Samuelson’s Model of Business Cycles: Interaction between Multiplier and Accelerator!
Keynes made an important contribution to the understanding of the cyclical fluctuations by pointing out that it is the ups and downs in investment demand, depending as it is on the profit expectations of the entrepreneurs, that causes changes in aggregate demand which affect the levels of income, output and employment.
Further, by putting forward the theory of multiplier, Keynes has shown how the effect of increase and decrease in investment on output and employment get magnified when multiplier is working during either the upswing or downswing of a business cycle.
However, Keynes did not explain the cyclical and cumulative nature of the fluctuations in economic activity. This is because Keynes did not give any importance to the accelerator in his explanation of business cycles. Samuelson in his seminal paper convincingly showed that it is the interaction between the multiplier and accelerator that gives rise to cyclical fluctuations in economic activity.
The multiplier alone cannot adequately explain the cyclical and cumulative nature of the economic fluct
This increase in income further induces the increases in investment through the acceleration effect. The increase in income brings about increase in aggregate demand for goods and services. To produce more goods we require more capital goods for which extra investment is undertaken.
Thus the relationship between investment and income is one of mutual interaction; investment affects income which in turn affects investment demand and in this process income and employment fluctuate in a cyclical manner.
We have shown below in Fig. 27.4 how income and output will increase by even larger amount when accelerator is combined with the Keynesian multiplier,
Where ∆Ia = Increase in Autonomous InvestmentCombining Accelerator with Keynesian Multiplier∆Y = Increase in Income.
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1/ 1 – MPC = Size of Multiplier where MPC = Marginal Propensity to Consume.
∆ld = Increase in Induced Investment
v = Size of accelerator.
Fluctuations in investment are the main cause of instability in a free private-enterprise economy. This instability further increases due to the interaction of the multiplier and accelerator The changes in any component of aggregate demand produce a multiplier effect whose magnitude depends upon the marginal propensity to consume.
When consumption, income and output increase under the influence of multiplier effect, they induce further changes in investment and the extent of this induced investment in capital goods industries depends on the capital-output ratio, that is, the interaction between the multiplier and accelerator without any external shocks can give rise to the business cycles whose pattern differs depending upon the magnitudes of the marginal propensity to consume and capital-output ratio.
The model of interaction between multiplier and accelerator can be mathematically represented as under:
Yt = Ct + It …(i)
Ct= Ca + c (Yt – 1) …(ii)
It = Ia + v (Y t – 1 – Y t – 2)….(iii)
Where Yt Ct It stand for income, consumption and investment respectively for a period t, Ca stands for autonomous consumption, la for autonomous investment, c for marginal propensity to consume and v for the capital-output ratio or accelerator.
From the above equations it is evident that consumption in a period t is a function of income of the previous period Yt-1. That is, one period lag has been assumed for income to determine the consumption of a period. As regards induced investment in period t, it is taken to be the function of the change in income in the previous period.
This means that there is two periods gap for changes in income to determine induced investment. In the equation (iii) above, induced investment equals v(Y t – 1 – Y t – 2) or v (∆Yt – 1). Substituting equations (ii) and (iii) in equation (i) we have the following income equation which states how changes in income are dependent on the values of marginal propensity to consume (c) and capital-output ratio v (i.e., accelerator).
Yt = Ca + c (Yt – 1) + Ia + v (Y t – 1 – Y t – 2) …(iv)
In static equilibrium, the level of income determined will be:
Y = Ca f cY + I
This is due to the fact that in static equilibrium, given the data of the determining factors-, the equilibrium level of income remains unchanged, that is, in this case, Yt = Y t – 1 = Y t – 2 = Y t – n so that period lags have no influence at all and accelerator is reduced to zero.
Thus, in a dynamic state when autonomous investment changes, the equation (iv) describes the path which a disequilibrium system follows to reach either a final equilibrium state or moves away from it. But whether the economy moves towards a new equilibrium or deviates away from it depends on the values of marginal propensity to consume (c) and capital-output ratio v (i.e., accelerator).
