Rate of economic growth is defined as ratio of addition to the stock of capital goods during a year to the stock of these at the beginning of the year. Let g be the rate of economic growth and I investment which is difference in the stock of capital goods at the beginning and end of a year. We, thus, have

KP

or g = I g P

or g = s (µ + R) (5.1.1)

where (µ + R) KP = g P

g P g P

g P = output [g P = ∑ ∑ g _ij P_j]

j i

KP = capital input [KP = ∑ ∑ K_ij P_j]

j i

S = total savings

We, thus, have growth rate expressed in terms of the Standard Ratio, the rate of capital depreciation and the fraction of gross income (in prices at w = 0) saved. There is some advantage in expressing it in this way. Since prices depend on the rates of profit and wage and the rate of profit is determined by investment and saving functions, it is convenient to express investment and saving in prices at w = 0 in order to find the equilibrium rate of profit and then to proceed on to the determination of relative prices at the equilibrium rate of profit. It is for this convenience that investment and saving functions have been expressed in prices at w = 0 in the last chapter. The growth rate in terms of the Standard Ratio is an expression cognate to such investment and saving functions. Besides this procedural convenience, there is independent advantage of the expression of growth rate in terms of the Standard Ratio which shows productivity of the technology employed and, as key to the distribution of income which influences relative prices, provides a link between economic growth and mechanism of resource-allocation.

The higher is the fraction of gross income saved and the larger is the Standard Ratio the greater is the rate of economic growth. The lower the rate of profit and the smaller the length of life of capital equipments, the higher is the rate of capital depreciation. In course of economic growth life of capital equipments may lengthen and the rate of capital depreciation may fall. The increasing amount of capital and the rising ratio of fixed capital to variable capital in the growth process may, however, augment the size of depreciation funds and provide opportunity for ‘deepening’ of capital-employment of increasingly capital-intensive technology. If the Standard Ratio declines, the growth rate would rise only when the savings ratio increases sufficiently to more than compensate the fall in the Standard Ratio.

We now express rate of economic growth in conditions of prices corresponding to positive wage rate. We, thus, have

g = I

KP

or g = I g P

or g = I (µ + r) KP +L ω

g P KP

or g = s′ ( µ + r + L ω ) (5.1.2) KP

where (µ + r) KP + L ω = g P

g P g P

In this form the growth rate is found to depend on the rate of profit, the ratio of wage capital to non-wage capital, the rate of capital depreciation and the fraction of gross income (in price at positive wage rate) saved. The Standard Ratio in the previous expression is replaced by the rate of profit and the ratio of wage and non-wage capital with corresponding change in the savings ratio. In process of economic growth it is crucial whether wage capital increases more or less than non-wage capital, for it shows short-term labour-absorption capacity of the process.

Let us now express growth rate in terms of the fraction of net income (in prices corresponding to positive wage rate) saved. We then have

g = I

KP

or g = I g P- µ KP

g P- µ KP KP

or g = I KPr + L ω

g P - µ KP KP

or g = s′′ ( r + L ω ) (5.2.1)

KP

where (µ + r) KP + L ω = g P

g P- µ KP g P - µ KP

We may now split in the equation above the overall savings ratio, s′′, into s₁, the portion of profits saved, and s₂, the portion of wage income saved. We, thus, have

The equilibrium rate of growth is equal to the optimum growth rate in the

optimum state and less than the optimum growth rate in super-optimum and sub-optimum states.

In conditions of equilibrium growth every industry earns equilibrium rate of profit¹ and normal prices² prevail notwithstanding whether the economy is in sub-optimum, optimum or super-optimum state. In conditions of optimum growth production capacity is in full use. Expressions (5.4.1) and (5.4.2) give us rate of optimum growth in initial conditions of underemployment and unemployment and in those of full employment respectively.

The paths of economic growth described by the various expressions of equilibrium economic growth depend, except in one case (5.1.1), on the rate of profit determined by the wage-profit curve and investment curve. If we have extreme classical saving postulate for net income saved, the rate of economic growth is equal to the rate of profit. The growth path in this case is given by the intersections of the wage-profit and investment curves, as in figure 4.3.1 in the last chapter. If we have proportional or distributive saving postulate for gross or net income saved in terms of prices at positive wage rate, the growth path is described with the help of the rate of profit determined by the wage-profit and investment curves. We can do without wage-profit and investment curves if we take proportional saving postulate for the fraction of gross income saved where income is expressed in prices at zero wage rate. In this case the Standard Ratio is calculated directly from the production equation, the rate of capital depreciation is calculated at a rate of profit equal to the Standard Ratio and the savings ratio implies equality between desired saving and desired investment. The optimum growth rate, on the other hand, gives us a trend line determined by growth of output-capacity and growth of labour, posited equal for the optimum. The following diagram shows the relation between equilibrium and optimum growth rate:

1. as defined in chapter 4.

