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### The Theory of Acceleration

The multiplier and accelerator are not rivals but parallel concepts. While the multiplier shows the effect of investment on consumption (and employment), the accelerator shows the effect of a change in consumption on investment. According to Hayek, “Since the production of any given amount of final output usually requires an amount of capital several times larger than the output produced with it during any short period (say a year) any increase to final demand will give rise to an additional demand for capital goods several times larger than the new final demand.”
 The principle of acceleration states that if demand for consumption goods rises, there will be an increase in the demand for the product. The accelerator therefore makes the level of investment a function of the rate of change in consumption. In other words, the accelerator measures the changes in investment goods industries as a result of changes in consumption goods industries.

The idea underlying the accelerator is not so much one or ever-rising demand as of a functional relationship between the demand for consumption goods and the demand for the machines, which make them. The acceleration coefficient is the ratio between the induced investments to a net change in consumption expenditure.

Symbolically α = ∆I/∆C where α stands for acceleration coefficient; ∆I denotes the net changes in investment outlays, and ∆C denotes the net change in consumption outlays.

Suppose an expenditure of 10\$ billion on consumption goods leads to an investment of 20\$ billion in investment industries, and then the accelerator is 2. It could be one or even less than that. In actual practice, however, increased expenditures on consumption goods always lead in increased expenditures on capital goods. Hence accelerator is usually more than zero. Where a good deal of capital equipment is needed per unit of output, acceleration coefficient is positive and more than unity.

Sometimes, the production of increased consumer goods does not lead to an increase in the capital equipment producing these goods. The existing machinery also wears out on account of the constant use, with the result than the increased demand for consumer goods cannot be met. In the absence of induced investment and the acceleration effects, the increase demand of consumption goods leveled off and the accelerator, which measures the effects of induced investment as a result of changes in consumption, did not seem to work during these years.

The actual basis of the acceleration principle is the knowledge that the fluctuations in output and employment in investment goods industries are greater than in consumption goods industries. Acceleration has greater applicability to the industrial sector of the economy and as such it seeks to analyze the problem as to why fluctuations in employment in the capital goods industries are more violent than those in the consumption goods industries. There would be no acceleration effects in an economy that used no capital goods. Hence, acceleration principle has been widely used to explain fluctuations in economic activity, especially in the investment goods industries.

Thus, the acceleration principles hold that investment demand in dependent on increases in output, because such increases put pressures on firms to expand their stocks of capital goods. In this theory, investment occurs to enlarge the stock of capital because more capital is needed to produce more output. Firms may be able to produce more output with existing capital through more intensive use, but there is, at any time, a particular ratio of capital to output that firms consider optimum. At any time, there is a particular ratio that is the desired ratio for the economy as a whole over time, this ratio will changes as the mix or output changes. In order to reduce the complication it is assumed that this ratio remains unchanged or constant over time. With K representing the capital stock, Y the level of output and W the capital output ratio, we have

K = W Y

K (the desired stock of capital) will change over successive time periods only with changes in output (Y). Denoting a particular time period by t, preceding time periods are t – 1 and t – 2 and future of subsequent time periods are t + 1 and t + 2. Assume that in the preceding period (t – 1) the desired capital stock was enough to produce the level of output of the period t – 1. That is

Kt-1 = W Yt-1

Its output rises from Yt-1 to Yt, the desired capital stock would also rise from Kt-1 to Kt that is:

Kt = WYt

This increase in the desired stock of capital is Kt – Kt-1

To get this increase in capital stock, additional net investment is needed – this net increase in net investment expenditure is equal to the change in capital stock, that is,

It = Kt – kt-1 ………………………. (i)

Where, It is not investment in period t. By substituting WYt for Kt and WYt-1 for Kt-1, we get

It = WYt – WYt-1 = W(Yt – Yt-1) …………………………….. (ii)

This equation simply means that investment during a particular time period (t) depends on the changes in output from ‘t-1’ to t multiplied by capital output ratio (W). If Yt > Yt-1, the equation shows that there is positive and investment during the period ‘t’. If, however, we want to show gross rather than net investment, all that it needed is to add replacement investment to both sides of the equation. This replacement investment is taken to be equal to depreciation and in shown by Dt, we have thus;

It + Dt = W (Yt – Yt-1) + Dt

The sum of It and Dt cannot be less than zero. If Ig represents gross investment in period t. We have,

Igt = W (YtYt-1) + Dt …………………………. (iii)

Investment will respond to changes in the level of output shown by the equation only if certain assumption are satisfied, the most important being the absence of excess capacity, if Xt shows the excess capacity at the beginning of the period t, we may rewrite equation (iii) as;

Igt = W (YtYt-1) + DtXt

Whatever the level of gross investment might otherwise be in t, it will be reduced by the amount of Xt. It’s the value of W(Yt – Yt-1) + Dt happened to be equal to an less than Xt then Igt would be zero, the minimum possible for gross investment in plant equipment.

## Limitations and Assumptions

The acceleration value may be considerably reduced as account of some practical limitations and assumptions:
1. No excess capacity: If there is already excess capacity in the consumer goods sector, a rise in demand for consumer goods will not lead to any induced investment or acceleration effects, because the increased demand may be met from the existing capital and machinery without producing additional capital goods. This will be a case of zero gross investment and is the typical case during the initial period of recovery phase of the trade cycle.
2. Surplus capacity: The operation of the principle depends upon the presumption that there in surplus capacity in the investment goods industries. If it were not so and no excess capacity existed in machine making industries, an increase in the derived demand for machines could not result in an increased supply of machines. Hence, the principle of acceleration depends upon very tough conditions that there shall be excess capacity in one industry (investment industry) but no excess capacity in other (consumer goods industries).
3. Capital output ratio: It is based on the assumption that there is a constant ratio of the output of consumer goods and capital equipment needed for their production. In reality thus ratio is not constant. Apart from the inventions and improvements in the technique of production, existing capital equipment may be worked more intensively. The capital output ratio also varies in different phases of the business cycle and does not remain constant.
4. Nature of demand: An increase in the demand for consumption goods must be more or less permanent in nature to have acceleration effects. A purely temporary increase in the demand for consumer goods will not lead to any addition in the capital goods. Durable capital goods are expensive and no producer will order capital goods that increases in demand in short level. This also shows that the acceleration is not based on merely technological factors but also no profit expectation.
5. Availability of resources: The working of acceleration principle is further impaired by the availability of resources and the ability of the machine making industry to produce more machines. In order that the increased demand for capital goods is followed by an increase in the production, there must be enough unemployed factors available for employment in the capital goods industries. This is possible only when there is widespread unemployment in the economy.
6. Elastic credit supply: The elastic supply of money and credit is another factor, which helps, in the smooth working of the acceleration principle. Whenever there is induced investment as a result of induced consumption, enough money and credit should be forthcoming for investment in investment goods industries. A scarcity of money and credit will raise the rate of interest and will make investment financed by borrowed funds easier. It is, therefore, essential that the rate of interest not be allowed to rise and that there is enough credit to allow for the acceleration effects to flow.