By taking different combinations of the values of marginal propensity to consume (c) and capital-output ratio (v), Samuelson has described different paths which the economy will follow. The various combinations of the values of marginal propensity to consume and capital-output ratio (which respectively determine the magnitudes of multiplier and accelerator) are shown in Fig. 27.5.004
The four paths or patterns of movements which the economic activity (as measured by gross national product or income) can have depending upon various combinations of the values of marginal propensity to consume (c) and capital-output ratio (v) are depicted in Fig. 27.6.
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27.6.Interaction between Multiplier and Accelerator: Different Patterns of Income (Output) Movement for Various Values of C and VWhen the combinations of the value of marginal propensity to consume (c) and capital-output ratio (v) lie within the region marked A, with a change in autonomous investment, the gross national product or income moves upward or downward at a decreasing rate and finally reaches a new equilibrium as is shown in panel a) of Fig. 27.6.
If the values of c and v are such that they lie within the region B, the change in autonomous investment or autonomous consumption will generate fluctuations in income which follow the pattern of a series of damped cycles whose amplitudes go on declining until the cycles disappear as is shown in panel (b) of Fig. 27.6.
The region C in Fig. 27.6 represents the combinations of c and v which are relatively high as compared to the region B and determine such values of multiplier and accelerator that bring about explosive cycles, that is, the fluctuations of income with successively greater and greater amplitude.
The situation is depicted in panel (c) of Fig. 27.6 which shows that the system tends to explode and diverges greatly from the equilibrium level. The region D provides the combinations of c and v which cause income to move upward or downward at an increasing rate which has somehow to be restrained if the cyclical movements are to occur.
This is depicted in panel (d) of Fig. 27.6. Like the values of multiplier and accelerator of region C, their values in region D cause the system to explode and diverge from the equilibrium state by an increasing amount.
In a special case when values of C and V (and therefore the magnitudes of multiplier and accelerator) lie in region E, they produce fluctuations in income of constant amplitude as is shown in panel (e) of Fig. 27.6.
It follows from above that region A and B are alike, they after a disturbance caused by a change in autonomous investment or consumption finally bring about stable equilibrium in the system. On the other hand, the values of c and v and therefore the magnitudes of multiplier and accelerator of region C and D resemble each other but are such that they cause great instability in the system as both of these values cause successively greater divergence from the equilibrium level and the system tends to explode. The case of region E lies in between the two as the combinations of values of c and v in it are such that cause cyclical movements of income which neither move toward nor away from the equilibrium.
It is worth noting that all the above five cases do not give rise to cyclical fluctuations or business cycles. It is only combinations of c and v lying in regions B, C and E that produce business cycles. The values of accelerator and multiplier in the region A are such that with a disturbance caused by a change in autonomous investment or autonomous consumption, the economic activity (as measured by the level of income or Gross National Product) moves smoothly from an initial equilibrium to a new equilibrium with no cyclical fluctuations or oscillations.
On the other hand, the values of c and v (and therefore of multiplier and accelerator) of the region B produce cyclical fluctuations which are of the type of damped oscillations that tend to disappear over time, that is, the amplitude of the cycles shrinks to zero over a period of time. However, this contradicts the historical experience which reveals that there is no tendency for the cyclical movements to disappear or die out over time.
However, it is worth noting that the case B explains the impact of a single disturbance on income and employment. For example, the effect of a onetime increase in autonomous investment goes on diminishing over time if no other disturbance takes place.
However, in reality, further disturbances such as technological advances, innovations, natural disasters and man-made disasters such as security scam in India in 1991-92 do take place quite frequently and at random intervals and in a way they provide shocks to the system.
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Thus, the values of c and v of region B can generate cyclical fluctuations over time without dying out if the above-mentioned disturbances are occurring frequently at random. This results in business cycles whose duration and amplitude are quite irregular and not uniform.
As a matter of fact, the business cycles in the real world also reveal such irregular pattern. To sum up, “what otherwise shows up as a tendency for the cycle to disappear in case B may be converted into unending sequence of cycles by the addition of randomly disturbed erratic shock system.”
