Question

Derive the expression for government purchase multiplier in Keynesian model. If the value of G
multiplier = 2.67 How do you interpret this?

Answer 1

Answer :

Government Spending Multiplier

Deriving the Government Spending Multiplier, G_M:

From the equilibrium condition:

AD = AS = Y = Income = RGDP Y = C + I + G + NX (1)

Let Consumption, C, be dependent on disposable income as follows:

C = C₀ + MPCx(Y — T), (2)

Where:

C₀ = autonomous consumption (consumption that does not depend on income)

MPC = marginal propensity to consume

T = Taxes on personal income.

MPC is a positive number greater than 0 and less than 1, which captures the proportion (or percentage) of disposable income, (Y – T), that goes for consumption spending. The rest of income that is not consumed is saved.

Thus,

MPC + MPS = 1

Where MPS is the marginal propensity to save.

By plugging (2) into (1), we get:

Y = C₀ + MPCx(Y — T) + I + G + NX

If we assume that T, I, G and NX do not depend on level of income, or RGDP, Y (thus are fixed terms), we can group them together with C0 under the same fixed term A, as shown below.

Y = C₀ + MPCxY — MPCxT + I + G + NX = MPCxY + A

Y — MPCxY = A

(1 — MPC)xY = A

Dividing both sides by 1-MPC (or solving for Y) we get:

Y = {1÷(1—MPC)}xA

The term inside the brackets is the multiplier: 1÷(1—MPC)

Notice that since MPC is less than 1, then 1÷(1—MPC) will be greater than 1. Also, the higher MPC, the higher the multiplier.

If G is the component of A that changes, then the government spending multiplier GM is given by the multiplier we derived above:

1÷(1—MPC) = G_M

The Government Spending Multiplier and the Tax Multiplier

The following formula gives the impact on RGDP of a change in G.

Change in RGDP = 1÷(1—MPC) x (change in G)

Implication : Fiscal policy is more effective in countries with greater MPC (because these countries tend to have a greater G _M , all else equal).

In a similar way, we can derive the Tax multiplier, T _M :

Change in RGDP = —MPC÷(1—MPC) x (change in T)

Let’s compare G _M with T _M :

The magnitude (size) of G _M is greater than T _M .

• Implication 1 : An increase in G is more effective than the same size decrease in T. G _M has the opposite sign compared to T _M .
• Implication 2 : Remember that the same directional result can be achieved if G is increased, or taxes are decreased and vice versa.
• Final Implication : If both G and T increase by same amount (thus the government is running a balanced budget), the net effect on RGDP is not zero, because of the different magnitude (size) of the G _M and T _M

Interpretation :

Government spending results in an increase in national income. Thus, its effect on national income is expansionary. There is a limit to private investment. Thus, to stimulate income the gap has to be filled up by government expenditure. However, the increase in income is greater than the increase in government spending. The impact of a change in income following a change in government spending is called government expenditure multiplier.

Government expenditure multiplier is the ratio of change in income (∆Y) to a change in government spending (∆G). In other words, an autonomous increase in government spending generates a multiple expansion of income. How much income would expand depends on the value of MPC or its reciprocal, MPS.

Government spending has expansionary effect on income is that the increase in public expenditure constitutes an increase in income, thereby triggering successive increases in consumption, which also constitutes increase in income. However, greater the MPC, greater will be the increase in income.

Therefore, If the value of G multiplier = 2.67, then it means that the increase in national income would increase by 2.67 times due to increase in government expenditure by 1 (one).

Government expenditure multiplier = G_M

G_M = ∆Y/∆G and ∆Y = G_M. ∆G

If change in government expenditure is 1, then change in national income will be

∆Y = G_M. ∆G = 2.67*1 = 2.67

This is the multiplier effect.

∆Y= Change in national income

∆G = Change in government expenditure