Chapter 6 Income Determination II --- The IS-LM Model
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Transcript of Chapter 6 Income Determination II --- The IS-LM Model
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Chapter 6
Income Determination II ---
The IS-LM Model
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• Assumptions of the IS-LM Model• What is the IS curve? • Graphical derivation of the IS curve• Slope of the IS curve • Shift of the IS curve • What is the LM curve? • Graphical derivation of the LM curve• Slope of the LM curve
Contents:
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• Shift of the LM curve•
Equilibrium in both the goods market and money market
• Change in equilibrium caused by an autonomous change in injection or withdrawal
• Change in equilibrium caused by an autonomous change in money demandor money supply
Contents:
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Assumptions of the IS-LM Model
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Assumptions of the IS-LM model:
1. Yf is a constant.
2. An unemployment of resources.-- The model is to find out determinants of the equilibrium GNP and ways to eliminate unemployment.
3. GNP = GDP = Y.
4. Price level is kept constant.-- Nominal variables = real variables and nominal r = real r.
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Keynesian modelsKeynesian
modelMarkets involved Endogenous
variablesY-E model (elementary Keynesian
model)
• Goods market Y
IS-LM model • Goods market• Money market
Y and r
AD-AS model • Goods market• Money market• Factor market
Y, r and P
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What is the IS Curve?
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IS Curve is a line relating real national income (Y) to real interest rate (r) at which the goods market is in equilibrium (i.e., with Y = E or J = W).
What is the IS Curve?
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Graphical Derivation of the IS Curve
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r
Y
W
J0
Quadrant II (NW quadrant)
Quadrant III (SW quadrant)
Quadrant IV (SE quadrant)
Four-quadrant diagram
Quadrant I (NE quadrant)
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r
JJ=I+G+X
rJ
Injection function: J = I + G + X
Quadrant II (NW quadrant)
G and X are autonomous.
Thus, J is negatively related to r.
0
I is negatively related to r.
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W
W=S+T+M
Y
W
Withdrawal function W = S + T + M
S, T and M are positively related to Y
Thus, W is positively related to Y
Quadrant IV (SE quadrant)
Y
0
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J1
J=W
45o
W
J
W1 (= J1)
Quadrant III (SW quadrant)0
J and W of points on the 45° line are equal. Hence they represent equilibrium in the goods market, where J = W.
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45o
r
Y
W
J0
Ar0
J0
W0
Y0
B
IS
When interest rate = r0 injection = J0
Corresponding national income = Y0
Equilibrium in goods market
To achieve equilibrium: J0
= W0
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Slope of the IS Curve
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IS
Y1
r1
r
YY0
r0
0
Initial goods market equil.: (Y0, r0)Downward sloping 1. If r ( r0 to r1) J ( J0 to J1)
So at point Z, J > W
2. To restore equil., Y should (Y0 to Y1) to raise W until W = J again.
Z r & Y are negatively related.
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Shift of the IS Curve
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r
Y0IS0
A (Y0,r0)
B (Y1,r0)
Assumption: An autonomous in J or in W (not caused by changes in r or Y)
IS1
J > W
If r remains constant, Y has to Y has to to Y to Y11 until W & equates with J again.
IS curve shifts rightward to restore equilibrium.
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Assumption: An autonomous in J or in W (not caused by changes in r or Y)
J > W
If Y remains constant, r has to r has to to r to r11 until J & equates with W again.
IS curve shifts upward to restore equilibrium.
r
Y0IS0
IS1
C (Y0,r1)
A (Y0,r0)
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IS2
r
Y0
IS0
A (Y0,r0)
E (Y0, r2)
D (Y2, r0)
J < W
IS curve shifts leftward or downward to restore equilibrium.
Assumption: An autonomous in J or in W (not caused by changes in r or Y)
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Q6.2: Derive from four-quadrant diagrams the change in the IS curve, (a) when there is an autonomous rise in J. (b) when there is an autonomous fall in J. (c) when there is an autonomous rise in W. (d) when there is an autonomous fall in W.
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Result Possible causes
Rightward (upward) shift of the IS curve
Leftward (downward) shift of the IS curve
Q6.3: Fill in the following table.
W autonomously
J autonomously
W autonomously
J autonomously
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What is the LM Curve?
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What is the LM Curve?
LM curve is a line relating real national income (Y) to real interest rate (r) at which the money market is in equilibrium [i.e.,with real money demand (Md) or real liquidity preference (L) = real money supply (Ms)].
