The value of multiplier, i.e., the number by which the amount of change in investment has to be multiplied to arrive at the figure of total increase in income, depends upon the value of the marginal propensity to consume. If the marginal propensity to consume is high, then a higher proportion of additional income will be spent upon consumption and thus give a larger increase in the total income. With a low marginal propensity to consume, the expenditure on consumption in each round will be spend upon consumption and thus give a larger increase in the total income. with a low marginal propensity to consume, the expenditure on consumption in each round will be smaller, which will give smaller increase in income in the successive rounds and thus generate only a small increase in the total income. The value of multiplier K, which is equal to ΔY / ΔI, is thus intimately linked up with the value of the marginal propensity to consumed (MPC). The value of the multiplier can be easily found out with the help of the following formulae.
K = 1/1-MPC
or K = 1/1-b
where b is the symbol used to denote MPC.
Since 1-MPC or (1-b) = MPS, which is marginal propensity to save, also denoted by the symbol (s) we can also write
K = 1/ MPS
K = 1/S
This formula for finding out the value of multiplier can be derived as follows. We know that
ΔY = ΔC + ΔI
Diving both sides by ΔY, we get
ΔY/ΔY = ΔC/ΔY + ΔI/ΔY
or 1 = ΔC/ΔY + ΔI/ΔY
or ΔI/ΔY = 1- ΔC/ΔY -------(i)
Since, the two sides of equation are equal; their reciprocal must also be equal. Therefore,
1/ΔI = 1/ 1-ΔC/ΔY
or ΔY/ΔI = 1/1-ΔC/ΔY ----------(ii)
Hence, K = 1/1-ΔC/ΔY
or = 1/1-MPC or 1/1-b
or = 1/MPC or 1/1-S.