Chapter 6
Income Determination II --The IS-LM Model
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Contents:
• 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
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Contents:
• 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
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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 models
Keynesian
model
Markets involved
Y-E model
• Goods market
(elementary
Keynesian model)
IS-LM model • Goods market
• Money market
AD-AS model • Goods market
• Money market
• Factor market
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Endogenous
variables
Y
Y and r
Y, r and P
What is the IS Curve?
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What is the IS Curve?
 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).
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Graphical Derivation of
the IS Curve
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Four-quadrant diagram
r
Quadrant II
(NW quadrant)
J
Quadrant I
(NE quadrant)
0
Quadrant III
(SW quadrant)
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Y
Quadrant IV
(SE quadrant)
W
Quadrant II (NW quadrant)
Injection function:
r
J=I+G+X
J
G and X are autonomous.

I is negatively related to r.
Thus, J is negatively related to r.
r

J=I+G+X
J
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0
Quadrant IV (SE quadrant)
0
Y

Withdrawal function W
=S+T+M
Y
W

S, T and M are
positively related to Y
Thus, W is positively
related to Y
W=S+T+M
W
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Quadrant III (SW quadrant)
J1
J
J and W of points on
the 45° line are equal.
Hence they represent
equilibrium in the
0
45o
W1 (= J1)
goods market, where
J = W.
J=W
W
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r
When interest rate = r0
injection = J0
r0
Equilibrium in
goods market
A
B
J
To achieve
equilibrium:
J0 = W0
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0
J0
IS
o
45
W0
W
Y0
Y
Corresponding
national income = Y0
Slope of the IS Curve
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Initial goods market equil.: (Y0, r0)
r
Downwar
d 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.
r0
r1
 r & Y are negatively related.
Z
IS
0
Y0
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Y1
Y
Shift of the IS Curve
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Assumption: An autonomous  in J or  in W
(not caused by changes in r or Y)
r
J>W

If r remains constant, Y has to  to Y1
until W  & equates with J again.

B (Y1,r0)
A (Y0,r0)

IS0
0
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
IS curve shifts rightward
to restore equilibrium.
IS1
Y
r
Assumption: An autonomous  in J or  in W
(not caused by changes in r or Y)
J>W
C (Y0,r1)


If Y remains constant, r has to  to r1
until J  & equates with W again.

A (Y0,r0)
IS0
0
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IS1

IS curve shifts upward
to restore equilibrium.
Y
r
Assumption: An autonomous  in J or  in W
(not caused by changes in r or Y)
J<W

A (Y0,r0)

D (Y2, r0)
E (Y0, r2)
IS0

0
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
IS curve shifts leftward
or downward to
restore equilibrium.
IS2
Y
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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Q6.3: Fill in the following table.
Result
Rightward (upward)
shift of the IS curve
Possible causes
J autonomously
W autonomously
Leftward (downward)
shift of the IS curve
J  autonomously
W 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:
Mt = dY
Income elasticity of 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.
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Asset demand for money:
Ma = e r + Ma*
Interest elasticity of
asset demand for money
Autonomous asset
demand for money
 e is the change in Ma resulting from a unit change in r
e<0
 Ma* is the autonomous asset demand for money
 Ma* > 0
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Money supply function
M1 is the sum of legal
tender in public circulation
and demand deposits
M1
Ms 
P
General price level
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Four-quadrant diagram
r
Quadrant II
(NW quadrant)
Ma
Quadrant I
(NE quadrant)
0
Quadrant III
(SW quadrant)
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Y
Quadrant IV
(SE quadrant)
Mt
Quadrant II (NW quadrant)
r
Asset demand for money
Ma is negatively related to r
Ma

r

Ma = er + Ma*
Ma
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0
Quadrant IV (SE quadrant)
Y

Y
Mt

Transactions demand
for money
Mt is positively related to Y
Mt = dY
Mt
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Quadrant III (SW quadrant)
Ma
M1/P

Equil. Condition: (Md=Ms )
If Mt = M1/P - Ma, then
Mt + Ma = Md = M1/P = Ms
Mt
Ma
= M1/P - Ma
Mt
45o
M1/P = Ms = Ma+Mt
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M1/P
Mt
r
LM
When interest rate = r0
assets demand = Ma0
Ma
B
r0
A
0
Ma0
In Equilibrium:
Mt0 = Ms – Ma0
Ma0 + Mt0 = Ms
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Y0
Mt0
45o
Mt
Equilibrium in
money market
Y
Corresponding
national income = Y0
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)
r
r1
Upward
sloping
r0
0
 At point Z, Md > Ms
LM
Z
Y0
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Y1
To restore equilibrium,
r should  ( r0 to r1 )
such that Ma & Mt + Ma
equates with Ms again.
 r & Y are positively related
Y
Shift of the LM Curve
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Assumption: An autonomous  in Md or  in Ms
(not caused by changes in r or Y)
r
LM1
 Md > M s
LM0 If r remains constant,
B (Y1,r0)