In the diagram above the points of tangent indicate expanded scale of operations. The initial upward movement of the equilibrium growth path shows that the economy moves from a sub-optimum equilibrium position through continuously higher sub-optimum positions to the optimum-equilibrium position at the first peak. Moving with the optimum path for some distance the economy moves down to super-optimum equilibrium positions of the quantity of capital until it reaches a sub-optimum equilibrium position of an increased quantity of capital and then moves up through higher sub-optimum equilibrium positions to a new higher optimum. It goes down and up again in the same way as in the previous round and reaches a higher optimum, the third in the diagram. Unless the economy is highly planned to keep continuously to the optimum growth path, the equilibrium growth path is bound to deviate from the optimum for short periods. If the economy is maintained on the equilibrium path, it is certain to converge to the optimum path. Only when an economy fails to maintain itself on the equilibrium path, it oscillates about the latter and in course of oscillation either overshoots the optimum or fails to reach it.

Technological change has not yet been considered so that the upward trend of the optimum growth path shows only expansion of scale consequent on rise in the rate of accumulation, s. The Standard Ratio has the same, unique value at convergence of the two growth paths. In diverging parts of equilibrium growth path the Standard Ratio is less than its optimum value: in upward or downward movement of the equilibrium path it rises or declines because of rising sub-optimum and falling super-optimum positions. If technological change is introduced and indicated at the second and third peaks, the optimum growth path would have upward links at these two peaks with its slope rising or falling according as the optimum value, unique value, of the Standard Ratio associated with a given technology rises or falls by the changes in technology at these two points. We discuss this towards the end of chapter 9 on technology.

The two growth paths described in fig. 5.1.1 may be shown in relation to time:

The constant rate of optimum growth rests on the assumption of constant growth of the working population and absorption of unemployed and underemployed labour at constant annual rate.

If the rate of capacity growth is not equal to that of growth of labour, the optimum path as defined does not exist. The equilibrium growth path would then have higher hunches if there is no labour shortage and lower one in case of labour shortage. Its troughs are effects of time lags in installation and full use of new capital goods. These would not be affected, therefore, by disparity between rates of capital growth and labour growth. Adequacy of demand is ensured by equilibrium prices and equilibrium rate of investment all along the equilibrium path.

An optimal growth path has unique Standard Ratio determined by the technology in use. Non-optimal growth paths have varying Standard Ratio even when the technology is given. There are two types of changes in R, one determined by technology, the other by effective demand. In the first it is the profile of durable capital goods which changes with change in technology. In the second the profile remains unchanged while output capacity of plants is used in varying degrees. In the first case the value of R can be calculated from blue prints of the technology. In the second its value depends on the quantum of effective demand. Keynesian analysis is relevant here.

Aggregate demand consists of consumption and investment. Let aggregate demand we denoted by D, consumption by D₁ and investment by D₂. For D₂ we have the investment function given in (4.1.2). We can derive consumption function from the saving function given in (4.2.2):

D₁ = β′ ₀ + β′ ₁ (1-s₁) KPr + β′ ₂ (1-s₂) Lω (5.5.1)

Where β′ ₀, β′ ₁ and β′ ₂ are parameters corresponding to β₀, β₁ and β₂ of the saving function, other quantities and ratios being the same as in (4.2.2).

Taking investment function (4.1.2) and consumption function (5.5.1) together we have

^ ^K Pr

D = (a ₀ + β′ ₀ ) + [a₁ + β′ ₁ (1-s₁)] KPr + a₂ ^ + β′ ₂ (1-s₂) Lω (5.5.2) L ⁿ √1-ω

We can have aggregate supply function:

^^X = x₀ + x₁ KP + x₂ KP + x₃L + x₄K²P + x₅L² (5.6.1)

^L

where X is aggregate supply, x₀, x₁, x₂, x₃, x₄ and x₅ are parameters of which x_{4 and} x₅ are zero when output is below or upto optimum capacity of plants and positive when output is beyond their optimum capacity. The aggregate supply function is, thus, linear upto optimum capacity and non-linear beyond it.

The aggregate demand and supply functions between them determine output which, in relation to the stock of capital goods, determines the Standard Ratio in non-optimal states. When aggregate demand and supply are in equilibrium at the optimum output, the economy is in the optimum state.

The growth model presented here is related to other outstanding growth models in various ways.

