## The generation of the accelerator coefficient

Even though the accelerator coefficient has been used in a considerable number of business cycle models, there has been relatively little in the literature on the mechanism by which its value is determined. In this chapter I wish to do the following: (a) examine statements by a number of authors as to the mechanism by which the value of the accelerator is determined, and (b) set out the relevant factors which enter into the determination of the value of the accelerator coefficient. These relevant factors will form the basis for the theoretical analysis of the behavior of business firms which follows. The examination of the statements of the mechanism which determines the value of the accelerator will not be comprehensive; rather it will be limited to the remarks of a few authors.

As was mentioned earlier, at least three different concepts as to the nature of the accelerator coefficient can be identified. These are:

1. The accelerator as a structural parameter.

2. The accelerator as a coefficient of induced investment.

3. The accelerator as a coefficient of realized investment.

Prior to taking up the rationalizations for the use of the accelerator which have been put forward, it is necessary to clarify these differing concepts.

If the aggregate accelerator coefficient is interpreted to be a structural parameter, then it is derived from the production function. Typically in such models the production function states that output is a constant times the capital stock, for example Y= uK, where u is the capital coefficient. Hence dy/dt = u [dk/dt]; I = dkldt = 1/u [dy/dt] and therefore, in this case, (3 (the accelerator coefficient), is the reciprocal of -, the capital coefficient. For this form of the accelerator coefficient to be accepted there can be no possibility of substitution among factors in production. In addition either the production function for every sector of the economy must have the same capital coefficient or the distribution of a change in income among the various sectors (the income elasticities of demand for the various outputs) must always be the same. If the distribution of a change in income among different outputs is always the same, then the aggregate accelerator coefficient is a weighted sum of the particular sectoral accelerator coefficients with constant weights.

Perhaps we should do well to get into the habit of thinking not of one accelerator coefficient, relating total output to total investment, but rather - as a more tenable approximation - of a series of such coefficients operating in each of the major industries. The task of summation will then present itself more clearly to our minds as a further complication of some importance which must be dealt with before the behavior of total investment and total output can be adequately explained. Or to make the point in a different way, the aggregate accelerator is the weighted average of the accelerator coefficients in each sector of the economy: not only are these component coefficients liable to vary themselves, but their respective weights in the national average will not be even approximately constant over the cycle.1

This implies that even if sectoral production functions can be written in a form which yields the accelerator as a structural parameter, the aggregate inducement to invest can vary.

'Suppose that within each firm the accelerator principle is working smoothly: I = u[do/dt]. Even in this highly simplified case care must be exercised in moving from the firms investment to aggregate business investment, there are likely to be wide variations in the capital coefficients between different branches of production, and if the income elasticities of demand are also markedly different with the result that output expands more in some lines than in others, the aggregate capital coefficients which relate total output to total investment will be altered accordingly. That is to say, constancy in the accelerator coefficient within each firm does not necessarily imply constancy in the aggregate accelerator coefficient.'2

Wilson's comments show the difficulties involved in taking the aggregate accelerator coefficient to be a constant even if the assumption of constant structural coefficients in the various sectors is made. If we assume that the production function of the particular firms is such that their investment coefficient can vary, then the derivation of the accelerator coefficient becomes even more complicated. In order to derive the sectoral accelerator coefficients it is necessary to assume that investment is taking place on the basis of 'given' conditions. These conditions include the relative prices of the different factors of production. Assume that firm plans are made on the basis of such a set of given conditions. Then the effective capital coefficient in each sector is determined. This, together with the weights of each sector, derived from the income elasticities of demand, is sufficient to yield a quantitative inducement to invest. But the sectoral coefficients' values depend upon relative prices - that is, upon the existing equilibria in the relevant markets. All the inducement to invest represents the quantity of investment goods demanded under given conditions. This is the second concept of the accelerator coefficient: that of a demand for investment goods.

