Harrod-Domar Models of Economic Growth

The Harrod Model

Professor R.F. Harrod tries to show in his model how steady (i.e., equilibrium) growth may occur in the economy. Once the steady growth rate is interrupted and the economy falls into disequilibrium, cumulative forces tend to perpetuate this divergence thereby leading to either secular deflation or secular inflation.

The Harrod model is based upon three distinct rate of growth. First, there is the actual growth rate represented by G which is determined by the saving ratio and the capital-output ratio. It shows short-run cyclical variation in the rate of growth. Secondly, there is warranted growth rate represented by Gw which is the full capacity growth rate of the income of an economy. Lastly, there is the natural growth rate represented by Gn which is regarded as the ‘the welfare optimum’ by Harrod. It may also be called the potential or the full employment rate of growth.

The Actual Growth Rate: In the Harrodian model the fist fundamental equation is:

                      GC = s

Where G is the rate of growth of output in a given period of time and can be expressed as ΔY/Y; C is the net addition to capital and is defined as the ratio of investment to the increase in income, i.e., I/ΔY and s is the average propensity to save, i.e., S/Y. Substituting these ratios in the above equation we get:

                      ΔY/Y * I/ΔY = S/Y or I/Y = S/Y or I=S

The equation is simply a re-statement of the truism that expost (actual, realized) saving equals expost investment.

The above relationship is disclosed by the behavior of income. Whereas S depends on Y, ‘I’ depends on the increment in income (ΔY), the later is nothing but the acceleration principle

The Warranted Rate of Growth: The warranted rate of growth is, according to Harrod, the rate “at which producers will be content with what they are doing”. It is the “entrepreneurial equilibrium; it is the line of advance which, if achieved, will satisfy profit takers that thy have done the right thing”. Thus this growth rate is primarily related to the behavior of businessmen to sell what they have produced and they will continue to produce at the same percentage rate of growth. Thus it is the path on which the supply and demand for goods and services will remain in equilibrium, given the propensity to save. The equation for the warranted rate is

GwCr= s

Where Gw is the “warranted rate of growth” or the full capacity rate of growth of income which will fully utilize a growing stock of capital that will satisfy the entrepreneurs with the amount of investment actually made. It is the value of ΔY/Y. Cr. The ‘capital retirements’ denotes the amount of capital needed to maintain the warranted rate i.e., required capital out-put ratio. It is the value of I/ ΔY or C, s is the same as in the first equation i.e., S/Y.

The equation, therefore, states that if the economy is to advance at the steady rate of Gw that will fully utilize its capacity, income must grow at the rate of s/Cr per year, i.e., Gw = s/Cr.

If income grows at the warranted rate, the capital stock of the economy will be fully utilized and entrepreneurs will be willing to continue to invest the amount of saving generated at full potential income. Gw is, therefore, a self-sustaining rate of growth and if the economy continues to grow at this rate it will follow the equilibrium path shown in fig 40.1. In the figure, the horizontal axis represents income and the vertical axis saving and investment. The change in income from Y1 to Y2 induces investment I1 to equal saving S1 at A. This investment, in turn, raises income to Y3 which induces I2 to equal S2 at B. I2 in turn raises income to Y4 which induces I3 to equal S3 at C (Y4 income level). In this way the economy moves on the growth path. The point of intersection of the investment line (I) and the line parallel to the Y-axis indicated the required investment that is forthcoming. ‘The greater the proportion of saving, the greater must the rate of increase in output be to induce sufficient investment to maintain equilibrium if we assume no change in the investment coefficient’.

Genesis of Long-run Disequilibria:

For full employment equilibrium growth, the actual growth rate G must equals Gw, the warranted rate of growth that would give steady advance to the economy and C (the actual capital goods) must equal Cr (the required capital goods ) for steady growth.

If G and Gw are not equal, the economy will be in disequilibrium. For instance, if G exceeds Gw, then C will be less than Cr. When G>Gw, shortages result. “There will be insufficient goods in the pipeline and/ or insufficient equipment”. Such a situation leads to secular inflation because actual income grows at a faster rate than that allowed by the growth in the productive capacity of the economy. It will further lead to a deficiency of capital goods, the actual amount of capital goods being less than the required capital goods (C<Cr). Under the circumstances, desired (planned, intended or ex-ante) investment Cr would be greater than realized (expost) investment C, and production would fall short of aggregate demand. There would thus be chronic inflation.

