Solow model of economic growth
Professor R.M. Solow builds his model of economic growth as an alternative to the Harrod- Domar line of thought without its crucial assumption of fixed proportions in production. Solow postulates a continuous production function linking output to the inputs of capital and labour which are substitutable.
Solow builds his model around the following assumptions:
One composite commodity is produced.
Output is regarded as net output after making allowance for the depreciation of capital.
There are constant returns to scale. In other words, the production function is homogeneous of the first degree.
The two factors of production labour and capital are paid according to their marginal physical productivities.
Prices and wages are flexible.
There is perpetual full employment of labour.
There is also full employment of the available stock of capital.
Labour and capital are substitutable for each other.
There is neutral technical progress.
The saving ratio is constant.
Given these assumptions, Solow shows in his model that with variable technical coefficient there would be a tendency for capital-labour ratio to adjust itself through time in the direction of equilibrium ratio.
If the initial ratio of capital to labour is more, capital and output would grow more slowly than labour force and vice versa. Solow’s analysis is convergent to equilibrium path (steady state) to start with any capital-labour ratio.
Solow taxes output as a whole, the only commodity, in the economy. Its annual rate of production is designated as Y (t) which represents the real income of the community, part of it is consumed and the rest is saved and invested.
That which is saved is a constant s, and the rate of saving is s Y (t). K (t) is the stock of capital. Thus net investment is the fate of increase of this stock of capital, i.e., dk/dt or K.
Since output is produced with capital and labour, technological possibilities are represented by the production function that shows constant returns to scale. Inserting equation (2) in (1), we have in equation (3), L represents total employment.
Since population is growing exogenously, the labour force increases at a constant relative rate n.
Solow regards n as Harrod’s natural rate of growth in the absence of technological change; and L (t) as the available supply of labour at time (t). The right hand side of equation (4) shows the compound rate of the growth of labour force from period o to period t.
Alternatively equation (4) can be regarded as a supply curve of labour. “It says that the exponentially growing labour force is offered for employment completely inelastically. The labour supply curve is a vertical line, which shifts to the right in time as the labour force grows according to (4).
Then the real wage rate adjusts so that all available labour is employed, and the marginal productivity equation determines the wage rate which will actually rule.”
He regards this basic equation as determining the time path of capital accumulation, K, which must be followed if all available labour is to be fully employed. It provides the time profile of the community’s capital stock which will fully employ the available labour.
Once the time paths of capital stock and of the labour force are known, the corresponding time path of real output can be computed from the production function.
Professor Solow sums up the growth process thus: “At any moment of time the available labour supply is given by equation (4) and the available stock of capital is also a datum.
Since the real return to factors will adjust to bring about full employment of labour and capital we can use the production function of equation (2) to find the current rate of output.
Then the propensity to save tells us how much of net output will be saved and invested. Hence we know the net accumulation of capital during the current period. Added to the already accumulated stock this gives the capital available for the next period, and the whole process can be repeated.”
Possible Growth Patterns. In order to find out if there is always a capital accumulation path consistent with any rate of growth of the labour force towards steady state, Professor Solow introduces his fundamental equation.
In this equation r is the ratio of capital to labour (K/L), n is the relative rate of change of the labour force (LIL). The function sF(r, 1) represents output per worker as a function of capital per worker. In other words, it is the total product curve as varying amounts r of capital are employed with one unit of labour.
The equation (6) itself states that the rate of change of capital-labour ratio (r) is the difference of two terms, one representing the increment of capital [sF(r, 1)] and the other increment of labour (nr).
Solow illustrates diagrammatically possible growth patterns based on his fundamental equation (6).
Once the capital-labour ratio is established, it will be maintained, and capital and labour will grow in proportion. Assuming constant returns to scale, real output will also grow at the same relative rate n, and output per head of labour force will be constant. At r’ there will be the balanced growth equilibrium.
Thus the equilibrium value is stable. “Whatever the initial value of the capital-labour ratio, the system will develop toward a state of balanced growth at the natural rate if the initial capital stock is below the equilibrium ratio, capital and output will grow at a faster pace than the labour force until the equilibrium ratio is approached.
If the initial ratio is above the equilibrium value, capital and output will grow more slowly than the labour force. The growth of output is always intermediate between those of labour and capital.”