To sharpen the issue let us suppose that for technical reasons u is not merely the ideal relationship between capital and output, but the only possible one, so that no change in capital intensity can be made by an individual concern. Sales are rising and are expected to rise for some time, but the funds needed for expansion are everywhere becoming more difficult to obtain. The expansion will then be retarded, but it will be retarded to a greater extent in the more capitalistic than in the less capitalistic lines of production. Thus the composition of output will change in such a way that the ratio of total capital to total output tends to decline: the aggregate ratio may then fall as Professor Hayek said it would, and its fall may weaken the stimulus for continued widening.3

The terms under which the funds needed for expansion can be obtained are part of the supply conditions of capital. We could also include the repercussions of a limited productive capacity of capital goods industries upon the price of capital goods as a determinant of the effect of a given change in income upon realized investment. This clearly leads to a third concept of the relation between a change in income and investment, one in which the effects of the inducement to invest upon the other markets is taken into account. In this concept the resultant investment depends upon the effects of the induced investment upon the equilibria in the relevant markets. An accelerator coefficient defined in such a way is a coefficient of realized investment.

It is obvious that the time path of income depends upon realized investment: not in the sense that realized investment determines productive capacity, but in the sense that realized investment is a portion of income. Accelerator models in which the accelerator is a coefficient of realized investment are not vulnerable to the criticism that they are solely a demand theory of the business cycle. However, with such an accelerator the simplicity of the accelerator model vanishes. The most that the formal accelerator model can be in this case is a framework for analysis: the content of the model for purposes of prediction and control depends upon the determinants of the coefficient of realized investment.

Accelerator models based upon these three alternative formulations investigate different problems. In the first type, where the accelerator is a structural parameter, the implications of assuming a constant ratio between capital and output are explored. In the second type the parameters other than income and the change in income which affect investment by a particular firm are assumed to be constant. Induced investment (aggregate) can vary due to the effects of changes in the structure of demand and in the level and rate of change of income. Such a model investigates the effects upon national income of the allowed changes, and investigates the effects of allowing the accelerator to vary. Both the first and the second types of model assume that income depends upon aggregate demand. Slight modifications of both the first and the second types of model which depend upon exogenously determined ceilings and floors have been produced. The time series that such a modified accelerator generates is altered, even though the content of the accelerator has not been changed.

In the third version of the accelerator, the supply conditions of capital are taken into account. The accelerator coefficient represents the relation between a change in output and investment, allowing these parameters of the individual firm's investment functions which are affected by the repercussions of the inducement to invest to vary. The third type is the broadest accelerator concept and such an accelerator does not result in the simple models to which the accelerator concepts lead.

Typically, the accelerator is introduced into a model by means of a simple assertion that the accelerator exists. 'There is a well-established relation, vouched for by experience and the laws of arithmetic, between the demand for consumable goods and the demand for durable goods, the essence of which is that the absolute amount of the latter depends primarily on the rate of increase of the former.'4 The experience to which Harrod referred boils down to the well-known greater relative amplitude of fluctuations in investment as compared with the fluctuations of consumption in the trade cycle. The laws of arithmetic are examples of stock and flow relations such as the textbook example of shoe and shoe machinery demand. The statistical foundations for the use of a fixed value accelerator coefficient are flimsy. Harrod's volume on the trade cycle contains no statement on the necessary conditions to be imposed upon the behavior of business firms for the relation between the change in income and investment which the accelerator implies to be valid.

An analysis of the conditions necessary for the accelerator to operate was set out by J. Tinbergen in 1938:

In this simplest form the (acceleration) principle states that percentage changes in the production of consumers goods are equal to percentage changes in the stock of capital goods. As the latter is usually considerably larger than the annual production of capital goods, the corresponding percentage changes in the latter are much larger than the percentage change in the production of consumer goods. The principle has two aspects between which it is useful to distinguish:

a) the correlation aspect: there must be correlation between new investment in durable capital goods and the rate of increase in consumer goods production;

b) the regression aspect: the percentage fluctuations in consumer goods production are equal to the percentage fluctuation in the stock of capital goods.5

Tinbergen asserted that:

In its more rigorous form, the acceleration principle can only be true if the following conditions are fulfilled:

a) Very strong decreases in consumer goods production must not occur. If the principle were right, they would lead to a corresponding disinvestment, and this can only take place to the extent of replacement.

