To derive the government spending multiplier, we start with the consumption function:
C = a + bY
where b is the marginal propensity to consume.
Y = C + I + G
Y = a + b(Y T) + I + G
Y = a + bY bT + I + G
and rearranging terms yields
Y bY = a + I + G - bT
Y (1 b) = a + I + G bT
solving for Y
Y = [1/(1 b)] (a + I + G + bT)
and so [1/(1 b)] is the government spending multiplier. Since b is the MPC, 1 b is the MPS, and the multiplier is 1/MPS.
We can also derive the tax multiplier from the above; the equation tells us that if T increases by $1, income will decrease by b/(1 b) dollars. Thus the tax multiplier is b/(1 b); recalling again that b is the MPC, it is - MPC/MPS.
The balanced budget multiplier can be shown to be equal to one of the following:
Because it is a balanced budget, G and T will increase by the same amount. The increase in T will reduce Y and reduce C by a fraction of the reduction in Y (that fraction being the MPC). Thus the net change in spending will be:
DG DC = DG DT(MPC) and since DG and DT are equal, this is DG DG(MPC) or
but remember that 1- MPC is MPS, so this is
That is the net change in aggregate expenditures, but to find its impact on Y, we must apply the spending multiplier, which is (1/MPS).
DG(MPS) [1/MPS] = DY
DG = DY
So whatever the amount of the equal changes in G and T, the resulting change in Y will be in the same amount, and the multiplier is 1.