

To derive the government spending multiplier, we start with the consumption function: C = a + bY where b is the marginal propensity to consume. In equilibrium, Y = C + I + G therefore 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 DG(1MPC) but remember that 1 MPC is MPS, so this is DG(MPS) 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). Thus: 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.
