Derivation of Multiplier
Another way to derive multiplier is based on the functional relation between consumption and income.
We start with the basis equilibrium condition, i.e.,
Y = C + I (1)
We known that consumption (C) is the function of income (Y). This functional relationship can be expresses as
C = a + bY (2)
Substituting equation (2) in equation (1), we get
Y = a + bY + I
or, Y - bY = a + I
or, (1 - b) Y = a + I
If we denote change in investment by ∆I and change in income by ∆Y, the equilibrium condition becomes
Dropping brackets, the first and last terms cancel out,
For a given change in investment, the change in income is equal to 1/(1 - b) times the change in investment. Thus 1/(1 - b) is the value of multiplier. If we divide both sides of the Equation (3) by ∆I, we get
The ratio ∆Y/∆I is the ratio of change in income to the change in investment which is the definition of the multiplier.
In equation (4), b =MPC
We know MPC+MPS = 1
K = 1 - MPC = 1 - b
Multiplier = K = 1/MPS
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