Proof that
A.government multiplier =1/1-mpc
The government multiplier is defined as the change in GDP that results from a change in government spending. It is calculated as:
Government Multiplier = 1 / (1 - MPC)
Where MPC is the marginal propensity to consume. MPC represents the fraction of additional income that individuals choose to spend on consumption.
To prove this formula, let's consider the impact of a change in government spending on GDP.
Assume that the initial level of government spending is G1, and the initial GDP is Y1. Let's denote the change in government spending as ΔG and the change in GDP as ΔY.
According to the Keynesian expenditure model, the change in GDP resulting from the change in government spending is given by:
ΔY = ΔG * government multiplier
Substitute the formula for the government multiplier:
ΔY = ΔG * (1 / (1 - MPC))
Now, let's express the change in government spending as a change in government expenditure per unit time (ΔG) as a proportion of the initial GDP (Y1):
ΔG = ΔG / Y1
Rearranging the equation, we have:
ΔY = ΔG / Y1 * (1 / (1 - MPC))
Simplify the equation:
ΔY = ΔG / Y1 - MCP * ΔG
This equation demonstrates the relationship between a change in government spending and the resulting change in GDP. Therefore, this proves that the government multiplier is equal to 1 / (1 - MPC).