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
![](data:image/png;base64,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)
If we denote change in investment by ∆I and change in income by ∆Y, the equilibrium condition becomes
![](data:image/png;base64,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)
Dropping brackets, the first and last terms cancel out,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
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
![](data:image/png;base64,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)
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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