The concept of Super Multiplier Theory

The multiplier that emerges from macroeconomic model in which is both induced consumption and induced investment is called the super-multiplier. The value of the super-multiplier is necessarily greater than the simple multiplier. It is believed that H.R. Hicks first used the term ‘super-multiplier’ in explaining his business cycle theory.

The concept of super-multiplier develops from the point that increase in autonomous investment originates the increase in income. The initial increase in income will give rise to induced investment via the marginal propensity to invest (MPI). The increase in income/output indicates a condition of growing economic activities. So, the business community takes the increase in income as a positive incentive to increase further investment. The increase in income will also affect consumption via marginal propensity to consume (MPC). Induced investment and induced consumption produce further increases in income, and these increases in income still induce/motivate more investment and more consumption spending, and this process continues until a new equilibrium is established.

The super-multiplier is derived by using the consumption and investment function along with national income accounting identity. In formulating the models it is assumed that the economy is operating below full employment level (which means there are unemployed resource of production in the economy), general price level in the economy is more or less constant, effect of interest rate on investment demand as well as on consumption is neutral/unbiased, the economy is simple two sector closed economy.

Aggregate expenditure/demand (AE or AD) is the sum of households’ expenditure on the goods and services (C) and business sectors’ investment expenditure (I). This relationship is mathematically written as

AE = C + I …………………………. (i)

Macroeconomic equilibrium needs that aggregate expenditure/demand should be equal to the total value of income (Y).

Therefore, Y = AE ……………….. (ii)

By combining equations (i) and (ii), we get

Y = C + I ………………………….. (iii)

Equation (iii) is a national income accounting identity (unique condition hold true by definition). It simply tells that total income in equilibrium is equal to the sum of household consumption expenditure and business investment expenditure. In other words, represents macroeconomic equilibrium condition when aggregate demand/expenditure (C+I) is equal to the value of aggregate supply/income (Y).

Household consumption expenditure totally depends upon the level of current income (Y). Consumption is linearly dependent upon income. Keynes in explaining his consumption-affecting factors is insignificant. So, the relation of consumption (C) with income (Y) is algebraically expressed as

C = Ca + bY (0 < b < 1) ……………………. (iv)

Where, Ca is autonomous consumption, b is marginal propensity to consume (MPC) and Y is income level, it is the disposable income or after-tax income (income left after the payment of tax). We have been assuming a simple two sector private economy where there is no government tax, so Y is both national income and disposable income.

Autonomous consumption is independent of the level of income, it is the minimum level of consumption to be made to keep one’s life alive even if the current level of income earning is zero; Ca is the intercept term of the consumption function which represents the effect of all factors other than current income on consumption. Marginal propensity to consume (MPC) or ‘b’ in equation (iv) is the coefficient of Y, and it measures the change in consumption resulting from some change in the level of income. MPC is expressed as a ratio of the change in consumption to the change in income. So, 

MPC = (Change in consumption)/(Change in income) = ∆C/∆Y

In equation (iv) the restriction/limitation on the MPC is that it is positive but less than one (0 < b < 1). This is fully justified by Keynes’ Fundamental Psychological Law of Consumption according to which human beings have a tendency of increasing consumption with the increase in income but not by as much as the increase in income. This means that consumption expenditure increases with the increase in income but all the increases in income is not spent in consumption. The relation of consumption to income is geometrically shown in figure.
Keynesian Consumption Function
The consumption function slopes upward indicating that consumption rises with the rise in income. In the equation “C = Ca + bY”, the term ‘b’ or MPC also measures the slope of the consumption function. Every point on the consumption line represents a combination of the value of Y and C, and hence the ratio of C and Y is a measure of the average propensity to consume (APC = C/Y). Along the consumption increases with the increase in income but by less than the increase in income; some part of the increase in income goes for saving.

Total investment (I) is made of the sum of autonomous investment (Ia) and induced investment. Autonomous investment is not influenced by the change in income but induced investment is totally dependent upon the level of income. Then the investment equation is algebraically written as

I = Ia + βY (0 < β < 1) ………………………. (v)

Where β (Greek small letter beta) is the marginal propensity to invest (MPI); it measures the change in investment with the change in income/output (i.e. MPI = ∆C/∆Y = β). The restriction on the MPI is that it is also positive but less than zero. The increase in induced investment due to the change in income is normally less than the increase in income.

