11A

This appendix shows how to derive the multiplier algebraically. First, recall that in this chapter planned aggregate expenditure, AE_Planned, is the sum of consumer spending, C, which is determined by the consumption function, and planned investment spending, I_Planned. That is, AE_Planned = C + I_Planned. 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 level of GDP, Y*, is equal to planned aggregate expenditure:

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 expenditure—a change in either AC or I_Planned—causes a $1/(1 − MPC) change in the income–expenditure equilibrium level of GDP, Y*. The multiplier in our simple model is therefore: