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**In this article we will discuss about Interaction of Multiplier and Accelerator:- 1. ****Combined Effect of Multiplier and Accelerator**** 2. ****Super-Multiplier**** 3. Significance of Multiplier – Accelerator Interaction**

**Combined Effect of Multiplier and Accelerator****: **

In order to measure the total effect of initial investment expenditure on income, it is necessary to combine the effects of multiplier and accelerator. Economists like P A. Samuelson, J.R., Hicks, A.H. Hansen, R.F. Harrod have attempted to integrate the two parallel concepts of multiplier and accelerator and analyse their combined effects.

An autonomous increase in investment by the government increases income through induced consumption expenditure because of multiplier effect. This increase in income and consumption expenditure will induce an increase in private investment due to acceleration effect. This induced increase in investment will start generating more income through multiplier process.

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Once again, die increased income will further induce private investment though acceleration effect. Thus we find an endless sequence of changes in income and output, and in consumption and investment as a result of the interaction between the multiplier and acceleration principles.

Hansen calls the total effect of initial increase in investment on income through the interaction of multiplier and accelerator as leverage effect. The multiplier and the accelerator mutually tend to strengthen each other and their interaction produces a leverage effect, causing very large changes in income.

Similarly, J.R. Hicks has combined the principles of multiplier and acceleration and arrived at the super-multiplier. The effect of super-multiplier (i.e., multiplier and accelerator combined) is much more than the simple multiplier.

**Paul Samuelson has developed the following multiplier-accelerator model which gives the combined effect of the two principles, multiplier and accelerator: **

Investment in period t is partly autonomous (I_{a}) and partly induced (I_{i}) by changes in consumption or by changes in income, because changes in consumption, in turn, depend upon income changes. Here l_{a} is constant and v denotes capital-output ratio or accelerator.

If the incomes of the periods t-1 and t-2 are known, the income for period t can be estimated as a weighted sum, the weights depending upon the values of b and v.

The combined effect of multiplier-accelerator interaction is illustrated through a numerical example given in Table-4.

(i) Suppose that the marginal propensity to consume, b = .5 and the acceleration coefficient, v = 2. The initial investment I_{a} = Rs.100 crores as shown in Column 2.

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(ii) In period zero, the income will increase by the amount of initial investment (i.e., Rs.100 crores).

(iii) This increased income of Rs. 100 crores leads to a rise in consumption of Rs. 50 crores (Column 3) in period 1 because MPC =.5 (i.e., C = b(Y_{t-1}) = .5 x 100 =50).

(iv) This increased consumption of Rs. 50 crores induces investment of Rs. 100 crores (Column 4) because acceleration coefficient is 2 (i.e., I_{i} = v (C_{t }– C_{t-1}) = 2(50) = 100).

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(v) Thus, income increases to Rs. 250 crores (Column 5 = Columns 2 + 3 + 4) in period 1 (i.e., Y = I_{a} + C + I_{i} = 100 + 50 + 100 = 250).

(vi) In period 2, additional consumption expenditure is Rs. 125 crores (i.e., C = b Y_{t-1} = .5 x 250 = 125); the induced investment is Rs.150 crores (i.e., I_{i} = v (C_{t} – C_{t-1}) = 2 (125 – 50) = 2 x 75 = 150); and the total income generated is Rs. 375 (i.e., Y = l_{a} + C + I_{i} = 100 + 125 + 150 = 375).

(vii) The behaviour of income as a result of combined operation of multiplier and accelerator reveals itself in a recurring trade cycle, repeating itself indefinitely. The increase in income is the highest (Rs. 412.5 crores) in period 3, which shows the peak of the cycle.

Thereafter it goes on falling till it reaches bottom of the cycle in period 7 when income is minus 11.8. From period 8, income again starts rising, indicating the revival phase of the cycle. The actual behaviour of the trade cycle, however depends upon the values of the multiplier and the accelerator.

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A less than unity marginal Propensity to consume provides an answer to the question- Why does the cumulative process comes to a stop before a complete collapse or before full employment?

According to Hansen, this is due to the fact that a large part of the increase in income in each period is not spent in each successive period. This eventually leads to a decline in the volume of induced investment.

And when such a decline exceeds the increase in induced consumption, a decline in income sets in. Thus, Hansen remarks, “It is the marginal propensity to save which calls a halt to the expansion process even when the expansion is intensified by the process of acceleration on top of the multiplier process”.

**Super-Multiplier****: **

J.R. Hicks, in his book: A Contribution to the Theory of Trade Cycle (1950), has combined multiplier and accelerator mathematically and called it super-multiplier. In other words, super-multiplier refers to the mathematical derivation of the interaction of multiplier and accelerator. If we know the value of super- multiplier, we can easily work out the equilibrium income level that corresponds to any given autonomous investment.

**The derivation of the formula of super-multiplier is as follows: **

Total income (Y) is equal to total consumption (C) plus total investment (I).

Y = C + I

Total investment (I) again is divided into two parts- autonomous investment (I_{a}) and induced investment (I_{i}).

Thus we get-

where,

K_{s} = Super-multiplier;

b = Marginal propensity to consume;

s = Marginal propensity to save;

v = Marginal propensity to invest or ∆I/∆Y

Super-multiplier tells that total income in the economy will increase by K_{s} times the initial increase in the autonomous investment.

**Thus, in the general form, the super-multiplier is expressed as: **

This effect of super-multiplier (or the combined effect of simple multiplier and accelerator) on income can also be graphically illustrated through Figure-4. Initially, the economy is in equilibrium at point E_{0}, where the saving curve SS intersects the investment curve I_{0}I_{0}. The equilibrium level of income is OY_{0}.

With an increased autonomous investment from I_{0}I_{0} to I_{1}I_{1} the equilibrium level shifts to E_{1}, where I_{1}l_{1} intersects SS and the income rises to OY_{1}. The increase in income from OY_{0} to OY_{1} is the multiplier effect.

The increased income will have its effect on private or induced investment. The induced investment carve is I_{i} I_{i} which intersects the SS curve at E_{2}, Thus increasing income further to OY_{2}. The increase in income from OY_{1} to OY_{2} is acceleration effect.

In this way, an increase in autonomous investment by I_{0}I_{1} has raised the income from OY_{0} to OY_{2}. This is super-multiplier effect or the combined effect of multiplier and accelerator. Thus,

Super-Multiplier Effect = Multiplier Effect + Acceleration Effect

or (Y_{0}Y_{2}) = (Y_{0}Y_{1}) + (Y_{1} Y_{2})

**Significance of Multiplier – Accelerator Interaction: **

(i) The interaction of multiplier and accelerator provides a comprehensive analysis of the process of income generation. It shows how an initial autonomous investment leads to a cumulative rise in output and income.

(ii) Hicks utilised the multiplier-accelerator interaction to explain the dynamics of business cycle more accurately.

(iii) The interaction of multiplier and accelerator provides a better explanation of the turning points of business cycle than the multiplier alone.

(iv) The government can make use of the super-multiplier to influence the level of economic activity and national income through government expenditure.

As regards the relative importance of multiplier and accelerator, it is to be noted that it is the multiplier alone that determines the ultimate levels of national income to be generalized. The accelerator tends to speed up the rate at which the national income is generated through multiplier effect.

Moreover, the accelerator is also responsible for causing fluctuations in national income while reaching the level as indicated by the multiplier. Thus, it is wrong to conclude that it is possible to continuously raise the national income to higher and higher levels on the basis of multiplier-accelerator interaction.

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