b) There should be no abrupt change in technique leading to a sudden increase in the amount of capital goods necessary to the production of one unit of consumer goods.6

The limitations to the value of induced disinvestment pointed out by Tinbergen has been incorporated in most of the later models. The acceptance of such a ceiling to disinvestment tends to 'draw out' the length of the depression phase of the resulting business cycle which leads to difficulties for cycle analysis. The assumption of 'strict complementarity' among the factors of production has usually been made, so that the capital coefficient is constant which in turn implies a constant accelerator, for example the accelerator is a structural parameter. This implies an assumption that whatever technological changes occur are, on the average, neither capital saving nor capital consuming. 'In its more rigorous form, the principle is equivalent to saying that a constant part of productive capacity is idle and that enterprises never increase production of consumer goods before having increased correspondingly their capacity. In the case of the constant part being zero - i.e. full occupation of capacity at any moment - that is, at least for increases a necessity, in all other cases this policy would have to be followed deliberately, and there are hardly enough reasons to suppose this occurs in reality.'7 The possibility of output increasing without inducing any investment has been accepted by some authors who use such accelerator models, and Chenery in Econometrica has analysed the operation of the accelerator coefficient under conditions where 'excess capacity' exists.8

Tinbergen modified the accelerator coefficient to allow for the above shortcomings, and in so doing he indicated a mechanism of firm behavior which had to operate.

The acceleration principle may, however, be given a less rigorous form. Instead of equality of percentage changes in consumers goods production and capital goods stock there may be assumed to be only proportionality or even only a linear relationship Two reasons exist for giving the principle its less rigorous form:

a) During a period of increasing production, not all firms and not all branches attain at the same moment, the point of full capacity. Suppose that for the individual firms the principle acts only - and of course in its rigorous form - when full capacity is reached, then to a given increase in total production for all firms a smaller percentage increase in total stock of capital goods may correspond.

b) A second reason for the less rigorous form of the acceleration principle might be that even with idle capacity a firm would expand its plant proportionately to the rate of increase in consumer goods production, but not by an equal percentage. This means that there would not be an immediate necessity for investment but that the willingness to invest would depend chiefly on the rate of increase in consumer goods production.9

These two factors which Tinbergen identified: that not all firms would reach 'full capacity' at the same moment of time and that a firm may, with idle capacity, 'anticipate' full capacity production and therefore 'expand its plant proportionately to the rate of increase in consumer goods production' lead naturally to a variable accelerator coefficient. The accelerator coefficient's value would be a function of the level of income and the rate of change of the level of income. The recognition that with less than full capacity production induced investment depends upon 'anticipations' would lead to a stochastic formulation of the accelerator coefficient. This concept of the accelerator is as a coefficient of induced investment.

Tinbergen offers as a substitute for the accelerator investment relation that 'an explanation of investment fluctuations by profit fluctuations is more natural'.10 The use of profits in the investment function, rather than the change in income, is not, in and of itself, inconsistent with an accelerator formulation. If we recognize that profits are a function of income, and perhaps the change in income, and have an investment-profits relation we get:

which is the accelerator formulation of induced investment. However, profit is a variable that directly impinges upon a firm's behavior; as is the relation between its capacity output and its present output. These factors therefore can be considered as the immediate determinants of investment behavior. If the change in profits, the level of profits, and the relation between output and capacity for individual firms are functions of income then, within our framework, the income investment relation belongs in the core of the model and the relation between profits and capacity and firms' investment decisions belongs in the supplementary relations. As such, the profit-income and capacity-income relations are appropriate places in which the mechanism which generates the accelerator coefficient allows for a variable accelerator coefficient.

An article by Sho-Chieh Tsiang contains a critique of the use of a constant accelerator coefficient over the business cycle. In his paper Tsiang recognizes the need to relate the value of the accelerator coefficient to the behavior of particular firms and realizes that the behavior of firms depends upon the structure of the market within which they are working.11

Tsiang emphasizes the inability to measure the accelerator coefficient, the dependence of the value of the accelerator coefficient upon industrial structure and the relation between the value of the accelerator coefficient and the supply of 'finance' to a particular firm. He uses these elements to advocate the use of a profit-income relation in the analysis of investment. The elements which he presents, and their similarity to the doctrine to be put forward here, is recognized. However, the use to which they are put seems unduly restricted. At this point I wish to take up his analysis of two phenomena: the relation between the value of the accelerator coefficient and industrial structure and the relation between the value of accelerator coefficient and the supply conditions of finance.