The first equation of our model, as given in (5.1.4), is

g = s (µ + R)

or g = s g P

This may be compared to Harrod’s and Domar’s :

g = s

C

In our model s is the same as s in the Harrod equation and a in the Domar equation. And g P corresponds to l in the Harrod and σ in the Domar model. Our model is, however,

KP C expressed as

g = s (µ + R)

The problem to which Harrod and Domar addressed themselves loses much of its importance in the analysis here. Divergence between aggregate demand and output capacity arises from disequilibrium in relative prices and investment flow. When relative prices are brought back to their position of importance with renewed prospect of equilibrium rate of investment, the problem of effective demand ceases to dominate analytical efforts.

The second equation of our model, as given in (5.4.2), is

λ = λ _p

This is comparable to the Harrod equation

G _{ω =} G _n

The natural rate of growth, G _n, in the Harrod model is practically the same as λ _p, the rate of population growth in our model except that Harrod’s concept includes technological progress as well. There are two important differences, again, First, Harrod’s natural rate is a narrow concept. It assumes initial full employment which implies existence of optimum output capacity initially sufficient to employ all the labour available. Our concept of optimum growth is free from this shortcoming. As given in (5.4.1) it is comprehensive:

λ = λp + Index (1_n) + Index (1_u)

Ί Ί

where 1_n and 1_u represent the number of underemployed and unemployed and Ί is the period during which full employment could be achieved. The relation, λ = λ _p is derived by making 1_n and 1_u equal to zero. Secondly, Harrod’s warranted rate of growth is a rigid, technological concept thrown in balance against economic forces without any essential link between the two. In our model, however, the optimum growth is a technological concept while R, the Standard Ratio, in the expression for equilibrium growth is economically as well as technologically determined so that demand-determined R may be less than technology-determined R, and equilibrium growth is equal to optimum growth only when demand-determined R is equal to technology- determined R.

In our equation (5.4.1 or 5.4.2) technological progress has been taken care of by the left hand side, that is, the optimum rate of growth. Harrod’s warranted rate of growth excludes technological change. Technological improvement is, therefore, compounded with the rate of growth of the working population to define the natural rate of growth, the maximum possible rate of growth, the rate which accords with the potential of the economy. It appears logical, however, to combine technological improvement with increase in the quantity of capital, for, if we leave aside ‘learning by doing’ technological change is embodied in capital goods.

Let y′ represent g P, output per worker, and k denote KP, capital per worker,

L L

when (µ + R) KP = g P. The equation (5.1.1) then takes the form:

g = sy′ (5.7.1)

k

From (5.4.2) and (5.4.3) we have

g = λ_p (5.7.2)

if we exclude the inequality (5.4.3) arising from sub-optimum and super-optimum states.

From (5.7.1) and (5.7.2) we, thus, have

y′ = λp (5.7.3)

k s

If we now assume that an infinite continuum of alternative techniques are available so that y′ in (5.7.3) is continuously variable, we have expression (5.7.3) as the neo-classical model.

k

It is evident that the main difference between our model and the neo-classical model is in respect to their assumptions about technology and relative prices. First, our model is built on the assumption of a limited number of technologies being available while the neo-classical model assumes an infinite continuum of alternative techniques. In this our model is more akin to the linear programming model than to the neo-classical. In our model R, thus, varies intermittently while in the neo-calssical model y′ changes continuously. Secondly, the economy is approached

k

in our model mainly as irreducible into separate, independent parts (with provision for repair of structural breaks, if any) while the neo-classical model treats it as reducible into different parts, each of which is amenable to analysis independently of other parts. Thirdly, our model lays emphasis on technological progress- replacement of one technology by another, more productive one- while the neo-classical model confines itself to the consideration of existing alternative techniques, infinite in number, and without a gap between any two of these in order to enable producers to choose the right capital - labour ratio and so satisfy the equilibrium condition,

k = sy′ . Fourthly, the neo-classical model does not take into consideration change in relative

λ _p

prices and impact of such a change on growth while our model treats change in relative prices as very significant for growth. For consistency as well as convenience the first of our five growth equations and the one used here has, for example, growth rates and saving and investment expressed in terms of prices corresponding to r = R and ω = 0.

For comparison with input-output model we can have our model in the form:

m m

Σ y _{i j} + l _i ω = yi - Σ y _{j i} (5.8.1)

j =1 j =1

where y _{i j} = output of the j th industry bought by the i th industry

y _{i j} = output of the i th industry sold to the j th industry

y_i = output of the i th industry

l_i = homogeneous labour units employed in the i th industry

i = 1,2 ……………,n

j = 1,2 ……………,m

ω = wage rate

If we introduce commodity prices on both sides of the equation above, the left hand side will be equal to the discounted value of the right hand side. Let g be the discount factor and p _i or p _j the commodity prices.

m m

Σ y _{i j} p _j + l_i ω = (y_i p_i - Σ y _{j i} p_j ) g

j = 1 j = 1

or m

g ( Σ y _{i j} p _j + l_i ω) = y_i p_i - Σ y i _j p_j (5.8.2)

j =1

(i = 1, 2, … n)