Therefore, r is an unstable equilibrium position. “Depending on the initial observed capital-labour ratio, the system will develop either to balanced growth at capital-labour ratio r, or ry In either case labour supply, capital stock and real output will asymptomatically expand at rate n but around r there is less capital than around r3, hence the level of output per head will be lower in the former case than in the latter.
The relevant balanced growth equilibrium is at r, for an initial ratio anywhere between O and r2 it is at r3 for any initial ratio greater than r2. The ratio r2 is itself an equilibrium growth ratio, but an unstable one, any accidental disturbance will be magnified over time.
On the other hand, the curve depicts a highly unproductive system in which the full employment path leads to ever diminishing per capita income.
However, aggregate income rises in this system because net investment is always positive and the
Professor Solow concludes model thus: “When production takes place under the usual neoclassical conditions of variable proportions and constant returns to scale, no simple opposition between natural and warranted rates of growth is possible.
There may not be…any knife-edge. The system can adjust to any given rate of growth of the labour force, and eventually approach a state of steady proportional expansion,” i.e.
The Solow model is a major improvement over the Harrod-Domar model. The Harrod-Domar model is at best a knife-edge balance in a long-run economic system where the saving ratio, the capital-output ratio, and the rate of increase of the labour force are the key parameters.
If the magnitudes of these parameters were to slip even slightly from the dead centre, the consequences would be either growing unemployment or chronic inflation. In Harrod’s terminology this balance is poised on the equality of Gw (which depends on the saving and investing habits of households and firms) and Gn (which depends, in the absence of technical change, on the increase of the labour force).
According to Solow, this delicate balance between Gw and Gn flows from the crucial assumption of fixed proportions in production whereby there is no possibility of substituting labour for capital. If this assumption is abandoned, the knife- edge balance between Gw and Gn also disappears with it.
He, therefore, builds a model of long-run growth without the assumption of fixed proportions in production demonstrating steady state growth. Solow is a pioneer in constructing the basic neoclassical model where he retains the main features of the Harrod-Domar model like homogeneous capital, proportional saving function and a given growth rate in the labour force.
He takes a continuous production function, which has come to be known as the neoclassical production function, in analysing the process of growth. The assumption of substitutability between labour and capital gives the growth process adjustability and provides a touch of realism. Unlike the Harrod-Domar model, he demonstrates steady-state growth paths.
Last but not the least, the long-run rate of growth is determined by an expanding labour force and technical progress. Thus Professor Solow has successfully shunted aside all the difficulties and rigidities which go into the modern Keynesian income analysis.
His “purpose was to examine what might be called the tight-rope view of economic growth and to see where more flexible assumptions about production would lead a simple model.” Despite this assertion by Solow, his model is weak in many respects, as pointed out by Professor Sen.
First, the Solow model takes up only the problem of balance between Harrod’s Gw and G” and leaves out the problem of balance between G and Gw Second, there is the absence of an investment function in Solow’s model and once it is introduced, the Harrodian problem of instability quickly reappears by the Solow model.
Thus, according to Sen, the assumption of substitutability between labour and capital does not seem to be a key difference between neoclassical and neo-Keynesian studies of growth, and the main difference seems to lie in the investment function and the consequent failure to assign a major role to entrepreneurial expectations about the future.
Third, the Sojow model is based on the assumption of labour- augmenting technical progress. It is, however, a special case of Harrod-neutral technical progress of the Cobb-Douglas production function type which does not possess any empirical justification.
Fourth, Solow assumed flexibility of factor prices which may bring difficulties in the path towards steady growth. For instance, the rate of interest may be prevented from falling below a certain minimum level due to the problem of liquidity trap.
This may, in turn, prevent the capital-output ratio from rising to a level necessary for attaining the path of equilibrium growth. Fifth, the Solow model is based on the unrealistic assumption of homogeneous and malleable capital.
As a matter of fact, capital goods are highly heterogeneous and thus pose the problem of aggregation. Consequently, it is not easy to arrive at the steady growth path when there are varieties of capital goods.
Lastly, Solow leaves out the causative of technical progress and treats the latter as an exogenous factor in the growth process. He thus ignores the problems of inducing technical progress through the process of learning, investment in research, and capital accumulation.