The investment concepts used in deriving the super-multiplier is shown in figure:
Autonomous, Induced and Total Investment
Autonomous investment is constant at some value (equal to Ia in the example) and totally unresponsive to income; the autonomous investment line is parallel to the horizontal income axis; it tells us that autonomous investment remains fixed for a particular period of time whatever may be the level of income. On the other hand, the induced investment line (βY) starts from the origin and is fully dependent upon income level. If income level is zero, induced investment is also zero, and as income rises induced investment also rises. As, total investment is the sum of autonomous and induced investments (I = Ia + βY), the total investment line (I) starts from above the origin with some positive value equal to the autonomous investment (Ia).

In order to derive the value of super-multiplier, we first of all find the equilibrium level of income. For this, we substitute equations (iv) and (v) into equation (iii) which gives:

Y = Ca + bY + Ia + βY

Or, Y –bY – βY = Ca + Ia (collecting Y terms in one side)

Or, Y (1 – b – β) = C0 + Ia

Then the equilibrium level of income (Ye) is

Ye = (Ca + Ia)/(1 – b – β) ………………………. (vi)

Now let us suppose that there is some increase in autonomous investment equal to ‘∆Ia’. This increase in autonomous investment will bring increase in the equilibrium level of income/output by some amount ∆Y. Then the new equilibrium is

Ye + ∆Y = (Ca + Ia + ∆Ia)/(1 – b – β) ……………………. (vii)

Then subtracting equation (vii) from equation (vi), 

Ye + ∆Y – Ye = (Ca + Ia + ∆Ia)/ (1 – b – β) - (Ca + Ia)/ (1 – b – β)

Simplifying,

∆Y = (Ca + Ia + ∆Ia – Ca – Ia)/ (1 – b – β)

Finally, we get,

∆Y = ∆Ia/(1 – b – β)

That is, ∆Y/∆Ia = 1/(1 – b – β)

= 1/{1 – ( b + β)}

= 1/1 – (MPC+MPI) ………………… (viii)

The right hand expression “= 1/{1 – ( b + β)} = 1/1 – (MPC+MPI)” in equation (viii) is what is called the super-multiplier. The restriction in the super-multiplier is that the sum of marginal propensity to consume (MPC = b) and marginal propensity to invest (MPI = β) is not greater than one, that is b + β = MPC + MPI ≠ 1. If b + β > 1 (i.e., MPC + MPI > 1) the value of the super-multiplier will be negative which would mean that increase in investment reduces income/output and this is nonsensical/ illogical in economic explanation because increase in investment is assumed to increase income. If b + β = 0 (MPC + MPI = 0), the value of the super-multiplier will be one which would mean that the increase in equilibrium income is exactly equal to the increase in investment; and if b + β = 1 (MPC + MPI = 1) the value of the super-multiplier tends to infinity (it becomes very large), this would mean that the increase in autonomous investment brings very large increase in the equilibrium level of income, and increased investment is highly productive.

We can now make a comparison of the simple and super-multipliers. The multiplier that emerges in a model where investment includes only the autonomous investment and no induced investment is called simple investment multiplier

∆Y/∆Ia = 1/1 – b = 1/1 – MPC

By comparison, it is obvious that as long as the marginal propensity to invest is positive (i.e., MPI > 0), the super-multiplier is greater than the simple multiplier because an initial rise in income created by a rise in autonomous investment leads to both induced consumption spending and induced investment spending and for this reason the overall rise in income will be greater than if only consumption spending had responded to the rise in income. Symbolically, we can write this as

1/{1 – (MPC + MPI)} > 1/(1 – MPC)

Or, Using the symbols of the equation, 1/{1 – (b + β)} > 1/(1 – b)

We can check this by taking some hypothetical values for MPC and MPI. Suppose MPC = 0.60 and MPI = 0.20. Then,

1/ {1 – (MPC + MPI)} = 1/{1 – (0.60 + 0.20)} = 1/1-0.80 = 1/0.20 = 5

And, 1/(1 – MPC) = 1/(1 – 0.60) = 1/0.40 = 2.5

It is obvious that 5 > 2.5 or super-multiplier is greater than simple multiplier. This would mean that the combined effect of autonomous and induced investment on the rise/increase in income (output) is more powerful than the effect of autonomous investment alone.

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