In case of the values of multiplier and accelerator falling within the region C, though they generate continued oscillations, the cycles produced by them tend to become ‘explosive’ (i.e. their amplitude tends to increase greatly). But they are not consistent with the real world situation where oscillations do not become explosive.
However, the values of multiplier and accelerator falling within region C can be made consistent with the actual world situation by incorporating in the analysis the so called buffers. Buffers are the factors which impose upper limit or ceiling on the expansion of income and output on the one hand or impose a lower limit or floor on the contraction of output and income on the other.
With the inclusion of these buffers the otherwise explosive upward and downward fluctuations arising out of values of multiplier (or MPC) and accelerator (or capital-output ratio) of the region C can become limited cyclical fluctuations, characteristic of the real world situation.
What has been said about case C above also applies to region D where the values of multiplier and accelerator are such that give rise to directly explosive upward or downward movement which can be restrained by the factors determining the ceiling and floor.
However, the adequate explanation of the business cycles in this case would require the reasons why the system starts moving in the reverse direction, say, after striking the ceiling. Hicks in his famous theory of the business cycles provide the reasons which cause movement of the system in the reverse direction after it hits the ceiling or the floor as the case may be. Hick’s theory of business cycles will be explained below at length.
Lastly, the case E represents a situation where the business cycles neither try to disappear, nor try to explode, they go on continually with constant amplitude. This however contradicts the real world situation and is quite impossible. This is because in the real world situation, business cycles differ a good deal in amplitude and duration.
Summing Up:
We have explained the interaction of multiplier and accelerator in case of various values of marginal propensity to consume (c) and capital-output ratio (v). On the basis of the interaction of the multiplier and accelerator the two categories of business cycle theories have been put forward.
One category of these business cycle theories assumes the values of multiplier and accelerator which generate explosive cycles. For example, Hicks’ theory of business cycles falls in this category. On the other hand, Hansen has propounded a business cycle theory based on the interaction of multiplier with a weak accelerator which produces only damped oscillations.
Further, as indicated above, the interaction theories have been modified either by incorporating in the analysis erratic shocks or random disturbances or by including so called buffers which check the upward movement of income and output by imposing ceiling of expansion and checking a downward movement by imposing a floor on the contraction of output.
One of the famous theories of business cycles based on the interaction of multiplier and accelerator which also incorporate buffers in his analysis of fluctuations is that put forward by the noted English economist J.R. Hicks. We discuss below his theory of business cycles in detail.
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A Numerical Example of the Interaction of the Multiplier and Accelerator:
How the interaction between the multiplier and accelerator gives rise to the cyclical movements in economic activity (as measured by income or output) will become clear from Table 27.1. In formulating this table we have assumed that marginal propensity to consume (c) being equal to 2/3 or 0.66 and capital-output ratio (v) or accelerator being equal to 2. Further, one period time lag has been assumed which implies that an increase in income in a period induces the increase in consumption in the next period.
It is assumed that initially in period t + 1, autonomous investment is of Rs. 10 crores. In period t + 3, with autonomous investment being maintained constant at Rs. 10 crores, the deviation of total income in the period t + 3 as compared to the base period will be equal to 10 + 20 + 26.6 = Rs. 56.6 crores.
Similarly, the changes in induced consumption and induced investment and hence in income brought about by the initial increase in autonomous investment of Rs. 10 crores which is maintained throughout, can be found out. It will be seen from column 5 of the Table 27.1 that there are large fluctuations in income.
Under the influence of the interaction between the multiplier and accelerator, the income increases up to the period t + 6. In other words, period up to t + 6 represents the expansion phase or upswing of the business cycle. Therefore, the period t + 6 is the upper turning point of the business cycle beyond which the contraction phase or downswing of the business cycle begins. It will be further observed that beyond the period t + 13, income again starts rising that is, recovery from the depression begins.
Thus, t + 13 represents the lower turning point of the business cycle. In this way we see that the interaction between the multiplier and accelerator can give rise to the cyclical movements of the economic activity and its various phases.
