Chapter 6 Income Determination II --- The IS-LM Model

© Pilot Publishing Company Ltd. 2005

Chapter 6

Income Determination II ---

The IS-LM Model


• 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:


• 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:


Assumptions of the IS-LM Model


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.


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


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).

What is the IS Curve?


Graphical Derivation of the IS Curve


r

Y

W

J0

Quadrant II (NW quadrant)

Quadrant III (SW quadrant)

Quadrant IV (SE quadrant)

Four-quadrant diagram

Quadrant I (NE quadrant)


r

JJ=I+G+X

rJ

Injection function: J = I + G + X


G and X are autonomous.

Thus, J is negatively related to r.

0

I is negatively related to r.


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


Y

0


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.


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


Slope of the IS Curve


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.


Shift of the IS Curve


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.



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)


IS2

r

Y0

IS0

A (Y0,r0)

E (Y0, r2)

D (Y2, r0)

J < W

IS curve shifts leftward or downward to restore equilibrium.



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.


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


What is the LM Curve?


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)].


Graphical derivation of the LM Curve


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


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


Money supply function

PM1Ms

M1 is the sum of legal tender in public circulation

and demand deposits

General price level


r

Y

Mt

Ma

0


Quadrant III(SW quadrant)


Four-quadrant diagram

Quadrant I (NE quadrant)


rMa

r

Ma

Ma = er + Ma*

0

Asset demand for money

Ma is negatively related to r



Y

Mt

Y

Mt

Mt = dY

Transactions demand for money

Mt is positively related to Y



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


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


Slope of the LM Curve


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


Shift of the LM Curve


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.




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


r

Y0

A(Y0,r0)

LM0

LM2

E (Y0,r2)

D (Y2,r0)

Md < Ms

LM curve shifts rightward or downward




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.


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


Equilibrium in Both

the Goods Market and

Money Market


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


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


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


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


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.


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


Change in Equilibrium Caused by

an Autonomous Change in Injection or Withdrawal

© 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


Q6.6:

When J < W, both income and interest rate fall. Describe the transmission mechanism.


Change in Equilibrium Caused by

an Autonomous Change in Money Demand or Money Supply


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.


Q6.7:When Md < Ms, income rises and interest rate falls. Describe the transmission mechanism.


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.


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*


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)


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)


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*


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


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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Chapter 6 Income Determination II --- The IS-LM Model