Tsiang's analysis of the manner in which investment is induced in a competitive industry by means of a change in income is, in its essentials, the same as the one to be presented later in this volume.

For the increase in aggregate effective demand that is registered in the minds of the individual entrepreneurs may add up to a sum quite different from the real increase in aggregate output. This is particularly obvious for a competitive industry. There an increase in aggregate demand would generally lead first to a rise in the price of the product, although the tendency to rise in price might be held in check somewhat by speculators. It is the rise in the market price of this product that would be registered in the mind of an individual entrepreneur; but he would have no idea of the magnitude of the increase in aggregate demand, let alone his share of the increase. If he assumes that the increase in demand, that is to say the price of his product, is permanent, the increase in output and the requisite optimum investment to produce it, which he would plan in response, would be that which would make his own long-run marginal cost curve equal to the new price. It is evident that there is no guarantee that the planned increase in the output of all the entrepreneurs concerned would add up exactly to the original increase in demand, had there been no increase in price. It is therefore quite unlikely that the induced investment would bear any rigid relation to the original increase in demand.12

The importance of the impact effect upon firms of a rise in aggregate demand, the rise in the market price of a competitively produced product, and the significance of the reaction of the entrepreneurs to this impact (including the importance of the entrepreneur's price expectations) is correctly emphasized. That, in time, the long run equilibrium of the industry is consistent with a long run normal profit situation for a representative firm in the industry is overlooked by Tsiang when he questions whether or not the change in output will 'add up exactly to the original increase in demand'. As is well known, the long run supply curve of the industry will result in an equilibrium output equal to, less than or greater than the output 'had there been no increase in price', depending upon the existence of external economies or diseconomies. If such long run equilibrium considerations are to be relevant to business cycle analysis, the problem of the relevance of long run adjustments to cyclical phenomena has to be analysed. The assumption that ceteris paribus, the development within the industry would be in the direction consistent with long run equilibrium must be modified in business cycle analysis on two grounds.

(a) Inherent in the nature of the business cycle is the statement that the relevant parameters for the firm's behavior do not remain the same over the cycle.

(b) The relation between the time periods of the business cycle and the time necessary for long run adjustments to work themselves out must be considered.

We therefore are left with the problem of how to actually use the theory of the firm in order to generate the accelerator coefficient, which Tsiang does not adequately handle.

A problem recognized by Tsiang is that the short run change in output is not the same as the long run change in output for a given shift in the demand curve for a product. Therefore the accelerator coefficient relating investment to a short run change in output would be different from the accelerator coefficient relating investment to a long run change in output. However, the short run change in output and the long run change in output can be the same if 'the demand curve is kinked because of the firm's asymmetric expectation with respect to other competitors' reaction to its own price changes, or if a firm simply adheres to a conventional price'.13 Figure 3.1 represents the discontinuous oligopoly demand curve analysis as given by Tsiang. If the demand curve as visualized by the firm shifts to D1 from D0, the marginal revenue curve shifts from MR0 to MR1. The short run marginal cost curve (SRMC) and the long run marginal cost curve (LRMC) both pass through the kink in the marginal revenue curve. Output therefore increases from O0 to O1, and the induced investment which takes place is designed to produce a fixed output O1 at a lower cost. Induced investment, therefore, is designed to reduce the cost for a particular output, rather than to expand output. The Tsiang critique of the use of output in the accelerator relation indicates that it may be preferable to use a change in expenditures upon the product, that is to use money income rather than real income in accelerator business cycle analysis. In fact such a use of money income rather than real income in the accelerator relation can make the cycle models which use the accelerator relation consistent with the

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### Responses

• patrycja
How to derive an accelerator coefficient?
11 days ago