It is worth mentioning that we have taken particular values of marginal propensity to consume (which determine the size of the multiplier) and capital-output ratio (which determines the size of the accelerator). The other values of multiplier and accelerator that have been explained above would give rise to the different patterns of fluctuations.
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Hick’s theory of trade cycle.
Salient Features:
It is quite true that the principle of acceleration has got quite a few limitations, despite it is accepted as the most effective too) for analyzing the complicated phenomenon of trade cycle.
Professor J.R. Hicks has done a commendable job by analyzing it in his book ‘A Contribution to the Theory of the Trade Cycle’ and showing how far is accelerator responsible for fluctuations and how far trade cycles can be explained in terms of accelerator.
Salient Features:
It is quite true that the principle of acceleration has got quite a few limitations, despite it is accepted as the most effective too) for analyzing the complicated phenomenon of trade cycle.
Professor J.R. Hicks has done a commendable job by analyzing it in his book ‘A Contribution to the Theory of the Trade Cycle’ and showing how far is accelerator responsible for fluctuations and how far trade cycles can be explained in terms of accelerator.
A distinction is made between autonomous investment and induced investment—the latter is a function of changes in the level of output and the former a function of the current levels of output. Under autonomous investment Hicks includes “public investment, investment which occurs in direct response to inventions and much of the ‘long range’ investment (as Harrod calls it) which is only expected to pay for itself over a long period.”
He assumes that investment increases at a regular rate so that it remains in progressive equilibrium if it were not disturbed by extraneous forces. On the other hand, induced investment depends upon change in the level of output or income and is a function of an economy’s growth rate.
Hicks agrees that, whereas, the monetary mechanism may greatly influence the course of the cycle, the fundamental causation of the cycle lies in the multiplier-accelerator relationship, and expect in rare instances, the effective ceiling is the full employment level and the effective floor, the trend levels of autonomous investment.
In short, according to Hicks, trade cycle is an explanation in real terms of a mechanical technological sort in which monetary factors are left out or admitted as a modifying factor and where, apparently, human judgment or varying business expectations and decisions play little or no part. Investment plays the leading role but is based on formula, not judgment. This would appear as the most serious limitations of such a theory.
Assumptions:
The following assumptions were made to develop his theory of the trade cycle:
(i) In Hicksian analysis, a progressive economy is assumed in which autonomous investment is increasing at a regular rate, so that system is such which could remain in progressive equilibrium.
(ii) The saving and investment coefficients are such that an upward displacement from the equilibrium path will tend to cause a movement away from equilibrium, though this movement may be lagged.
(iii) There is no direct restraint upon upward expansion in the form of a scarcity of employable resources provided by the full employment ceiling i.e., it is impossible for the output to expand beyond full employment level.
(iv) Though there is no direct constraint on the contraction yet the transformation of accelerator in the downswing (i.e., disinvestment cannot exceed depreciation) provides an indirect constraint.
(v) There are fixed values of the multiplier and accelerator throughout the different phases of a cycle, i.e., consumption function and investment function are both assumed to be constant.
(vi) However, in Hicksian analysis both the multiplier and accelerator are treated with a lag. He treats multiplier as a lagged relation, so that consumption in period t is regarded as a function of income of the previous period t – 1 and not of current period t. He also uses accelerator with a time lag i.e., induced investment in present period also responds to output changes in the previous period.
Upswing Downswing—Upper and Lower Turning Points:
In the upswing of cycle income rises as a result of the combined action of the multiplier and accelerator. A super cumulative process of income propagation and investment expansion based on the ‘interaction’ of the multiplier and accelerator is attained in the economy called ‘leverage effects’. These two tools of multiplier and accelerator work hand in hand to make expansion cumulative in character.
This continues till the economy touches the ‘full employment ceiling point’. In a dynamic economy, there will be an expanding or rising ceiling and, therefore, it may take much longer than in a static set up to reach the ceiling but once the ceiling is touched the cycle takes the downward swing.
