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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 289 to express all its terms fully, we have:
(28A1)
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 28A1 becomes:
(28A2)
AE
Planned
=
A
+
MPC
×
GDP
+
I
Planned
The income–expenditure equilibrium GDP,
Y
*, is equal to planned aggregate spending:
(28A3)
Y
*
=
AE
Planned
=
A
+
MPC
×
Y
*
+
I
Planned
in income–expenditure equilibrium
Just two more steps. Subtract
MPC
×
Y
* from both sides of Equation 28A3:
(28A4)
Y
*
−
MPC
×
Y
*
=
Y
*
×
(1
−
MPC
)
=
A
+
I
Planned
Finally, divide both sides by (1
−
MPC
):
(28A5)
Y*
=
Equation 28A5 tells us that a $1 autonomous change in planned aggregate spending—
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 Fall '10
 Kyle
 Macroeconomics, Algebra, $10 million, $25 million, $20 million, $2.5 million

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