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The below mentioned article provides a note on super-multiplier.

The term ‘super-multiplier’ was first coined by J.R. Hicks in his business cycle theory. The object was to show the relationship between change in induced investment and the corresponding change in income. To be more specific, it indicates the ratio between the two changes, i.e., in investment and in equilibrium output. In fact, if we know the super-multiplier, we can easily calculate the level of income that would correspond to any fixed (exogenous) level of autonomous investment.

**Therefore, total income will be: **

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Y = C + I_{a} + I_{P}

where, Y = total income, C = consumption expenditure, I_{a} = autonomous expenditure and I_{p} = induced (private) investment.

Induced investment, like private consumption, may now be taken to be a function of income. As we have noted earlier, the major driving force behind any change in induced investment is the demand for consumer goods or household expenditure on consumption.

But, consumption, as Keynes specified, is a stable function of the income level — i.e., C = bY, where b = m.p.c. = dC/dY, i.e., change in consumption induced by a change in income. Likewise, the marginal propensity to invest (MPI), to be denoted by i, is the ratio of the change in the aggregate induced investment to a given unit change in aggregate income (dl/dY).

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**We may now write the equilibrium condition of income as: **

Y = C + I_{a} + I_{p}

= bY + I_{a} + iY

**or, by rearranging, we get: **

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Y – bY – iY = I_{a}

Y (1 – b – i) = I_{a}

or, Y = I_{a}/1 – b – i = 1/1 – b – i (I_{a}) = K’ (I_{a}) … (1)

Here, 1 / (1 – b – i) is the super-multiplier (K’).

**Now, if autonomous investment increases by a given amount (ΔI _{a}) the corresponding increase in income would be: **

ΔY = ΔI_{a} /1 – b – i = ΔI_{a}. 1/1 – b – a

Here, 0 ≤ b ≤ 1 and 0 ≤ i ≤ 1. Moreover, 0 ≤ b + i ≤ 1. Equation (1) above makes it abundantly clear that the change in income will be equal to the autonomous investments multiplied by the super-multiplier. Thus, induced investment makes the value of the multiplier larger.

**In the words of G. Ackley:**

“If a rise in income not only leads to increased consumption but also to increased investment (creating the basis for further expansion of income, consumption and investment in endless but diminishing chain), the ultimate increase in income will be greater than if only consumption so responded”.

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This model is actually the source of the so-called paradox of thrift.

**In the language of Ackley again: **

“An increased propensity to save, in the simpler Keynesian model, left total saving (and investment) unchanged, although reducing income. Yet in the present model the effort of the community to save more is actually self-defeating; the new equilibrium involves not only lower income but lower saving also. Yet, a community that loses its thrifty habits succeeds in saving more than it did previously”.

Example, suppose, marginal propensity of induced investment is 0.2 and marginal propensity to consume is 0.6. If autonomous investment increases by Rs. 20 crore, what will be the final increase in income?

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**Solution: **

Here, the super-multiplier

k’ = 1/1 – b – i

= 1/1 – 0.2 – 0.6 = 1/0.2

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Therefore, the increment in income,

ΔY = ΔI_{a} × 1/0.2

= 20 × 1/0.2 = Rs. 100 crore

But, under simple multiplier,

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Δ Y = Δ I_{a} × 1/1 – 0.6 = 20 ×1/0.4 = Rs. 50 crore