The upper turning point of income is determined by the availability of resources like population, technology, capital stock, etc. The process of expansion hits against the ceiling and turns down or in some cases when the interaction of the multiplier and accelerator is not strong enough, the downswing starts even before the ceiling is touched. The decline in investment in the downswing also operates cumulatively but the decline cannot continue indefinitely because of the lower limit which depends upon the fact that gross investment cannot fall below zero.
At the lower level, some essential and basic investment for replacing inventories and equipment becomes inevitable; the autonomous investment starts asserting itself once more at this stage and is higher than the amount of disinvestment. The increment of net investment causes an upturn of aggregate income and takes the economy along an upward phase. Hicks has expressed the opinion that while the upswing is the result of the interaction of multiplier and accelerator, the downswing is largely a product of the multiplier (the accelerator remaining inoperative for the most part).
Thus, the lower turning point during depression is caused when the amount of disinvestment turns out to be less than the amount of autonomous investment, so that, there is increase in net investment turning the cycle on a path to prosperity. Hicks theory of the cycle is shown in the Fig. 42.4.
. In this figure Line AA shows autonomous investment, which is assumed to be growing at a constant rate ‘g’.
2. EE shows the equilibrium path of output which depends upon AA and is deduced by applying ‘super multiplier’ to it.
3. LL represents the lower equilibrium path of output or floor or bottom or lower limit.
4. FF represents the full employment ceiling showing maximum expansion when scarcity of resources occurs. It is supposed to be above the equilibrium path EE and is assumed to grow at the same rate at which AA is growing.
Up to P0 the economy moves along equilibrium path of output and employment EE. Suppose at P0 there is a burst of autonomous investment following, say, an invention. This will cause a disturbance and the path of output moves steadily away from EE to FF. This happens despite the fact that although the burst is short lived and may be over and autonomous investment falls back to its old level, yet on account of explosive S and I coefficients (as assumed above) the multiplier and acceleration interaction takes the economy from P0 to P1.
However, the upward expansion cannot continue indefinitely and must finally reach the ceiling FF at some point as P1. As soon as the expansion of output hits the point P1 the cycle reaches the top of the boom and the output hits the hump. It has got to come down but it does not fall with a crash immediately but creeps along the ceiling for some time on account of lagged effects and adjustments of induced investments.
After creeping along for a while and it will creep as long as the lagged effects of induced investments are there; afterwards it moves down and the downward trend of cycle begins. But once the output starts falling it can no longer remain even along the equilibrium path EE.
Once a fall starts it is interesting to note that it does not halt at the equilibrium level on account of the effects of past investments and because current investments are below the level at which output can be maintained at equilibrium level, hence the fall doesn’t stop at equilibrium level and it moves down.
Now, the downturn is not abrupt or sudden or quick as shown in Q1P2q without any floor or bottom but slow and gradual along Q1P2q with a bottom beyond which it cannot go because the multiplier is less than unity and accelerator (or disinvestment) is limited by replacement or depreciation— so it must have a floor.
Hick has shown that the downward trend of the accelerator is not the same as upward, while moving up it goes very fast. In fast on the downward path, there is a change in the working of the accelerator. If upward and downward functions of accelerator were the same, economy would have a steep fall along Q1P2q; but in reality since disinvestment is limited by the rate of depreciation, the fall in output is slower but prolonged as indicated by Q1Q2 (Investment now consists of autonomous investment minus the constant rate of depreciation).
Since gross investment cannot fall below zero, the fall in output cannot go on indefinitely as in Q1P2q. The slump must have a bottom which is provided by EL. After it reaches LL it does not go up immediately, but it creeps along LL for some time on account of the existence of excess capacity. Once this excess capacity is exhausted, the positive acceleration effect becomes operative again and the cycle will be repeated.
Evaluation:
Thus, we find that Hicks provides a satisfactory explanation of turning points of trade cycle through accelerator and also sheds light as to the periodicity of the cycle which may not be regular. Since the system has a hump or a ceiling and a floor or a bottom it must oscillate between these two limits like the pendulum of a clock.