Let G be the growth factor. The condition then for equilibrium growth is

m m

y _i - Σ y j _j > G Σ y _{i j} (5.8.3)

j = 1 j = 1

If we include in (5.8.3) the equilibrium growth condition for labour also, we have

m m

y _i - Σ y j _i ≥ G ( Σ y i _j + l_i ω) (5.8.4)

j = 1 j = 1

We now introduce prices into (5.8.4) and assume that there is no excess supply. We then have

m m

y_i p_i - Σ y j _i p_j = G ( Σ y _{i j} p_j + l_i ω ) (5.8.5)

j =1 j =1

From (5.8.2) and (5.9.5) together we have

G = g (5.8.6)

Since G = 1 + g

And g = 1 + i

Where i is the rate of interest,

we have

g = i (5.8.7)

From (5.3.1) and (5.8.7) we have

g = r = i (5.8.8)

Let us assume the rate of capital depreciation, µ, to be equal to unity in our equation (5.1.1). We, thus, have

g = s (1 + R) (5.9.1)

This we may regard as Adam Smith’s growth model based on purely circulating capital¹.

If in (5.9.1) we replace R by its equivalent from the wage-profit equation and take wage rate in place of wage proportion, we have

g = [y′ (1 + r) – ω] (5.9.2)

y′- ω

where ω is wage rate and y′ output per unit of labour in terms of prices corresponding to zero wage.

In this form the model brings out crucial significance of two elements in economic growth: rate of accumulation, s, and profit factor, 1 + r, by which the excess of output per labour unit over the wage rate increases in the process of economic growth.

Our net output, YP - µKP, is what Ricardo calls ‘gross produce’.¹ He divides the latter into two parts: ‘net produce’ and ‘circulating capital’. Ricardo’s ‘net produce’ could be represented by YP - µKP - L ω and his ‘circulating capital’ by L ω.

Let us have the equation for equilibrium growth (5.1.2) rewritten here:

g = s′ (µ + r + L ω )

KP

Where s′ is the ratio of saving to gross output at prices corresponding to positive wage rate.

In the equation above L ω represents the ratio between wage capital and non-

KP

wage capital. It corresponds closely to the ratio between circulating capital and fixed capital, a crucial ratio in what we consider the model of Ricardo who, taking of course the optimum

1. Hicks: Capital and Growth, London 1965 p. 40.

capacity of fixed capital and optimum level of circulating capital, treats the ratio as constant in his chapter on wages and as variable in his chapter on machinery. The implication is that the ratio is constant in the short run with technique of production unchanged and variable in the long run on account of technological change.

With rates of profit, accumulation and wage all high, production increases and demand for labour rises. The wage rate goes up. Production expands. Land being fixed in quantity, average returns to additional units of labour diminish and reduce the 'net produce’, YP-µKP-Lω. Since rate of profit is ‘net produce’ divided by the amount of capital, we have

r = g P-µKP-Lω (5.10.2)

KP

KP/Lω (5.11.1)

In the equation above KP/Lω denotes organic composition of capital, KPr/Lω rate of surplus value and s’ the degree of accumulation.

Capitalists raise the degree of accumulation, s′, in order to acquire increased quantity of capital goods of given technology and increased quantity of labour power. They do this by ploughing back their profits. The rate of profit would fall as the economy moves to super-optimum states or capital saturation. To halt decline in the amount of profits a new technology is adopted. Organic composition of capital, KP/Lω, rises. There is direct long term relation between the degree of accumulation and organic composition of capital. Rise in the former leads to rise in the latter. Marx posited an inverse relation between organic composition of capital and the rate of profit. This is certain only in case of a single switch-point between two technologies. In case of multiple switch-points the inverse relation may not hold nor is certain the inverse relation between organic composition of capital and the Standard Ratio.

As shown in (2.3.17) in chapter 2 the Standard Ratio declines only when rise in net output per unit of labour is less than the rise in capital per unit of labour. If net-output labour ratio rises at the same rate as capital-labour ratio, the Standard Ratio is constant. If net-output-labour ratio rises more than capital-labour ratio, the Standard Ratio increases. The Marxian analysis of growth process is true in only one of the three possible cases and even in that case there is inverse relation between organic composition of capital and the Standard Ratio, not the rate of profit. For that relevant case, the one in which net-output-labour ratio rises less than capital labour ratio, we may rewrite the equation (5.1.1):

g = s (µ + R)

In the particular case in which the Marxian analysis comes true in terms of the maximum rate of profit (the Standard Ratio), the rate of economic growth would, thus, fall unless the degree of accumulation, s, out of increasing amounts of profit or surplus value appropriated, more than counteracts the decline in the Standard Ratio, R.

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