26 APPENDIX: Deriving the Multiplier Algebraically

This appendix shows how to derive the multiplier algebraically. First, recall that in this chapter planned aggregate spending, AEPlanned, is the sum of consumer spending, C, which is determined by the consumption function, and planned investment spending, IPlanned. That is, AEPlanned = C + IPlanned. Re­writing this equation to express all its terms fully, we have:

Because there are no taxes or government transfers in this model, disposable income is equal to GDP, so Equation 11A-1 becomes:

The income-expenditure equilibrium GDP, Y*, is equal to planned aggregate spending:

Just two more steps. Subtract MPC × Y* from both sides of Equation 11A-3:

Finally, divide both sides by (1 − MPC):

Equation 11A-5 tells us that a $1 autonomous change in planned aggregate spending—a change in either A or IPlanned—causes a $1/(1 − MPC) change in income–expenditure equilibrium GDP, Y*. The multiplier in our simple model is therefore: