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The multiplier formula can be derived by using the basic income expenditure identity for the two sector economy. Y = C + I   … (6.1)
If ΔY, ΔC and ΔI denote the changes in the levels of income, consumption and investment respectively, we have  Y+ ΔY = C + ΔC +I + Δ I   …   (6.2)  
Subtracting equation (6.1) from (6.2), we get, ΔY = ΔC + ΔI … (6.3)                                                                           
The above equation indicates the change in the level of income required to attain the new equilibrium level of income. Since the consumption function is assumed to be linear, C = a + by. Therefore, ΔC = bΔY.  In other words, the consumption expenditure changes by an amount equal to ‘b’ times the change in the income, where ‘b’ is the marginal propensity to consume (MPC). Thus, we have the following.

MPS is the marginal propensity to save. Alternatively, the value of the multiplier can be derived by dividing both sides of the equation (6.3) by ΔY. We have,

Thus, for given any change in the investment (ΔI), the change in the income (ΔY) required to restore the equilibrium is known, if the value of the multiplier is known. Further, the value of the multiplier can be known, if the value of MPC or MPS is known. From the formula of the multiplier, it is clear that the multiplier is the reciprocal of MPS or (1- MPC). The larger/smaller  the value of the MPC/MPS, the larger /smaller will be the value of the multiplier (as the case may be) and vice-versa. In the developing countries, the MPC is relatively higher. So, the value of the multiplier is higher in these countries. In other words, the rise in the income induced by a given increase in the investment will be larger, when a greater proportion of the additional income is spent on consumption and vice versa. For example, if MPC = 0.5, then the value of the multiplier (K) is 1/(1-0.5) or2. If the value of the MPC increases to 0.o75, the value of the multiplier increases and vice – versa. It is clear from the following Table 6.2.
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The value of the multiplier, thus, ranges front 1 to infinity. On the one extreme, when the marginal propensity to consume (MPC) is zero, the economy decides to save the whole of the additional income. Here, the value of the multiplier is equal to 1 [K = 1/ (1-0) = 1], since whole of the increase in the income (ΔY) leaks out of the spending stream in the second round itself. Hence, there will not be any further increase in the income. As a result, the total increase in the income will be same as the increase in the investment., i.e., Δ Y = ΔI.

On the other extreme, if MPC = 1, the economy decides to consume the whole of additional income. Here, the value of the multiplier is infinity. [K= 1/(1-1) = ∞]. In this case, even a very small increment in the investment will make a tremendous impact on the consumption expenditure and hence the income. Here, not even a bit of the additional income is diverted into the saving. In this exceptional situation, the expansion process is very fast and so the aggregate demand always exceeds the aggregate supply (or the investment exceeds the saving), round after round. This will produce extreme instability in the levels of income and output on account of the continuously rising consumption expenditure after each round. But, the expansion process will come to an end, the moment unemployed resources are fully  employed. Any pressure of the rise in demand, thereafter, will only generate inflationary tendencies. 
Both of the above extreme cases are only rare possibilities. MPC can never be zero or one,irrespective of the level of development in the economy. MPC is the highest in the third world countries. Even there, the saving rate is not zero, so MPC is not equal to 1. Hence, the limiting values of 1 and infinity for the multiplier are completely ruled out. In actual practice, MPC assumes value between 0 and 1. Therefore, the multiplier takes values only between 1 and ∞.
Most of the economists are of the opinion that the actual value of the MPC must lie between 0.33 to 0.9, and so the multiplier values will fall within the range of 1.5 and 10.
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