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Hicks by showing how the excess capacity delays the upswing make an important contribution to the theory of trade cycle. Hicks model, while highly simplified as presented here serves as a useful framework of analysis, which with modification, yields a fairly good picture of cyclical fluctuation within a framework of growth.
It serves specially to emphasize that, in a capitalist economy characterized by substantial amounts of durable equipment, a period of contraction almost inevitably follows expansion. His model also pinpoints the fact that in the absence of technological development and other powerful growth factors, the economy will tend to languish in depression for long periods of time.
Kaldor’s model of the trade cycle.
Kaldor’s theory of the trade cycle appeared in 1940 just four years after the publication of the General Theory in 1936.
It is a comparatively simple and very neat theory built directly on Keynes’ saving- investment analysis.
Although Keynes did devote a lot in the General Theory ‘Notes on the Trade Cycle’ and laid the basis for further discussion on the subject yet he did not develop a systematic theory of the trade cycle as such.
His theory of the determination of the level of income did not take into consideration the theory of the fluctuations of income, which received at his end a passing and scant attention.
It is important and interesting to note at the very outset that Kaidor’s theory of the trade cycle emerges essentially from the substitution of his particular non-linear saving and investment functions for the linear functions used by Keynes in his income model and from his intelligent tracing of the implications that follow from the quite different saving and investment relationships given by the nonlinear functions.
Kaldor in his trade cycle theory does not make use of the acceleration principle in a rigid form. In his model, investment is related directly to the level of income and inversely to the stock of capital. This approach, which is also associated with names like Kalecki and Goodwin, breaks the unrealistic, inflexible tying (or dependence) of investment to changes in output that is implied by the rigid acceleration principle (at the same time retaining the basic idea of the accelerator). Kaldor introduces an important variable that plays a major role in cyclical changes in saving and investment and this variable is the capital stock (K) in the economy.
Saving is a direct function of the capital stock, for any level of income, the greater the capital stock, the larger is the amount of saving. On the other hand, investment is an inverse function of the capital stock, for any given level of income, the greater the capital stock, the smaller is the amount of investment. In Kaidor’s cycle theory we try to trace out how the changes in the capital stock, that occur over time, alter the equilibrium situations.
In other words, and in short, instead of the investment function incorporating the strict acceleration principle It, + Ia+ w(Y,t-1 – Y,t-2), this approach gives us an investment function, which is like this: It = la + hY t-1 – jK1; where K is the stock of capital at the beginning of the period t and where h and j are constants. The new equation simply means that if output or income (Y) increases while the capital stock (K) remains constant—investment will rise to increase the capital stock (other things being equal).
If, on the other hand, the capital stock increases while output or income remains constant—investment will fall as the desired stock of capital is (or has been) reached. The main difference between Hicks’ model of the trade cycle and Kaidor’s model is that the former uses the acceleration principle in its rigid form; while the latter uses it in a way as to avoid some of the shortcomings of the rigid acceleration principle. This is implied by the two equations given above on which investment at a time depends.
Kaidor’s model relying on Keynes’ model of income determination assumes that the process of change in the business activity is related to the difference between ex-ante saving and investment in the economy. If S > I, the savings are more than investments and there is a decline in consumer spending which through multiplier will bring a fall in income and business activity. If, on the other hand, I > S, then the income rises due to increased spending and higher investment. Thus, a discrepancy between ex-ante saving and investment induce a chain of reactions in the level of income till the equilibrium is restored.
Kaldor, thus, makes both S and I depend upon income (Y) and stock of capital (K), that is:
I = I (Y, K)
S = S (Y, K).
Both S and I are usually related to the level of income except in case of deep depression or extreme inflation, so that ∆I/∆Y and ∆S/∆Y are normally greater than zero. The behaviour of S and I in relation to the stock of capital, however, shows that saving is related positively with the accumulation of the stock of capital and vice-versa; while investment generally bears an inverse relationship with the stock of capital. The fluctuations (cycle) in the economic system can be traced to the movements of the variables like, I, S, Y and K. Now, if we suppose that S and I functions are linear (straight line curves), Kaldor, then, points out two possibilities as shown in the Fig 42.5.