Y has to  to Y1
such that Mt and
Md = Ms again.
A (Y0,r0)  LM curve shifts leftward
to restore equilibrium.
0
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Y
r
Assumption: An autonomous  in Md or  in Ms
(not caused by changes in r or Y)
 Md > M s
LM1
C (Y0,r1)


LM0
If Y remains constant,
r has to  to r1
such that Ma and
Md = Ms again.
A (Y0,r0)  LM curve shifts upward
to restore equilibrium
0
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Y
Assumption: An autonomous  in Md or  in Ms
(not
caused
by
changes
in
r
or
Y)
r
LM0
LM2
A(Y0,r0)


 Md < M s

D (Y2,r0)

LM curve shifts
rightward or
downward
E (Y0,r2)
0
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Y
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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Q6.5: Fill in the following table.
Result
Possible causes
Rightward (downward)
shift of the LM curve
Md  autonomously
Leftward (upward)
shift of the LM curve
Md  autonomously
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Ms  autonomously
Ms  autonomously
Equilibrium in Both
the Goods Market and
Money Market
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Goods market equilibrium and the IS curve
r


B(J < W)
A(J = W)
Pt. B lies above pt. A. At
pt. B, r   J   At
pt. B, J < W
There is an unintended
inventory investment.
Firms cut production &
Y.
IS
0
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Y
Goods market equilibrium and the IS curve
r
Pt. C lies below pt. A.
At pt. C, r   J 
 At pt. C, J > W

A(J = W)
There is an unintended
inventory disinvestment.
Firms raise production &
Y.
C(J > W) 
IS
0
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Y
Money market equilibrium and the LM curve
r
Pt. B lies above pt. A.
At pt. B, r   Ma 
 At pt. B, Md < Ms
B (Md < Ms)


A (Md=Ms)
With excess money
balance, people buy
bonds to earn interest.
Bond price   r 
LM
0
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Y
Money market equilibrium and the LM curve
r
LM
With excess money
demand, people sell
bonds for liquidity.
Bond price   r 
A (Md=Ms)

 C(
0
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Pt. C lies below pt. A.
At pt. C, r   Ma 
 At pt. C, Md > Ms
Md > M s)
Y
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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Adjustment towards twin equilibrium
r
J<W  Y
Md<Ms r
LM

J>W  Y
Md<Ms r


J<W  Y
Md>Ms r

0
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J>W  Y
Md>Ms r
IS
Y
Change in Equilibrium
Caused by
an Autonomous Change in
Injection or Withdrawal
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Transmission mechanism:
r
LM0
Autonomous J or W

IS curve shifts rightward
r1
As J > W  Y  W & Mt
 Md > Ms  r  Ma & J
r0

Until both markets
restore equil.
IS1
IS0
0
Y0
Y1
Y’
Y
Crowding-out effect
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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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r
Transmission Mechanism:
LM1
Autonomous Md or Ms

LM0
LM curve shifts leftward
r1
As Md > Ms  r  Ma & J
r0
 J < W  Y  W & Mt
IS0
0
Y1
Y0
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Y

Until both markets
restore equil.
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
1. Mathematical derivation of the IS curve
In a four sector economy,
E=C+I+G+X–M
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
1. Mathematical derivation of the IS curve
Equation of IS curve: Y = E = C+I+G+X-M

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
r=
1 - c - i  ct - cq  m
b
(C *  I *  G *  X *  M * cT *  cQ*) (1)
Y
b
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2. Mathematical derivation of the LM curve
Given Md = Mt + Ma; Ms = M1/P
where Mt = dY & Ma = er + Ma*
Equation of a LM curve: Md = Mt + Ma = Ms


dY + er +Ma* = M1/P
er = -dY + (M1/P – Ma*)
d
(M1/P  Ma*)


Y

r=
e
e
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(2)
Sub. equation (1) into (2), the equil. income can be found.
=A*
d
(M1/P  Ma*)
1 - c - i  ct - cq  m
(C *  I *  G *  X *  M * cT *  cQ*)


Y

Y

e
e
b
b

1 - c - i  ct - cq  m
Y
b

d
e
(1 - c - i  ct - cq  m)e  bd
Y 
A*
Y 
A*
be
Y
1
(1 - c - i  ct - cq  m)  bd/e
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b
b
 A*

(M1/P - Ma*)

(M1/P - Ma*)
e
e
1
(1 - c - i  ct - cq  m)e/b  d
(
M1
P
 Ma*)
IS-LM multiplier 
1
(1 - c - i  ct - cq  m)  bd
e
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.
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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.
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