>> 692 Chapter 28 Appendix: Deriving the Multiplier Algebraically This appendix shows how to derive the multiplier algebraically. First recall that in this chapter planned aggregate spending, AE Planned , is the sum of consumer spending, C, which is determined by the consumption function, and planned investment spend-ing, I Planned . Rewriting Equation 28-9 to express all its terms fully, we have: (28A-1) AE Planned = A + MPC × YD + I Planned Because there are no taxes or government transfers in this model, disposable income is equal to GDP, so Equation 28A-1 becomes: (28A-2) AE Planned = A + MPC × GDP + I Planned The income–expenditure equilibrium GDP, Y *, is equal to planned aggregate spending: (28A-3) Y * = AE Planned = A + MPC × Y * + I Planned in income–expenditure equilibrium Just two more steps. Subtract MPC × Y * from both sides of Equation 28A-3: (28A-4) Y * − MPC × Y * = Y * × (1 − MPC ) = A + I Planned Finally, divide both sides by (1 − MPC ): (28A-5) Y* = Equation 28A-5 tells us that a $1 autonomous change in planned aggregate spending—
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