Derivation of Investment Multiplier Formula

The multiplier formula can be derived by using the simple equilibrium condition for the two sector model i.e  

Y = C + I 

When there is an increase in investment by (∆I), it will lead to increase in income (∆Y) and this induces increase in consumption (∆C) i.e

∆Y = ∆C + ∆I 

Since, change in total consumption (∆C) equals change in income multiplied by MPC (marginal propensity to consume, “c”)

∆Y = c∆Y + ∆I

∆Y - c∆Y = ∆I

∆Y(1-c) = ∆I

∆Y = ( 1 / 1-c)  ∆I 

∆Y/ ∆I = 1 / 1-c

(Since, K = ∆Y / ∆I) 

K = 1 / 1-c

Thus, the term 1 / 1-c gives the value of investment multiplier. 

From the above formula of the multiplier, it is clear that the size of the multiplier depends on the Marginal Propensity to consume (c)

The larger the value of the MPC, larger is the size of the multiplier

The smaller the value of the MPC, the smaller is the size of the multiplier.