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Week 5 Lectures 9 & 10
Spending and Output in the Short-Run (continued)
Reference: Bernanke, Olekalns and Frank - Chapter 5
Key Issues
45-degree diagramEquilibrium and disequilibriumInjections and
withdrawalsMultiplier
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Review
In Week 4 we developed a simple algebraic model of
output. It consists of:
An equilibrium condition:PAEY =
output that equals planned aggregate expenditureA definition of
PAE:
NXGICPAEP
+++= An economic model for consumption:
)( TYcCC +=
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Two Sector Model
Assumptions (simplifying):
no government sectorno foreign sector (i.e. a closed
economy)
Planned aggregate expenditureP
ICPAE += Consumption function (no taxes, so consumption
depends on total, not disposable income)
cYCC += Planned investment is exogenous
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A Diagram Showing Consumption and Investment
C, I
cYCC +=
PI
Y
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Planned Aggregate Expenditure
cYICICPAE PP ++=+=
PICPAE +=
C, I, PAE
cYCC +=
PI
Y
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45-Degree Diagram
How can we represent equilibrium diagrammatically?
Equilibrium is where PAEY =
PAE
20
10
10 20 Y
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PAEY = for all points on the 45-degree line
PAEY = PAE
1PAE
045
1Y Y
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Equilibrium GDP in the 2-Sector Model
45-degree line
PAE
PAE
C
PI
eY Y
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Equilibrium GDP in the 2-Sector Model: The Algebra
Equilibrium Condition PAEY =
Definition ofPAE PICPAE +=
Consumption Function cYCC +=
(Substitution) PIcYCY ++=
(Collect terms in Y)P
ICcY += )1(
Equilibrium GDP ][11 Pe
ICc
Y +
=
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Injections and Withdrawals
There is an alternative way to look at the equilibrium
condition for GDP.
PAEY = P
ICY +=
Now subtract Cfrom both sidesP
ICY = or
PIS =
S = Withdrawals (WD): Part of income not consumed
PI = Injections ( PINJ ): All exogenous spending
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Short-Run Equilibrium Condition
WDINJP=
Planned Injections equals Withdrawals
Saving FunctionP
ICY = cYCC +=
PIcYCY =
PIYcC =+ )1(
PIS =
so YcCS )1( +=
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Injections and Withdrawals Diagram
WDINJP
= SI
P=
YcCIP )1( +=
SIP ,
YcCS )1( +=
PI
eY Y
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Equilibrium
We have two equivalent equilibrium conditions for the
level of GDP:
Y = PAE WDINJ
P=
If these conditions hold there will be no tendency for
GDP to change, i.e. eYY = .
What happens if these conditions are not met?
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Disequilibrium
Suppose that the level of GDP is such that either:
PAE > Y and WDINJP > or
PAE < Yand WDINJP < In either case there will be a
tendency for the level of
GDP to change.
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PAE > Y
45-degree linePAE
PAE
0PAE
0Y Y
Planned Expenditure exceeds Aggregate Production
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Adjustment to Equilibrium
Firms will experience an unplanned decline in their
inventories
To re-build their inventories firms will increase theirlevel of
production
This will cause GDP to increase and it will move towards
its equilibrium value, where PAEcuts the 45-degree line
GDP will increase until PAE=Y.
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45-degree line
PAE
PAE
0PAE
0Y
e
Y
Y
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WDINJP<
SIP ,
YcCS )1( +=
PI
eY 1Y Y
Withdrawals (saving) exceed Planned Injections
(investment)
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Adjustment to Equilibrium
Firms will experience an unplanned increase in their
inventories
To reduce their inventories firms will revise downwardtheir
production plans
This will cause GDP to fall and it will move towards its
equilibrium value, where WDINJP =
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Paradox of Thrift
Suppose there is an exogenous increase in agents desire
to save
This can be represented by an upward shift in the
savingfunction
S S S
Y
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Initial Equilibrium
S,I
S
PI
eY Y
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Prediction: The aggregate amount of saving is unchanged
S,I newS
S
PI
e
newY
eY Y
Prediction: The level of GDP will fall
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Numerical Example for 2-Sector Model
Equilibrium Condition PAEY =
Definition ofPAE PICPAE +=
Consumption Function cYCC +=
Some Numbers 200=
PI
YC 8.050+=
Find eY ?
YYPAE 8.02508.020050 +=++= YY 8.0250+=
250)8.01( =Y
125025052502.0
1250
)8.01(
1===
=
eY
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45-degree line
PAE
YY 8.0250+=
250,1 Y
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Four Sector Model
Re-introduce
government sectorGforeign sectorNX
NXGICPAEP
+++=
1. Consumption Function: )( TYcCC +=
Consumption depends on disposable income
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New Functions
2. Tax Function: tYTT += There is an exogenous component to
taxes T and apart that is proportional to income tY
We can define tY
T=
as the marginal tax rate (how
much tax is paid on an additional dollar of income).