If, on the other hand, if G is less than Gw, then C is greater than Cr. Such a situation leads to secular depression because actual income grows more slowly than what is required by the productive capacity of the economy leading to an excess of capital goods (C>Cr). This means that desired investment is less than realized investment and that the aggregate demand falls short of aggregate supply. The result is fall in output, employment and income. There would thus be chronic depression.

Harrod states that once G departs from Gw, it will depart farther and farther away form equilibrium. He writes: “Around that line of advance which if adhered to would alone give satisfaction, centrifugal forces are at work, causing the system to depart further and further from the required line of advance”. Thus the equilibrium between G and Gw is a knife-edge equilibrium. It follows that one of the major task of public policy is to bring G and Gw together in order to maintain long-run stability. For this purpose, Harrod introduces his third concept of natural rate of growth.

The natural Rate of Growth:

It “is the rate of advance which the increase of population and technological improvements allow”. The natural rate of growth depends on the macro variables like population, technology, natural resources and capital equipment. In other words, it is the rate of increase in output at full employment as determined by a growing population and the rate of technological progress. The equation for the natural rate of growth is :

Gn, Cr= or =/ s

Here Gn is the natural or full employment rate of growth.

Divergence of G, Gw, and Gn:

Now for full employment equilibrium growth Gn = Gw = G. But this is a knife-edge balance. For once there is any divergence between natural, warranted and actual rates of growth conditions of secular stagnation or inflation would be generated in the economy. If G>Gw, investment increases faster than saving and income rises faster than Gw. If G<Gw, saving increases faster than investment and rise of income is less than Gw. Thus Harrod points out that if Gw>Gn secular stagnation will develop. In such a situation Gw is also greater than G because the upper limit to the actual rate is set by the natural rate. When Gw exceeds Gn, C>Cr and there is an excess of capital goods due to a shortage of labor. The shortage of labor keeps the rate of increase in output to a level less than Gw. Machines become idle and there is excess capacity. This further dampens investment, output, employment and income. Thus the economy will be in the grip of chronic depression. Under such conditions saving is a vice.

If Gw< Gn, Gw is also less than G. The tendency is for secular inflation to develop in the economy. When Gw is less than Gn, C<Cr. There is a shortage of capital goods and labor is plentiful. Profits are high since desired investment is greater than realized investment and the businessmen have a tendency to increase their capital stock. This will lead to secular inflation. In such a situation saving is a virtue for it permits the warranted rate to increase.

This instability in Harrod’s model is due to the rigidity of its basic assumptions. They are a fixed production function, a fixed saving ratio, and a fixed growth rate of labor force. Economists have attempted to relieve this rigidity by permitting capital and labor substitution in the production function, by making saving ratio a function of the profit rate and the growth rate of labor force as a variable in the growth process.

The policy implication of the model are that saving is a virtue in any inflationary gap economy and vice in a deflationary gap economy. Thus in an advanced economy, s has to be moved up or down as the situation demands.

Points of similarity of two models:

Harrod’s s is Domar’ sα. Harrod’s warranted rate of growth (Gw) is Domar’s full employment rate of growth (σα). Harrod’s Gw= s/Cr = Domar’s σα

To prove it    Î± = S/Y or S = αY

σ   = ΔY/I or I = ΔY/ σ

S = I, and substituting S for I in the equation, we have

αY = ΔY/ σ         ( because S= αY)

Or, α σ = ΔY/Y

Therefore α σ = Gw (since Gw= ΔY/Y)

We have proved mathematically that Harrod’s Gw is the same as Domar’s ασ. But, in reality, Domar’s rate of growth r = αs is Harrod’s Gw and Domar’s  r=ασ is Harrod’s natural growth rate. In Domar’s model s is the annual productive capacity of newly created capital which is greater than  Ïƒ which is the net potential social average productivity of investment. It is the lack of labor and other factors of production which reduces Domar’s growth rate from r= αs to r = α σ. Since labor is involved in σ therefore Domar’s potential growth rate resembles Harrod’s natural rate. We may also say that the excess of s over σ in Domar’s model expresses the excess of Gw over Gn in Harrod’s model.

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