Proof that
A. Tax multiplier =-mpc/1-mpc
B. Government multiplier =1/1-mpc
To prove that the tax multiplier is equal to -mpc/1-mpc, we need to understand the key concepts involved.
The tax multiplier is the change in GDP that results from a change in government spending. It is calculated using the formula:
Tax multiplier = -MPC / 1-MPC
Where MPC is the marginal propensity to consume, which is the proportion of additional income that a household would spend on consumption.
Now, let's prove this:
Assume that a change in taxes of $1 leads to a change in consumption of -MPC, i.e., households decrease their consumption by MPC for every $1 increase in taxes.
This implies that the change in GDP resulting from the change in taxes is:
Change in GDP = (Change in taxes) * Tax multiplier
Change in GDP = $1 * (-MPC / 1-MPC)
Hence, the tax multiplier = -MPC / 1-MPC
To prove that the government multiplier is equal to 1/1-MPC, we need to understand that the government multiplier accounts for the change in GDP resulting from a change in government spending, rather than taxes.
The government multiplier is calculated using the formula:
Government multiplier = 1 / 1-MPC
This formula is derived from the same logic as above, but with government spending rather than taxes. Given that MPC represents how much of an increase in income is spent, 1-MPC represents how much is saved or taxed.
Therefore, the government multiplier = 1 / 1-MPC
Thus, the proofs for both statements are supported by sound economic reasoning and formulas.