3.Import Function: )( TYmM =
Imports are proportional to disposable incomeWe can define mTYM
= )( as the marginal propensity
to import
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Some Simplifications
)( TYcCC +=
tYTT += )( TYmM =
ConsumptionYtcTcCtYTYcCC )1()( +=+=
ImportsYtmTmtYTYmTYmM )1()()( +===
NB. BOF (page 149) write the import function as:YtmM )1( =
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Equation for PAE
NXGICPAEP
+++= YtcTcCC )1( += YtmTmM )1( +=
MXNX =
Substitute YtmTmXGIYtcTcCPAE P )1()1( +++++=
Collect the exogenous variables
YtcYtmTmXGITcCPAE P )1()1(][ +++++=
YtmcTmXGITcCPAEP )1)((][ +++++=
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Short-run Equilibrium
Equilibrium condition
PAEY =
YtmcTmXGITcCYP )1)((][ +++++=
Solve for equilibrium GDP
][)])1)([(1( TmXGITcCtmcY P ++++=
][)])1)([(1(
1TmXGITcC
tmcY
Pe++++
=
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Equilibrium in Four Sector Model
PAE 45-degree line
NXGICPAEP
+++=
eY Y
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Injections and Withdrawals
Not surprisingly, we can re-write the condition for
equilibrium in terms of injections and withdrawals.
PAEY = NXGICPAE
P+++=
NXGICY P +++= Subtract T and C from both sides;
TNXGICTYP
++= Now CTYS = and MXNX =
TMXGISP
++= or XGIMTS P ++=++
Withdrawals = (planned) Injections
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Withdrawals and Injections Diagram
WDINJP ,
MTS ++
XGIP
++
eY Y
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How Does the Keynesian Model Explain Fluctuations in
GDP?][
)])1)([(1(
1TmXGITcC
tmcY
Pe++++
=
2 possibilities
A change in one of the exogenous variables:XGITC
P ,,,,
A change in one of the parameters: c, m, t
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Example: Increase in PI
PAE 45 degree line
1PAE
0PAE
eY0
eY1 Y
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PAE and the Output Gap
Output gap = Actual output less Potential output
Output gap = *YY
We can use our model to understand contractionary
(negative) and expansionary (positive) output gaps
We will focus on a contractionary output gap since it is
relevant to current economic circumstances.
We need to indicate potential output in our model.
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Level ofPAEis Consistent with GDP equal to Potential
Output
PAE 0PAE
*Y Y
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Decline in Planned Spending leads to Contractionary Gap
PAE 0PAE
1PAE
eY1 *Y Y
Contractionary gap = 0*
1
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Investment and the 1990s Recession (Feb 90-Oct 91)
15000
17000
19000
21000
23000
25000
27000
29000
31000
33000
Mar
-1988
Jun-1988
Sep
-1988
Dec
-1988
Mar
-1989
Jun-1989
Sep
-1989
Dec
-1989
Mar
-1990
Jun-1990
Sep
-1990
Dec
-1990
Mar
-1991
Jun-1991
Sep
-1991
Dec
-1991
Mar
-1992
Jun-1992
Sep
-1992
Dec
-1992
Mar
-1993
Investment($mill)
140000
142000
144000
146000
148000150000
152000
154000
156000
158000
160000
GDP($mill)
Investment GDP
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The Multiplier
In the following diagram compare the relative sizes of the
change in Ycaused by the change in PAE.
PAE 1PAE
0PAE
PAE Y
eY0
eY1 Y
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The Multiplier
An additional dollar ofexogenous PAE generates more
that a dollars worth of GDP
How much more?
2-Sector Model
Equilibrium GDP ][1
1 PeIC
c
Y +
=
11
1>
=
cC
Ye
or 111
>
=
cI
Y
P
e
since 10
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Size of the Multiplier
Suppose MPC = 0.75
Multiplier = 425.01
75.01
1
1
1==
=
c
4-Sector Model
][)])1)([(1(
1TmXGITcC
tmcY
Pe++++
=
XGICP ,,, Multiplier = )])1)([(1(
1
tmc
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Economics of the Multiplier
2-Sector ModelcYICPAE
P++= ][
PAEY = Increase exogenous PAE by 100
Rounds
1 2 3 4 ..PAE 100 100c )100( cc )100( ccc
Y 100 100c )100( cc )100( ccc
Initial increase in GDP of 100 is paid out as income
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What is the Total Effect on GDP?
Rounds1 2 3 4 ..
PAE 100 100c )100( cc )100( ccc Y 100 100c )100( cc )100(
ccc
Total Increase in GDP
.........100100100100 32 ++++ ccc or
.........]1[100 32 ++++ ccc
or
]1
1[100
c
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Does .........]1[32++++ ccc Really Equal
c1
1?
Let
.........1 32 ++++= cccS Multiply both sides by c
.........1 32 ++++= ccccccccS
or.........432 ++++= cccccS
Subtract cSfrom S.....)(.........1 3232 +++++++= cccccccSS
1= cSS
cS
=1
1 Done
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Recap
We have a model that can determine the level of GDP (in
the short-run).
We have a model that can explain short-run fluctuations
in GDP.
Finally we now have a framework for thinking about
macroeconomic policy.
Well Done.