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Graphical derivation of the LM Curve
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Transactions demand for money:
d is the change in Mt resulting from a unit change in income.
Income & Mt are positively related & so d > 0.
Mt = dY
Income elasticity of transactions demand for money
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Asset demand for money:
e is the change in Ma resulting from a unit change in r e < 0
Ma = e r + Ma*
Interest elasticity of asset demand for money
Autonomous asset demand for money
Ma* is the autonomous asset demand for money Ma* > 0
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Money supply function
PM1Ms
M1 is the sum of legal tender in public circulation
and demand deposits
General price level
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r
Y
Mt
Ma
0
Quadrant II (NW quadrant)
Quadrant III(SW quadrant)
Quadrant IV (SE quadrant)
Four-quadrant diagram
Quadrant I (NE quadrant)
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rMa
r
Ma
Ma = er + Ma*
0
Asset demand for money
Ma is negatively related to r
Quadrant II (NW quadrant)
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Y
Mt
Y
Mt
Mt = dY
Transactions demand for money
Mt is positively related to Y
Quadrant IV (SE quadrant)
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M1/P = Ms = Ma+Mt
Ma
Mt
45o
M1/P
M1/P
Equil. Condition: (Md=Ms )
MaMt
= M1/P - Ma
Mt
Quadrant III (SW quadrant)
If Mt = M1/P - Ma, then Mt + Ma = Md = M1/P = Ms
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r
Y
Mt
Ma0
45o
LM
Ma0
Mt0
Y0
r0
When interest rate = r0 assets demand = Ma0
In Equilibrium: Mt0 = Ms – Ma0
Ma0 + Mt0 = Ms
Corresponding national income = Y0
AB
Equilibrium in money market
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Slope of the LM Curve
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Assume initial money market equil.: (Y0, r0)
If Y (Y0 to Y1) Mt ( Mt0 to Mt1)
At point Z, Md > Ms
To restore equilibrium, r should ( r0 to r1 ) such that Ma & Mt + Ma equates with Ms again.
r & Y are positively relatedr0
Y0
r
YY1
r1
0
Upward sloping
LM
Z
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Shift of the LM Curve
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B (Y1,r0)
r
Y0
A (Y0,r0)
LM0
LM1
Assumption: An autonomous in Md or in Ms
(not caused by changes in r or Y)
Md > Ms
If r remains constant, Y has to to Y1 such that Mt and Md = Ms again.
LM curve shifts leftward to restore equilibrium.
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Assumption: An autonomous in Md or in Ms
(not caused by changes in r or Y)
Md > Ms
If Y remains constant, r has to to r1 such that Ma and Md = Ms again.
LM curve shifts upward to restore equilibrium
C (Y0,r1)
r
Y0
A (Y0,r0)
LM0
LM1
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r
Y0
A(Y0,r0)
LM0
LM2
E (Y0,r2)
D (Y2,r0)
Md < Ms
LM curve shifts rightward or downward
Assumption: An autonomous in Md or in Ms
(not caused by changes in r or Y)
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Q6.4: Derive from four-quadrant diagrams, the change in the LM curve(a) when there is an autonomous rise in Ma.
(b) when there is an autonomous fall in Ma.
(c) when there is an autonomous rise in Mt.
(d) when there is an autonomous fall in Mt.
(e) when there is an autonomous rise in Ms.
(f) when there is an autonomous fall in Ms.
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Result Possible causes
Rightward (downward) shift of the LM curve
Leftward (upward) shift of the LM curve
Q6.5: Fill in the following table.
Md autonomouslyMs autonomously
Md autonomouslyMs autonomously
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Equilibrium in Both
the Goods Market and
Money Market
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r
Y0
IS
A(J = W)
B(J < W)
Goods market equilibrium and the IS curve
There is an unintended inventory investment. Firms cut production & Y.
Pt. B lies above pt. A. At pt. B, r J At pt. B, J < W
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r
Y0
IS
A(J = W)
C(J > W)
There is an unintended inventory disinvestment. Firms raise production & Y.
Goods market equilibrium and the IS curve
Pt. C lies below pt. A. At pt. C, r J
At pt. C, J > W
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r
Y0
Money market equilibrium and the LM curve
LM
B (Md < Ms)
A (Md=Ms)
Pt. B lies above pt. A. At pt. B, r Ma
At pt. B, Md < Ms
With excess money balance, people buy bonds to earn interest.