In part A of the Figure, the equilibrium level of Y is Ye—the only income level at which planned saving and planned investment are equal. With any given pair of linear S and I functions, there is a single equilibrium position and any disturbance that results in a shift in either function or both would tend to be followed by a movement to a new equilibrium position. But from the specific viewpoint of the business cycle, this model offers little help because it shows more stability than appears to be in the real world. In part B, there is again a single equilibrium position but it is unstable one.
Any disturbance producing a movement above, Ye means that I > S and that the income level may rise without limit, first to full employment and then beyond to hyper-inflation. Any disturbance leading to a movement below Ye means that S > 1 and that the income level would collapse to zero output or income. Part B gives us greater instability than the real world shows. But as an explanation of the business cycle both the cases pointed out by Kaldor are found wanting, one for too much stability and the other for too little. Kaldor, therefore, concludes from this analysis that S and I functions cannot both be linear, at least not over the full arrange of income during the business cycle. Nonlinear S and I functions appear to conform more closely with the behaviour of saving and investment during the course of cycle as shown in Fig. 42.6
In part A, the curve is almost flat for both relatively high and low income levels and the MPI is almost zero. The MPI is expected to reach zero at low income levels because there is already large excess capacity and rise in income at low point will not induce any investment spending.
Similarly, in case of high level of income, according to Kaldor, MPI will be small because of rising costs of business, construction, borrowing etc. which will discourage entrepreneurs to invest more. Thus, there is a range of income over which increases in income (∆Y) will be accompanied by small or zero increments to investment (∆Y) or ∆I/ ∆Y will be very small or zero over this range of Y.
Again in part B, at relatively high and low income levels, the MPS is relatively large compared to its magnitude at normal income levels. During recession when incomes fall to low levels, people cut saving to maintain their previous standards of living and at high income levels, people not only save a large amount but a larger proportion of their income, therefore, the MPS is high. This shifts the distribution of income in favour of profits and away from wages because the MPS of profit seekers is higher than the wage earners. This is reflected in a steep rise of the S function at high income levels.
The Fig. 42.7 has been derived by combining the nonlinear I and S functions as shown below. This figure shows multiple equilibria, with both A and B as stable positions. At income levels below Y1 or between Y2 and Y3 I > S, so the income level rises. At income levels between Y1 and Y2 or above Y3, S > I, so the income level falls. C is an unstable position and, therefore, the income level Y2 is not a possible equilibrium level.
Fig42.7
If income is between Y2 and Y3, it will rise to Y3, and if income is between Y1 and Y2, it will fall to Y1. It appears that the economy can reach stability only at some high level of income Y3, or at some low level of income Y1. This, however, does not give us a complete model of business cycle, because a business cycle is made up of alternating expansions and contractions and this figure shows simply two possible positions of stable equilibrium.
According to Kaldor, “The key to the explanation of the trade cycle is to be found in the fact that each of these two positions is stable only in the short period—that as activity continues at either one of these levels, forces gradually accumulate which sooner or later will render that particular position unstable”.
If indeed it can be shown that the stable equilibrium at A becomes unstable over time and forces a movement to B, we will have pushed ahead to a model of business cycle. This is apparent from the study of the models given in Fig. 42.8 by Kaldor.
The Kaldor ModelThe first stage of the Kaldor model given in Fig. 42.8 corresponds to the figure already given in the above paragraph. We start off in this with the assumption that the economy is in equilibrium position at B, which corresponds to a relatively high or above normal income, at which investment is also hi5
high but the higher the rate of investment, the more rapid is the increase in the size of the capital stock.
As the capital stock grows, it means MEC falls, which in turn, leads to a downward shift in the MEI curve, which is denoted here by a downward shift in the I curve (beyond point B). At the same time, the growth in the capital stock of the economy means a growth in the total wealth of the economy which in turn, will tend to push up the saving curve S (beyond point B in stage I of this Figure).
