrade on the variables of the model and deprived it of the element of reality.11. Neglects Demand Functions. The Mahalanobis model concentrates exclusively on the supplyfunctions and neglects the demand functions altogether. This is an unrealistic assumption andmakes the growth model incomplete. “Actually speaking, many important considerationsconnected with market forces, psychological environment, popular enthusiasm and theemergence of specific pressure points are unavoidably involved in the course of developmentplanning in a backward economy. The Mahalanobis model quietly ignores these importantproblems for the sake of mathematical simplicity.”12. Failure to Link up Investment Decisions with the Rates of Saving Required. According toK.N. Raj, one of the weaknesses of the Mahalanobis model is its failure to link up investmentdecisions with the rates of saving required. The necessity of high marginal rates of saving is oneof the main considerations in favour of capital-intensive techniques of production.13. Failure to Explain the Problem of Choice of Techniques. Prof. Raj further points out thatfrom the theoretical angle, the Mahalanobis model fails to explain the problem of choice oftechniques satisfactorily. He asks, if sector C is divided according to techniques of production,why should not sector K be similarly divided? Even in the manufacture of machine-tools thereare more or less capital-intensive techniques. The case of labour-intensive techniques couldhave been perhaps stated more pointedly.3

Conclusion. Despite these practical and theorectical weaknesses, the Mahalanobis model wasinstrumental in putting the Indian economy on the right path to development planning withthe Second Five-Year Plan and paved the way for the subsequent bolder Plans.

3. K.N. Raj “Growth Models and Indian Planning”, IER, February, 1961, pp. 260-61.

Chapter No.50

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