Bond price r
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C( Md > Ms)
Pt. C lies below pt. A. At pt. C, r Ma At pt. C, Md > Ms
With excess money demand, people sell bonds for liquidity.
r
Y0
LM
A (Md=Ms)
Money market equilibrium and the LM curve
Bond price r
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Adjustment towards twin equilibrium
1. Whenever a point is not on the IS curve, Y will change until it reaches the IS curve.
2. Whenever a point is not on the LM curve, r will change until it reaches the LM curve.
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r
Y0
LM
IS
J<W YMd<Ms r
J>W YMd>Ms r
J>W YMd<Ms r
J<W YMd>Ms r
Adjustment towards twin equilibrium
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Change in Equilibrium Caused by
an Autonomous Change in Injection or Withdrawal
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© Pilot Publishing Company Ltd. 2005Crowding-out effect
r
Y0
LM0
IS1
IS0
r1
r0
Y1Y0 Y’}
Transmission mechanism:Autonomous J or W
As J > W Y W & Mt
Md > Ms r Ma & J
Until both markets r
estore equil.
IS curve shifts rightward
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Q6.6:
When J < W, both income and interest rate fall. Describe the transmission mechanism.
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Change in Equilibrium Caused by
an Autonomous Change in Money Demand or Money Supply
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Transmission Mechanism:
Y1
r1
LM1r
Y0IS0
LM0
r0
Y0
As Md > Ms r Ma & J
Autonomous Md or Ms
LM curve shifts leftward
J < W Y W & Mt
Until both markets r
estore equil.
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Q6.7:When Md < Ms, income rises and interest rate falls. Describe the transmission mechanism.
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Q6.8:(a) Suppose there exist an autonomous increase in J and an autonomous increase in Ms. Predict the change in Y and r with the aid of an IS-LM diagram.
(b) Suppose there exist an autonomous increase in W and an autonomous increase in Ms. Predict the change in Y and r with the aid of an IS-LM diagram.
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Mathematical derivation of the IS-LM equilibrium
In a four sector economy, E = C + I + G + X – M
1. Mathematical derivation of the IS curve
where C = cYd+C*;
Yd= Y-tY-T*+qY+Q*;
I = br+I*+iY;
G= G*;
X = X*;
M = mY+M*
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Mathematical derivation of equil. income & multiplier
Y = cYd+C* +br+I*+iY + G* +X*-mY-M*
Y = c(Y-tY-T*+qY+Q*)+C* +br+I*+iY +G*+X*-mY-M* (1–c-i+ct-cq+m)•Y = C*+ I*+ G*+ X*-M*-cT*+cQ* +br
1. Mathematical derivation of the IS curveEquation of IS curve: Y = E = C+I+G+X-M
bcQ*)*cT*M*X*G*I*(C
Yb
mcq-cti-c-1
r =
(1)
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Given Md = Mt + Ma; Ms = M1/P
where Mt = dY & Ma = er + Ma*
dY + er +Ma* = M1/P
er = -dY + (M1/P – Ma*)
2. Mathematical derivation of the LM curve
Equation of a LM curve: Md = Mt + Ma = Ms
eMa*)(M1/PY
ed
r = (2)
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bcQ*)*cT*M*X*G*I*(C
Yb
mcq-cti-c-1
Sub. equation (1) into (2), the equil. income can be found.
eMa*)(M1/PY
ed
eMa*)-(M1/P
b*A
Yed
Yb
mcq-cti-c-1
eMa*)-(M1/P
b*A
Ybe
bdm)ecq-cti-c-(1
Ma*)P
M1(
dm)e/bcq-cti-c-(11
*Abd/em)cq-cti-c-(1
1Y
=A*
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IS-LM multiplier
Note: As b<0, d>0 and e<0 bd/e>0 IS-LM multiplier is smaller than Y-E multiplier, because
When there is a rise in autonomous E, Y With money market, Y Mt & disturb money market
As Md > Ms, r and crowds out private investment
Overall in equilibrium income is smaller.
ebdm)cq-cti-c-(1
1
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© Pilot Publishing Company Ltd. 2005
Correcting Misconceptions:
1. The IS curve represents situations with I = S.
2. Whenever J or W, the IS curve shifts rightward or upward.
3. Whenever Md or Ms, the LM curve shifts leftward or upward.