This means there is a rise in the average propensity to save in the economy induced by an increase in its wealth. As shown by stage 2 of the diagram, the downward movement of the I curve and the upward movement of S curve result in a gradual shift to the left in the position of B and a gradual shift to the right in the position of C so that B and C are brought closer to each other.
The critical point is reached when these gradual shifts of the I and 5 curves make the two curves tangential (tangent to each other at point B) and bring B and C together as is shown in stage 3 of the diagram. Now, at the position of B + C, S > I in both directions, and the equilibrium is unstable in a downward direction. The cyclical contraction, once started, reduces the income level until a new stable equilibrium is reached at the relatively low level that corresponds with A.
It may be noted that even A is a stable equilibrium only in the short-run. Over time, the S and I curves gradually shift, but now, with the system at a relatively low income level, the I curve shifts upward and the S curve shifts downward as shown by stage 4 in the diagram. If the level of investment corresponding to A is less than replacement requirements some inward shift in the I curve will occur sooner or later on account of replacement reasons alone.
Besides, as the time passes more and more investment opportunities develop, which means the MEC curve will rise and shift to the right pushing up the MEI curve which here would mean an upward shift in the I curve. At the same time, any decline in the capital stock or in the wealth of the economy that occurs during the period of low income will tend to lower the average propensity to save or push the 5 curve downward.
These shifts cause the position of A to move to the right and that of C to move to the left, thus, bringing A and C close together as is shown from stage 4 and stage 5 in diagrams. Again, the critical point is reached when these gradual shifts of the I and S curves makes the two curves tangential (tangent to each other at point A) and bring A and C together, as shown in stage 6 of the diagram.
The A + C position is unstable in an upward direction, since I > S on both sides of the position. The cyclical expansion, once started, raises the income level till a new stage of equilibrium is reached at the relatively high level that corresponds with B. The curves, thereafter, are likely to return gradually to the position shown in stage I of the diagram and another cycle begins.
The cyclical process, as described above by Kaldor is, thus, self-generating. The very movement to relatively high income levels brings into play forces, that after a period of time, produced a downward movement to relatively low income levels, and vice-versa. These forces, such as the changing size of the APS and the accumulation and de-cumulation of capital that occur over the cycle, are inherent in the economic process they are endogenous (within the system) forces in the full sense of the term.
One of the most important features of the Kaldor’s model of trade cycle is the impact or the importance of the distribution of income because the income of the society is distributed between different classes (Y – W + P i.e., wages plus profits), each of which has its own propensity to save, the equilibrium can be brought about only under a proper and appropriate distribution of income.
On the one hand, the relations of distribution determine the given level of social saving and, therefore, of investment, on the other hand, achievement of equilibrium (growth rate)’ requires a given level of investment and, therefore, of saving, which in turn, means corresponding distribution of income (provided the MPS of each class remains unchanged).
According to Kaldor, the introduction of the distribution mechanism (of income) into the model (with the proviso that Sp > 5V i.e., profit seekers savings are more than wage earners) makes the system more stable and more capable of automatically restoring equilibrium. Here we find Kaidor’s model differs materially from Harrod’s model. Kaldor believes that any change in I in relation to S—which in Harrod’s model will tend to produce cumulative processes of decline or growth in income and production—will set off (in Kaidor’s model) the mechanism of income redistribution, which adopts S to the new level of I.
Inflationary processes have an important part to play in this redistribution of income (necessitated by I > S or I < S). Kaldor assumes that when I > S, the rising investment and the general growth of demand under full employment will result in faster growth of prices than of wages, thereby, changing the distribution of income in favour of profit and reducing the share of the workers.
Because savings from profits are higher than the savings from wages (Sp > Sw), this will result in a growth of savings and the equality of S and I will be restored, if, on the other hand, investment and overall demand tends to decline, prices are likely to drop faster than wages, distribution will tend to change in favour of the workers, savings will decline, and the equality of S and I will be restored (though at low equilibrium level). In economic writings the equilibrium, thus, restored through the mechanism of income distribution is called ‘Kaldor Effect’.
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