Multiplier can be derived in two alternative ways : (i) Through Income-Expenditure Identity, and (ii) Through Functional Relationship.

Income-Expenditure Indenitity

Multiplier can be derived from the total income and total expenditure identity as follows:

Y = C + I

If we denote change in investment, consumption and income by ΔI, ΔC and ΔY respectively, the income-expenditure identity can take the following form:

               ΔY/ΔI = ΔC/ΔY + ΔI/ΔY

or             1 = ΔC/ΔY + ΔI/ΔY

or            ΔI/ΔY = 1-ΔC/ΔY

or            ΔY/ΔI = 1/1-ΔC/ΔY

or            ΔY = 1/ΔC/ΔY X ΔI

We know that

ΔC/ΔY = Marginal Propensity to Consume
          
          = MPC
So         ΔY              =          1/1-MPC        X          ΔI
         Total income will     the multiple        of a change in
           change by                                                 investment

Dividing equation (2) by ΔI on both sides, we get

Equation (3) tells us that an increase in investment by ΔI increases the income by 1 / 1-MPC times. Thus 1/1-MPC is the value of multiplier. We known that MPC + MPS = 1, i.e., 1-MPC = MPS. We have

Multiplier = 1 / 1- MPC = 1 / MPS

We denote multiplier by ‘K’ and hence write

K = 1 / MPS

Another way to derive multiplier si based nthe functional relation between consumtpion and income.

We start with the basic equilibrium condition, i.e,.

Y = C + 1

We know that consumption (C) is the function of income (Y). This functional relatinship can be expressed as

C = a + bY

Substituting equation (2) in equation (1), we get

                 Y = a + bY + 1

or        Y - by = a +I

or        (1-b)Y =a + I
or         Y = a + I /1-b

If we denote change in investment by ΔI and change in income by ΔY, the equilibrium condition becomes

Y + ΔY = [a+I / 1-b]

ΔY = [a+I / 1-b + ΔI / 1-b] - [a + I / 1-b]

Dropping brackets, the first and last terms cancel out,

ΔY = ΔI / 1-B

or       Δy = 1 / 1-b ΔI

For a given changne 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 equation (3) by ΔI, we get

K = ΔY/ΔI = 1/1-b

The ratio ΔY/ΔI is the ratio fo change in incoem to the change in investment which is the definition fo 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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